3.1456 \(\int \frac {A+B x}{\sqrt {d+e x} (a-c x^2)^2} \, dx\)

Optimal. Leaf size=250 \[ -\frac {\left (-3 \sqrt {a} A \sqrt {c} e+a B e+2 A c d\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{4 a^{3/2} c^{3/4} \left (\sqrt {c} d-\sqrt {a} e\right )^{3/2}}+\frac {\left (3 \sqrt {a} A \sqrt {c} e+a B e+2 A c d\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{4 a^{3/2} c^{3/4} \left (\sqrt {a} e+\sqrt {c} d\right )^{3/2}}+\frac {\sqrt {d+e x} (x (A c d-a B e)+a (B d-A e))}{2 a \left (a-c x^2\right ) \left (c d^2-a e^2\right )} \]

[Out]

-1/4*arctanh(c^(1/4)*(e*x+d)^(1/2)/(-e*a^(1/2)+d*c^(1/2))^(1/2))*(2*A*c*d+a*B*e-3*A*e*a^(1/2)*c^(1/2))/a^(3/2)
/c^(3/4)/(-e*a^(1/2)+d*c^(1/2))^(3/2)+1/4*arctanh(c^(1/4)*(e*x+d)^(1/2)/(e*a^(1/2)+d*c^(1/2))^(1/2))*(2*A*c*d+
a*B*e+3*A*e*a^(1/2)*c^(1/2))/a^(3/2)/c^(3/4)/(e*a^(1/2)+d*c^(1/2))^(3/2)+1/2*(a*(-A*e+B*d)+(A*c*d-B*a*e)*x)*(e
*x+d)^(1/2)/a/(-a*e^2+c*d^2)/(-c*x^2+a)

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Rubi [A]  time = 0.50, antiderivative size = 250, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {823, 827, 1166, 208} \[ -\frac {\left (-3 \sqrt {a} A \sqrt {c} e+a B e+2 A c d\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{4 a^{3/2} c^{3/4} \left (\sqrt {c} d-\sqrt {a} e\right )^{3/2}}+\frac {\left (3 \sqrt {a} A \sqrt {c} e+a B e+2 A c d\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{4 a^{3/2} c^{3/4} \left (\sqrt {a} e+\sqrt {c} d\right )^{3/2}}+\frac {\sqrt {d+e x} (x (A c d-a B e)+a (B d-A e))}{2 a \left (a-c x^2\right ) \left (c d^2-a e^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/(Sqrt[d + e*x]*(a - c*x^2)^2),x]

[Out]

(Sqrt[d + e*x]*(a*(B*d - A*e) + (A*c*d - a*B*e)*x))/(2*a*(c*d^2 - a*e^2)*(a - c*x^2)) - ((2*A*c*d + a*B*e - 3*
Sqrt[a]*A*Sqrt[c]*e)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]])/(4*a^(3/2)*c^(3/4)*(Sqrt[c]
*d - Sqrt[a]*e)^(3/2)) + ((2*A*c*d + a*B*e + 3*Sqrt[a]*A*Sqrt[c]*e)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[
c]*d + Sqrt[a]*e]])/(4*a^(3/2)*c^(3/4)*(Sqrt[c]*d + Sqrt[a]*e)^(3/2))

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rule 823

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(
m + 1)*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*(a + c*x^2)^(p + 1))/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), x] + Di
st[1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Simp[f*(c^2*d^2*(2*p + 3) + a*c*e^2*
(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x]
&& NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 827

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2, Subst[Int[(e*f
- d*g + g*x^2)/(c*d^2 + a*e^2 - 2*c*d*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e, f, g}, x]
&& NeQ[c*d^2 + a*e^2, 0]

Rule 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rubi steps

\begin {align*} \int \frac {A+B x}{\sqrt {d+e x} \left (a-c x^2\right )^2} \, dx &=\frac {\sqrt {d+e x} (a (B d-A e)+(A c d-a B e) x)}{2 a \left (c d^2-a e^2\right ) \left (a-c x^2\right )}-\frac {\int \frac {-\frac {1}{2} c \left (2 A c d^2+a B d e-3 a A e^2\right )-\frac {1}{2} c e (A c d-a B e) x}{\sqrt {d+e x} \left (a-c x^2\right )} \, dx}{2 a c \left (c d^2-a e^2\right )}\\ &=\frac {\sqrt {d+e x} (a (B d-A e)+(A c d-a B e) x)}{2 a \left (c d^2-a e^2\right ) \left (a-c x^2\right )}-\frac {\operatorname {Subst}\left (\int \frac {\frac {1}{2} c d e (A c d-a B e)-\frac {1}{2} c e \left (2 A c d^2+a B d e-3 a A e^2\right )-\frac {1}{2} c e (A c d-a B e) x^2}{-c d^2+a e^2+2 c d x^2-c x^4} \, dx,x,\sqrt {d+e x}\right )}{a c \left (c d^2-a e^2\right )}\\ &=\frac {\sqrt {d+e x} (a (B d-A e)+(A c d-a B e) x)}{2 a \left (c d^2-a e^2\right ) \left (a-c x^2\right )}-\frac {\left (2 A c d+a B e-3 \sqrt {a} A \sqrt {c} e\right ) \operatorname {Subst}\left (\int \frac {1}{c d-\sqrt {a} \sqrt {c} e-c x^2} \, dx,x,\sqrt {d+e x}\right )}{4 a^{3/2} \left (\sqrt {c} d-\sqrt {a} e\right )}+\frac {\left (2 A c d+a B e+3 \sqrt {a} A \sqrt {c} e\right ) \operatorname {Subst}\left (\int \frac {1}{c d+\sqrt {a} \sqrt {c} e-c x^2} \, dx,x,\sqrt {d+e x}\right )}{4 a^{3/2} \left (\sqrt {c} d+\sqrt {a} e\right )}\\ &=\frac {\sqrt {d+e x} (a (B d-A e)+(A c d-a B e) x)}{2 a \left (c d^2-a e^2\right ) \left (a-c x^2\right )}-\frac {\left (2 A c d+a B e-3 \sqrt {a} A \sqrt {c} e\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{4 a^{3/2} c^{3/4} \left (\sqrt {c} d-\sqrt {a} e\right )^{3/2}}+\frac {\left (2 A c d+a B e+3 \sqrt {a} A \sqrt {c} e\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {a} e}}\right )}{4 a^{3/2} c^{3/4} \left (\sqrt {c} d+\sqrt {a} e\right )^{3/2}}\\ \end {align*}

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Mathematica [A]  time = 0.45, size = 353, normalized size = 1.41 \[ \frac {-\frac {c^{3/4} \left (-3 a A e^2+2 a B d e+A c d^2\right ) \left (\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{\sqrt {\sqrt {a} e+\sqrt {c} d}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{2 \sqrt {a}}+\frac {c \sqrt {d+e x} (-a A e+a B (d-e x)+A c d x)}{c x^2-a}+\frac {\sqrt [4]{c} (A c d-a B e) \left (\sqrt {\sqrt {c} d-\sqrt {a} e} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )-\sqrt {\sqrt {a} e+\sqrt {c} d} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )\right )}{2 \sqrt {a}}}{2 a c \left (a e^2-c d^2\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/(Sqrt[d + e*x]*(a - c*x^2)^2),x]

[Out]

((c*Sqrt[d + e*x]*(-(a*A*e) + A*c*d*x + a*B*(d - e*x)))/(-a + c*x^2) - (c^(3/4)*(A*c*d^2 + 2*a*B*d*e - 3*a*A*e
^2)*(-(ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]]/Sqrt[Sqrt[c]*d - Sqrt[a]*e]) + ArcTanh[(c^
(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]]/Sqrt[Sqrt[c]*d + Sqrt[a]*e]))/(2*Sqrt[a]) + (c^(1/4)*(A*c*d
- a*B*e)*(Sqrt[Sqrt[c]*d - Sqrt[a]*e]*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]] - Sqrt[Sqrt
[c]*d + Sqrt[a]*e]*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]]))/(2*Sqrt[a]))/(2*a*c*(-(c*d^2
) + a*e^2))

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fricas [B]  time = 52.56, size = 7506, normalized size = 30.02 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(e*x+d)^(1/2)/(-c*x^2+a)^2,x, algorithm="fricas")

[Out]

