3.13.74 \(\int \frac {(A+B x) (d+e x)^{3/2}}{a-c x^2} \, dx\)
Optimal. Leaf size=202 \[ \frac {\left (\sqrt {a} B-A \sqrt {c}\right ) \left (\sqrt {c} d-\sqrt {a} e\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{\sqrt {a} c^{7/4}}+\frac {\left (\sqrt {a} B+A \sqrt {c}\right ) \left (\sqrt {a} e+\sqrt {c} d\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{\sqrt {a} c^{7/4}}-\frac {2 \sqrt {d+e x} (A e+B d)}{c}-\frac {2 B (d+e x)^{3/2}}{3 c} \]
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Rubi [A] time = 0.44, antiderivative size = 202, normalized size of antiderivative = 1.00,
number of steps used = 6, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used =
{825, 827, 1166, 208} \begin {gather*} \frac {\left (\sqrt {a} B-A \sqrt {c}\right ) \left (\sqrt {c} d-\sqrt {a} e\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{\sqrt {a} c^{7/4}}+\frac {\left (\sqrt {a} B+A \sqrt {c}\right ) \left (\sqrt {a} e+\sqrt {c} d\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{\sqrt {a} c^{7/4}}-\frac {2 \sqrt {d+e x} (A e+B d)}{c}-\frac {2 B (d+e x)^{3/2}}{3 c} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[((A + B*x)*(d + e*x)^(3/2))/(a - c*x^2),x]
[Out]
(-2*(B*d + A*e)*Sqrt[d + e*x])/c - (2*B*(d + e*x)^(3/2))/(3*c) + ((Sqrt[a]*B - A*Sqrt[c])*(Sqrt[c]*d - Sqrt[a]
*e)^(3/2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]])/(Sqrt[a]*c^(7/4)) + ((Sqrt[a]*B + A*Sq
rt[c])*(Sqrt[c]*d + Sqrt[a]*e)^(3/2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(Sqrt[a]*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 825
Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(g*(d + e*x)^m)/
(c*m), x] + Dist[1/c, Int[((d + e*x)^(m - 1)*Simp[c*d*f - a*e*g + (g*c*d + c*e*f)*x, x])/(a + c*x^2), x], x] /
; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && FractionQ[m] && GtQ[m, 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}}{a-c x^2} \, dx &=-\frac {2 B (d+e x)^{3/2}}{3 c}-\frac {\int \frac {\sqrt {d+e x} (-A c d-a B e-c (B d+A e) x)}{a-c x^2} \, dx}{c}\\ &=-\frac {2 (B d+A e) \sqrt {d+e x}}{c}-\frac {2 B (d+e x)^{3/2}}{3 c}+\frac {\int \frac {c \left (A c d^2+2 a B d e+a A e^2\right )+c \left (B c d^2+2 A c d e+a B e^2\right ) x}{\sqrt {d+e x} \left (a-c x^2\right )} \, dx}{c^2}\\ &=-\frac {2 (B d+A e) \sqrt {d+e x}}{c}-\frac {2 B (d+e x)^{3/2}}{3 c}+\frac {2 \operatorname {Subst}\left (\int \frac {c e \left (A c d^2+2 a B d e+a A e^2\right )-c d \left (B c d^2+2 A c d e+a B e^2\right )+c \left (B c d^2+2 A c d e+a B e^2\right ) x^2}{-c d^2+a e^2+2 c d x^2-c x^4} \, dx,x,\sqrt {d+e x}\right )}{c^2}\\ &=-\frac {2 (B d+A e) \sqrt {d+e x}}{c}-\frac {2 B (d+e x)^{3/2}}{3 c}+\frac {\left (\left (\sqrt {a} B-A \sqrt {c}\right ) \left (\sqrt {c} d-\sqrt {a} e\right )^2\right ) \operatorname {Subst}\left (\int \frac {1}{c d-\sqrt {a} \sqrt {c} e-c x^2} \, dx,x,\sqrt {d+e x}\right )}{\sqrt {a} c}+\frac {\left (\left (\sqrt {a} B+A \sqrt {c}\right ) \left (\sqrt {c} d+\sqrt {a} e\right )^2\right ) \operatorname {Subst}\left (\int \frac {1}{c d+\sqrt {a} \sqrt {c} e-c x^2} \, dx,x,\sqrt {d+e x}\right )}{\sqrt {a} c}\\ &=-\frac {2 (B d+A e) \sqrt {d+e x}}{c}-\frac {2 B (d+e x)^{3/2}}{3 c}+\frac {\left (\sqrt {a} B-A \sqrt {c}\right ) \left (\sqrt {c} d-\sqrt {a} e\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{\sqrt {a} c^{7/4}}+\frac {\left (\sqrt {a} B+A \sqrt {c}\right ) \left (\sqrt {c} d+\sqrt {a} e\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {a} e}}\right )}{\sqrt {a} c^{7/4}}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 194, normalized size = 0.96 \begin {gather*} \frac {-2 \sqrt {a} c^{3/4} \sqrt {d+e x} (3 A e+4 B d+B e x)-3 \left (A \sqrt {c}-\sqrt {a} B\right ) \left (\sqrt {c} d-\sqrt {a} e\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )+3 \left (\sqrt {a} B+A \sqrt {c}\right ) \left (\sqrt {a} e+\sqrt {c} d\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{3 \sqrt {a} c^{7/4}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[((A + B*x)*(d + e*x)^(3/2))/(a - c*x^2),x]
