3.13.80 \(\int \frac {(A+B x) (d+e x)^{3/2}}{(a-c x^2)^2} \, dx\)

Optimal. Leaf size=238 \[ -\frac {\sqrt {\sqrt {c} d-\sqrt {a} e} \left (\sqrt {a} A \sqrt {c} e-3 a B e+2 A c d\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{4 a^{3/2} c^{7/4}}+\frac {\sqrt {\sqrt {a} e+\sqrt {c} d} \left (-\sqrt {a} A \sqrt {c} e-3 a B e+2 A c d\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{4 a^{3/2} c^{7/4}}+\frac {\sqrt {d+e x} (x (a B e+A c d)+a (A e+B d))}{2 a c \left (a-c x^2\right )} \]

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Rubi [A]  time = 0.39, antiderivative size = 238, 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 = {819, 827, 1166, 208} \begin {gather*} -\frac {\sqrt {\sqrt {c} d-\sqrt {a} e} \left (\sqrt {a} A \sqrt {c} e-3 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^{7/4}}+\frac {\sqrt {\sqrt {a} e+\sqrt {c} d} \left (-\sqrt {a} A \sqrt {c} e-3 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^{7/4}}+\frac {\sqrt {d+e x} (x (a B e+A c d)+a (A e+B d))}{2 a c \left (a-c x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(d + e*x)^(3/2))/(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*(a - c*x^2)) - (Sqrt[Sqrt[c]*d - Sqrt[a]*e]*(2*A*c*
d - 3*a*B*e + 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^
(7/4)) + (Sqrt[Sqrt[c]*d + Sqrt[a]*e]*(2*A*c*d - 3*a*B*e - 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^(7/4))

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 819

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

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) (d+e x)^{3/2}}{\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 c \left (a-c x^2\right )}-\frac {\int \frac {\frac {1}{2} \left (-2 A c d^2+3 a B d e+a A e^2\right )-\frac {1}{2} e (A c d-3 a B e) x}{\sqrt {d+e x} \left (a-c x^2\right )} \, dx}{2 a c}\\ &=\frac {\sqrt {d+e x} (a (B d+A e)+(A c d+a B e) x)}{2 a c \left (a-c x^2\right )}-\frac {\operatorname {Subst}\left (\int \frac {\frac {1}{2} d e (A c d-3 a B e)+\frac {1}{2} e \left (-2 A c d^2+3 a B d e+a A e^2\right )-\frac {1}{2} e (A c d-3 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}\\ &=\frac {\sqrt {d+e x} (a (B d+A e)+(A c d+a B e) x)}{2 a c \left (a-c x^2\right )}+\frac {\left (\left (\sqrt {c} d+\sqrt {a} e\right ) \left (2 A c d-3 a B e-\sqrt {a} A \sqrt {c} e\right )\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} c}-\frac {\left (\left (\sqrt {c} d-\sqrt {a} e\right ) \left (2 A c d-3 a B e+\sqrt {a} A \sqrt {c} e\right )\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} c}\\ &=\frac {\sqrt {d+e x} (a (B d+A e)+(A c d+a B e) x)}{2 a c \left (a-c x^2\right )}-\frac {\sqrt {\sqrt {c} d-\sqrt {a} e} \left (2 A c d-3 a B e+\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^{7/4}}+\frac {\sqrt {\sqrt {c} d+\sqrt {a} e} \left (2 A c d-3 a B e-\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^{7/4}}\\ \end {align*}

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Mathematica [A]  time = 0.37, size = 244, normalized size = 1.03 \begin {gather*} \frac {2 \sqrt {a} c^{3/4} \sqrt {d+e x} (a A e+a B (d+e x)+A c d x)+\left (c x^2-a\right ) \sqrt {\sqrt {c} d-\sqrt {a} e} \left (\sqrt {a} A \sqrt {c} e-3 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 )-\left (c x^2-a\right ) \sqrt {\sqrt {a} e+\sqrt {c} d} \left (-\sqrt {a} A \sqrt {c} e-3 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^{7/4} \left (a-c x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(d + e*x)^(3/2))/(a - c*x^2)^2,x]

[Out]

(2*Sqrt[a]*c^(3/4)*Sqrt[d + e*x]*(a*A*e + A*c*d*x + a*B*(d + e*x)) + Sqrt[Sqrt[c]*d - Sqrt[a]*e]*(2*A*c*d - 3*
a*B*e + Sqrt[a]*A*Sqrt[c]*e)*(-a + c*x^2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]] - Sqrt[
Sqrt[c]*d + Sqrt[a]*e]*(2*A*c*d - 3*a*B*e - Sqrt[a]*A*Sqrt[c]*e)*(-a + c*x^2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/
Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(4*a^(3/2)*c^(7/4)*(a - c*x^2))