-1/8*((a^2*c*d^2 - a^3*e^2 - (a*c^2*d^2 - a^2*c*e^2)*x^2)*sqrt((4*A^2*c^3*d^5 + 4*A*B*a*c^2*d^4*e - 6*A*B*a^2*
c*d^2*e^3 - 6*A*B*a^3*e^5 + (B^2*a^2*c - 15*A^2*a*c^2)*d^3*e^2 + 3*(B^2*a^3 + 5*A^2*a^2*c)*d*e^4 + (a^3*c^4*d^
6 - 3*a^4*c^3*d^4*e^2 + 3*a^5*c^2*d^2*e^4 - a^6*c*e^6)*sqrt((36*A^2*B^2*c^4*d^6*e^4 + 12*(3*A*B^3*a*c^3 - 5*A^
3*B*c^4)*d^5*e^5 + (9*B^4*a^2*c^2 - 198*A^2*B^2*a*c^3 + 25*A^4*c^4)*d^4*e^6 - 8*(9*A*B^3*a^2*c^2 - 31*A^3*B*a*
c^3)*d^3*e^7 + 6*(B^4*a^3*c + 40*A^2*B^2*a^2*c^2 - 15*A^4*a*c^3)*d^2*e^8 - 28*(A*B^3*a^3*c + 9*A^3*B*a^2*c^2)*
d*e^9 + (B^4*a^4 + 18*A^2*B^2*a^3*c + 81*A^4*a^2*c^2)*e^10)/(a^3*c^9*d^12 - 6*a^4*c^8*d^10*e^2 + 15*a^5*c^7*d^
8*e^4 - 20*a^6*c^6*d^6*e^6 + 15*a^7*c^5*d^4*e^8 - 6*a^8*c^4*d^2*e^10 + a^9*c^3*e^12)))/(a^3*c^4*d^6 - 3*a^4*c^
3*d^4*e^2 + 3*a^5*c^2*d^2*e^4 - a^6*c*e^6))*log(-(24*A^3*B*c^4*d^5*e^2 + 4*(9*A^2*B^2*a*c^3 - 5*A^4*c^4)*d^4*e
^3 + 2*(9*A*B^3*a^2*c^2 - 65*A^3*B*a*c^3)*d^3*e^4 + 3*(B^4*a^3*c - 28*A^2*B^2*a^2*c^2 + 27*A^4*a*c^3)*d^2*e^5
- 2*(5*A*B^3*a^3*c - 81*A^3*B*a^2*c^2)*d*e^6 + (B^4*a^4 - 81*A^4*a^2*c^2)*e^7)*sqrt(e*x + d) + (6*A^2*B*a^2*c^
4*d^5*e^3 + 5*(3*A*B^2*a^3*c^3 - A^3*a^2*c^4)*d^4*e^4 + 6*(B^3*a^4*c^2 - 7*A^2*B*a^3*c^3)*d^3*e^5 - 12*(3*A*B^
2*a^4*c^2 - 2*A^3*a^3*c^3)*d^2*e^6 + 2*(B^3*a^5*c + 30*A^2*B*a^4*c^2)*d*e^7 - 3*(A*B^2*a^5*c + 9*A^3*a^4*c^2)*
e^8 - (2*A*a^3*c^7*d^9 + B*a^4*c^6*d^8*e - 10*A*a^4*c^6*d^7*e^2 - 2*B*a^5*c^5*d^6*e^3 + 18*A*a^5*c^5*d^5*e^4 -
14*A*a^6*c^4*d^3*e^6 + 2*B*a^7*c^3*d^2*e^7 + 4*A*a^7*c^3*d*e^8 - B*a^8*c^2*e^9)*sqrt((36*A^2*B^2*c^4*d^6*e^4
+ 12*(3*A*B^3*a*c^3 - 5*A^3*B*c^4)*d^5*e^5 + (9*B^4*a^2*c^2 - 198*A^2*B^2*a*c^3 + 25*A^4*c^4)*d^4*e^6 - 8*(9*A
*B^3*a^2*c^2 - 31*A^3*B*a*c^3)*d^3*e^7 + 6*(B^4*a^3*c + 40*A^2*B^2*a^2*c^2 - 15*A^4*a*c^3)*d^2*e^8 - 28*(A*B^3
*a^3*c + 9*A^3*B*a^2*c^2)*d*e^9 + (B^4*a^4 + 18*A^2*B^2*a^3*c + 81*A^4*a^2*c^2)*e^10)/(a^3*c^9*d^12 - 6*a^4*c^
8*d^10*e^2 + 15*a^5*c^7*d^8*e^4 - 20*a^6*c^6*d^6*e^6 + 15*a^7*c^5*d^4*e^8 - 6*a^8*c^4*d^2*e^10 + a^9*c^3*e^12)
))*sqrt((4*A^2*c^3*d^5 + 4*A*B*a*c^2*d^4*e - 6*A*B*a^2*c*d^2*e^3 - 6*A*B*a^3*e^5 + (B^2*a^2*c - 15*A^2*a*c^2)*
d^3*e^2 + 3*(B^2*a^3 + 5*A^2*a^2*c)*d*e^4 + (a^3*c^4*d^6 - 3*a^4*c^3*d^4*e^2 + 3*a^5*c^2*d^2*e^4 - a^6*c*e^6)*
sqrt((36*A^2*B^2*c^4*d^6*e^4 + 12*(3*A*B^3*a*c^3 - 5*A^3*B*c^4)*d^5*e^5 + (9*B^4*a^2*c^2 - 198*A^2*B^2*a*c^3 +
25*A^4*c^4)*d^4*e^6 - 8*(9*A*B^3*a^2*c^2 - 31*A^3*B*a*c^3)*d^3*e^7 + 6*(B^4*a^3*c + 40*A^2*B^2*a^2*c^2 - 15*A
^4*a*c^3)*d^2*e^8 - 28*(A*B^3*a^3*c + 9*A^3*B*a^2*c^2)*d*e^9 + (B^4*a^4 + 18*A^2*B^2*a^3*c + 81*A^4*a^2*c^2)*e
^10)/(a^3*c^9*d^12 - 6*a^4*c^8*d^10*e^2 + 15*a^5*c^7*d^8*e^4 - 20*a^6*c^6*d^6*e^6 + 15*a^7*c^5*d^4*e^8 - 6*a^8
*c^4*d^2*e^10 + a^9*c^3*e^12)))/(a^3*c^4*d^6 - 3*a^4*c^3*d^4*e^2 + 3*a^5*c^2*d^2*e^4 - a^6*c*e^6))) - (a^2*c*d
^2 - a^3*e^2 - (a*c^2*d^2 - a^2*c*e^2)*x^2)*sqrt((4*A^2*c^3*d^5 + 4*A*B*a*c^2*d^4*e - 6*A*B*a^2*c*d^2*e^3 - 6*
A*B*a^3*e^5 + (B^2*a^2*c - 15*A^2*a*c^2)*d^3*e^2 + 3*(B^2*a^3 + 5*A^2*a^2*c)*d*e^4 + (a^3*c^4*d^6 - 3*a^4*c^3*
d^4*e^2 + 3*a^5*c^2*d^2*e^4 - a^6*c*e^6)*sqrt((36*A^2*B^2*c^4*d^6*e^4 + 12*(3*A*B^3*a*c^3 - 5*A^3*B*c^4)*d^5*e
^5 + (9*B^4*a^2*c^2 - 198*A^2*B^2*a*c^3 + 25*A^4*c^4)*d^4*e^6 - 8*(9*A*B^3*a^2*c^2 - 31*A^3*B*a*c^3)*d^3*e^7 +
6*(B^4*a^3*c + 40*A^2*B^2*a^2*c^2 - 15*A^4*a*c^3)*d^2*e^8 - 28*(A*B^3*a^3*c + 9*A^3*B*a^2*c^2)*d*e^9 + (B^4*a
^4 + 18*A^2*B^2*a^3*c + 81*A^4*a^2*c^2)*e^10)/(a^3*c^9*d^12 - 6*a^4*c^8*d^10*e^2 + 15*a^5*c^7*d^8*e^4 - 20*a^6
*c^6*d^6*e^6 + 15*a^7*c^5*d^4*e^8 - 6*a^8*c^4*d^2*e^10 + a^9*c^3*e^12)))/(a^3*c^4*d^6 - 3*a^4*c^3*d^4*e^2 + 3*
a^5*c^2*d^2*e^4 - a^6*c*e^6))*log(-(24*A^3*B*c^4*d^5*e^2 + 4*(9*A^2*B^2*a*c^3 - 5*A^4*c^4)*d^4*e^3 + 2*(9*A*B^
3*a^2*c^2 - 65*A^3*B*a*c^3)*d^3*e^4 + 3*(B^4*a^3*c - 28*A^2*B^2*a^2*c^2 + 27*A^4*a*c^3)*d^2*e^5 - 2*(5*A*B^3*a
^3*c - 81*A^3*B*a^2*c^2)*d*e^6 + (B^4*a^4 - 81*A^4*a^2*c^2)*e^7)*sqrt(e*x + d) - (6*A^2*B*a^2*c^4*d^5*e^3 + 5*
(3*A*B^2*a^3*c^3 - A^3*a^2*c^4)*d^4*e^4 + 6*(B^3*a^4*c^2 - 7*A^2*B*a^3*c^3)*d^3*e^5 - 12*(3*A*B^2*a^4*c^2 - 2*
A^3*a^3*c^3)*d^2*e^6 + 2*(B^3*a^5*c + 30*A^2*B*a^4*c^2)*d*e^7 - 3*(A*B^2*a^5*c + 9*A^3*a^4*c^2)*e^8 - (2*A*a^3
*c^7*d^9 + B*a^4*c^6*d^8*e - 10*A*a^4*c^6*d^7*e^2 - 2*B*a^5*c^5*d^6*e^3 + 18*A*a^5*c^5*d^5*e^4 - 14*A*a^6*c^4*
d^3*e^6 + 2*B*a^7*c^3*d^2*e^7 + 4*A*a^7*c^3*d*e^8 - B*a^8*c^2*e^9)*sqrt((36*A^2*B^2*c^4*d^6*e^4 + 12*(3*A*B^3*
a*c^3 - 5*A^3*B*c^4)*d^5*e^5 + (9*B^4*a^2*c^2 - 198*A^2*B^2*a*c^3 + 25*A^4*c^4)*d^4*e^6 - 8*(9*A*B^3*a^2*c^2 -
31*A^3*B*a*c^3)*d^3*e^7 + 6*(B^4*a^3*c + 40*A^2*B^2*a^2*c^2 - 15*A^4*a*c^3)*d^2*e^8 - 28*(A*B^3*a^3*c + 9*A^3
*B*a^2*c^2)*d*e^9 + (B^4*a^4 + 18*A^2*B^2*a^3*c + 81*A^4*a^2*c^2)*e^10)/(a^3*c^9*d^12 - 6*a^4*c^8*d^10*e^2 + 1
5*a^5*c^7*d^8*e^4 - 20*a^6*c^6*d^6*e^6 + 15*a^7*c^5*d^4*e^8 - 6*a^8*c^4*d^2*e^10 + a^9*c^3*e^12)))*sqrt((4*A^2
*c^3*d^5 + 4*A*B*a*c^2*d^4*e - 6*A*B*a^2*c*d^2*e^3 - 6*A*B*a^3*e^5 + (B^2*a^2*c - 15*A^2*a*c^2)*d^3*e^2 + 3*(B
^2*a^3 + 5*A^2*a^2*c)*d*e^4 + (a^3*c^4*d^6 - 3*a^4*c^3*d^4*e^2 + 3*a^5*c^2*d^2*e^4 - a^6*c*e^6)*sqrt((36*A^2*B
^2*c^4*d^6*e^4 + 12*(3*A*B^3*a*c^3 - 5*A^3*B*c^4)*d^5*e^5 + (9*B^4*a^2*c^2 - 198*A^2*B^2*a*c^3 + 25*A^4*c^4)*d
^4*e^6 - 8*(9*A*B^3*a^2*c^2 - 31*A^3*B*a*c^3)*d^3*e^7 + 6*(B^4*a^3*c + 40*A^2*B^2*a^2*c^2 - 15*A^4*a*c^3)*d^2*
e^8 - 28*(A*B^3*a^3*c + 9*A^3*B*a^2*c^2)*d*e^9 + (B^4*a^4 + 18*A^2*B^2*a^3*c + 81*A^4*a^2*c^2)*e^10)/(a^3*c^9*
d^12 - 6*a^4*c^8*d^10*e^2 + 15*a^5*c^7*d^8*e^4 - 20*a^6*c^6*d^6*e^6 + 15*a^7*c^5*d^4*e^8 - 6*a^8*c^4*d^2*e^10
+ a^9*c^3*e^12)))/(a^3*c^4*d^6 - 3*a^4*c^3*d^4*e^2 + 3*a^5*c^2*d^2*e^4 - a^6*c*e^6))) + (a^2*c*d^2 - a^3*e^2 -