[Out]
(-2*Sqrt[a]*c^(3/4)*Sqrt[d + e*x]*(4*B*d + 3*A*e + B*e*x) - 3*(-(Sqrt[a]*B) + A*Sqrt[c])*(Sqrt[c]*d - Sqrt[a]*
e)^(3/2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]] + 3*(Sqrt[a]*B + A*Sqrt[c])*(Sqrt[c]*d +
Sqrt[a]*e)^(3/2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(3*Sqrt[a]*c^(7/4))
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IntegrateAlgebraic [A] time = 0.60, size = 275, normalized size = 1.36 \begin {gather*} -\frac {\left (A \sqrt {c}-\sqrt {a} B\right ) \left (\sqrt {c} d-\sqrt {a} e\right )^2 \tan ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {\sqrt {a} \sqrt {c} e-c d}}{\sqrt {c} d-\sqrt {a} e}\right )}{\sqrt {a} c^{3/2} \sqrt {-\sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right )}}+\frac {\left (\sqrt {a} B+A \sqrt {c}\right ) \left (\sqrt {a} e+\sqrt {c} d\right )^2 \tan ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {-\sqrt {a} \sqrt {c} e-c d}}{\sqrt {a} e+\sqrt {c} d}\right )}{\sqrt {a} c^{3/2} \sqrt {-\sqrt {c} \left (\sqrt {a} e+\sqrt {c} d\right )}}-\frac {2 \sqrt {d+e x} (3 A e+B (d+e x)+3 B d)}{3 c} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[((A + B*x)*(d + e*x)^(3/2))/(a - c*x^2),x]
[Out]
(-2*Sqrt[d + e*x]*(3*B*d + 3*A*e + B*(d + e*x)))/(3*c) + ((Sqrt[a]*B + A*Sqrt[c])*(Sqrt[c]*d + Sqrt[a]*e)^2*Ar
cTan[(Sqrt[-(c*d) - Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d + Sqrt[a]*e)])/(Sqrt[a]*c^(3/2)*Sqrt[-(Sqrt[c
]*(Sqrt[c]*d + Sqrt[a]*e))]) - ((-(Sqrt[a]*B) + A*Sqrt[c])*(Sqrt[c]*d - Sqrt[a]*e)^2*ArcTan[(Sqrt[-(c*d) + Sqr
t[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d - Sqrt[a]*e)])/(Sqrt[a]*c^(3/2)*Sqrt[-(Sqrt[c]*(Sqrt[c]*d - Sqrt[a]*
e))])
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fricas [B] time = 1.86, size = 4480, normalized size = 22.18
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),x, algorithm="fricas")
[Out]
1/6*(3*c*sqrt((6*A*B*a*c*d^2*e + 2*A*B*a^2*e^3 + a*c^3*sqrt((4*A^2*B^2*c^4*d^6 + 12*(A*B^3*a*c^3 + A^3*B*c^4)*
d^5*e + 3*(3*B^4*a^2*c^2 + 14*A^2*B^2*a*c^3 + 3*A^4*c^4)*d^4*e^2 + 40*(A*B^3*a^2*c^2 + A^3*B*a*c^3)*d^3*e^3 +
6*(B^4*a^3*c + 8*A^2*B^2*a^2*c^2 + A^4*a*c^3)*d^2*e^4 + 12*(A*B^3*a^3*c + A^3*B*a^2*c^2)*d*e^5 + (B^4*a^4 + 2*
A^2*B^2*a^3*c + A^4*a^2*c^2)*e^6)/(a*c^7)) + (B^2*a*c + A^2*c^2)*d^3 + 3*(B^2*a^2 + A^2*a*c)*d*e^2)/(a*c^3))*l
og((2*(A*B^3*a*c^3 - A^3*B*c^4)*d^5 + 3*(B^4*a^2*c^2 - A^4*c^4)*d^4*e + 4*(A*B^3*a^2*c^2 - A^3*B*a*c^3)*d^3*e^
2 - 2*(B^4*a^3*c - A^4*a*c^3)*d^2*e^3 - 6*(A*B^3*a^3*c - A^3*B*a^2*c^2)*d*e^4 - (B^4*a^4 - A^4*a^2*c^2)*e^5)*s
qrt(e*x + d) + (2*A*B^2*a*c^4*d^4 + (3*B^3*a^2*c^3 + 5*A^2*B*a*c^4)*d^3*e + 3*(3*A*B^2*a^2*c^3 + A^3*a*c^4)*d^
2*e^2 + (B^3*a^3*c^2 + 7*A^2*B*a^2*c^3)*d*e^3 + (A*B^2*a^3*c^2 + A^3*a^2*c^3)*e^4 - (A*a*c^6*d + B*a^2*c^5*e)*
sqrt((4*A^2*B^2*c^4*d^6 + 12*(A*B^3*a*c^3 + A^3*B*c^4)*d^5*e + 3*(3*B^4*a^2*c^2 + 14*A^2*B^2*a*c^3 + 3*A^4*c^4
)*d^4*e^2 + 40*(A*B^3*a^2*c^2 + A^3*B*a*c^3)*d^3*e^3 + 6*(B^4*a^3*c + 8*A^2*B^2*a^2*c^2 + A^4*a*c^3)*d^2*e^4 +
12*(A*B^3*a^3*c + A^3*B*a^2*c^2)*d*e^5 + (B^4*a^4 + 2*A^2*B^2*a^3*c + A^4*a^2*c^2)*e^6)/(a*c^7)))*sqrt((6*A*B
*a*c*d^2*e + 2*A*B*a^2*e^3 + a*c^3*sqrt((4*A^2*B^2*c^4*d^6 + 12*(A*B^3*a*c^3 + A^3*B*c^4)*d^5*e + 3*(3*B^4*a^2
*c^2 + 14*A^2*B^2*a*c^3 + 3*A^4*c^4)*d^4*e^2 + 40*(A*B^3*a^2*c^2 + A^3*B*a*c^3)*d^3*e^3 + 6*(B^4*a^3*c + 8*A^2
*B^2*a^2*c^2 + A^4*a*c^3)*d^2*e^4 + 12*(A*B^3*a^3*c + A^3*B*a^2*c^2)*d*e^5 + (B^4*a^4 + 2*A^2*B^2*a^3*c + A^4*
a^2*c^2)*e^6)/(a*c^7)) + (B^2*a*c + A^2*c^2)*d^3 + 3*(B^2*a^2 + A^2*a*c)*d*e^2)/(a*c^3))) - 3*c*sqrt((6*A*B*a*
