3.1447 \(\int \frac{(A+B x) (d+e x)^{5/2}}{a-c x^2} \, dx\)

Optimal. Leaf size=237 \[ \frac{\left (\sqrt{a} B-A \sqrt{c}\right ) \left (\sqrt{c} d-\sqrt{a} e\right )^{5/2} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{\sqrt{a} c^{9/4}}+\frac{\left (\sqrt{a} B+A \sqrt{c}\right ) \left (\sqrt{a} e+\sqrt{c} d\right )^{5/2} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{\sqrt{a} c^{9/4}}-\frac{2 \sqrt{d+e x} \left (a B e^2+2 A c d e+B c d^2\right )}{c^2}-\frac{2 (d+e x)^{3/2} (A e+B d)}{3 c}-\frac{2 B (d+e x)^{5/2}}{5 c} \]

[Out]

(-2*(B*c*d^2 + 2*A*c*d*e + a*B*e^2)*Sqrt[d + e*x])/c^2 - (2*(B*d + A*e)*(d + e*x
)^(3/2))/(3*c) - (2*B*(d + e*x)^(5/2))/(5*c) + ((Sqrt[a]*B - A*Sqrt[c])*(Sqrt[c]
*d - Sqrt[a]*e)^(5/2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e
]])/(Sqrt[a]*c^(9/4)) + ((Sqrt[a]*B + A*Sqrt[c])*(Sqrt[c]*d + Sqrt[a]*e)^(5/2)*A
rcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(Sqrt[a]*c^(9/4))

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Rubi [A]  time = 1.35681, antiderivative size = 237, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16 \[ \frac{\left (\sqrt{a} B-A \sqrt{c}\right ) \left (\sqrt{c} d-\sqrt{a} e\right )^{5/2} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{\sqrt{a} c^{9/4}}+\frac{\left (\sqrt{a} B+A \sqrt{c}\right ) \left (\sqrt{a} e+\sqrt{c} d\right )^{5/2} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{\sqrt{a} c^{9/4}}-\frac{2 \sqrt{d+e x} \left (a B e^2+2 A c d e+B c d^2\right )}{c^2}-\frac{2 (d+e x)^{3/2} (A e+B d)}{3 c}-\frac{2 B (d+e x)^{5/2}}{5 c} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(d + e*x)^(5/2))/(a - c*x^2),x]

[Out]

(-2*(B*c*d^2 + 2*A*c*d*e + a*B*e^2)*Sqrt[d + e*x])/c^2 - (2*(B*d + A*e)*(d + e*x
)^(3/2))/(3*c) - (2*B*(d + e*x)^(5/2))/(5*c) + ((Sqrt[a]*B - A*Sqrt[c])*(Sqrt[c]
*d - Sqrt[a]*e)^(5/2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e
]])/(Sqrt[a]*c^(9/4)) + ((Sqrt[a]*B + A*Sqrt[c])*(Sqrt[c]*d + Sqrt[a]*e)^(5/2)*A
rcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(Sqrt[a]*c^(9/4))

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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)**(5/2)/(-c*x**2+a),x)

[Out]

Timed out

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Mathematica [A]  time = 0.410771, size = 258, normalized size = 1.09 \[ -\frac{2 \sqrt{d+e x} \left (15 a B e^2+5 A c e (7 d+e x)+B c \left (23 d^2+11 d e x+3 e^2 x^2\right )\right )}{15 c^2}-\frac{\left (\sqrt{a} B-A \sqrt{c}\right ) \left (\sqrt{a} e-\sqrt{c} d\right )^3 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-\sqrt{a} \sqrt{c} e}}\right )}{\sqrt{a} c^2 \sqrt{c d-\sqrt{a} \sqrt{c} e}}+\frac{\left (\sqrt{a} B+A \sqrt{c}\right ) \left (\sqrt{a} e+\sqrt{c} d\right )^3 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} \sqrt{c} e+c d}}\right )}{\sqrt{a} c^2 \sqrt{\sqrt{a} \sqrt{c} e+c d}} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(d + e*x)^(5/2))/(a - c*x^2),x]