(a*c^2*d^2 - a^2*c*e^2)*x^2)*sqrt((4*A^2*c^3*d^5 + 4*A*B*a*c^2*d^4*e - 6*A*B*a^2*c*d^2*e^3 - 6*A*B*a^3*e^5 +
(B^2*a^2*c - 15*A^2*a*c^2)*d^3*e^2 + 3*(B^2*a^3 + 5*A^2*a^2*c)*d*e^4 - (a^3*c^4*d^6 - 3*a^4*c^3*d^4*e^2 + 3*a^
5*c^2*d^2*e^4 - a^6*c*e^6)*sqrt((36*A^2*B^2*c^4*d^6*e^4 + 12*(3*A*B^3*a*c^3 - 5*A^3*B*c^4)*d^5*e^5 + (9*B^4*a^
2*c^2 - 198*A^2*B^2*a*c^3 + 25*A^4*c^4)*d^4*e^6 - 8*(9*A*B^3*a^2*c^2 - 31*A^3*B*a*c^3)*d^3*e^7 + 6*(B^4*a^3*c
+ 40*A^2*B^2*a^2*c^2 - 15*A^4*a*c^3)*d^2*e^8 - 28*(A*B^3*a^3*c + 9*A^3*B*a^2*c^2)*d*e^9 + (B^4*a^4 + 18*A^2*B^
2*a^3*c + 81*A^4*a^2*c^2)*e^10)/(a^3*c^9*d^12 - 6*a^4*c^8*d^10*e^2 + 15*a^5*c^7*d^8*e^4 - 20*a^6*c^6*d^6*e^6 +
15*a^7*c^5*d^4*e^8 - 6*a^8*c^4*d^2*e^10 + a^9*c^3*e^12)))/(a^3*c^4*d^6 - 3*a^4*c^3*d^4*e^2 + 3*a^5*c^2*d^2*e^
4 - a^6*c*e^6))*log(-(24*A^3*B*c^4*d^5*e^2 + 4*(9*A^2*B^2*a*c^3 - 5*A^4*c^4)*d^4*e^3 + 2*(9*A*B^3*a^2*c^2 - 65
*A^3*B*a*c^3)*d^3*e^4 + 3*(B^4*a^3*c - 28*A^2*B^2*a^2*c^2 + 27*A^4*a*c^3)*d^2*e^5 - 2*(5*A*B^3*a^3*c - 81*A^3*
B*a^2*c^2)*d*e^6 + (B^4*a^4 - 81*A^4*a^2*c^2)*e^7)*sqrt(e*x + d) + (6*A^2*B*a^2*c^4*d^5*e^3 + 5*(3*A*B^2*a^3*c
^3 - A^3*a^2*c^4)*d^4*e^4 + 6*(B^3*a^4*c^2 - 7*A^2*B*a^3*c^3)*d^3*e^5 - 12*(3*A*B^2*a^4*c^2 - 2*A^3*a^3*c^3)*d
^2*e^6 + 2*(B^3*a^5*c + 30*A^2*B*a^4*c^2)*d*e^7 - 3*(A*B^2*a^5*c + 9*A^3*a^4*c^2)*e^8 + (2*A*a^3*c^7*d^9 + B*a
^4*c^6*d^8*e - 10*A*a^4*c^6*d^7*e^2 - 2*B*a^5*c^5*d^6*e^3 + 18*A*a^5*c^5*d^5*e^4 - 14*A*a^6*c^4*d^3*e^6 + 2*B*
a^7*c^3*d^2*e^7 + 4*A*a^7*c^3*d*e^8 - B*a^8*c^2*e^9)*sqrt((36*A^2*B^2*c^4*d^6*e^4 + 12*(3*A*B^3*a*c^3 - 5*A^3*
B*c^4)*d^5*e^5 + (9*B^4*a^2*c^2 - 198*A^2*B^2*a*c^3 + 25*A^4*c^4)*d^4*e^6 - 8*(9*A*B^3*a^2*c^2 - 31*A^3*B*a*c^
3)*d^3*e^7 + 6*(B^4*a^3*c + 40*A^2*B^2*a^2*c^2 - 15*A^4*a*c^3)*d^2*e^8 - 28*(A*B^3*a^3*c + 9*A^3*B*a^2*c^2)*d*
e^9 + (B^4*a^4 + 18*A^2*B^2*a^3*c + 81*A^4*a^2*c^2)*e^10)/(a^3*c^9*d^12 - 6*a^4*c^8*d^10*e^2 + 15*a^5*c^7*d^8*
e^4 - 20*a^6*c^6*d^6*e^6 + 15*a^7*c^5*d^4*e^8 - 6*a^8*c^4*d^2*e^10 + a^9*c^3*e^12)))*sqrt((4*A^2*c^3*d^5 + 4*A
*B*a*c^2*d^4*e - 6*A*B*a^2*c*d^2*e^3 - 6*A*B*a^3*e^5 + (B^2*a^2*c - 15*A^2*a*c^2)*d^3*e^2 + 3*(B^2*a^3 + 5*A^2
*a^2*c)*d*e^4 - (a^3*c^4*d^6 - 3*a^4*c^3*d^4*e^2 + 3*a^5*c^2*d^2*e^4 - a^6*c*e^6)*sqrt((36*A^2*B^2*c^4*d^6*e^4
+ 12*(3*A*B^3*a*c^3 - 5*A^3*B*c^4)*d^5*e^5 + (9*B^4*a^2*c^2 - 198*A^2*B^2*a*c^3 + 25*A^4*c^4)*d^4*e^6 - 8*(9*
A*B^3*a^2*c^2 - 31*A^3*B*a*c^3)*d^3*e^7 + 6*(B^4*a^3*c + 40*A^2*B^2*a^2*c^2 - 15*A^4*a*c^3)*d^2*e^8 - 28*(A*B^
3*a^3*c + 9*A^3*B*a^2*c^2)*d*e^9 + (B^4*a^4 + 18*A^2*B^2*a^3*c + 81*A^4*a^2*c^2)*e^10)/(a^3*c^9*d^12 - 6*a^4*c
^8*d^10*e^2 + 15*a^5*c^7*d^8*e^4 - 20*a^6*c^6*d^6*e^6 + 15*a^7*c^5*d^4*e^8 - 6*a^8*c^4*d^2*e^10 + a^9*c^3*e^12
)))/(a^3*c^4*d^6 - 3*a^4*c^3*d^4*e^2 + 3*a^5*c^2*d^2*e^4 - a^6*c*e^6))) - (a^2*c*d^2 - a^3*e^2 - (a*c^2*d^2 -
a^2*c*e^2)*x^2)*sqrt((4*A^2*c^3*d^5 + 4*A*B*a*c^2*d^4*e - 6*A*B*a^2*c*d^2*e^3 - 6*A*B*a^3*e^5 + (B^2*a^2*c - 1
5*A^2*a*c^2)*d^3*e^2 + 3*(B^2*a^3 + 5*A^2*a^2*c)*d*e^4 - (a^3*c^4*d^6 - 3*a^4*c^3*d^4*e^2 + 3*a^5*c^2*d^2*e^4
- a^6*c*e^6)*sqrt((36*A^2*B^2*c^4*d^6*e^4 + 12*(3*A*B^3*a*c^3 - 5*A^3*B*c^4)*d^5*e^5 + (9*B^4*a^2*c^2 - 198*A^
2*B^2*a*c^3 + 25*A^4*c^4)*d^4*e^6 - 8*(9*A*B^3*a^2*c^2 - 31*A^3*B*a*c^3)*d^3*e^7 + 6*(B^4*a^3*c + 40*A^2*B^2*a
^2*c^2 - 15*A^4*a*c^3)*d^2*e^8 - 28*(A*B^3*a^3*c + 9*A^3*B*a^2*c^2)*d*e^9 + (B^4*a^4 + 18*A^2*B^2*a^3*c + 81*A
^4*a^2*c^2)*e^10)/(a^3*c^9*d^12 - 6*a^4*c^8*d^10*e^2 + 15*a^5*c^7*d^8*e^4 - 20*a^6*c^6*d^6*e^6 + 15*a^7*c^5*d^
4*e^8 - 6*a^8*c^4*d^2*e^10 + a^9*c^3*e^12)))/(a^3*c^4*d^6 - 3*a^4*c^3*d^4*e^2 + 3*a^5*c^2*d^2*e^4 - a^6*c*e^6)
)*log(-(24*A^3*B*c^4*d^5*e^2 + 4*(9*A^2*B^2*a*c^3 - 5*A^4*c^4)*d^4*e^3 + 2*(9*A*B^3*a^2*c^2 - 65*A^3*B*a*c^3)*
d^3*e^4 + 3*(B^4*a^3*c - 28*A^2*B^2*a^2*c^2 + 27*A^4*a*c^3)*d^2*e^5 - 2*(5*A*B^3*a^3*c - 81*A^3*B*a^2*c^2)*d*e
^6 + (B^4*a^4 - 81*A^4*a^2*c^2)*e^7)*sqrt(e*x + d) - (6*A^2*B*a^2*c^4*d^5*e^3 + 5*(3*A*B^2*a^3*c^3 - A^3*a^2*c
^4)*d^4*e^4 + 6*(B^3*a^4*c^2 - 7*A^2*B*a^3*c^3)*d^3*e^5 - 12*(3*A*B^2*a^4*c^2 - 2*A^3*a^3*c^3)*d^2*e^6 + 2*(B^
3*a^5*c + 30*A^2*B*a^4*c^2)*d*e^7 - 3*(A*B^2*a^5*c + 9*A^3*a^4*c^2)*e^8 + (2*A*a^3*c^7*d^9 + B*a^4*c^6*d^8*e -
10*A*a^4*c^6*d^7*e^2 - 2*B*a^5*c^5*d^6*e^3 + 18*A*a^5*c^5*d^5*e^4 - 14*A*a^6*c^4*d^3*e^6 + 2*B*a^7*c^3*d^2*e^
7 + 4*A*a^7*c^3*d*e^8 - B*a^8*c^2*e^9)*sqrt((36*A^2*B^2*c^4*d^6*e^4 + 12*(3*A*B^3*a*c^3 - 5*A^3*B*c^4)*d^5*e^5
+ (9*B^4*a^2*c^2 - 198*A^2*B^2*a*c^3 + 25*A^4*c^4)*d^4*e^6 - 8*(9*A*B^3*a^2*c^2 - 31*A^3*B*a*c^3)*d^3*e^7 + 6
*(B^4*a^3*c + 40*A^2*B^2*a^2*c^2 - 15*A^4*a*c^3)*d^2*e^8 - 28*(A*B^3*a^3*c + 9*A^3*B*a^2*c^2)*d*e^9 + (B^4*a^4
+ 18*A^2*B^2*a^3*c + 81*A^4*a^2*c^2)*e^10)/(a^3*c^9*d^12 - 6*a^4*c^8*d^10*e^2 + 15*a^5*c^7*d^8*e^4 - 20*a^6*c
^6*d^6*e^6 + 15*a^7*c^5*d^4*e^8 - 6*a^8*c^4*d^2*e^10 + a^9*c^3*e^12)))*sqrt((4*A^2*c^3*d^5 + 4*A*B*a*c^2*d^4*e
- 6*A*B*a^2*c*d^2*e^3 - 6*A*B*a^3*e^5 + (B^2*a^2*c - 15*A^2*a*c^2)*d^3*e^2 + 3*(B^2*a^3 + 5*A^2*a^2*c)*d*e^4
- (a^3*c^4*d^6 - 3*a^4*c^3*d^4*e^2 + 3*a^5*c^2*d^2*e^4 - a^6*c*e^6)*sqrt((36*A^2*B^2*c^4*d^6*e^4 + 12*(3*A*B^3
*a*c^3 - 5*A^3*B*c^4)*d^5*e^5 + (9*B^4*a^2*c^2 - 198*A^2*B^2*a*c^3 + 25*A^4*c^4)*d^4*e^6 - 8*(9*A*B^3*a^2*c^2
- 31*A^3*B*a*c^3)*d^3*e^7 + 6*(B^4*a^3*c + 40*A^2*B^2*a^2*c^2 - 15*A^4*a*c^3)*d^2*e^8 - 28*(A*B^3*a^3*c + 9*A^
3*B*a^2*c^2)*d*e^9 + (B^4*a^4 + 18*A^2*B^2*a^3*c + 81*A^4*a^2*c^2)*e^10)/(a^3*c^9*d^12 - 6*a^4*c^8*d^10*e^2 +
15*a^5*c^7*d^8*e^4 - 20*a^6*c^6*d^6*e^6 + 15*a^7*c^5*d^4*e^8 - 6*a^8*c^4*d^2*e^10 + a^9*c^3*e^12)))/(a^3*c^4*d
^6 - 3*a^4*c^3*d^4*e^2 + 3*a^5*c^2*d^2*e^4 - a^6*c*e^6))) - 4*(B*a*d - A*a*e + (A*c*d - B*a*e)*x)*sqrt(e*x + d
))/(a^2*c*d^2 - a^3*e^2 - (a*c^2*d^2 - a^2*c*e^2)*x^2)