c*d^2*e + 2*A*B*a^2*e^3 + a*c^3*sqrt((4*A^2*B^2*c^4*d^6 + 12*(A*B^3*a*c^3 + A^3*B*c^4)*d^5*e + 3*(3*B^4*a^2*c^
2 + 14*A^2*B^2*a*c^3 + 3*A^4*c^4)*d^4*e^2 + 40*(A*B^3*a^2*c^2 + A^3*B*a*c^3)*d^3*e^3 + 6*(B^4*a^3*c + 8*A^2*B^
2*a^2*c^2 + A^4*a*c^3)*d^2*e^4 + 12*(A*B^3*a^3*c + A^3*B*a^2*c^2)*d*e^5 + (B^4*a^4 + 2*A^2*B^2*a^3*c + A^4*a^2
*c^2)*e^6)/(a*c^7)) + (B^2*a*c + A^2*c^2)*d^3 + 3*(B^2*a^2 + A^2*a*c)*d*e^2)/(a*c^3))*log((2*(A*B^3*a*c^3 - A^
3*B*c^4)*d^5 + 3*(B^4*a^2*c^2 - A^4*c^4)*d^4*e + 4*(A*B^3*a^2*c^2 - A^3*B*a*c^3)*d^3*e^2 - 2*(B^4*a^3*c - A^4*
a*c^3)*d^2*e^3 - 6*(A*B^3*a^3*c - A^3*B*a^2*c^2)*d*e^4 - (B^4*a^4 - A^4*a^2*c^2)*e^5)*sqrt(e*x + d) - (2*A*B^2
*a*c^4*d^4 + (3*B^3*a^2*c^3 + 5*A^2*B*a*c^4)*d^3*e + 3*(3*A*B^2*a^2*c^3 + A^3*a*c^4)*d^2*e^2 + (B^3*a^3*c^2 +
7*A^2*B*a^2*c^3)*d*e^3 + (A*B^2*a^3*c^2 + A^3*a^2*c^3)*e^4 - (A*a*c^6*d + B*a^2*c^5*e)*sqrt((4*A^2*B^2*c^4*d^6
+ 12*(A*B^3*a*c^3 + A^3*B*c^4)*d^5*e + 3*(3*B^4*a^2*c^2 + 14*A^2*B^2*a*c^3 + 3*A^4*c^4)*d^4*e^2 + 40*(A*B^3*a
^2*c^2 + A^3*B*a*c^3)*d^3*e^3 + 6*(B^4*a^3*c + 8*A^2*B^2*a^2*c^2 + A^4*a*c^3)*d^2*e^4 + 12*(A*B^3*a^3*c + A^3*
B*a^2*c^2)*d*e^5 + (B^4*a^4 + 2*A^2*B^2*a^3*c + A^4*a^2*c^2)*e^6)/(a*c^7)))*sqrt((6*A*B*a*c*d^2*e + 2*A*B*a^2*
e^3 + a*c^3*sqrt((4*A^2*B^2*c^4*d^6 + 12*(A*B^3*a*c^3 + A^3*B*c^4)*d^5*e + 3*(3*B^4*a^2*c^2 + 14*A^2*B^2*a*c^3
+ 3*A^4*c^4)*d^4*e^2 + 40*(A*B^3*a^2*c^2 + A^3*B*a*c^3)*d^3*e^3 + 6*(B^4*a^3*c + 8*A^2*B^2*a^2*c^2 + A^4*a*c^
3)*d^2*e^4 + 12*(A*B^3*a^3*c + A^3*B*a^2*c^2)*d*e^5 + (B^4*a^4 + 2*A^2*B^2*a^3*c + A^4*a^2*c^2)*e^6)/(a*c^7))
+ (B^2*a*c + A^2*c^2)*d^3 + 3*(B^2*a^2 + A^2*a*c)*d*e^2)/(a*c^3))) + 3*c*sqrt((6*A*B*a*c*d^2*e + 2*A*B*a^2*e^3
- a*c^3*sqrt((4*A^2*B^2*c^4*d^6 + 12*(A*B^3*a*c^3 + A^3*B*c^4)*d^5*e + 3*(3*B^4*a^2*c^2 + 14*A^2*B^2*a*c^3 +
3*A^4*c^4)*d^4*e^2 + 40*(A*B^3*a^2*c^2 + A^3*B*a*c^3)*d^3*e^3 + 6*(B^4*a^3*c + 8*A^2*B^2*a^2*c^2 + A^4*a*c^3)*
d^2*e^4 + 12*(A*B^3*a^3*c + A^3*B*a^2*c^2)*d*e^5 + (B^4*a^4 + 2*A^2*B^2*a^3*c + A^4*a^2*c^2)*e^6)/(a*c^7)) + (
B^2*a*c + A^2*c^2)*d^3 + 3*(B^2*a^2 + A^2*a*c)*d*e^2)/(a*c^3))*log((2*(A*B^3*a*c^3 - A^3*B*c^4)*d^5 + 3*(B^4*a
^2*c^2 - A^4*c^4)*d^4*e + 4*(A*B^3*a^2*c^2 - A^3*B*a*c^3)*d^3*e^2 - 2*(B^4*a^3*c - A^4*a*c^3)*d^2*e^3 - 6*(A*B
^3*a^3*c - A^3*B*a^2*c^2)*d*e^4 - (B^4*a^4 - A^4*a^2*c^2)*e^5)*sqrt(e*x + d) + (2*A*B^2*a*c^4*d^4 + (3*B^3*a^2
*c^3 + 5*A^2*B*a*c^4)*d^3*e + 3*(3*A*B^2*a^2*c^3 + A^3*a*c^4)*d^2*e^2 + (B^3*a^3*c^2 + 7*A^2*B*a^2*c^3)*d*e^3
+ (A*B^2*a^3*c^2 + A^3*a^2*c^3)*e^4 + (A*a*c^6*d + B*a^2*c^5*e)*sqrt((4*A^2*B^2*c^4*d^6 + 12*(A*B^3*a*c^3 + A^
3*B*c^4)*d^5*e + 3*(3*B^4*a^2*c^2 + 14*A^2*B^2*a*c^3 + 3*A^4*c^4)*d^4*e^2 + 40*(A*B^3*a^2*c^2 + A^3*B*a*c^3)*d
^3*e^3 + 6*(B^4*a^3*c + 8*A^2*B^2*a^2*c^2 + A^4*a*c^3)*d^2*e^4 + 12*(A*B^3*a^3*c + A^3*B*a^2*c^2)*d*e^5 + (B^4
*a^4 + 2*A^2*B^2*a^3*c + A^4*a^2*c^2)*e^6)/(a*c^7)))*sqrt((6*A*B*a*c*d^2*e + 2*A*B*a^2*e^3 - a*c^3*sqrt((4*A^2
*B^2*c^4*d^6 + 12*(A*B^3*a*c^3 + A^3*B*c^4)*d^5*e + 3*(3*B^4*a^2*c^2 + 14*A^2*B^2*a*c^3 + 3*A^4*c^4)*d^4*e^2 +
40*(A*B^3*a^2*c^2 + A^3*B*a*c^3)*d^3*e^3 + 6*(B^4*a^3*c + 8*A^2*B^2*a^2*c^2 + A^4*a*c^3)*d^2*e^4 + 12*(A*B^3*
a^3*c + A^3*B*a^2*c^2)*d*e^5 + (B^4*a^4 + 2*A^2*B^2*a^3*c + A^4*a^2*c^2)*e^6)/(a*c^7)) + (B^2*a*c + A^2*c^2)*d
^3 + 3*(B^2*a^2 + A^2*a*c)*d*e^2)/(a*c^3))) - 3*c*sqrt((6*A*B*a*c*d^2*e + 2*A*B*a^2*e^3 - a*c^3*sqrt((4*A^2*B^
2*c^4*d^6 + 12*(A*B^3*a*c^3 + A^3*B*c^4)*d^5*e + 3*(3*B^4*a^2*c^2 + 14*A^2*B^2*a*c^3 + 3*A^4*c^4)*d^4*e^2 + 40
*(A*B^3*a^2*c^2 + A^3*B*a*c^3)*d^3*e^3 + 6*(B^4*a^3*c + 8*A^2*B^2*a^2*c^2 + A^4*a*c^3)*d^2*e^4 + 12*(A*B^3*a^3
*c + A^3*B*a^2*c^2)*d*e^5 + (B^4*a^4 + 2*A^2*B^2*a^3*c + A^4*a^2*c^2)*e^6)/(a*c^7)) + (B^2*a*c + A^2*c^2)*d^3