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IntegrateAlgebraic [A]  time = 1.19, size = 378, normalized size = 1.59 \begin {gather*} \frac {\left (-3 a^{3/2} B e^2+\sqrt {a} A c d e-a A \sqrt {c} e^2-3 a B \sqrt {c} d e+2 A c^{3/2} d^2\right ) \tan ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {-\sqrt {a} \sqrt {c} e-c d}}{\sqrt {a} e+\sqrt {c} d}\right )}{4 a^{3/2} c^{3/2} \sqrt {-\sqrt {c} \left (\sqrt {a} e+\sqrt {c} d\right )}}+\frac {\left (-3 a^{3/2} B e^2+\sqrt {a} A c d e+a A \sqrt {c} e^2+3 a B \sqrt {c} d e-2 A c^{3/2} d^2\right ) \tan ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {\sqrt {a} \sqrt {c} e-c d}}{\sqrt {c} d-\sqrt {a} e}\right )}{4 a^{3/2} c^{3/2} \sqrt {-\sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right )}}+\frac {e \sqrt {d+e x} \left (a A e^2+a B e (d+e x)-A c d^2+A c d (d+e x)\right )}{2 a c \left (a e^2-c d^2+2 c d (d+e x)-c (d+e x)^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((A + B*x)*(d + e*x)^(3/2))/(a - c*x^2)^2,x]

[Out]

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

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fricas [B]  time = 0.65, size = 2327, normalized size = 9.78

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(3/2)/(-c*x^2+a)^2,x, algorithm="fricas")

[Out]