[Out]

(-2*Sqrt[d + e*x]*(15*a*B*e^2 + 5*A*c*e*(7*d + e*x) + B*c*(23*d^2 + 11*d*e*x + 3
*e^2*x^2)))/(15*c^2) - ((Sqrt[a]*B - A*Sqrt[c])*(-(Sqrt[c]*d) + Sqrt[a]*e)^3*Arc
Tanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - Sqrt[a]*Sqrt[c]*e]])/(Sqrt[a]*c^2*Sqrt[c
*d - Sqrt[a]*Sqrt[c]*e]) + ((Sqrt[a]*B + A*Sqrt[c])*(Sqrt[c]*d + Sqrt[a]*e)^3*Ar
cTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d + Sqrt[a]*Sqrt[c]*e]])/(Sqrt[a]*c^2*Sqrt[
c*d + Sqrt[a]*Sqrt[c]*e])

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Maple [B]  time = 0.061, size = 981, normalized size = 4.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(e*x+d)^(5/2)/(-c*x^2+a),x)

[Out]

-2/5*B*(e*x+d)^(5/2)/c-2/3/c*A*(e*x+d)^(3/2)*e-2/3/c*B*(e*x+d)^(3/2)*d-4/c*A*d*e
*(e*x+d)^(1/2)-2/c^2*a*B*e^2*(e*x+d)^(1/2)-2/c*B*d^2*(e*x+d)^(1/2)+3/(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*d*e^3+c/(a*c*e^2)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctan
h(c*(e*x+d)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2))*A*d^3*e+1/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))*B*e^4*a^2+3/(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))*B*a*d^2*e^2+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))*A*a*e^3+3/
((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+3/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*d*e^2+1/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctan
h(c*(e*x+d)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2))*B*d^3+3/(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*d*e^3+c/(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*d^3*e+1/c/(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))
*B*e^4*a^2+3/(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))*B*a*d^2*e^2-1/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))*A*a*e^3-3/((-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^2*e-3/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*d*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^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\int \frac{{\left (B x + A\right )}{\left (e x + d\right )}^{\frac{5}{2}}}{c x^{2} - a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(B*x + A)*(e*x + d)^(5/2)/(c*x^2 - a),x, algorithm="maxima")

[Out]

-integrate((B*x + A)*(e*x + d)^(5/2)/(c*x^2 - a), x)

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Fricas [A]  time = 9.36161, size = 10004, normalized size = 42.21 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(B*x + A)*(e*x + d)^(5/2)/(c*x^2 - a),x, algorithm="fricas")

[Out]