+ 3*(B^2*a^2 + A^2*a*c)*d*e^2)/(a*c^3))*log((2*(A*B^3*a*c^3 - A^3*B*c^4)*d^5 + 3*(B^4*a^2*c^2 - A^4*c^4)*d^4*e
+ 4*(A*B^3*a^2*c^2 - A^3*B*a*c^3)*d^3*e^2 - 2*(B^4*a^3*c - A^4*a*c^3)*d^2*e^3 - 6*(A*B^3*a^3*c - A^3*B*a^2*c^
2)*d*e^4 - (B^4*a^4 - A^4*a^2*c^2)*e^5)*sqrt(e*x + d) - (2*A*B^2*a*c^4*d^4 + (3*B^3*a^2*c^3 + 5*A^2*B*a*c^4)*d
^3*e + 3*(3*A*B^2*a^2*c^3 + A^3*a*c^4)*d^2*e^2 + (B^3*a^3*c^2 + 7*A^2*B*a^2*c^3)*d*e^3 + (A*B^2*a^3*c^2 + A^3*
a^2*c^3)*e^4 + (A*a*c^6*d + B*a^2*c^5*e)*sqrt((4*A^2*B^2*c^4*d^6 + 12*(A*B^3*a*c^3 + A^3*B*c^4)*d^5*e + 3*(3*B
^4*a^2*c^2 + 14*A^2*B^2*a*c^3 + 3*A^4*c^4)*d^4*e^2 + 40*(A*B^3*a^2*c^2 + A^3*B*a*c^3)*d^3*e^3 + 6*(B^4*a^3*c +
8*A^2*B^2*a^2*c^2 + A^4*a*c^3)*d^2*e^4 + 12*(A*B^3*a^3*c + A^3*B*a^2*c^2)*d*e^5 + (B^4*a^4 + 2*A^2*B^2*a^3*c
+ A^4*a^2*c^2)*e^6)/(a*c^7)))*sqrt((6*A*B*a*c*d^2*e + 2*A*B*a^2*e^3 - a*c^3*sqrt((4*A^2*B^2*c^4*d^6 + 12*(A*B^
3*a*c^3 + A^3*B*c^4)*d^5*e + 3*(3*B^4*a^2*c^2 + 14*A^2*B^2*a*c^3 + 3*A^4*c^4)*d^4*e^2 + 40*(A*B^3*a^2*c^2 + A^
3*B*a*c^3)*d^3*e^3 + 6*(B^4*a^3*c + 8*A^2*B^2*a^2*c^2 + A^4*a*c^3)*d^2*e^4 + 12*(A*B^3*a^3*c + A^3*B*a^2*c^2)*
d*e^5 + (B^4*a^4 + 2*A^2*B^2*a^3*c + A^4*a^2*c^2)*e^6)/(a*c^7)) + (B^2*a*c + A^2*c^2)*d^3 + 3*(B^2*a^2 + A^2*a
*c)*d*e^2)/(a*c^3))) - 4*(B*e*x + 4*B*d + 3*A*e)*sqrt(e*x + d))/c
________________________________________________________________________________________
giac [B] time = 0.41, size = 528, normalized size = 2.61 \begin {gather*} -\frac {{\left (2 \, \sqrt {a c} B a c^{3} d^{2} e - 2 \, \sqrt {a c} A a c^{3} d e^{2} - {\left (\sqrt {a c} a c d^{2} e + \sqrt {a c} a^{2} e^{3}\right )} B c^{2} + {\left (a c^{3} d^{2} e - a^{2} c^{2} e^{3}\right )} A {\left | c \right |} + {\left (a c^{3} d^{3} - a^{2} c^{2} d e^{2}\right )} B {\left | c \right |} + {\left (\sqrt {a c} c^{4} d^{3} + \sqrt {a c} a c^{3} d e^{2}\right )} A\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {c^{4} d + \sqrt {c^{8} d^{2} - {\left (c^{4} d^{2} - a c^{3} e^{2}\right )} c^{4}}}{c^{4}}}}\right )}{{\left (a c^{4} d - \sqrt {a c} a c^{3} e\right )} \sqrt {-c^{2} d - \sqrt {a c} c e}} + \frac {{\left (2 \, \sqrt {a c} B a c^{3} d^{2} e - 2 \, \sqrt {a c} A a c^{3} d e^{2} - {\left (\sqrt {a c} a c d^{2} e + \sqrt {a c} a^{2} e^{3}\right )} B c^{2} - {\left (a c^{3} d^{2} e - a^{2} c^{2} e^{3}\right )} A {\left | c \right |} - {\left (a c^{3} d^{3} - a^{2} c^{2} d e^{2}\right )} B {\left | c \right |} + {\left (\sqrt {a c} c^{4} d^{3} + \sqrt {a c} a c^{3} d e^{2}\right )} A\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {c^{4} d - \sqrt {c^{8} d^{2} - {\left (c^{4} d^{2} - a c^{3} e^{2}\right )} c^{4}}}{c^{4}}}}\right )}{{\left (a c^{4} d + \sqrt {a c} a c^{3} e\right )} \sqrt {-c^{2} d + \sqrt {a c} c e}} - \frac {2 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} B c^{2} + 3 \, \sqrt {x e + d} B c^{2} d + 3 \, \sqrt {x e + d} A c^{2} e\right )}}{3 \, c^{3}} \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),x, algorithm="giac")
[Out]
-(2*sqrt(a*c)*B*a*c^3*d^2*e - 2*sqrt(a*c)*A*a*c^3*d*e^2 - (sqrt(a*c)*a*c*d^2*e + sqrt(a*c)*a^2*e^3)*B*c^2 + (a
*c^3*d^2*e - a^2*c^2*e^3)*A*abs(c) + (a*c^3*d^3 - a^2*c^2*d*e^2)*B*abs(c) + (sqrt(a*c)*c^4*d^3 + sqrt(a*c)*a*c
^3*d*e^2)*A)*arctan(sqrt(x*e + d)/sqrt(-(c^4*d + sqrt(c^8*d^2 - (c^4*d^2 - a*c^3*e^2)*c^4))/c^4))/((a*c^4*d -
sqrt(a*c)*a*c^3*e)*sqrt(-c^2*d - sqrt(a*c)*c*e)) + (2*sqrt(a*c)*B*a*c^3*d^2*e - 2*sqrt(a*c)*A*a*c^3*d*e^2 - (s
qrt(a*c)*a*c*d^2*e + sqrt(a*c)*a^2*e^3)*B*c^2 - (a*c^3*d^2*e - a^2*c^2*e^3)*A*abs(c) - (a*c^3*d^3 - a^2*c^2*d*
e^2)*B*abs(c) + (sqrt(a*c)*c^4*d^3 + sqrt(a*c)*a*c^3*d*e^2)*A)*arctan(sqrt(x*e + d)/sqrt(-(c^4*d - sqrt(c^8*d^
2 - (c^4*d^2 - a*c^3*e^2)*c^4))/c^4))/((a*c^4*d + sqrt(a*c)*a*c^3*e)*sqrt(-c^2*d + sqrt(a*c)*c*e)) - 2/3*((x*e
+ d)^(3/2)*B*c^2 + 3*sqrt(x*e + d)*B*c^2*d + 3*sqrt(x*e + d)*A*c^2*e)/c^3