1/8*((a*c^2*x^2 - a^2*c)*sqrt((4*A^2*c^2*d^3 - 12*A*B*a*c*d^2*e + 6*A*B*a^2*e^3 + a^3*c^3*sqrt((36*A^2*B^2*c^2
*d^2*e^4 - 12*(9*A*B^3*a*c + A^3*B*c^2)*d*e^5 + (81*B^4*a^2 + 18*A^2*B^2*a*c + A^4*c^2)*e^6)/(a^3*c^7)) + 3*(3
*B^2*a^2 - A^2*a*c)*d*e^2)/(a^3*c^3))*log((24*A^3*B*c^3*d^3*e^2 - 4*(27*A^2*B^2*a*c^2 + A^4*c^3)*d^2*e^3 + 6*(
27*A*B^3*a^2*c + A^3*B*a*c^2)*d*e^4 - (81*B^4*a^3 - A^4*a*c^2)*e^5)*sqrt(e*x + d) + (6*A^2*B*a^2*c^3*d*e^3 - (
9*A*B^2*a^3*c^2 + A^3*a^2*c^3)*e^4 - (2*A*a^3*c^6*d - 3*B*a^4*c^5*e)*sqrt((36*A^2*B^2*c^2*d^2*e^4 - 12*(9*A*B^
3*a*c + A^3*B*c^2)*d*e^5 + (81*B^4*a^2 + 18*A^2*B^2*a*c + A^4*c^2)*e^6)/(a^3*c^7)))*sqrt((4*A^2*c^2*d^3 - 12*A
*B*a*c*d^2*e + 6*A*B*a^2*e^3 + a^3*c^3*sqrt((36*A^2*B^2*c^2*d^2*e^4 - 12*(9*A*B^3*a*c + A^3*B*c^2)*d*e^5 + (81
*B^4*a^2 + 18*A^2*B^2*a*c + A^4*c^2)*e^6)/(a^3*c^7)) + 3*(3*B^2*a^2 - A^2*a*c)*d*e^2)/(a^3*c^3))) - (a*c^2*x^2
- a^2*c)*sqrt((4*A^2*c^2*d^3 - 12*A*B*a*c*d^2*e + 6*A*B*a^2*e^3 + a^3*c^3*sqrt((36*A^2*B^2*c^2*d^2*e^4 - 12*(
9*A*B^3*a*c + A^3*B*c^2)*d*e^5 + (81*B^4*a^2 + 18*A^2*B^2*a*c + A^4*c^2)*e^6)/(a^3*c^7)) + 3*(3*B^2*a^2 - A^2*
a*c)*d*e^2)/(a^3*c^3))*log((24*A^3*B*c^3*d^3*e^2 - 4*(27*A^2*B^2*a*c^2 + A^4*c^3)*d^2*e^3 + 6*(27*A*B^3*a^2*c
+ A^3*B*a*c^2)*d*e^4 - (81*B^4*a^3 - A^4*a*c^2)*e^5)*sqrt(e*x + d) - (6*A^2*B*a^2*c^3*d*e^3 - (9*A*B^2*a^3*c^2
+ A^3*a^2*c^3)*e^4 - (2*A*a^3*c^6*d - 3*B*a^4*c^5*e)*sqrt((36*A^2*B^2*c^2*d^2*e^4 - 12*(9*A*B^3*a*c + A^3*B*c
^2)*d*e^5 + (81*B^4*a^2 + 18*A^2*B^2*a*c + A^4*c^2)*e^6)/(a^3*c^7)))*sqrt((4*A^2*c^2*d^3 - 12*A*B*a*c*d^2*e +
6*A*B*a^2*e^3 + a^3*c^3*sqrt((36*A^2*B^2*c^2*d^2*e^4 - 12*(9*A*B^3*a*c + A^3*B*c^2)*d*e^5 + (81*B^4*a^2 + 18*A
^2*B^2*a*c + A^4*c^2)*e^6)/(a^3*c^7)) + 3*(3*B^2*a^2 - A^2*a*c)*d*e^2)/(a^3*c^3))) + (a*c^2*x^2 - a^2*c)*sqrt(
(4*A^2*c^2*d^3 - 12*A*B*a*c*d^2*e + 6*A*B*a^2*e^3 - a^3*c^3*sqrt((36*A^2*B^2*c^2*d^2*e^4 - 12*(9*A*B^3*a*c + A
^3*B*c^2)*d*e^5 + (81*B^4*a^2 + 18*A^2*B^2*a*c + A^4*c^2)*e^6)/(a^3*c^7)) + 3*(3*B^2*a^2 - A^2*a*c)*d*e^2)/(a^
3*c^3))*log((24*A^3*B*c^3*d^3*e^2 - 4*(27*A^2*B^2*a*c^2 + A^4*c^3)*d^2*e^3 + 6*(27*A*B^3*a^2*c + A^3*B*a*c^2)*
d*e^4 - (81*B^4*a^3 - A^4*a*c^2)*e^5)*sqrt(e*x + d) + (6*A^2*B*a^2*c^3*d*e^3 - (9*A*B^2*a^3*c^2 + A^3*a^2*c^3)
*e^4 + (2*A*a^3*c^6*d - 3*B*a^4*c^5*e)*sqrt((36*A^2*B^2*c^2*d^2*e^4 - 12*(9*A*B^3*a*c + A^3*B*c^2)*d*e^5 + (81
*B^4*a^2 + 18*A^2*B^2*a*c + A^4*c^2)*e^6)/(a^3*c^7)))*sqrt((4*A^2*c^2*d^3 - 12*A*B*a*c*d^2*e + 6*A*B*a^2*e^3 -
a^3*c^3*sqrt((36*A^2*B^2*c^2*d^2*e^4 - 12*(9*A*B^3*a*c + A^3*B*c^2)*d*e^5 + (81*B^4*a^2 + 18*A^2*B^2*a*c + A^
4*c^2)*e^6)/(a^3*c^7)) + 3*(3*B^2*a^2 - A^2*a*c)*d*e^2)/(a^3*c^3))) - (a*c^2*x^2 - a^2*c)*sqrt((4*A^2*c^2*d^3
- 12*A*B*a*c*d^2*e + 6*A*B*a^2*e^3 - a^3*c^3*sqrt((36*A^2*B^2*c^2*d^2*e^4 - 12*(9*A*B^3*a*c + A^3*B*c^2)*d*e^5
+ (81*B^4*a^2 + 18*A^2*B^2*a*c + A^4*c^2)*e^6)/(a^3*c^7)) + 3*(3*B^2*a^2 - A^2*a*c)*d*e^2)/(a^3*c^3))*log((24
*A^3*B*c^3*d^3*e^2 - 4*(27*A^2*B^2*a*c^2 + A^4*c^3)*d^2*e^3 + 6*(27*A*B^3*a^2*c + A^3*B*a*c^2)*d*e^4 - (81*B^4
*a^3 - A^4*a*c^2)*e^5)*sqrt(e*x + d) - (6*A^2*B*a^2*c^3*d*e^3 - (9*A*B^2*a^3*c^2 + A^3*a^2*c^3)*e^4 + (2*A*a^3
*c^6*d - 3*B*a^4*c^5*e)*sqrt((36*A^2*B^2*c^2*d^2*e^4 - 12*(9*A*B^3*a*c + A^3*B*c^2)*d*e^5 + (81*B^4*a^2 + 18*A
^2*B^2*a*c + A^4*c^2)*e^6)/(a^3*c^7)))*sqrt((4*A^2*c^2*d^3 - 12*A*B*a*c*d^2*e + 6*A*B*a^2*e^3 - a^3*c^3*sqrt((
36*A^2*B^2*c^2*d^2*e^4 - 12*(9*A*B^3*a*c + A^3*B*c^2)*d*e^5 + (81*B^4*a^2 + 18*A^2*B^2*a*c + A^4*c^2)*e^6)/(a^
3*c^7)) + 3*(3*B^2*a^2 - A^2*a*c)*d*e^2)/(a^3*c^3))) - 4*(B*a*d + A*a*e + (A*c*d + B*a*e)*x)*sqrt(e*x + d))/(a
*c^2*x^2 - a^2*c)

