3.1448 \(\int \frac{(A+B x) (d+e x)^{3/2}}{a-c x^2} \, dx\)

Optimal. Leaf size=202 \[ \frac{\left (\sqrt{a} B-A \sqrt{c}\right ) \left (\sqrt{c} d-\sqrt{a} e\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{\sqrt{a} c^{7/4}}+\frac{\left (\sqrt{a} B+A \sqrt{c}\right ) \left (\sqrt{a} e+\sqrt{c} d\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{\sqrt{a} c^{7/4}}-\frac{2 \sqrt{d+e x} (A e+B d)}{c}-\frac{2 B (d+e x)^{3/2}}{3 c} \]

[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[Sqr
t[c]*d - Sqrt[a]*e]])/(Sqrt[a]*c^(7/4)) + ((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))

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Rubi [A]  time = 0.993054, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16 \[ \frac{\left (\sqrt{a} B-A \sqrt{c}\right ) \left (\sqrt{c} d-\sqrt{a} e\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{\sqrt{a} c^{7/4}}+\frac{\left (\sqrt{a} B+A \sqrt{c}\right ) \left (\sqrt{a} e+\sqrt{c} d\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{\sqrt{a} c^{7/4}}-\frac{2 \sqrt{d+e x} (A e+B d)}{c}-\frac{2 B (d+e x)^{3/2}}{3 c} \]

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[Sqr
t[c]*d - Sqrt[a]*e]])/(Sqrt[a]*c^(7/4)) + ((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))

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [A]  time = 0.268715, size = 232, normalized size = 1.15 \[ \frac{\left (\sqrt{a} B-A \sqrt{c}\right ) \left (\sqrt{c} d-\sqrt{a} e\right )^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-\sqrt{a} \sqrt{c} e}}\right )}{\sqrt{a} c^{3/2} \sqrt{c d-\sqrt{a} \sqrt{c} e}}+\frac{\left (\sqrt{a} B+A \sqrt{c}\right ) \left (\sqrt{a} e+\sqrt{c} d\right )^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} \sqrt{c} e+c d}}\right )}{\sqrt{a} c^{3/2} \sqrt{\sqrt{a} \sqrt{c} e+c d}}-\frac{2 \sqrt{d+e x} (3 A e+4 B d+B e x)}{3 c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.049, size = 689, normalized size = 3.4 \[ \text{result too large to display} \]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\int \frac{{\left (B x + A\right )}{\left (e x + d\right )}^{\frac{3}{2}}}{c x^{2} - a}\,{d x} \]

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)

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Fricas [A]  time = 1.66405, size = 6048, normalized size = 29.94 \[ \text{result too large to display} \]

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))*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)))*s
qrt((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

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

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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GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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]

Timed out