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maple [B] time = 0.08, size = 689, normalized size = 3.41 \begin {gather*} \frac {A a \,e^{3} \arctanh \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {A a \,e^{3} \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}+\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 )}{\sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\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 )}{\sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {2 B a d \,e^{2} \arctanh \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {2 B a d \,e^{2} \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {2 A d e \arctanh \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}-\frac {2 A d e \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {B a \,e^{2} \arctanh \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, c}-\frac {B a \,e^{2} \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, c}+\frac {B \,d^{2} \arctanh \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}-\frac {B \,d^{2} \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}-\frac {2 \sqrt {e x +d}\, A e}{c}-\frac {2 \sqrt {e x +d}\, B d}{c}-\frac {2 \left (e x +d \right )^{\frac {3}{2}} B}{3 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),x)
[Out]
-2/3*(e*x+d)^(3/2)*B/c-2/c*A*e*(e*x+d)^(1/2)-2/c*B*d*(e*x+d)^(1/2)+1/(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*A*e^3+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*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)*a*B*d*e^2+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*e+1/c/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arc
tanh((e*x+d)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)*B*a*e^2+1/((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^2+1/(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*A*e^3+c/(a*c*e^2)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arct
an((e*x+d)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)*A*d^2*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)*a*B*d*e^2-2/((-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*e-1/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*a*e^2-1/((-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^2
________________________________________________________________________________________
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}}}{c x^{2} - a}\,{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),x, algorithm="maxima")
[Out]
-integrate((B*x + A)*(e*x + d)^(3/2)/(c*x^2 - a), x)
________________________________________________________________________________________
mupad [B] time = 2.71, size = 7560, normalized size = 37.43
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),x)
[Out]
- ((2*A*e - 2*B*d)/c + (4*B*d)/c)*(d + e*x)^(1/2) - atan(((((8*(4*A*a^2*c^4*e^5 - 4*A*a*c^5*d^2*e^3 - 4*B*a*c^
5*d^3*e^2 + 4*B*a^2*c^4*d*e^4))/c^2 - 64*a*c^4*d*e^2*(d + e*x)^(1/2)*((B^2*a^2*c^5*d^3 + B^2*a^2*e^3*(a^3*c^7)
^(1/2) + A^2*a*c^6*d^3 + 2*A*B*a^3*c^4*e^3 + 3*A^2*c^2*d^2*e*(a^3*c^7)^(1/2) + 2*A*B*c^2*d^3*(a^3*c^7)^(1/2) +
3*A^2*a^2*c^5*d*e^2 + 3*B^2*a^3*c^4*d*e^2 + A^2*a*c*e^3*(a^3*c^7)^(1/2) + 3*B^2*a*c*d^2*e*(a^3*c^7)^(1/2) + 6
*A*B*a^2*c^5*d^2*e + 6*A*B*a*c*d*e^2*(a^3*c^7)^(1/2))/(4*a^2*c^7))^(1/2))*((B^2*a^2*c^5*d^3 + B^2*a^2*e^3*(a^3
*c^7)^(1/2) + A^2*a*c^6*d^3 + 2*A*B*a^3*c^4*e^3 + 3*A^2*c^2*d^2*e*(a^3*c^7)^(1/2) + 2*A*B*c^2*d^3*(a^3*c^7)^(1
/2) + 3*A^2*a^2*c^5*d*e^2 + 3*B^2*a^3*c^4*d*e^2 + A^2*a*c*e^3*(a^3*c^7)^(1/2) + 3*B^2*a*c*d^2*e*(a^3*c^7)^(1/2
) + 6*A*B*a^2*c^5*d^2*e + 6*A*B*a*c*d*e^2*(a^3*c^7)^(1/2))/(4*a^2*c^7))^(1/2) + (d + e*x)^(1/2)*(16*B^2*a^3*e^
6 + 16*A^2*c^3*d^4*e^2 + 16*A^2*a^2*c*e^6 + 96*A^2*a*c^2*d^2*e^4 + 16*B^2*a*c^2*d^4*e^2 + 96*B^2*a^2*c*d^2*e^4