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giac [B]  time = 0.71, size = 533, normalized size = 2.24 \begin {gather*} \frac {{\left (3 \, \sqrt {a c} B a^{2} c^{3} d^{2} e + \sqrt {a c} A a^{2} c^{3} d e^{2} - 3 \, \sqrt {a c} B a^{3} c^{2} e^{3} + {\left (a c^{3} d^{2} e - a^{2} c^{2} e^{3}\right )} A {\left | a \right |} {\left | c \right |} - {\left (2 \, \sqrt {a c} a c^{4} d^{3} - \sqrt {a c} a^{2} c^{3} d e^{2}\right )} A\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {a c^{2} d + \sqrt {a^{2} c^{4} d^{2} - {\left (a c^{2} d^{2} - a^{2} c e^{2}\right )} a c^{2}}}{a c^{2}}}}\right )}{4 \, {\left (a^{2} c^{4} d - \sqrt {a c} a^{2} c^{3} e\right )} \sqrt {-c^{2} d - \sqrt {a c} c e} {\left | a \right |}} - \frac {{\left (3 \, B a^{2} c^{3} d^{2} e + A a^{2} c^{3} d e^{2} - 3 \, B a^{3} c^{2} e^{3} - {\left (\sqrt {a c} c^{2} d^{2} e - \sqrt {a c} a c e^{3}\right )} A {\left | a \right |} {\left | c \right |} - {\left (2 \, a c^{4} d^{3} - a^{2} c^{3} d e^{2}\right )} A\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {a c^{2} d - \sqrt {a^{2} c^{4} d^{2} - {\left (a c^{2} d^{2} - a^{2} c e^{2}\right )} a c^{2}}}{a c^{2}}}}\right )}{4 \, {\left (a^{2} c^{3} e + \sqrt {a c} a c^{3} d\right )} \sqrt {-c^{2} d + \sqrt {a c} c e} {\left | a \right |}} - \frac {{\left (x e + d\right )}^{\frac {3}{2}} A c d e - \sqrt {x e + d} A c d^{2} e + {\left (x e + d\right )}^{\frac {3}{2}} B a e^{2} + \sqrt {x e + d} A a e^{3}}{2 \, {\left ({\left (x e + d\right )}^{2} c - 2 \, {\left (x e + d\right )} c d + c d^{2} - a e^{2}\right )} a c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(3/2)/(-c*x^2+a)^2,x, algorithm="giac")

[Out]

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

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maple [B]  time = 0.11, size = 694, normalized size = 2.92 \begin {gather*} \frac {A c \,d^{2} 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}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, a}+\frac {A c \,d^{2} 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}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, a}-\frac {A \,e^{3} \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}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}-\frac {A \,e^{3} \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}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}-\frac {3 B d \,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}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}-\frac {3 B d \,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}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {\sqrt {e x +d}\, A \,d^{2} e}{2 \left (c \,e^{2} x^{2}-a \,e^{2}\right ) a}+\frac {A d e \arctanh \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{4 \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, a}-\frac {A d e \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{4 \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, a}-\frac {\sqrt {e x +d}\, A \,e^{3}}{2 \left (c \,e^{2} x^{2}-a \,e^{2}\right ) c}-\frac {3 B \,e^{2} \arctanh \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{4 \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, c}+\frac {3 B \,e^{2} \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{4 \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, c}-\frac {\left (e x +d \right )^{\frac {3}{2}} A d e}{2 \left (c \,e^{2} x^{2}-a \,e^{2}\right ) a}-\frac {\left (e x +d \right )^{\frac {3}{2}} B \,e^{2}}{2 \left (c \,e^{2} x^{2}-a \,e^{2}\right ) c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^(3/2)/(-c*x^2+a)^2,x)

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (B x + A\right )} {\left (e x + d\right )}^{\frac {3}{2}}}{{\left (c x^{2} - a\right )}^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(3/2)/(-c*x^2+a)^2,x, algorithm="maxima")

[Out]

integrate((B*x + A)*(e*x + d)^(3/2)/(c*x^2 - a)^2, x)

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mupad [B]  time = 0.92, size = 5212, normalized size = 21.90

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(d + e*x)^(3/2))/(a - c*x^2)^2,x)