-1/30*(15*c^2*sqrt((10*A*B*a*c^2*d^4*e + 20*A*B*a^2*c*d^2*e^3 + 2*A*B*a^3*e^5 +
(B^2*a*c^2 + A^2*c^3)*d^5 + 10*(B^2*a^2*c + A^2*a*c^2)*d^3*e^2 + 5*(B^2*a^3 + A^
2*a^2*c)*d*e^4 + a*c^4*sqrt((4*A^2*B^2*c^6*d^10 + 20*(A*B^3*a*c^5 + A^3*B*c^6)*d
^9*e + 5*(5*B^4*a^2*c^4 + 26*A^2*B^2*a*c^5 + 5*A^4*c^6)*d^8*e^2 + 240*(A*B^3*a^2
*c^4 + A^3*B*a*c^5)*d^7*e^3 + 20*(5*B^4*a^3*c^3 + 32*A^2*B^2*a^2*c^4 + 5*A^4*a*c
^5)*d^6*e^4 + 504*(A*B^3*a^3*c^3 + A^3*B*a^2*c^4)*d^5*e^5 + 10*(11*B^4*a^4*c^2 +
62*A^2*B^2*a^3*c^3 + 11*A^4*a^2*c^4)*d^4*e^6 + 240*(A*B^3*a^4*c^2 + A^3*B*a^3*c
^3)*d^3*e^7 + 20*(B^4*a^5*c + 7*A^2*B^2*a^4*c^2 + A^4*a^3*c^3)*d^2*e^8 + 20*(A*B
^3*a^5*c + A^3*B*a^4*c^2)*d*e^9 + (B^4*a^6 + 2*A^2*B^2*a^5*c + A^4*a^4*c^2)*e^10
)/(a*c^9)))/(a*c^4))*log(-(2*(A*B^3*a*c^5 - A^3*B*c^6)*d^9 + 5*(B^4*a^2*c^4 - A^
4*c^6)*d^8*e + 16*(A*B^3*a^2*c^4 - A^3*B*a*c^5)*d^7*e^2 - 28*(A*B^3*a^3*c^3 - A^
3*B*a^2*c^4)*d^5*e^4 - 14*(B^4*a^4*c^2 - A^4*a^2*c^4)*d^4*e^5 + 8*(B^4*a^5*c - A
^4*a^3*c^3)*d^2*e^7 + 10*(A*B^3*a^5*c - A^3*B*a^4*c^2)*d*e^8 + (B^4*a^6 - A^4*a^
4*c^2)*e^9)*sqrt(e*x + d) + (2*A*B^2*a*c^6*d^7 + (5*B^3*a^2*c^5 + 9*A^2*B*a*c^6)
*d^6*e + 2*(16*A*B^2*a^2*c^5 + 5*A^3*a*c^6)*d^5*e^2 + 5*(3*B^3*a^3*c^4 + 11*A^2*
B*a^2*c^5)*d^4*e^3 + 10*(5*A*B^2*a^3*c^4 + 2*A^3*a^2*c^5)*d^3*e^4 + (11*B^3*a^4*
c^3 + 31*A^2*B*a^3*c^4)*d^2*e^5 + 2*(6*A*B^2*a^4*c^3 + A^3*a^3*c^4)*d*e^6 + (B^3
*a^5*c^2 + A^2*B*a^4*c^3)*e^7 - (A*a*c^8*d^2 + 2*B*a^2*c^7*d*e + A*a^2*c^7*e^2)*
sqrt((4*A^2*B^2*c^6*d^10 + 20*(A*B^3*a*c^5 + A^3*B*c^6)*d^9*e + 5*(5*B^4*a^2*c^4
+ 26*A^2*B^2*a*c^5 + 5*A^4*c^6)*d^8*e^2 + 240*(A*B^3*a^2*c^4 + A^3*B*a*c^5)*d^7
*e^3 + 20*(5*B^4*a^3*c^3 + 32*A^2*B^2*a^2*c^4 + 5*A^4*a*c^5)*d^6*e^4 + 504*(A*B^
3*a^3*c^3 + A^3*B*a^2*c^4)*d^5*e^5 + 10*(11*B^4*a^4*c^2 + 62*A^2*B^2*a^3*c^3 + 1