+ 128*A*B*a^2*c*d*e^5 + 128*A*B*a*c^2*d^3*e^3))*((B^2*a^2*c^5*d^3 + B^2*a^2*e^3*(a^3*c^7)^(1/2) + A^2*a*c^6*d
^3 + 2*A*B*a^3*c^4*e^3 + 3*A^2*c^2*d^2*e*(a^3*c^7)^(1/2) + 2*A*B*c^2*d^3*(a^3*c^7)^(1/2) + 3*A^2*a^2*c^5*d*e^2
+ 3*B^2*a^3*c^4*d*e^2 + A^2*a*c*e^3*(a^3*c^7)^(1/2) + 3*B^2*a*c*d^2*e*(a^3*c^7)^(1/2) + 6*A*B*a^2*c^5*d^2*e +
6*A*B*a*c*d*e^2*(a^3*c^7)^(1/2))/(4*a^2*c^7))^(1/2)*1i - (((8*(4*A*a^2*c^4*e^5 - 4*A*a*c^5*d^2*e^3 - 4*B*a*c^
5*d^3*e^2 + 4*B*a^2*c^4*d*e^4))/c^2 + 64*a*c^4*d*e^2*(d + e*x)^(1/2)*((B^2*a^2*c^5*d^3 + B^2*a^2*e^3*(a^3*c^7)
^(1/2) + A^2*a*c^6*d^3 + 2*A*B*a^3*c^4*e^3 + 3*A^2*c^2*d^2*e*(a^3*c^7)^(1/2) + 2*A*B*c^2*d^3*(a^3*c^7)^(1/2) +
3*A^2*a^2*c^5*d*e^2 + 3*B^2*a^3*c^4*d*e^2 + A^2*a*c*e^3*(a^3*c^7)^(1/2) + 3*B^2*a*c*d^2*e*(a^3*c^7)^(1/2) + 6
*A*B*a^2*c^5*d^2*e + 6*A*B*a*c*d*e^2*(a^3*c^7)^(1/2))/(4*a^2*c^7))^(1/2))*((B^2*a^2*c^5*d^3 + B^2*a^2*e^3*(a^3
*c^7)^(1/2) + A^2*a*c^6*d^3 + 2*A*B*a^3*c^4*e^3 + 3*A^2*c^2*d^2*e*(a^3*c^7)^(1/2) + 2*A*B*c^2*d^3*(a^3*c^7)^(1
/2) + 3*A^2*a^2*c^5*d*e^2 + 3*B^2*a^3*c^4*d*e^2 + A^2*a*c*e^3*(a^3*c^7)^(1/2) + 3*B^2*a*c*d^2*e*(a^3*c^7)^(1/2
) + 6*A*B*a^2*c^5*d^2*e + 6*A*B*a*c*d*e^2*(a^3*c^7)^(1/2))/(4*a^2*c^7))^(1/2) - (d + e*x)^(1/2)*(16*B^2*a^3*e^
6 + 16*A^2*c^3*d^4*e^2 + 16*A^2*a^2*c*e^6 + 96*A^2*a*c^2*d^2*e^4 + 16*B^2*a*c^2*d^4*e^2 + 96*B^2*a^2*c*d^2*e^4
+ 128*A*B*a^2*c*d*e^5 + 128*A*B*a*c^2*d^3*e^3))*((B^2*a^2*c^5*d^3 + B^2*a^2*e^3*(a^3*c^7)^(1/2) + A^2*a*c^6*d
^3 + 2*A*B*a^3*c^4*e^3 + 3*A^2*c^2*d^2*e*(a^3*c^7)^(1/2) + 2*A*B*c^2*d^3*(a^3*c^7)^(1/2) + 3*A^2*a^2*c^5*d*e^2
+ 3*B^2*a^3*c^4*d*e^2 + A^2*a*c*e^3*(a^3*c^7)^(1/2) + 3*B^2*a*c*d^2*e*(a^3*c^7)^(1/2) + 6*A*B*a^2*c^5*d^2*e +
6*A*B*a*c*d*e^2*(a^3*c^7)^(1/2))/(4*a^2*c^7))^(1/2)*1i)/((16*(B^3*a^4*e^8 - 2*A^3*c^4*d^5*e^3 - B^3*a^2*c^2*d
^4*e^4 - A^2*B*a^3*c*e^8 - A^2*B*c^4*d^6*e^2 + 4*A^3*a*c^3*d^3*e^5 - 2*A^3*a^2*c^2*d*e^7 + B^3*a*c^3*d^6*e^2 -
B^3*a^3*c*d^2*e^6 - 4*A*B^2*a^2*c^2*d^3*e^5 + A^2*B*a^2*c^2*d^2*e^6 + 2*A*B^2*a^3*c*d*e^7 + 2*A*B^2*a*c^3*d^5
*e^3 + A^2*B*a*c^3*d^4*e^4))/c^2 + (((8*(4*A*a^2*c^4*e^5 - 4*A*a*c^5*d^2*e^3 - 4*B*a*c^5*d^3*e^2 + 4*B*a^2*c^4
*d*e^4))/c^2 - 64*a*c^4*d*e^2*(d + e*x)^(1/2)*((B^2*a^2*c^5*d^3 + B^2*a^2*e^3*(a^3*c^7)^(1/2) + A^2*a*c^6*d^3
+ 2*A*B*a^3*c^4*e^3 + 3*A^2*c^2*d^2*e*(a^3*c^7)^(1/2) + 2*A*B*c^2*d^3*(a^3*c^7)^(1/2) + 3*A^2*a^2*c^5*d*e^2 +
3*B^2*a^3*c^4*d*e^2 + A^2*a*c*e^3*(a^3*c^7)^(1/2) + 3*B^2*a*c*d^2*e*(a^3*c^7)^(1/2) + 6*A*B*a^2*c^5*d^2*e + 6*
A*B*a*c*d*e^2*(a^3*c^7)^(1/2))/(4*a^2*c^7))^(1/2))*((B^2*a^2*c^5*d^3 + B^2*a^2*e^3*(a^3*c^7)^(1/2) + A^2*a*c^6
*d^3 + 2*A*B*a^3*c^4*e^3 + 3*A^2*c^2*d^2*e*(a^3*c^7)^(1/2) + 2*A*B*c^2*d^3*(a^3*c^7)^(1/2) + 3*A^2*a^2*c^5*d*e
^2 + 3*B^2*a^3*c^4*d*e^2 + A^2*a*c*e^3*(a^3*c^7)^(1/2) + 3*B^2*a*c*d^2*e*(a^3*c^7)^(1/2) + 6*A*B*a^2*c^5*d^2*e
+ 6*A*B*a*c*d*e^2*(a^3*c^7)^(1/2))/(4*a^2*c^7))^(1/2) + (d + e*x)^(1/2)*(16*B^2*a^3*e^6 + 16*A^2*c^3*d^4*e^2
+ 16*A^2*a^2*c*e^6 + 96*A^2*a*c^2*d^2*e^4 + 16*B^2*a*c^2*d^4*e^2 + 96*B^2*a^2*c*d^2*e^4 + 128*A*B*a^2*c*d*e^5
+ 128*A*B*a*c^2*d^3*e^3))*((B^2*a^2*c^5*d^3 + B^2*a^2*e^3*(a^3*c^7)^(1/2) + A^2*a*c^6*d^3 + 2*A*B*a^3*c^4*e^3
+ 3*A^2*c^2*d^2*e*(a^3*c^7)^(1/2) + 2*A*B*c^2*d^3*(a^3*c^7)^(1/2) + 3*A^2*a^2*c^5*d*e^2 + 3*B^2*a^3*c^4*d*e^2
+ A^2*a*c*e^3*(a^3*c^7)^(1/2) + 3*B^2*a*c*d^2*e*(a^3*c^7)^(1/2) + 6*A*B*a^2*c^5*d^2*e + 6*A*B*a*c*d*e^2*(a^3*c
^7)^(1/2))/(4*a^2*c^7))^(1/2) + (((8*(4*A*a^2*c^4*e^5 - 4*A*a*c^5*d^2*e^3 - 4*B*a*c^5*d^3*e^2 + 4*B*a^2*c^4*d*
e^4))/c^2 + 64*a*c^4*d*e^2*(d + e*x)^(1/2)*((B^2*a^2*c^5*d^3 + B^2*a^2*e^3*(a^3*c^7)^(1/2) + A^2*a*c^6*d^3 + 2
*A*B*a^3*c^4*e^3 + 3*A^2*c^2*d^2*e*(a^3*c^7)^(1/2) + 2*A*B*c^2*d^3*(a^3*c^7)^(1/2) + 3*A^2*a^2*c^5*d*e^2 + 3*B
^2*a^3*c^4*d*e^2 + A^2*a*c*e^3*(a^3*c^7)^(1/2) + 3*B^2*a*c*d^2*e*(a^3*c^7)^(1/2) + 6*A*B*a^2*c^5*d^2*e + 6*A*B