[Out]

atan(((((64*A*a^4*c^4*e^5 - 64*A*a^3*c^5*d^2*e^3)/(8*a^3*c^2) - 64*a*c^4*d*e^2*(d + e*x)^(1/2)*((4*A^2*a^3*c^6
*d^3 + 9*B^2*a*e^3*(a^9*c^7)^(1/2) + A^2*c*e^3*(a^9*c^7)^(1/2) + 6*A*B*a^5*c^4*e^3 - 3*A^2*a^4*c^5*d*e^2 + 9*B
^2*a^5*c^4*d*e^2 - 12*A*B*a^4*c^5*d^2*e - 6*A*B*c*d*e^2*(a^9*c^7)^(1/2))/(64*a^6*c^7))^(1/2))*((4*A^2*a^3*c^6*
d^3 + 9*B^2*a*e^3*(a^9*c^7)^(1/2) + A^2*c*e^3*(a^9*c^7)^(1/2) + 6*A*B*a^5*c^4*e^3 - 3*A^2*a^4*c^5*d*e^2 + 9*B^
2*a^5*c^4*d*e^2 - 12*A*B*a^4*c^5*d^2*e - 6*A*B*c*d*e^2*(a^9*c^7)^(1/2))/(64*a^6*c^7))^(1/2) + ((d + e*x)^(1/2)
*(9*B^2*a^3*e^6 + 4*A^2*c^3*d^4*e^2 + A^2*a^2*c*e^6 - 3*A^2*a*c^2*d^2*e^4 + 9*B^2*a^2*c*d^2*e^4 - 12*A*B*a*c^2
*d^3*e^3))/a^2)*((4*A^2*a^3*c^6*d^3 + 9*B^2*a*e^3*(a^9*c^7)^(1/2) + A^2*c*e^3*(a^9*c^7)^(1/2) + 6*A*B*a^5*c^4*
e^3 - 3*A^2*a^4*c^5*d*e^2 + 9*B^2*a^5*c^4*d*e^2 - 12*A*B*a^4*c^5*d^2*e - 6*A*B*c*d*e^2*(a^9*c^7)^(1/2))/(64*a^
6*c^7))^(1/2)*1i - (((64*A*a^4*c^4*e^5 - 64*A*a^3*c^5*d^2*e^3)/(8*a^3*c^2) + 64*a*c^4*d*e^2*(d + e*x)^(1/2)*((
4*A^2*a^3*c^6*d^3 + 9*B^2*a*e^3*(a^9*c^7)^(1/2) + A^2*c*e^3*(a^9*c^7)^(1/2) + 6*A*B*a^5*c^4*e^3 - 3*A^2*a^4*c^
5*d*e^2 + 9*B^2*a^5*c^4*d*e^2 - 12*A*B*a^4*c^5*d^2*e - 6*A*B*c*d*e^2*(a^9*c^7)^(1/2))/(64*a^6*c^7))^(1/2))*((4
*A^2*a^3*c^6*d^3 + 9*B^2*a*e^3*(a^9*c^7)^(1/2) + A^2*c*e^3*(a^9*c^7)^(1/2) + 6*A*B*a^5*c^4*e^3 - 3*A^2*a^4*c^5
*d*e^2 + 9*B^2*a^5*c^4*d*e^2 - 12*A*B*a^4*c^5*d^2*e - 6*A*B*c*d*e^2*(a^9*c^7)^(1/2))/(64*a^6*c^7))^(1/2) - ((d
+ e*x)^(1/2)*(9*B^2*a^3*e^6 + 4*A^2*c^3*d^4*e^2 + A^2*a^2*c*e^6 - 3*A^2*a*c^2*d^2*e^4 + 9*B^2*a^2*c*d^2*e^4 -
12*A*B*a*c^2*d^3*e^3))/a^2)*((4*A^2*a^3*c^6*d^3 + 9*B^2*a*e^3*(a^9*c^7)^(1/2) + A^2*c*e^3*(a^9*c^7)^(1/2) + 6
*A*B*a^5*c^4*e^3 - 3*A^2*a^4*c^5*d*e^2 + 9*B^2*a^5*c^4*d*e^2 - 12*A*B*a^4*c^5*d^2*e - 6*A*B*c*d*e^2*(a^9*c^7)^
(1/2))/(64*a^6*c^7))^(1/2)*1i)/((((64*A*a^4*c^4*e^5 - 64*A*a^3*c^5*d^2*e^3)/(8*a^3*c^2) - 64*a*c^4*d*e^2*(d +
e*x)^(1/2)*((4*A^2*a^3*c^6*d^3 + 9*B^2*a*e^3*(a^9*c^7)^(1/2) + A^2*c*e^3*(a^9*c^7)^(1/2) + 6*A*B*a^5*c^4*e^3 -
3*A^2*a^4*c^5*d*e^2 + 9*B^2*a^5*c^4*d*e^2 - 12*A*B*a^4*c^5*d^2*e - 6*A*B*c*d*e^2*(a^9*c^7)^(1/2))/(64*a^6*c^7