1*A^4*a^2*c^4)*d^4*e^6 + 240*(A*B^3*a^4*c^2 + A^3*B*a^3*c^3)*d^3*e^7 + 20*(B^4*a
^5*c + 7*A^2*B^2*a^4*c^2 + A^4*a^3*c^3)*d^2*e^8 + 20*(A*B^3*a^5*c + A^3*B*a^4*c^
2)*d*e^9 + (B^4*a^6 + 2*A^2*B^2*a^5*c + A^4*a^4*c^2)*e^10)/(a*c^9)))*sqrt((10*A*
B*a*c^2*d^4*e + 20*A*B*a^2*c*d^2*e^3 + 2*A*B*a^3*e^5 + (B^2*a*c^2 + A^2*c^3)*d^5
+ 10*(B^2*a^2*c + A^2*a*c^2)*d^3*e^2 + 5*(B^2*a^3 + A^2*a^2*c)*d*e^4 + a*c^4*sq
rt((4*A^2*B^2*c^6*d^10 + 20*(A*B^3*a*c^5 + A^3*B*c^6)*d^9*e + 5*(5*B^4*a^2*c^4 +
26*A^2*B^2*a*c^5 + 5*A^4*c^6)*d^8*e^2 + 240*(A*B^3*a^2*c^4 + A^3*B*a*c^5)*d^7*e
^3 + 20*(5*B^4*a^3*c^3 + 32*A^2*B^2*a^2*c^4 + 5*A^4*a*c^5)*d^6*e^4 + 504*(A*B^3*
a^3*c^3 + A^3*B*a^2*c^4)*d^5*e^5 + 10*(11*B^4*a^4*c^2 + 62*A^2*B^2*a^3*c^3 + 11*
A^4*a^2*c^4)*d^4*e^6 + 240*(A*B^3*a^4*c^2 + A^3*B*a^3*c^3)*d^3*e^7 + 20*(B^4*a^5
*c + 7*A^2*B^2*a^4*c^2 + A^4*a^3*c^3)*d^2*e^8 + 20*(A*B^3*a^5*c + A^3*B*a^4*c^2)
*d*e^9 + (B^4*a^6 + 2*A^2*B^2*a^5*c + A^4*a^4*c^2)*e^10)/(a*c^9)))/(a*c^4))) - 1
5*c^2*sqrt((10*A*B*a*c^2*d^4*e + 20*A*B*a^2*c*d^2*e^3 + 2*A*B*a^3*e^5 + (B^2*a*c
^2 + A^2*c^3)*d^5 + 10*(B^2*a^2*c + A^2*a*c^2)*d^3*e^2 + 5*(B^2*a^3 + A^2*a^2*c)
*d*e^4 + a*c^4*sqrt((4*A^2*B^2*c^6*d^10 + 20*(A*B^3*a*c^5 + A^3*B*c^6)*d^9*e + 5
*(5*B^4*a^2*c^4 + 26*A^2*B^2*a*c^5 + 5*A^4*c^6)*d^8*e^2 + 240*(A*B^3*a^2*c^4 + A
^3*B*a*c^5)*d^7*e^3 + 20*(5*B^4*a^3*c^3 + 32*A^2*B^2*a^2*c^4 + 5*A^4*a*c^5)*d^6*
e^4 + 504*(A*B^3*a^3*c^3 + A^3*B*a^2*c^4)*d^5*e^5 + 10*(11*B^4*a^4*c^2 + 62*A^2*
B^2*a^3*c^3 + 11*A^4*a^2*c^4)*d^4*e^6 + 240*(A*B^3*a^4*c^2 + A^3*B*a^3*c^3)*d^3*
e^7 + 20*(B^4*a^5*c + 7*A^2*B^2*a^4*c^2 + A^4*a^3*c^3)*d^2*e^8 + 20*(A*B^3*a^5*c
+ A^3*B*a^4*c^2)*d*e^9 + (B^4*a^6 + 2*A^2*B^2*a^5*c + A^4*a^4*c^2)*e^10)/(a*c^9
)))/(a*c^4))*log(-(2*(A*B^3*a*c^5 - A^3*B*c^6)*d^9 + 5*(B^4*a^2*c^4 - A^4*c^6)*d