*a*c*d*e^2*(a^3*c^7)^(1/2))/(4*a^2*c^7))^(1/2))*((B^2*a^2*c^5*d^3 + B^2*a^2*e^3*(a^3*c^7)^(1/2) + A^2*a*c^6*d^
3 + 2*A*B*a^3*c^4*e^3 + 3*A^2*c^2*d^2*e*(a^3*c^7)^(1/2) + 2*A*B*c^2*d^3*(a^3*c^7)^(1/2) + 3*A^2*a^2*c^5*d*e^2
+ 3*B^2*a^3*c^4*d*e^2 + A^2*a*c*e^3*(a^3*c^7)^(1/2) + 3*B^2*a*c*d^2*e*(a^3*c^7)^(1/2) + 6*A*B*a^2*c^5*d^2*e +
6*A*B*a*c*d*e^2*(a^3*c^7)^(1/2))/(4*a^2*c^7))^(1/2) - (d + e*x)^(1/2)*(16*B^2*a^3*e^6 + 16*A^2*c^3*d^4*e^2 + 1
6*A^2*a^2*c*e^6 + 96*A^2*a*c^2*d^2*e^4 + 16*B^2*a*c^2*d^4*e^2 + 96*B^2*a^2*c*d^2*e^4 + 128*A*B*a^2*c*d*e^5 + 1
28*A*B*a*c^2*d^3*e^3))*((B^2*a^2*c^5*d^3 + B^2*a^2*e^3*(a^3*c^7)^(1/2) + A^2*a*c^6*d^3 + 2*A*B*a^3*c^4*e^3 + 3
*A^2*c^2*d^2*e*(a^3*c^7)^(1/2) + 2*A*B*c^2*d^3*(a^3*c^7)^(1/2) + 3*A^2*a^2*c^5*d*e^2 + 3*B^2*a^3*c^4*d*e^2 + A
^2*a*c*e^3*(a^3*c^7)^(1/2) + 3*B^2*a*c*d^2*e*(a^3*c^7)^(1/2) + 6*A*B*a^2*c^5*d^2*e + 6*A*B*a*c*d*e^2*(a^3*c^7)
^(1/2))/(4*a^2*c^7))^(1/2)))*((B^2*a^2*c^5*d^3 + B^2*a^2*e^3*(a^3*c^7)^(1/2) + A^2*a*c^6*d^3 + 2*A*B*a^3*c^4*e
^3 + 3*A^2*c^2*d^2*e*(a^3*c^7)^(1/2) + 2*A*B*c^2*d^3*(a^3*c^7)^(1/2) + 3*A^2*a^2*c^5*d*e^2 + 3*B^2*a^3*c^4*d*e
^2 + A^2*a*c*e^3*(a^3*c^7)^(1/2) + 3*B^2*a*c*d^2*e*(a^3*c^7)^(1/2) + 6*A*B*a^2*c^5*d^2*e + 6*A*B*a*c*d*e^2*(a^
3*c^7)^(1/2))/(4*a^2*c^7))^(1/2)*2i - atan(((((8*(4*A*a^2*c^4*e^5 - 4*A*a*c^5*d^2*e^3 - 4*B*a*c^5*d^3*e^2 + 4*
B*a^2*c^4*d*e^4))/c^2 - 64*a*c^4*d*e^2*(d + e*x)^(1/2)*((B^2*a^2*c^5*d^3 - B^2*a^2*e^3*(a^3*c^7)^(1/2) + A^2*a
*c^6*d^3 + 2*A*B*a^3*c^4*e^3 - 3*A^2*c^2*d^2*e*(a^3*c^7)^(1/2) - 2*A*B*c^2*d^3*(a^3*c^7)^(1/2) + 3*A^2*a^2*c^5
*d*e^2 + 3*B^2*a^3*c^4*d*e^2 - A^2*a*c*e^3*(a^3*c^7)^(1/2) - 3*B^2*a*c*d^2*e*(a^3*c^7)^(1/2) + 6*A*B*a^2*c^5*d
^2*e - 6*A*B*a*c*d*e^2*(a^3*c^7)^(1/2))/(4*a^2*c^7))^(1/2))*((B^2*a^2*c^5*d^3 - B^2*a^2*e^3*(a^3*c^7)^(1/2) +
A^2*a*c^6*d^3 + 2*A*B*a^3*c^4*e^3 - 3*A^2*c^2*d^2*e*(a^3*c^7)^(1/2) - 2*A*B*c^2*d^3*(a^3*c^7)^(1/2) + 3*A^2*a^
2*c^5*d*e^2 + 3*B^2*a^3*c^4*d*e^2 - A^2*a*c*e^3*(a^3*c^7)^(1/2) - 3*B^2*a*c*d^2*e*(a^3*c^7)^(1/2) + 6*A*B*a^2*
c^5*d^2*e - 6*A*B*a*c*d*e^2*(a^3*c^7)^(1/2))/(4*a^2*c^7))^(1/2) + (d + e*x)^(1/2)*(16*B^2*a^3*e^6 + 16*A^2*c^3
*d^4*e^2 + 16*A^2*a^2*c*e^6 + 96*A^2*a*c^2*d^2*e^4 + 16*B^2*a*c^2*d^4*e^2 + 96*B^2*a^2*c*d^2*e^4 + 128*A*B*a^2
*c*d*e^5 + 128*A*B*a*c^2*d^3*e^3))*((B^2*a^2*c^5*d^3 - B^2*a^2*e^3*(a^3*c^7)^(1/2) + A^2*a*c^6*d^3 + 2*A*B*a^3
*c^4*e^3 - 3*A^2*c^2*d^2*e*(a^3*c^7)^(1/2) - 2*A*B*c^2*d^3*(a^3*c^7)^(1/2) + 3*A^2*a^2*c^5*d*e^2 + 3*B^2*a^3*c
^4*d*e^2 - A^2*a*c*e^3*(a^3*c^7)^(1/2) - 3*B^2*a*c*d^2*e*(a^3*c^7)^(1/2) + 6*A*B*a^2*c^5*d^2*e - 6*A*B*a*c*d*e
^2*(a^3*c^7)^(1/2))/(4*a^2*c^7))^(1/2)*1i - (((8*(4*A*a^2*c^4*e^5 - 4*A*a*c^5*d^2*e^3 - 4*B*a*c^5*d^3*e^2 + 4*
B*a^2*c^4*d*e^4))/c^2 + 64*a*c^4*d*e^2*(d + e*x)^(1/2)*((B^2*a^2*c^5*d^3 - B^2*a^2*e^3*(a^3*c^7)^(1/2) + A^2*a
*c^6*d^3 + 2*A*B*a^3*c^4*e^3 - 3*A^2*c^2*d^2*e*(a^3*c^7)^(1/2) - 2*A*B*c^2*d^3*(a^3*c^7)^(1/2) + 3*A^2*a^2*c^5
*d*e^2 + 3*B^2*a^3*c^4*d*e^2 - A^2*a*c*e^3*(a^3*c^7)^(1/2) - 3*B^2*a*c*d^2*e*(a^3*c^7)^(1/2) + 6*A*B*a^2*c^5*d
^2*e - 6*A*B*a*c*d*e^2*(a^3*c^7)^(1/2))/(4*a^2*c^7))^(1/2))*((B^2*a^2*c^5*d^3 - B^2*a^2*e^3*(a^3*c^7)^(1/2) +
A^2*a*c^6*d^3 + 2*A*B*a^3*c^4*e^3 - 3*A^2*c^2*d^2*e*(a^3*c^7)^(1/2) - 2*A*B*c^2*d^3*(a^3*c^7)^(1/2) + 3*A^2*a^
2*c^5*d*e^2 + 3*B^2*a^3*c^4*d*e^2 - A^2*a*c*e^3*(a^3*c^7)^(1/2) - 3*B^2*a*c*d^2*e*(a^3*c^7)^(1/2) + 6*A*B*a^2*
c^5*d^2*e - 6*A*B*a*c*d*e^2*(a^3*c^7)^(1/2))/(4*a^2*c^7))^(1/2) - (d + e*x)^(1/2)*(16*B^2*a^3*e^6 + 16*A^2*c^3
*d^4*e^2 + 16*A^2*a^2*c*e^6 + 96*A^2*a*c^2*d^2*e^4 + 16*B^2*a*c^2*d^4*e^2 + 96*B^2*a^2*c*d^2*e^4 + 128*A*B*a^2
*c*d*e^5 + 128*A*B*a*c^2*d^3*e^3))*((B^2*a^2*c^5*d^3 - B^2*a^2*e^3*(a^3*c^7)^(1/2) + A^2*a*c^6*d^3 + 2*A*B*a^3