))^(1/2))*((4*A^2*a^3*c^6*d^3 + 9*B^2*a*e^3*(a^9*c^7)^(1/2) + A^2*c*e^3*(a^9*c^7)^(1/2) + 6*A*B*a^5*c^4*e^3 -
3*A^2*a^4*c^5*d*e^2 + 9*B^2*a^5*c^4*d*e^2 - 12*A*B*a^4*c^5*d^2*e - 6*A*B*c*d*e^2*(a^9*c^7)^(1/2))/(64*a^6*c^7)
)^(1/2) + ((d + e*x)^(1/2)*(9*B^2*a^3*e^6 + 4*A^2*c^3*d^4*e^2 + A^2*a^2*c*e^6 - 3*A^2*a*c^2*d^2*e^4 + 9*B^2*a^
2*c*d^2*e^4 - 12*A*B*a*c^2*d^3*e^3))/a^2)*((4*A^2*a^3*c^6*d^3 + 9*B^2*a*e^3*(a^9*c^7)^(1/2) + A^2*c*e^3*(a^9*c
^7)^(1/2) + 6*A*B*a^5*c^4*e^3 - 3*A^2*a^4*c^5*d*e^2 + 9*B^2*a^5*c^4*d*e^2 - 12*A*B*a^4*c^5*d^2*e - 6*A*B*c*d*e
^2*(a^9*c^7)^(1/2))/(64*a^6*c^7))^(1/2) + (((64*A*a^4*c^4*e^5 - 64*A*a^3*c^5*d^2*e^3)/(8*a^3*c^2) + 64*a*c^4*d
*e^2*(d + e*x)^(1/2)*((4*A^2*a^3*c^6*d^3 + 9*B^2*a*e^3*(a^9*c^7)^(1/2) + A^2*c*e^3*(a^9*c^7)^(1/2) + 6*A*B*a^5
*c^4*e^3 - 3*A^2*a^4*c^5*d*e^2 + 9*B^2*a^5*c^4*d*e^2 - 12*A*B*a^4*c^5*d^2*e - 6*A*B*c*d*e^2*(a^9*c^7)^(1/2))/(
64*a^6*c^7))^(1/2))*((4*A^2*a^3*c^6*d^3 + 9*B^2*a*e^3*(a^9*c^7)^(1/2) + A^2*c*e^3*(a^9*c^7)^(1/2) + 6*A*B*a^5*
c^4*e^3 - 3*A^2*a^4*c^5*d*e^2 + 9*B^2*a^5*c^4*d*e^2 - 12*A*B*a^4*c^5*d^2*e - 6*A*B*c*d*e^2*(a^9*c^7)^(1/2))/(6
4*a^6*c^7))^(1/2) - ((d + e*x)^(1/2)*(9*B^2*a^3*e^6 + 4*A^2*c^3*d^4*e^2 + A^2*a^2*c*e^6 - 3*A^2*a*c^2*d^2*e^4
+ 9*B^2*a^2*c*d^2*e^4 - 12*A*B*a*c^2*d^3*e^3))/a^2)*((4*A^2*a^3*c^6*d^3 + 9*B^2*a*e^3*(a^9*c^7)^(1/2) + A^2*c*
e^3*(a^9*c^7)^(1/2) + 6*A*B*a^5*c^4*e^3 - 3*A^2*a^4*c^5*d*e^2 + 9*B^2*a^5*c^4*d*e^2 - 12*A*B*a^4*c^5*d^2*e - 6
*A*B*c*d*e^2*(a^9*c^7)^(1/2))/(64*a^6*c^7))^(1/2) + (27*B^3*a^4*e^8 + 4*A^3*c^4*d^5*e^3 - 3*A^2*B*a^3*c*e^8 -
5*A^3*a*c^3*d^3*e^5 + A^3*a^2*c^2*d*e^7 - 27*B^3*a^3*c*d^2*e^6 + 45*A*B^2*a^2*c^2*d^3*e^5 + 27*A^2*B*a^2*c^2*d
^2*e^6 - 45*A*B^2*a^3*c*d*e^7 - 24*A^2*B*a*c^3*d^4*e^4)/(4*a^3*c^2)))*((4*A^2*a^3*c^6*d^3 + 9*B^2*a*e^3*(a^9*c
^7)^(1/2) + A^2*c*e^3*(a^9*c^7)^(1/2) + 6*A*B*a^5*c^4*e^3 - 3*A^2*a^4*c^5*d*e^2 + 9*B^2*a^5*c^4*d*e^2 - 12*A*B
*a^4*c^5*d^2*e - 6*A*B*c*d*e^2*(a^9*c^7)^(1/2))/(64*a^6*c^7))^(1/2)*2i - (((B*a*e^2 + A*c*d*e)*(d + e*x)^(3/2)
)/(2*a*c) + ((A*a*e^3 - A*c*d^2*e)*(d + e*x)^(1/2))/(2*a*c))/(c*(d + e*x)^2 - a*e^2 + c*d^2 - 2*c*d*(d + e*x))