^8*e + 16*(A*B^3*a^2*c^4 - A^3*B*a*c^5)*d^7*e^2 - 28*(A*B^3*a^3*c^3 - A^3*B*a^2*
c^4)*d^5*e^4 - 14*(B^4*a^4*c^2 - A^4*a^2*c^4)*d^4*e^5 + 8*(B^4*a^5*c - A^4*a^3*c
^3)*d^2*e^7 + 10*(A*B^3*a^5*c - A^3*B*a^4*c^2)*d*e^8 + (B^4*a^6 - A^4*a^4*c^2)*e
^9)*sqrt(e*x + d) - (2*A*B^2*a*c^6*d^7 + (5*B^3*a^2*c^5 + 9*A^2*B*a*c^6)*d^6*e +
2*(16*A*B^2*a^2*c^5 + 5*A^3*a*c^6)*d^5*e^2 + 5*(3*B^3*a^3*c^4 + 11*A^2*B*a^2*c^
5)*d^4*e^3 + 10*(5*A*B^2*a^3*c^4 + 2*A^3*a^2*c^5)*d^3*e^4 + (11*B^3*a^4*c^3 + 31
*A^2*B*a^3*c^4)*d^2*e^5 + 2*(6*A*B^2*a^4*c^3 + A^3*a^3*c^4)*d*e^6 + (B^3*a^5*c^2
+ A^2*B*a^4*c^3)*e^7 - (A*a*c^8*d^2 + 2*B*a^2*c^7*d*e + A*a^2*c^7*e^2)*sqrt((4*
A^2*B^2*c^6*d^10 + 20*(A*B^3*a*c^5 + A^3*B*c^6)*d^9*e + 5*(5*B^4*a^2*c^4 + 26*A^
2*B^2*a*c^5 + 5*A^4*c^6)*d^8*e^2 + 240*(A*B^3*a^2*c^4 + A^3*B*a*c^5)*d^7*e^3 + 2
0*(5*B^4*a^3*c^3 + 32*A^2*B^2*a^2*c^4 + 5*A^4*a*c^5)*d^6*e^4 + 504*(A*B^3*a^3*c^
3 + A^3*B*a^2*c^4)*d^5*e^5 + 10*(11*B^4*a^4*c^2 + 62*A^2*B^2*a^3*c^3 + 11*A^4*a^
2*c^4)*d^4*e^6 + 240*(A*B^3*a^4*c^2 + A^3*B*a^3*c^3)*d^3*e^7 + 20*(B^4*a^5*c + 7
*A^2*B^2*a^4*c^2 + A^4*a^3*c^3)*d^2*e^8 + 20*(A*B^3*a^5*c + A^3*B*a^4*c^2)*d*e^9
+ (B^4*a^6 + 2*A^2*B^2*a^5*c + A^4*a^4*c^2)*e^10)/(a*c^9)))*sqrt((10*A*B*a*c^2*
d^4*e + 20*A*B*a^2*c*d^2*e^3 + 2*A*B*a^3*e^5 + (B^2*a*c^2 + A^2*c^3)*d^5 + 10*(B
^2*a^2*c + A^2*a*c^2)*d^3*e^2 + 5*(B^2*a^3 + A^2*a^2*c)*d*e^4 + a*c^4*sqrt((4*A^
2*B^2*c^6*d^10 + 20*(A*B^3*a*c^5 + A^3*B*c^6)*d^9*e + 5*(5*B^4*a^2*c^4 + 26*A^2*
B^2*a*c^5 + 5*A^4*c^6)*d^8*e^2 + 240*(A*B^3*a^2*c^4 + A^3*B*a*c^5)*d^7*e^3 + 20*
(5*B^4*a^3*c^3 + 32*A^2*B^2*a^2*c^4 + 5*A^4*a*c^5)*d^6*e^4 + 504*(A*B^3*a^3*c^3
+ A^3*B*a^2*c^4)*d^5*e^5 + 10*(11*B^4*a^4*c^2 + 62*A^2*B^2*a^3*c^3 + 11*A^4*a^2*