*c^4*e^3 - 3*A^2*c^2*d^2*e*(a^3*c^7)^(1/2) - 2*A*B*c^2*d^3*(a^3*c^7)^(1/2) + 3*A^2*a^2*c^5*d*e^2 + 3*B^2*a^3*c
^4*d*e^2 - A^2*a*c*e^3*(a^3*c^7)^(1/2) - 3*B^2*a*c*d^2*e*(a^3*c^7)^(1/2) + 6*A*B*a^2*c^5*d^2*e - 6*A*B*a*c*d*e
^2*(a^3*c^7)^(1/2))/(4*a^2*c^7))^(1/2)*1i)/((16*(B^3*a^4*e^8 - 2*A^3*c^4*d^5*e^3 - B^3*a^2*c^2*d^4*e^4 - A^2*B
*a^3*c*e^8 - A^2*B*c^4*d^6*e^2 + 4*A^3*a*c^3*d^3*e^5 - 2*A^3*a^2*c^2*d*e^7 + B^3*a*c^3*d^6*e^2 - B^3*a^3*c*d^2
*e^6 - 4*A*B^2*a^2*c^2*d^3*e^5 + A^2*B*a^2*c^2*d^2*e^6 + 2*A*B^2*a^3*c*d*e^7 + 2*A*B^2*a*c^3*d^5*e^3 + A^2*B*a
*c^3*d^4*e^4))/c^2 + (((8*(4*A*a^2*c^4*e^5 - 4*A*a*c^5*d^2*e^3 - 4*B*a*c^5*d^3*e^2 + 4*B*a^2*c^4*d*e^4))/c^2 -
64*a*c^4*d*e^2*(d + e*x)^(1/2)*((B^2*a^2*c^5*d^3 - B^2*a^2*e^3*(a^3*c^7)^(1/2) + A^2*a*c^6*d^3 + 2*A*B*a^3*c^
4*e^3 - 3*A^2*c^2*d^2*e*(a^3*c^7)^(1/2) - 2*A*B*c^2*d^3*(a^3*c^7)^(1/2) + 3*A^2*a^2*c^5*d*e^2 + 3*B^2*a^3*c^4*
d*e^2 - A^2*a*c*e^3*(a^3*c^7)^(1/2) - 3*B^2*a*c*d^2*e*(a^3*c^7)^(1/2) + 6*A*B*a^2*c^5*d^2*e - 6*A*B*a*c*d*e^2*
(a^3*c^7)^(1/2))/(4*a^2*c^7))^(1/2))*((B^2*a^2*c^5*d^3 - B^2*a^2*e^3*(a^3*c^7)^(1/2) + A^2*a*c^6*d^3 + 2*A*B*a
^3*c^4*e^3 - 3*A^2*c^2*d^2*e*(a^3*c^7)^(1/2) - 2*A*B*c^2*d^3*(a^3*c^7)^(1/2) + 3*A^2*a^2*c^5*d*e^2 + 3*B^2*a^3
*c^4*d*e^2 - A^2*a*c*e^3*(a^3*c^7)^(1/2) - 3*B^2*a*c*d^2*e*(a^3*c^7)^(1/2) + 6*A*B*a^2*c^5*d^2*e - 6*A*B*a*c*d
*e^2*(a^3*c^7)^(1/2))/(4*a^2*c^7))^(1/2) + (d + e*x)^(1/2)*(16*B^2*a^3*e^6 + 16*A^2*c^3*d^4*e^2 + 16*A^2*a^2*c
*e^6 + 96*A^2*a*c^2*d^2*e^4 + 16*B^2*a*c^2*d^4*e^2 + 96*B^2*a^2*c*d^2*e^4 + 128*A*B*a^2*c*d*e^5 + 128*A*B*a*c^
2*d^3*e^3))*((B^2*a^2*c^5*d^3 - B^2*a^2*e^3*(a^3*c^7)^(1/2) + A^2*a*c^6*d^3 + 2*A*B*a^3*c^4*e^3 - 3*A^2*c^2*d^
2*e*(a^3*c^7)^(1/2) - 2*A*B*c^2*d^3*(a^3*c^7)^(1/2) + 3*A^2*a^2*c^5*d*e^2 + 3*B^2*a^3*c^4*d*e^2 - A^2*a*c*e^3*
(a^3*c^7)^(1/2) - 3*B^2*a*c*d^2*e*(a^3*c^7)^(1/2) + 6*A*B*a^2*c^5*d^2*e - 6*A*B*a*c*d*e^2*(a^3*c^7)^(1/2))/(4*
a^2*c^7))^(1/2) + (((8*(4*A*a^2*c^4*e^5 - 4*A*a*c^5*d^2*e^3 - 4*B*a*c^5*d^3*e^2 + 4*B*a^2*c^4*d*e^4))/c^2 + 64
*a*c^4*d*e^2*(d + e*x)^(1/2)*((B^2*a^2*c^5*d^3 - B^2*a^2*e^3*(a^3*c^7)^(1/2) + A^2*a*c^6*d^3 + 2*A*B*a^3*c^4*e
^3 - 3*A^2*c^2*d^2*e*(a^3*c^7)^(1/2) - 2*A*B*c^2*d^3*(a^3*c^7)^(1/2) + 3*A^2*a^2*c^5*d*e^2 + 3*B^2*a^3*c^4*d*e
^2 - A^2*a*c*e^3*(a^3*c^7)^(1/2) - 3*B^2*a*c*d^2*e*(a^3*c^7)^(1/2) + 6*A*B*a^2*c^5*d^2*e - 6*A*B*a*c*d*e^2*(a^
3*c^7)^(1/2))/(4*a^2*c^7))^(1/2))*((B^2*a^2*c^5*d^3 - B^2*a^2*e^3*(a^3*c^7)^(1/2) + A^2*a*c^6*d^3 + 2*A*B*a^3*
c^4*e^3 - 3*A^2*c^2*d^2*e*(a^3*c^7)^(1/2) - 2*A*B*c^2*d^3*(a^3*c^7)^(1/2) + 3*A^2*a^2*c^5*d*e^2 + 3*B^2*a^3*c^
4*d*e^2 - A^2*a*c*e^3*(a^3*c^7)^(1/2) - 3*B^2*a*c*d^2*e*(a^3*c^7)^(1/2) + 6*A*B*a^2*c^5*d^2*e - 6*A*B*a*c*d*e^
2*(a^3*c^7)^(1/2))/(4*a^2*c^7))^(1/2) - (d + e*x)^(1/2)*(16*B^2*a^3*e^6 + 16*A^2*c^3*d^4*e^2 + 16*A^2*a^2*c*e^
6 + 96*A^2*a*c^2*d^2*e^4 + 16*B^2*a*c^2*d^4*e^2 + 96*B^2*a^2*c*d^2*e^4 + 128*A*B*a^2*c*d*e^5 + 128*A*B*a*c^2*d
^3*e^3))*((B^2*a^2*c^5*d^3 - B^2*a^2*e^3*(a^3*c^7)^(1/2) + A^2*a*c^6*d^3 + 2*A*B*a^3*c^4*e^3 - 3*A^2*c^2*d^2*e
*(a^3*c^7)^(1/2) - 2*A*B*c^2*d^3*(a^3*c^7)^(1/2) + 3*A^2*a^2*c^5*d*e^2 + 3*B^2*a^3*c^4*d*e^2 - A^2*a*c*e^3*(a^
3*c^7)^(1/2) - 3*B^2*a*c*d^2*e*(a^3*c^7)^(1/2) + 6*A*B*a^2*c^5*d^2*e - 6*A*B*a*c*d*e^2*(a^3*c^7)^(1/2))/(4*a^2
*c^7))^(1/2)))*((B^2*a^2*c^5*d^3 - B^2*a^2*e^3*(a^3*c^7)^(1/2) + A^2*a*c^6*d^3 + 2*A*B*a^3*c^4*e^3 - 3*A^2*c^2
*d^2*e*(a^3*c^7)^(1/2) - 2*A*B*c^2*d^3*(a^3*c^7)^(1/2) + 3*A^2*a^2*c^5*d*e^2 + 3*B^2*a^3*c^4*d*e^2 - A^2*a*c*e
^3*(a^3*c^7)^(1/2) - 3*B^2*a*c*d^2*e*(a^3*c^7)^(1/2) + 6*A*B*a^2*c^5*d^2*e - 6*A*B*a*c*d*e^2*(a^3*c^7)^(1/2))/
(4*a^2*c^7))^(1/2)*2i - (2*B*(d + e*x)^(3/2))/(3*c)
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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),x)
[Out]
Timed out
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