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giac [B]  time = 0.91, size = 1156, normalized size = 4.62 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(e*x+d)^(1/2)/(-c*x^2+a)^2,x, algorithm="giac")

[Out]

-1/4*((a*c*d^2*e - a^2*e^3)^2*A*c*d*abs(c)*e - (a*c*d^2*e - a^2*e^3)^2*B*a*abs(c)*e^2 + (sqrt(a*c)*c^2*d^4*e -
4*sqrt(a*c)*a*c*d^2*e^3 + 3*sqrt(a*c)*a^2*e^5)*A*abs(a*c*d^2*e - a^2*e^3)*abs(c) + 2*(sqrt(a*c)*a*c*d^3*e^2 -
sqrt(a*c)*a^2*d*e^4)*B*abs(a*c*d^2*e - a^2*e^3)*abs(c) - (2*a*c^4*d^7*e - 7*a^2*c^3*d^5*e^3 + 8*a^3*c^2*d^3*e
^5 - 3*a^4*c*d*e^7)*A*abs(c) - (a^2*c^3*d^6*e^2 - 2*a^3*c^2*d^4*e^4 + a^4*c*d^2*e^6)*B*abs(c))*arctan(sqrt(x*e
+ d)/sqrt(-(a*c^2*d^3 - a^2*c*d*e^2 + sqrt((a*c^2*d^3 - a^2*c*d*e^2)^2 - (a*c^2*d^4 - 2*a^2*c*d^2*e^2 + a^3*e
^4)*(a*c^2*d^2 - a^2*c*e^2)))/(a*c^2*d^2 - a^2*c*e^2)))/((a^2*c^3*d^4*e - sqrt(a*c)*a*c^3*d^5 + 2*sqrt(a*c)*a^
2*c^2*d^3*e^2 - 2*a^3*c^2*d^2*e^3 - sqrt(a*c)*a^3*c*d*e^4 + a^4*c*e^5)*sqrt(-c^2*d - sqrt(a*c)*c*e)*abs(a*c*d^
2*e - a^2*e^3)) - 1/4*((a*c*d^2*e - a^2*e^3)^2*sqrt(a*c)*A*c*d*abs(c)*e - (a*c*d^2*e - a^2*e^3)^2*sqrt(a*c)*B*
a*abs(c)*e^2 - (a*c^3*d^4*e - 4*a^2*c^2*d^2*e^3 + 3*a^3*c*e^5)*A*abs(a*c*d^2*e - a^2*e^3)*abs(c) - 2*(a^2*c^2*
d^3*e^2 - a^3*c*d*e^4)*B*abs(a*c*d^2*e - a^2*e^3)*abs(c) - (2*sqrt(a*c)*a*c^4*d^7*e - 7*sqrt(a*c)*a^2*c^3*d^5*
e^3 + 8*sqrt(a*c)*a^3*c^2*d^3*e^5 - 3*sqrt(a*c)*a^4*c*d*e^7)*A*abs(c) - (sqrt(a*c)*a^2*c^3*d^6*e^2 - 2*sqrt(a*
c)*a^3*c^2*d^4*e^4 + sqrt(a*c)*a^4*c*d^2*e^6)*B*abs(c))*arctan(sqrt(x*e + d)/sqrt(-(a*c^2*d^3 - a^2*c*d*e^2 -
sqrt((a*c^2*d^3 - a^2*c*d*e^2)^2 - (a*c^2*d^4 - 2*a^2*c*d^2*e^2 + a^3*e^4)*(a*c^2*d^2 - a^2*c*e^2)))/(a*c^2*d^
2 - a^2*c*e^2)))/((a^2*c^4*d^5 + sqrt(a*c)*a^2*c^3*d^4*e - 2*a^3*c^3*d^3*e^2 - 2*sqrt(a*c)*a^3*c^2*d^2*e^3 + a
^4*c^2*d*e^4 + sqrt(a*c)*a^4*c*e^5)*sqrt(-c^2*d + sqrt(a*c)*c*e)*abs(a*c*d^2*e - a^2*e^3)) - 1/2*((x*e + d)^(3
/2)*A*c*d*e - sqrt(x*e + d)*A*c*d^2*e - (x*e + d)^(3/2)*B*a*e^2 + 2*sqrt(x*e + d)*B*a*d*e^2 - sqrt(x*e + d)*A*
a*e^3)/((a*c*d^2 - a^2*e^2)*((x*e + d)^2*c - 2*(x*e + d)*c*d + c*d^2 - a*e^2))

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maple [B]  time = 0.29, size = 635, normalized size = 2.54 \[ \frac {A \,c^{2} d e \arctanh \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \left (c d +\sqrt {a c \,e^{2}}\right ) \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, a}-\frac {A \,c^{2} d e \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \left (-c d +\sqrt {a c \,e^{2}}\right ) \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, a}+\frac {B c \,e^{2} \arctanh \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{4 \sqrt {a c \,e^{2}}\, \left (c d +\sqrt {a c \,e^{2}}\right ) \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}-\frac {B c \,e^{2} \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{4 \sqrt {a c \,e^{2}}\, \left (-c d +\sqrt {a c \,e^{2}}\right ) \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {3 A c e \arctanh \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{4 \left (c d +\sqrt {a c \,e^{2}}\right ) \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, a}+\frac {3 A c e \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{4 \left (-c d +\sqrt {a c \,e^{2}}\right ) \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, a}-\frac {\sqrt {e x +d}\, B \,e^{2}}{4 \sqrt {a c \,e^{2}}\, \left (c d +\sqrt {a c \,e^{2}}\right ) \left (e x -\frac {\sqrt {a c \,e^{2}}}{c}\right )}+\frac {\sqrt {e x +d}\, B \,e^{2}}{4 \sqrt {a c \,e^{2}}\, \left (c d -\sqrt {a c \,e^{2}}\right ) \left (e x +\frac {\sqrt {a c \,e^{2}}}{c}\right )}-\frac {\sqrt {e x +d}\, A e}{4 \left (c d +\sqrt {a c \,e^{2}}\right ) \left (e x -\frac {\sqrt {a c \,e^{2}}}{c}\right ) a}-\frac {\sqrt {e x +d}\, A e}{4 \left (c d -\sqrt {a c \,e^{2}}\right ) \left (e x +\frac {\sqrt {a c \,e^{2}}}{c}\right ) a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/(e*x+d)^(1/2)/(-c*x^2+a)^2,x)

[Out]