c^4)*d^4*e^6 + 240*(A*B^3*a^4*c^2 + A^3*B*a^3*c^3)*d^3*e^7 + 20*(B^4*a^5*c + 7*A
^2*B^2*a^4*c^2 + A^4*a^3*c^3)*d^2*e^8 + 20*(A*B^3*a^5*c + A^3*B*a^4*c^2)*d*e^9 +
(B^4*a^6 + 2*A^2*B^2*a^5*c + A^4*a^4*c^2)*e^10)/(a*c^9)))/(a*c^4))) + 15*c^2*sq
rt((10*A*B*a*c^2*d^4*e + 20*A*B*a^2*c*d^2*e^3 + 2*A*B*a^3*e^5 + (B^2*a*c^2 + A^2
*c^3)*d^5 + 10*(B^2*a^2*c + A^2*a*c^2)*d^3*e^2 + 5*(B^2*a^3 + A^2*a^2*c)*d*e^4 -
a*c^4*sqrt((4*A^2*B^2*c^6*d^10 + 20*(A*B^3*a*c^5 + A^3*B*c^6)*d^9*e + 5*(5*B^4*
a^2*c^4 + 26*A^2*B^2*a*c^5 + 5*A^4*c^6)*d^8*e^2 + 240*(A*B^3*a^2*c^4 + A^3*B*a*c
^5)*d^7*e^3 + 20*(5*B^4*a^3*c^3 + 32*A^2*B^2*a^2*c^4 + 5*A^4*a*c^5)*d^6*e^4 + 50
4*(A*B^3*a^3*c^3 + A^3*B*a^2*c^4)*d^5*e^5 + 10*(11*B^4*a^4*c^2 + 62*A^2*B^2*a^3*
c^3 + 11*A^4*a^2*c^4)*d^4*e^6 + 240*(A*B^3*a^4*c^2 + A^3*B*a^3*c^3)*d^3*e^7 + 20
*(B^4*a^5*c + 7*A^2*B^2*a^4*c^2 + A^4*a^3*c^3)*d^2*e^8 + 20*(A*B^3*a^5*c + A^3*B
*a^4*c^2)*d*e^9 + (B^4*a^6 + 2*A^2*B^2*a^5*c + A^4*a^4*c^2)*e^10)/(a*c^9)))/(a*c
^4))*log(-(2*(A*B^3*a*c^5 - A^3*B*c^6)*d^9 + 5*(B^4*a^2*c^4 - A^4*c^6)*d^8*e + 1
6*(A*B^3*a^2*c^4 - A^3*B*a*c^5)*d^7*e^2 - 28*(A*B^3*a^3*c^3 - A^3*B*a^2*c^4)*d^5
*e^4 - 14*(B^4*a^4*c^2 - A^4*a^2*c^4)*d^4*e^5 + 8*(B^4*a^5*c - A^4*a^3*c^3)*d^2*
e^7 + 10*(A*B^3*a^5*c - A^3*B*a^4*c^2)*d*e^8 + (B^4*a^6 - A^4*a^4*c^2)*e^9)*sqrt
(e*x + d) + (2*A*B^2*a*c^6*d^7 + (5*B^3*a^2*c^5 + 9*A^2*B*a*c^6)*d^6*e + 2*(16*A
*B^2*a^2*c^5 + 5*A^3*a*c^6)*d^5*e^2 + 5*(3*B^3*a^3*c^4 + 11*A^2*B*a^2*c^5)*d^4*e
^3 + 10*(5*A*B^2*a^3*c^4 + 2*A^3*a^2*c^5)*d^3*e^4 + (11*B^3*a^4*c^3 + 31*A^2*B*a
^3*c^4)*d^2*e^5 + 2*(6*A*B^2*a^4*c^3 + A^3*a^3*c^4)*d*e^6 + (B^3*a^5*c^2 + A^2*B
*a^4*c^3)*e^7 + (A*a*c^8*d^2 + 2*B*a^2*c^7*d*e + A*a^2*c^7*e^2)*sqrt((4*A^2*B^2*
c^6*d^10 + 20*(A*B^3*a*c^5 + A^3*B*c^6)*d^9*e + 5*(5*B^4*a^2*c^4 + 26*A^2*B^2*a*