+ atan(((((64*A*a^4*c^4*e^5 - 64*A*a^3*c^5*d^2*e^3)/(8*a^3*c^2) - 64*a*c^4*d*e^2*(d + e*x)^(1/2)*((4*A^2*a^3*
c^6*d^3 - 9*B^2*a*e^3*(a^9*c^7)^(1/2) - A^2*c*e^3*(a^9*c^7)^(1/2) + 6*A*B*a^5*c^4*e^3 - 3*A^2*a^4*c^5*d*e^2 +
9*B^2*a^5*c^4*d*e^2 - 12*A*B*a^4*c^5*d^2*e + 6*A*B*c*d*e^2*(a^9*c^7)^(1/2))/(64*a^6*c^7))^(1/2))*((4*A^2*a^3*c
^6*d^3 - 9*B^2*a*e^3*(a^9*c^7)^(1/2) - A^2*c*e^3*(a^9*c^7)^(1/2) + 6*A*B*a^5*c^4*e^3 - 3*A^2*a^4*c^5*d*e^2 + 9
*B^2*a^5*c^4*d*e^2 - 12*A*B*a^4*c^5*d^2*e + 6*A*B*c*d*e^2*(a^9*c^7)^(1/2))/(64*a^6*c^7))^(1/2) + ((d + e*x)^(1
/2)*(9*B^2*a^3*e^6 + 4*A^2*c^3*d^4*e^2 + A^2*a^2*c*e^6 - 3*A^2*a*c^2*d^2*e^4 + 9*B^2*a^2*c*d^2*e^4 - 12*A*B*a*
c^2*d^3*e^3))/a^2)*((4*A^2*a^3*c^6*d^3 - 9*B^2*a*e^3*(a^9*c^7)^(1/2) - A^2*c*e^3*(a^9*c^7)^(1/2) + 6*A*B*a^5*c
^4*e^3 - 3*A^2*a^4*c^5*d*e^2 + 9*B^2*a^5*c^4*d*e^2 - 12*A*B*a^4*c^5*d^2*e + 6*A*B*c*d*e^2*(a^9*c^7)^(1/2))/(64
*a^6*c^7))^(1/2)*1i - (((64*A*a^4*c^4*e^5 - 64*A*a^3*c^5*d^2*e^3)/(8*a^3*c^2) + 64*a*c^4*d*e^2*(d + e*x)^(1/2)
*((4*A^2*a^3*c^6*d^3 - 9*B^2*a*e^3*(a^9*c^7)^(1/2) - A^2*c*e^3*(a^9*c^7)^(1/2) + 6*A*B*a^5*c^4*e^3 - 3*A^2*a^4
*c^5*d*e^2 + 9*B^2*a^5*c^4*d*e^2 - 12*A*B*a^4*c^5*d^2*e + 6*A*B*c*d*e^2*(a^9*c^7)^(1/2))/(64*a^6*c^7))^(1/2))*
((4*A^2*a^3*c^6*d^3 - 9*B^2*a*e^3*(a^9*c^7)^(1/2) - A^2*c*e^3*(a^9*c^7)^(1/2) + 6*A*B*a^5*c^4*e^3 - 3*A^2*a^4*
c^5*d*e^2 + 9*B^2*a^5*c^4*d*e^2 - 12*A*B*a^4*c^5*d^2*e + 6*A*B*c*d*e^2*(a^9*c^7)^(1/2))/(64*a^6*c^7))^(1/2) -
((d + e*x)^(1/2)*(9*B^2*a^3*e^6 + 4*A^2*c^3*d^4*e^2 + A^2*a^2*c*e^6 - 3*A^2*a*c^2*d^2*e^4 + 9*B^2*a^2*c*d^2*e^
4 - 12*A*B*a*c^2*d^3*e^3))/a^2)*((4*A^2*a^3*c^6*d^3 - 9*B^2*a*e^3*(a^9*c^7)^(1/2) - A^2*c*e^3*(a^9*c^7)^(1/2)
+ 6*A*B*a^5*c^4*e^3 - 3*A^2*a^4*c^5*d*e^2 + 9*B^2*a^5*c^4*d*e^2 - 12*A*B*a^4*c^5*d^2*e + 6*A*B*c*d*e^2*(a^9*c^
7)^(1/2))/(64*a^6*c^7))^(1/2)*1i)/((((64*A*a^4*c^4*e^5 - 64*A*a^3*c^5*d^2*e^3)/(8*a^3*c^2) - 64*a*c^4*d*e^2*(d
+ e*x)^(1/2)*((4*A^2*a^3*c^6*d^3 - 9*B^2*a*e^3*(a^9*c^7)^(1/2) - A^2*c*e^3*(a^9*c^7)^(1/2) + 6*A*B*a^5*c^4*e^
3 - 3*A^2*a^4*c^5*d*e^2 + 9*B^2*a^5*c^4*d*e^2 - 12*A*B*a^4*c^5*d^2*e + 6*A*B*c*d*e^2*(a^9*c^7)^(1/2))/(64*a^6*