-1/4*e^2/(a*c*e^2)^(1/2)/(c*d+(a*c*e^2)^(1/2))*(e*x+d)^(1/2)/(e*x-(a*c*e^2)^(1/2)/c)*B-1/4*e/a/(c*d+(a*c*e^2)^
(1/2))*(e*x+d)^(1/2)/(e*x-(a*c*e^2)^(1/2)/c)*A+1/2*c^2*e/(a*c*e^2)^(1/2)/a/(c*d+(a*c*e^2)^(1/2))/((c*d+(a*c*e^
2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)*A*d+1/4*c*e^2/(a*c*e^2)^(1/2)/(c*d
+(a*c*e^2)^(1/2))/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)*B+3
/4*c*e/a/(c*d+(a*c*e^2)^(1/2))/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)
^(1/2)*c)*A+1/4*e^2/(a*c*e^2)^(1/2)/(c*d-(a*c*e^2)^(1/2))*(e*x+d)^(1/2)/(e*x+(a*c*e^2)^(1/2)/c)*B-1/4*e/a/(c*d
-(a*c*e^2)^(1/2))*(e*x+d)^(1/2)/(e*x+(a*c*e^2)^(1/2)/c)*A-1/2*c^2*e/(a*c*e^2)^(1/2)/a/(-c*d+(a*c*e^2)^(1/2))/(
(-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)*A*d-1/4*c*e^2/(a*c*e^
2)^(1/2)/(-c*d+(a*c*e^2)^(1/2))/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*
c)^(1/2)*c)*B+3/4*c*e/a/(-c*d+(a*c*e^2)^(1/2))/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)/((-c*d+(a
*c*e^2)^(1/2))*c)^(1/2)*c)*A

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {B x + A}{{\left (c x^{2} - a\right )}^{2} \sqrt {e x + d}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(e*x+d)^(1/2)/(-c*x^2+a)^2,x, algorithm="maxima")

[Out]

integrate((B*x + A)/((c*x^2 - a)^2*sqrt(e*x + d)), x)

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mupad [B]  time = 6.24, size = 10862, normalized size = 43.45 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)/((a - c*x^2)^2*(d + e*x)^(1/2)),x)

[Out]

atan(((((192*A*a^5*c^3*e^7 - 128*B*a^5*c^3*d*e^6 + 64*A*a^3*c^5*d^4*e^3 - 256*A*a^4*c^4*d^2*e^5 + 128*B*a^4*c^
4*d^3*e^4)/(8*(a^5*e^4 + a^3*c^2*d^4 - 2*a^4*c*d^2*e^2)) + ((d + e*x)^(1/2)*((4*A^2*a^3*c^5*d^5 + B^2*a^2*e^5*
(a^9*c^3)^(1/2) - 15*A^2*a^4*c^4*d^3*e^2 + B^2*a^5*c^3*d^3*e^2 - 6*A*B*a^6*c^2*e^5 - 5*A^2*c^2*d^2*e^3*(a^9*c^
3)^(1/2) + 15*A^2*a^5*c^3*d*e^4 + 3*B^2*a^6*c^2*d*e^4 + 9*A^2*a*c*e^5*(a^9*c^3)^(1/2) + 6*A*B*c^2*d^3*e^2*(a^9
*c^3)^(1/2) + 4*A*B*a^4*c^4*d^4*e + 3*B^2*a*c*d^2*e^3*(a^9*c^3)^(1/2) - 6*A*B*a^5*c^3*d^2*e^3 - 14*A*B*a*c*d*e
^4*(a^9*c^3)^(1/2))/(64*(a^6*c^6*d^6 - a^9*c^3*e^6 - 3*a^7*c^5*d^4*e^2 + 3*a^8*c^4*d^2*e^4)))^(1/2)*(64*a^5*c^
4*d*e^6 + 64*a^3*c^6*d^5*e^2 - 128*a^4*c^5*d^3*e^4))/(a^4*e^4 + a^2*c^2*d^4 - 2*a^3*c*d^2*e^2))*((4*A^2*a^3*c^
5*d^5 + B^2*a^2*e^5*(a^9*c^3)^(1/2) - 15*A^2*a^4*c^4*d^3*e^2 + B^2*a^5*c^3*d^3*e^2 - 6*A*B*a^6*c^2*e^5 - 5*A^2
*c^2*d^2*e^3*(a^9*c^3)^(1/2) + 15*A^2*a^5*c^3*d*e^4 + 3*B^2*a^6*c^2*d*e^4 + 9*A^2*a*c*e^5*(a^9*c^3)^(1/2) + 6*
A*B*c^2*d^3*e^2*(a^9*c^3)^(1/2) + 4*A*B*a^4*c^4*d^4*e + 3*B^2*a*c*d^2*e^3*(a^9*c^3)^(1/2) - 6*A*B*a^5*c^3*d^2*
e^3 - 14*A*B*a*c*d*e^4*(a^9*c^3)^(1/2))/(64*(a^6*c^6*d^6 - a^9*c^3*e^6 - 3*a^7*c^5*d^4*e^2 + 3*a^8*c^4*d^2*e^4
)))^(1/2) - ((d + e*x)^(1/2)*(9*A^2*a^2*c^3*e^6 + B^2*a^3*c^2*e^6 + 4*A^2*c^5*d^4*e^2 + B^2*a^2*c^3*d^2*e^4 -
11*A^2*a*c^4*d^2*e^4 + 4*A*B*a*c^4*d^3*e^3 - 8*A*B*a^2*c^3*d*e^5))/(a^4*e^4 + a^2*c^2*d^4 - 2*a^3*c*d^2*e^2))*
((4*A^2*a^3*c^5*d^5 + B^2*a^2*e^5*(a^9*c^3)^(1/2) - 15*A^2*a^4*c^4*d^3*e^2 + B^2*a^5*c^3*d^3*e^2 - 6*A*B*a^6*c
^2*e^5 - 5*A^2*c^2*d^2*e^3*(a^9*c^3)^(1/2) + 15*A^2*a^5*c^3*d*e^4 + 3*B^2*a^6*c^2*d*e^4 + 9*A^2*a*c*e^5*(a^9*c
^3)^(1/2) + 6*A*B*c^2*d^3*e^2*(a^9*c^3)^(1/2) + 4*A*B*a^4*c^4*d^4*e + 3*B^2*a*c*d^2*e^3*(a^9*c^3)^(1/2) - 6*A*
B*a^5*c^3*d^2*e^3 - 14*A*B*a*c*d*e^4*(a^9*c^3)^(1/2))/(64*(a^6*c^6*d^6 - a^9*c^3*e^6 - 3*a^7*c^5*d^4*e^2 + 3*a
^8*c^4*d^2*e^4)))^(1/2)*1i - (((192*A*a^5*c^3*e^7 - 128*B*a^5*c^3*d*e^6 + 64*A*a^3*c^5*d^4*e^3 - 256*A*a^4*c^4
*d^2*e^5 + 128*B*a^4*c^4*d^3*e^4)/(8*(a^5*e^4 + a^3*c^2*d^4 - 2*a^4*c*d^2*e^2)) - ((d + e*x)^(1/2)*((4*A^2*a^3
*c^5*d^5 + B^2*a^2*e^5*(a^9*c^3)^(1/2) - 15*A^2*a^4*c^4*d^3*e^2 + B^2*a^5*c^3*d^3*e^2 - 6*A*B*a^6*c^2*e^5 - 5*
A^2*c^2*d^2*e^3*(a^9*c^3)^(1/2) + 15*A^2*a^5*c^3*d*e^4 + 3*B^2*a^6*c^2*d*e^4 + 9*A^2*a*c*e^5*(a^9*c^3)^(1/2) +
6*A*B*c^2*d^3*e^2*(a^9*c^3)^(1/2) + 4*A*B*a^4*c^4*d^4*e + 3*B^2*a*c*d^2*e^3*(a^9*c^3)^(1/2) - 6*A*B*a^5*c^3*d
^2*e^3 - 14*A*B*a*c*d*e^4*(a^9*c^3)^(1/2))/(64*(a^6*c^6*d^6 - a^9*c^3*e^6 - 3*a^7*c^5*d^4*e^2 + 3*a^8*c^4*d^2*
e^4)))^(1/2)*(64*a^5*c^4*d*e^6 + 64*a^3*c^6*d^5*e^2 - 128*a^4*c^5*d^3*e^4))/(a^4*e^4 + a^2*c^2*d^4 - 2*a^3*c*d
^2*e^2))*((4*A^2*a^3*c^5*d^5 + B^2*a^2*e^5*(a^9*c^3)^(1/2) - 15*A^2*a^4*c^4*d^3*e^2 + B^2*a^5*c^3*d^3*e^2 - 6*
A*B*a^6*c^2*e^5 - 5*A^2*c^2*d^2*e^3*(a^9*c^3)^(1/2) + 15*A^2*a^5*c^3*d*e^4 + 3*B^2*a^6*c^2*d*e^4 + 9*A^2*a*c*e
^5*(a^9*c^3)^(1/2) + 6*A*B*c^2*d^3*e^2*(a^9*c^3)^(1/2) + 4*A*B*a^4*c^4*d^4*e + 3*B^2*a*c*d^2*e^3*(a^9*c^3)^(1/
2) - 6*A*B*a^5*c^3*d^2*e^3 - 14*A*B*a*c*d*e^4*(a^9*c^3)^(1/2))/(64*(a^6*c^6*d^6 - a^9*c^3*e^6 - 3*a^7*c^5*d^4*
e^2 + 3*a^8*c^4*d^2*e^4)))^(1/2) + ((d + e*x)^(1/2)*(9*A^2*a^2*c^3*e^6 + B^2*a^3*c^2*e^6 + 4*A^2*c^5*d^4*e^2 +
B^2*a^2*c^3*d^2*e^4 - 11*A^2*a*c^4*d^2*e^4 + 4*A*B*a*c^4*d^3*e^3 - 8*A*B*a^2*c^3*d*e^5))/(a^4*e^4 + a^2*c^2*d
^4 - 2*a^3*c*d^2*e^2))*((4*A^2*a^3*c^5*d^5 + B^2*a^2*e^5*(a^9*c^3)^(1/2) - 15*A^2*a^4*c^4*d^3*e^2 + B^2*a^5*c^
3*d^3*e^2 - 6*A*B*a^6*c^2*e^5 - 5*A^2*c^2*d^2*e^3*(a^9*c^3)^(1/2) + 15*A^2*a^5*c^3*d*e^4 + 3*B^2*a^6*c^2*d*e^4
+ 9*A^2*a*c*e^5*(a^9*c^3)^(1/2) + 6*A*B*c^2*d^3*e^2*(a^9*c^3)^(1/2) + 4*A*B*a^4*c^4*d^4*e + 3*B^2*a*c*d^2*e^3
*(a^9*c^3)^(1/2) - 6*A*B*a^5*c^3*d^2*e^3 - 14*A*B*a*c*d*e^4*(a^9*c^3)^(1/2))/(64*(a^6*c^6*d^6 - a^9*c^3*e^6 -
3*a^7*c^5*d^4*e^2 + 3*a^8*c^4*d^2*e^4)))^(1/2)*1i)/((B^3*a^3*c*e^6 - 4*A^3*c^4*d^3*e^3 + 9*A^3*a*c^3*d*e^5 - 9
*A^2*B*a^2*c^2*e^6 + 3*A*B^2*a^2*c^2*d*e^5)/(4*(a^5*e^4 + a^3*c^2*d^4 - 2*a^4*c*d^2*e^2)) + (((192*A*a^5*c^3*e
^7 - 128*B*a^5*c^3*d*e^6 + 64*A*a^3*c^5*d^4*e^3 - 256*A*a^4*c^4*d^2*e^5 + 128*B*a^4*c^4*d^3*e^4)/(8*(a^5*e^4 +
a^3*c^2*d^4 - 2*a^4*c*d^2*e^2)) + ((d + e*x)^(1/2)*((4*A^2*a^3*c^5*d^5 + B^2*a^2*e^5*(a^9*c^3)^(1/2) - 15*A^2
*a^4*c^4*d^3*e^2 + B^2*a^5*c^3*d^3*e^2 - 6*A*B*a^6*c^2*e^5 - 5*A^2*c^2*d^2*e^3*(a^9*c^3)^(1/2) + 15*A^2*a^5*c^
3*d*e^4 + 3*B^2*a^6*c^2*d*e^4 + 9*A^2*a*c*e^5*(a^9*c^3)^(1/2) + 6*A*B*c^2*d^3*e^2*(a^9*c^3)^(1/2) + 4*A*B*a^4*
c^4*d^4*e + 3*B^2*a*c*d^2*e^3*(a^9*c^3)^(1/2) - 6*A*B*a^5*c^3*d^2*e^3 - 14*A*B*a*c*d*e^4*(a^9*c^3)^(1/2))/(64*
(a^6*c^6*d^6 - a^9*c^3*e^6 - 3*a^7*c^5*d^4*e^2 + 3*a^8*c^4*d^2*e^4)))^(1/2)*(64*a^5*c^4*d*e^6 + 64*a^3*c^6*d^5
*e^2 - 128*a^4*c^5*d^3*e^4))/(a^4*e^4 + a^2*c^2*d^4 - 2*a^3*c*d^2*e^2))*((4*A^2*a^3*c^5*d^5 + B^2*a^2*e^5*(a^9
*c^3)^(1/2) - 15*A^2*a^4*c^4*d^3*e^2 + B^2*a^5*c^3*d^3*e^2 - 6*A*B*a^6*c^2*e^5 - 5*A^2*c^2*d^2*e^3*(a^9*c^3)^(
1/2) + 15*A^2*a^5*c^3*d*e^4 + 3*B^2*a^6*c^2*d*e^4 + 9*A^2*a*c*e^5*(a^9*c^3)^(1/2) + 6*A*B*c^2*d^3*e^2*(a^9*c^3
)^(1/2) + 4*A*B*a^4*c^4*d^4*e + 3*B^2*a*c*d^2*e^3*(a^9*c^3)^(1/2) - 6*A*B*a^5*c^3*d^2*e^3 - 14*A*B*a*c*d*e^4*(
a^9*c^3)^(1/2))/(64*(a^6*c^6*d^6 - a^9*c^3*e^6 - 3*a^7*c^5*d^4*e^2 + 3*a^8*c^4*d^2*e^4)))^(1/2) - ((d + e*x)^(
1/2)*(9*A^2*a^2*c^3*e^6 + B^2*a^3*c^2*e^6 + 4*A^2*c^5*d^4*e^2 + B^2*a^2*c^3*d^2*e^4 - 11*A^2*a*c^4*d^2*e^4 + 4
*A*B*a*c^4*d^3*e^3 - 8*A*B*a^2*c^3*d*e^5))/(a^4*e^4 + a^2*c^2*d^4 - 2*a^3*c*d^2*e^2))*((4*A^2*a^3*c^5*d^5 + B^
2*a^2*e^5*(a^9*c^3)^(1/2) - 15*A^2*a^4*c^4*d^3*e^2 + B^2*a^5*c^3*d^3*e^2 - 6*A*B*a^6*c^2*e^5 - 5*A^2*c^2*d^2*e
^3*(a^9*c^3)^(1/2) + 15*A^2*a^5*c^3*d*e^4 + 3*B^2*a^6*c^2*d*e^4 + 9*A^2*a*c*e^5*(a^9*c^3)^(1/2) + 6*A*B*c^2*d^