c^7))^(1/2))*((4*A^2*a^3*c^6*d^3 - 9*B^2*a*e^3*(a^9*c^7)^(1/2) - A^2*c*e^3*(a^9*c^7)^(1/2) + 6*A*B*a^5*c^4*e^3
- 3*A^2*a^4*c^5*d*e^2 + 9*B^2*a^5*c^4*d*e^2 - 12*A*B*a^4*c^5*d^2*e + 6*A*B*c*d*e^2*(a^9*c^7)^(1/2))/(64*a^6*c
^7))^(1/2) + ((d + e*x)^(1/2)*(9*B^2*a^3*e^6 + 4*A^2*c^3*d^4*e^2 + A^2*a^2*c*e^6 - 3*A^2*a*c^2*d^2*e^4 + 9*B^2
*a^2*c*d^2*e^4 - 12*A*B*a*c^2*d^3*e^3))/a^2)*((4*A^2*a^3*c^6*d^3 - 9*B^2*a*e^3*(a^9*c^7)^(1/2) - A^2*c*e^3*(a^
9*c^7)^(1/2) + 6*A*B*a^5*c^4*e^3 - 3*A^2*a^4*c^5*d*e^2 + 9*B^2*a^5*c^4*d*e^2 - 12*A*B*a^4*c^5*d^2*e + 6*A*B*c*
d*e^2*(a^9*c^7)^(1/2))/(64*a^6*c^7))^(1/2) + (((64*A*a^4*c^4*e^5 - 64*A*a^3*c^5*d^2*e^3)/(8*a^3*c^2) + 64*a*c^
4*d*e^2*(d + e*x)^(1/2)*((4*A^2*a^3*c^6*d^3 - 9*B^2*a*e^3*(a^9*c^7)^(1/2) - A^2*c*e^3*(a^9*c^7)^(1/2) + 6*A*B*
a^5*c^4*e^3 - 3*A^2*a^4*c^5*d*e^2 + 9*B^2*a^5*c^4*d*e^2 - 12*A*B*a^4*c^5*d^2*e + 6*A*B*c*d*e^2*(a^9*c^7)^(1/2)
)/(64*a^6*c^7))^(1/2))*((4*A^2*a^3*c^6*d^3 - 9*B^2*a*e^3*(a^9*c^7)^(1/2) - A^2*c*e^3*(a^9*c^7)^(1/2) + 6*A*B*a
^5*c^4*e^3 - 3*A^2*a^4*c^5*d*e^2 + 9*B^2*a^5*c^4*d*e^2 - 12*A*B*a^4*c^5*d^2*e + 6*A*B*c*d*e^2*(a^9*c^7)^(1/2))
/(64*a^6*c^7))^(1/2) - ((d + e*x)^(1/2)*(9*B^2*a^3*e^6 + 4*A^2*c^3*d^4*e^2 + A^2*a^2*c*e^6 - 3*A^2*a*c^2*d^2*e
^4 + 9*B^2*a^2*c*d^2*e^4 - 12*A*B*a*c^2*d^3*e^3))/a^2)*((4*A^2*a^3*c^6*d^3 - 9*B^2*a*e^3*(a^9*c^7)^(1/2) - A^2
*c*e^3*(a^9*c^7)^(1/2) + 6*A*B*a^5*c^4*e^3 - 3*A^2*a^4*c^5*d*e^2 + 9*B^2*a^5*c^4*d*e^2 - 12*A*B*a^4*c^5*d^2*e
+ 6*A*B*c*d*e^2*(a^9*c^7)^(1/2))/(64*a^6*c^7))^(1/2) + (27*B^3*a^4*e^8 + 4*A^3*c^4*d^5*e^3 - 3*A^2*B*a^3*c*e^8
- 5*A^3*a*c^3*d^3*e^5 + A^3*a^2*c^2*d*e^7 - 27*B^3*a^3*c*d^2*e^6 + 45*A*B^2*a^2*c^2*d^3*e^5 + 27*A^2*B*a^2*c^
2*d^2*e^6 - 45*A*B^2*a^3*c*d*e^7 - 24*A^2*B*a*c^3*d^4*e^4)/(4*a^3*c^2)))*((4*A^2*a^3*c^6*d^3 - 9*B^2*a*e^3*(a^
9*c^7)^(1/2) - A^2*c*e^3*(a^9*c^7)^(1/2) + 6*A*B*a^5*c^4*e^3 - 3*A^2*a^4*c^5*d*e^2 + 9*B^2*a^5*c^4*d*e^2 - 12*
A*B*a^4*c^5*d^2*e + 6*A*B*c*d*e^2*(a^9*c^7)^(1/2))/(64*a^6*c^7))^(1/2)*2i

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)**(3/2)/(-c*x**2+a)**2,x)

[Out]

Timed out

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