3*e^2*(a^9*c^3)^(1/2) + 4*A*B*a^4*c^4*d^4*e + 3*B^2*a*c*d^2*e^3*(a^9*c^3)^(1/2) - 6*A*B*a^5*c^3*d^2*e^3 - 14*A
*B*a*c*d*e^4*(a^9*c^3)^(1/2))/(64*(a^6*c^6*d^6 - a^9*c^3*e^6 - 3*a^7*c^5*d^4*e^2 + 3*a^8*c^4*d^2*e^4)))^(1/2)
+ (((192*A*a^5*c^3*e^7 - 128*B*a^5*c^3*d*e^6 + 64*A*a^3*c^5*d^4*e^3 - 256*A*a^4*c^4*d^2*e^5 + 128*B*a^4*c^4*d^
3*e^4)/(8*(a^5*e^4 + a^3*c^2*d^4 - 2*a^4*c*d^2*e^2)) - ((d + e*x)^(1/2)*((4*A^2*a^3*c^5*d^5 + B^2*a^2*e^5*(a^9
*c^3)^(1/2) - 15*A^2*a^4*c^4*d^3*e^2 + B^2*a^5*c^3*d^3*e^2 - 6*A*B*a^6*c^2*e^5 - 5*A^2*c^2*d^2*e^3*(a^9*c^3)^(
1/2) + 15*A^2*a^5*c^3*d*e^4 + 3*B^2*a^6*c^2*d*e^4 + 9*A^2*a*c*e^5*(a^9*c^3)^(1/2) + 6*A*B*c^2*d^3*e^2*(a^9*c^3
)^(1/2) + 4*A*B*a^4*c^4*d^4*e + 3*B^2*a*c*d^2*e^3*(a^9*c^3)^(1/2) - 6*A*B*a^5*c^3*d^2*e^3 - 14*A*B*a*c*d*e^4*(
a^9*c^3)^(1/2))/(64*(a^6*c^6*d^6 - a^9*c^3*e^6 - 3*a^7*c^5*d^4*e^2 + 3*a^8*c^4*d^2*e^4)))^(1/2)*(64*a^5*c^4*d*
e^6 + 64*a^3*c^6*d^5*e^2 - 128*a^4*c^5*d^3*e^4))/(a^4*e^4 + a^2*c^2*d^4 - 2*a^3*c*d^2*e^2))*((4*A^2*a^3*c^5*d^
5 + B^2*a^2*e^5*(a^9*c^3)^(1/2) - 15*A^2*a^4*c^4*d^3*e^2 + B^2*a^5*c^3*d^3*e^2 - 6*A*B*a^6*c^2*e^5 - 5*A^2*c^2
*d^2*e^3*(a^9*c^3)^(1/2) + 15*A^2*a^5*c^3*d*e^4 + 3*B^2*a^6*c^2*d*e^4 + 9*A^2*a*c*e^5*(a^9*c^3)^(1/2) + 6*A*B*
c^2*d^3*e^2*(a^9*c^3)^(1/2) + 4*A*B*a^4*c^4*d^4*e + 3*B^2*a*c*d^2*e^3*(a^9*c^3)^(1/2) - 6*A*B*a^5*c^3*d^2*e^3
- 14*A*B*a*c*d*e^4*(a^9*c^3)^(1/2))/(64*(a^6*c^6*d^6 - a^9*c^3*e^6 - 3*a^7*c^5*d^4*e^2 + 3*a^8*c^4*d^2*e^4)))^
(1/2) + ((d + e*x)^(1/2)*(9*A^2*a^2*c^3*e^6 + B^2*a^3*c^2*e^6 + 4*A^2*c^5*d^4*e^2 + B^2*a^2*c^3*d^2*e^4 - 11*A
^2*a*c^4*d^2*e^4 + 4*A*B*a*c^4*d^3*e^3 - 8*A*B*a^2*c^3*d*e^5))/(a^4*e^4 + a^2*c^2*d^4 - 2*a^3*c*d^2*e^2))*((4*
A^2*a^3*c^5*d^5 + B^2*a^2*e^5*(a^9*c^3)^(1/2) - 15*A^2*a^4*c^4*d^3*e^2 + B^2*a^5*c^3*d^3*e^2 - 6*A*B*a^6*c^2*e
^5 - 5*A^2*c^2*d^2*e^3*(a^9*c^3)^(1/2) + 15*A^2*a^5*c^3*d*e^4 + 3*B^2*a^6*c^2*d*e^4 + 9*A^2*a*c*e^5*(a^9*c^3)^
(1/2) + 6*A*B*c^2*d^3*e^2*(a^9*c^3)^(1/2) + 4*A*B*a^4*c^4*d^4*e + 3*B^2*a*c*d^2*e^3*(a^9*c^3)^(1/2) - 6*A*B*a^
5*c^3*d^2*e^3 - 14*A*B*a*c*d*e^4*(a^9*c^3)^(1/2))/(64*(a^6*c^6*d^6 - a^9*c^3*e^6 - 3*a^7*c^5*d^4*e^2 + 3*a^8*c
^4*d^2*e^4)))^(1/2)))*((4*A^2*a^3*c^5*d^5 + B^2*a^2*e^5*(a^9*c^3)^(1/2) - 15*A^2*a^4*c^4*d^3*e^2 + B^2*a^5*c^3
*d^3*e^2 - 6*A*B*a^6*c^2*e^5 - 5*A^2*c^2*d^2*e^3*(a^9*c^3)^(1/2) + 15*A^2*a^5*c^3*d*e^4 + 3*B^2*a^6*c^2*d*e^4
+ 9*A^2*a*c*e^5*(a^9*c^3)^(1/2) + 6*A*B*c^2*d^3*e^2*(a^9*c^3)^(1/2) + 4*A*B*a^4*c^4*d^4*e + 3*B^2*a*c*d^2*e^3*
(a^9*c^3)^(1/2) - 6*A*B*a^5*c^3*d^2*e^3 - 14*A*B*a*c*d*e^4*(a^9*c^3)^(1/2))/(64*(a^6*c^6*d^6 - a^9*c^3*e^6 - 3
*a^7*c^5*d^4*e^2 + 3*a^8*c^4*d^2*e^4)))^(1/2)*2i + atan(((((192*A*a^5*c^3*e^7 - 128*B*a^5*c^3*d*e^6 + 64*A*a^3
*c^5*d^4*e^3 - 256*A*a^4*c^4*d^2*e^5 + 128*B*a^4*c^4*d^3*e^4)/(8*(a^5*e^4 + a^3*c^2*d^4 - 2*a^4*c*d^2*e^2)) +
((d + e*x)^(1/2)*((4*A^2*a^3*c^5*d^5 - B^2*a^2*e^5*(a^9*c^3)^(1/2) - 15*A^2*a^4*c^4*d^3*e^2 + B^2*a^5*c^3*d^3*
e^2 - 6*A*B*a^6*c^2*e^5 + 5*A^2*c^2*d^2*e^3*(a^9*c^3)^(1/2) + 15*A^2*a^5*c^3*d*e^4 + 3*B^2*a^6*c^2*d*e^4 - 9*A
^2*a*c*e^5*(a^9*c^3)^(1/2) - 6*A*B*c^2*d^3*e^2*(a^9*c^3)^(1/2) + 4*A*B*a^4*c^4*d^4*e - 3*B^2*a*c*d^2*e^3*(a^9*
c^3)^(1/2) - 6*A*B*a^5*c^3*d^2*e^3 + 14*A*B*a*c*d*e^4*(a^9*c^3)^(1/2))/(64*(a^6*c^6*d^6 - a^9*c^3*e^6 - 3*a^7*
c^5*d^4*e^2 + 3*a^8*c^4*d^2*e^4)))^(1/2)*(64*a^5*c^4*d*e^6 + 64*a^3*c^6*d^5*e^2 - 128*a^4*c^5*d^3*e^4))/(a^4*e
^4 + a^2*c^2*d^4 - 2*a^3*c*d^2*e^2))*((4*A^2*a^3*c^5*d^5 - B^2*a^2*e^5*(a^9*c^3)^(1/2) - 15*A^2*a^4*c^4*d^3*e^
2 + B^2*a^5*c^3*d^3*e^2 - 6*A*B*a^6*c^2*e^5 + 5*A^2*c^2*d^2*e^3*(a^9*c^3)^(1/2) + 15*A^2*a^5*c^3*d*e^4 + 3*B^2
*a^6*c^2*d*e^4 - 9*A^2*a*c*e^5*(a^9*c^3)^(1/2) - 6*A*B*c^2*d^3*e^2*(a^9*c^3)^(1/2) + 4*A*B*a^4*c^4*d^4*e - 3*B
^2*a*c*d^2*e^3*(a^9*c^3)^(1/2) - 6*A*B*a^5*c^3*d^2*e^3 + 14*A*B*a*c*d*e^4*(a^9*c^3)^(1/2))/(64*(a^6*c^6*d^6 -
a^9*c^3*e^6 - 3*a^7*c^5*d^4*e^2 + 3*a^8*c^4*d^2*e^4)))^(1/2) - ((d + e*x)^(1/2)*(9*A^2*a^2*c^3*e^6 + B^2*a^3*c
^2*e^6 + 4*A^2*c^5*d^4*e^2 + B^2*a^2*c^3*d^2*e^4 - 11*A^2*a*c^4*d^2*e^4 + 4*A*B*a*c^4*d^3*e^3 - 8*A*B*a^2*c^3*
d*e^5))/(a^4*e^4 + a^2*c^2*d^4 - 2*a^3*c*d^2*e^2))*((4*A^2*a^3*c^5*d^5 - B^2*a^2*e^5*(a^9*c^3)^(1/2) - 15*A^2*
a^4*c^4*d^3*e^2 + B^2*a^5*c^3*d^3*e^2 - 6*A*B*a^6*c^2*e^5 + 5*A^2*c^2*d^2*e^3*(a^9*c^3)^(1/2) + 15*A^2*a^5*c^3
*d*e^4 + 3*B^2*a^6*c^2*d*e^4 - 9*A^2*a*c*e^5*(a^9*c^3)^(1/2) - 6*A*B*c^2*d^3*e^2*(a^9*c^3)^(1/2) + 4*A*B*a^4*c
^4*d^4*e - 3*B^2*a*c*d^2*e^3*(a^9*c^3)^(1/2) - 6*A*B*a^5*c^3*d^2*e^3 + 14*A*B*a*c*d*e^4*(a^9*c^3)^(1/2))/(64*(
a^6*c^6*d^6 - a^9*c^3*e^6 - 3*a^7*c^5*d^4*e^2 + 3*a^8*c^4*d^2*e^4)))^(1/2)*1i - (((192*A*a^5*c^3*e^7 - 128*B*a
^5*c^3*d*e^6 + 64*A*a^3*c^5*d^4*e^3 - 256*A*a^4*c^4*d^2*e^5 + 128*B*a^4*c^4*d^3*e^4)/(8*(a^5*e^4 + a^3*c^2*d^4
- 2*a^4*c*d^2*e^2)) - ((d + e*x)^(1/2)*((4*A^2*a^3*c^5*d^5 - B^2*a^2*e^5*(a^9*c^3)^(1/2) - 15*A^2*a^4*c^4*d^3
*e^2 + B^2*a^5*c^3*d^3*e^2 - 6*A*B*a^6*c^2*e^5 + 5*A^2*c^2*d^2*e^3*(a^9*c^3)^(1/2) + 15*A^2*a^5*c^3*d*e^4 + 3*
B^2*a^6*c^2*d*e^4 - 9*A^2*a*c*e^5*(a^9*c^3)^(1/2) - 6*A*B*c^2*d^3*e^2*(a^9*c^3)^(1/2) + 4*A*B*a^4*c^4*d^4*e -
3*B^2*a*c*d^2*e^3*(a^9*c^3)^(1/2) - 6*A*B*a^5*c^3*d^2*e^3 + 14*A*B*a*c*d*e^4*(a^9*c^3)^(1/2))/(64*(a^6*c^6*d^6
- a^9*c^3*e^6 - 3*a^7*c^5*d^4*e^2 + 3*a^8*c^4*d^2*e^4)))^(1/2)*(64*a^5*c^4*d*e^6 + 64*a^3*c^6*d^5*e^2 - 128*a
^4*c^5*d^3*e^4))/(a^4*e^4 + a^2*c^2*d^4 - 2*a^3*c*d^2*e^2))*((4*A^2*a^3*c^5*d^5 - B^2*a^2*e^5*(a^9*c^3)^(1/2)
- 15*A^2*a^4*c^4*d^3*e^2 + B^2*a^5*c^3*d^3*e^2 - 6*A*B*a^6*c^2*e^5 + 5*A^2*c^2*d^2*e^3*(a^9*c^3)^(1/2) + 15*A^
2*a^5*c^3*d*e^4 + 3*B^2*a^6*c^2*d*e^4 - 9*A^2*a*c*e^5*(a^9*c^3)^(1/2) - 6*A*B*c^2*d^3*e^2*(a^9*c^3)^(1/2) + 4*
A*B*a^4*c^4*d^4*e - 3*B^2*a*c*d^2*e^3*(a^9*c^3)^(1/2) - 6*A*B*a^5*c^3*d^2*e^3 + 14*A*B*a*c*d*e^4*(a^9*c^3)^(1/
2))/(64*(a^6*c^6*d^6 - a^9*c^3*e^6 - 3*a^7*c^5*d^4*e^2 + 3*a^8*c^4*d^2*e^4)))^(1/2) + ((d + e*x)^(1/2)*(9*A^2*
a^2*c^3*e^6 + B^2*a^3*c^2*e^6 + 4*A^2*c^5*d^4*e^2 + B^2*a^2*c^3*d^2*e^4 - 11*A^2*a*c^4*d^2*e^4 + 4*A*B*a*c^4*d
^3*e^3 - 8*A*B*a^2*c^3*d*e^5))/(a^4*e^4 + a^2*c^2*d^4 - 2*a^3*c*d^2*e^2))*((4*A^2*a^3*c^5*d^5 - B^2*a^2*e^5*(a
^9*c^3)^(1/2) - 15*A^2*a^4*c^4*d^3*e^2 + B^2*a^5*c^3*d^3*e^2 - 6*A*B*a^6*c^2*e^5 + 5*A^2*c^2*d^2*e^3*(a^9*c^3)
^(1/2) + 15*A^2*a^5*c^3*d*e^4 + 3*B^2*a^6*c^2*d*e^4 - 9*A^2*a*c*e^5*(a^9*c^3)^(1/2) - 6*A*B*c^2*d^3*e^2*(a^9*c
^3)^(1/2) + 4*A*B*a^4*c^4*d^4*e - 3*B^2*a*c*d^2*e^3*(a^9*c^3)^(1/2) - 6*A*B*a^5*c^3*d^2*e^3 + 14*A*B*a*c*d*e^4
*(a^9*c^3)^(1/2))/(64*(a^6*c^6*d^6 - a^9*c^3*e^6 - 3*a^7*c^5*d^4*e^2 + 3*a^8*c^4*d^2*e^4)))^(1/2)*1i)/((B^3*a^
3*c*e^6 - 4*A^3*c^4*d^3*e^3 + 9*A^3*a*c^3*d*e^5 - 9*A^2*B*a^2*c^2*e^6 + 3*A*B^2*a^2*c^2*d*e^5)/(4*(a^5*e^4 + a
^3*c^2*d^4 - 2*a^4*c*d^2*e^2)) + (((192*A*a^5*c^3*e^7 - 128*B*a^5*c^3*d*e^6 + 64*A*a^3*c^5*d^4*e^3 - 256*A*a^4
*c^4*d^2*e^5 + 128*B*a^4*c^4*d^3*e^4)/(8*(a^5*e^4 + a^3*c^2*d^4 - 2*a^4*c*d^2*e^2)) + ((d + e*x)^(1/2)*((4*A^2
*a^3*c^5*d^5 - B^2*a^2*e^5*(a^9*c^3)^(1/2) - 15*A^2*a^4*c^4*d^3*e^2 + B^2*a^5*c^3*d^3*e^2 - 6*A*B*a^6*c^2*e^5
+ 5*A^2*c^2*d^2*e^3*(a^9*c^3)^(1/2) + 15*A^2*a^5*c^3*d*e^4 + 3*B^2*a^6*c^2*d*e^4 - 9*A^2*a*c*e^5*(a^9*c^3)^(1/
2) - 6*A*B*c^2*d^3*e^2*(a^9*c^3)^(1/2) + 4*A*B*a^4*c^4*d^4*e - 3*B^2*a*c*d^2*e^3*(a^9*c^3)^(1/2) - 6*A*B*a^5*c
^3*d^2*e^3 + 14*A*B*a*c*d*e^4*(a^9*c^3)^(1/2))/(64*(a^6*c^6*d^6 - a^9*c^3*e^6 - 3*a^7*c^5*d^4*e^2 + 3*a^8*c^4*
d^2*e^4)))^(1/2)*(64*a^5*c^4*d*e^6 + 64*a^3*c^6*d^5*e^2 - 128*a^4*c^5*d^3*e^4))/(a^4*e^4 + a^2*c^2*d^4 - 2*a^3
*c*d^2*e^2))*((4*A^2*a^3*c^5*d^5 - B^2*a^2*e^5*(a^9*c^3)^(1/2) - 15*A^2*a^4*c^4*d^3*e^2 + B^2*a^5*c^3*d^3*e^2
- 6*A*B*a^6*c^2*e^5 + 5*A^2*c^2*d^2*e^3*(a^9*c^3)^(1/2) + 15*A^2*a^5*c^3*d*e^4 + 3*B^2*a^6*c^2*d*e^4 - 9*A^2*a
*c*e^5*(a^9*c^3)^(1/2) - 6*A*B*c^2*d^3*e^2*(a^9*c^3)^(1/2) + 4*A*B*a^4*c^4*d^4*e - 3*B^2*a*c*d^2*e^3*(a^9*c^3)
^(1/2) - 6*A*B*a^5*c^3*d^2*e^3 + 14*A*B*a*c*d*e^4*(a^9*c^3)^(1/2))/(64*(a^6*c^6*d^6 - a^9*c^3*e^6 - 3*a^7*c^5*
d^4*e^2 + 3*a^8*c^4*d^2*e^4)))^(1/2) - ((d + e*x)^(1/2)*(9*A^2*a^2*c^3*e^6 + B^2*a^3*c^2*e^6 + 4*A^2*c^5*d^4*e
^2 + B^2*a^2*c^3*d^2*e^4 - 11*A^2*a*c^4*d^2*e^4 + 4*A*B*a*c^4*d^3*e^3 - 8*A*B*a^2*c^3*d*e^5))/(a^4*e^4 + a^2*c
^2*d^4 - 2*a^3*c*d^2*e^2))*((4*A^2*a^3*c^5*d^5 - B^2*a^2*e^5*(a^9*c^3)^(1/2) - 15*A^2*a^4*c^4*d^3*e^2 + B^2*a^
5*c^3*d^3*e^2 - 6*A*B*a^6*c^2*e^5 + 5*A^2*c^2*d^2*e^3*(a^9*c^3)^(1/2) + 15*A^2*a^5*c^3*d*e^4 + 3*B^2*a^6*c^2*d
*e^4 - 9*A^2*a*c*e^5*(a^9*c^3)^(1/2) - 6*A*B*c^2*d^3*e^2*(a^9*c^3)^(1/2) + 4*A*B*a^4*c^4*d^4*e - 3*B^2*a*c*d^2
*e^3*(a^9*c^3)^(1/2) - 6*A*B*a^5*c^3*d^2*e^3 + 14*A*B*a*c*d*e^4*(a^9*c^3)^(1/2))/(64*(a^6*c^6*d^6 - a^9*c^3*e^
6 - 3*a^7*c^5*d^4*e^2 + 3*a^8*c^4*d^2*e^4)))^(1/2) + (((192*A*a^5*c^3*e^7 - 128*B*a^5*c^3*d*e^6 + 64*A*a^3*c^5
*d^4*e^3 - 256*A*a^4*c^4*d^2*e^5 + 128*B*a^4*c^4*d^3*e^4)/(8*(a^5*e^4 + a^3*c^2*d^4 - 2*a^4*c*d^2*e^2)) - ((d
+ e*x)^(1/2)*((4*A^2*a^3*c^5*d^5 - B^2*a^2*e^5*(a^9*c^3)^(1/2) - 15*A^2*a^4*c^4*d^3*e^2 + B^2*a^5*c^3*d^3*e^2
- 6*A*B*a^6*c^2*e^5 + 5*A^2*c^2*d^2*e^3*(a^9*c^3)^(1/2) + 15*A^2*a^5*c^3*d*e^4 + 3*B^2*a^6*c^2*d*e^4 - 9*A^2*a
*c*e^5*(a^9*c^3)^(1/2) - 6*A*B*c^2*d^3*e^2*(a^9*c^3)^(1/2) + 4*A*B*a^4*c^4*d^4*e - 3*B^2*a*c*d^2*e^3*(a^9*c^3)
^(1/2) - 6*A*B*a^5*c^3*d^2*e^3 + 14*A*B*a*c*d*e^4*(a^9*c^3)^(1/2))/(64*(a^6*c^6*d^6 - a^9*c^3*e^6 - 3*a^7*c^5*
d^4*e^2 + 3*a^8*c^4*d^2*e^4)))^(1/2)*(64*a^5*c^4*d*e^6 + 64*a^3*c^6*d^5*e^2 - 128*a^4*c^5*d^3*e^4))/(a^4*e^4 +
a^2*c^2*d^4 - 2*a^3*c*d^2*e^2))*((4*A^2*a^3*c^5*d^5 - B^2*a^2*e^5*(a^9*c^3)^(1/2) - 15*A^2*a^4*c^4*d^3*e^2 +
B^2*a^5*c^3*d^3*e^2 - 6*A*B*a^6*c^2*e^5 + 5*A^2*c^2*d^2*e^3*(a^9*c^3)^(1/2) + 15*A^2*a^5*c^3*d*e^4 + 3*B^2*a^6
*c^2*d*e^4 - 9*A^2*a*c*e^5*(a^9*c^3)^(1/2) - 6*A*B*c^2*d^3*e^2*(a^9*c^3)^(1/2) + 4*A*B*a^4*c^4*d^4*e - 3*B^2*a
*c*d^2*e^3*(a^9*c^3)^(1/2) - 6*A*B*a^5*c^3*d^2*e^3 + 14*A*B*a*c*d*e^4*(a^9*c^3)^(1/2))/(64*(a^6*c^6*d^6 - a^9*
c^3*e^6 - 3*a^7*c^5*d^4*e^2 + 3*a^8*c^4*d^2*e^4)))^(1/2) + ((d + e*x)^(1/2)*(9*A^2*a^2*c^3*e^6 + B^2*a^3*c^2*e
^6 + 4*A^2*c^5*d^4*e^2 + B^2*a^2*c^3*d^2*e^4 - 11*A^2*a*c^4*d^2*e^4 + 4*A*B*a*c^4*d^3*e^3 - 8*A*B*a^2*c^3*d*e^
5))/(a^4*e^4 + a^2*c^2*d^4 - 2*a^3*c*d^2*e^2))*((4*A^2*a^3*c^5*d^5 - B^2*a^2*e^5*(a^9*c^3)^(1/2) - 15*A^2*a^4*
c^4*d^3*e^2 + B^2*a^5*c^3*d^3*e^2 - 6*A*B*a^6*c^2*e^5 + 5*A^2*c^2*d^2*e^3*(a^9*c^3)^(1/2) + 15*A^2*a^5*c^3*d*e
^4 + 3*B^2*a^6*c^2*d*e^4 - 9*A^2*a*c*e^5*(a^9*c^3)^(1/2) - 6*A*B*c^2*d^3*e^2*(a^9*c^3)^(1/2) + 4*A*B*a^4*c^4*d
^4*e - 3*B^2*a*c*d^2*e^3*(a^9*c^3)^(1/2) - 6*A*B*a^5*c^3*d^2*e^3 + 14*A*B*a*c*d*e^4*(a^9*c^3)^(1/2))/(64*(a^6*
c^6*d^6 - a^9*c^3*e^6 - 3*a^7*c^5*d^4*e^2 + 3*a^8*c^4*d^2*e^4)))^(1/2)))*((4*A^2*a^3*c^5*d^5 - B^2*a^2*e^5*(a^
9*c^3)^(1/2) - 15*A^2*a^4*c^4*d^3*e^2 + B^2*a^5*c^3*d^3*e^2 - 6*A*B*a^6*c^2*e^5 + 5*A^2*c^2*d^2*e^3*(a^9*c^3)^
(1/2) + 15*A^2*a^5*c^3*d*e^4 + 3*B^2*a^6*c^2*d*e^4 - 9*A^2*a*c*e^5*(a^9*c^3)^(1/2) - 6*A*B*c^2*d^3*e^2*(a^9*c^
3)^(1/2) + 4*A*B*a^4*c^4*d^4*e - 3*B^2*a*c*d^2*e^3*(a^9*c^3)^(1/2) - 6*A*B*a^5*c^3*d^2*e^3 + 14*A*B*a*c*d*e^4*
(a^9*c^3)^(1/2))/(64*(a^6*c^6*d^6 - a^9*c^3*e^6 - 3*a^7*c^5*d^4*e^2 + 3*a^8*c^4*d^2*e^4)))^(1/2)*2i - (((B*a*e
^2 - A*c*d*e)*(d + e*x)^(3/2))/(2*a*(a*e^2 - c*d^2)) + ((d + e*x)^(1/2)*(A*a*e^3 - 2*B*a*d*e^2 + A*c*d^2*e))/(
2*a*(a*e^2 - c*d^2)))/(c*(d + e*x)^2 - a*e^2 + c*d^2 - 2*c*d*(d + e*x))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(e*x+d)**(1/2)/(-c*x**2+a)**2,x)

[Out]

Timed out

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