c^5 + 5*A^4*c^6)*d^8*e^2 + 240*(A*B^3*a^2*c^4 + A^3*B*a*c^5)*d^7*e^3 + 20*(5*B^4
*a^3*c^3 + 32*A^2*B^2*a^2*c^4 + 5*A^4*a*c^5)*d^6*e^4 + 504*(A*B^3*a^3*c^3 + A^3*
B*a^2*c^4)*d^5*e^5 + 10*(11*B^4*a^4*c^2 + 62*A^2*B^2*a^3*c^3 + 11*A^4*a^2*c^4)*d
^4*e^6 + 240*(A*B^3*a^4*c^2 + A^3*B*a^3*c^3)*d^3*e^7 + 20*(B^4*a^5*c + 7*A^2*B^2
*a^4*c^2 + A^4*a^3*c^3)*d^2*e^8 + 20*(A*B^3*a^5*c + A^3*B*a^4*c^2)*d*e^9 + (B^4*
a^6 + 2*A^2*B^2*a^5*c + A^4*a^4*c^2)*e^10)/(a*c^9)))*sqrt((10*A*B*a*c^2*d^4*e +
20*A*B*a^2*c*d^2*e^3 + 2*A*B*a^3*e^5 + (B^2*a*c^2 + A^2*c^3)*d^5 + 10*(B^2*a^2*c
+ A^2*a*c^2)*d^3*e^2 + 5*(B^2*a^3 + A^2*a^2*c)*d*e^4 - a*c^4*sqrt((4*A^2*B^2*c^
6*d^10 + 20*(A*B^3*a*c^5 + A^3*B*c^6)*d^9*e + 5*(5*B^4*a^2*c^4 + 26*A^2*B^2*a*c^
5 + 5*A^4*c^6)*d^8*e^2 + 240*(A*B^3*a^2*c^4 + A^3*B*a*c^5)*d^7*e^3 + 20*(5*B^4*a
^3*c^3 + 32*A^2*B^2*a^2*c^4 + 5*A^4*a*c^5)*d^6*e^4 + 504*(A*B^3*a^3*c^3 + A^3*B*
a^2*c^4)*d^5*e^5 + 10*(11*B^4*a^4*c^2 + 62*A^2*B^2*a^3*c^3 + 11*A^4*a^2*c^4)*d^4
*e^6 + 240*(A*B^3*a^4*c^2 + A^3*B*a^3*c^3)*d^3*e^7 + 20*(B^4*a^5*c + 7*A^2*B^2*a
^4*c^2 + A^4*a^3*c^3)*d^2*e^8 + 20*(A*B^3*a^5*c + A^3*B*a^4*c^2)*d*e^9 + (B^4*a^
6 + 2*A^2*B^2*a^5*c + A^4*a^4*c^2)*e^10)/(a*c^9)))/(a*c^4))) - 15*c^2*sqrt((10*A
*B*a*c^2*d^4*e + 20*A*B*a^2*c*d^2*e^3 + 2*A*B*a^3*e^5 + (B^2*a*c^2 + A^2*c^3)*d^
5 + 10*(B^2*a^2*c + A^2*a*c^2)*d^3*e^2 + 5*(B^2*a^3 + A^2*a^2*c)*d*e^4 - a*c^4*s
qrt((4*A^2*B^2*c^6*d^10 + 20*(A*B^3*a*c^5 + A^3*B*c^6)*d^9*e + 5*(5*B^4*a^2*c^4
+ 26*A^2*B^2*a*c^5 + 5*A^4*c^6)*d^8*e^2 + 240*(A*B^3*a^2*c^4 + A^3*B*a*c^5)*d^7*
e^3 + 20*(5*B^4*a^3*c^3 + 32*A^2*B^2*a^2*c^4 + 5*A^4*a*c^5)*d^6*e^4 + 504*(A*B^3
*a^3*c^3 + A^3*B*a^2*c^4)*d^5*e^5 + 10*(11*B^4*a^4*c^2 + 62*A^2*B^2*a^3*c^3 + 11
*A^4*a^2*c^4)*d^4*e^6 + 240*(A*B^3*a^4*c^2 + A^3*B*a^3*c^3)*d^3*e^7 + 20*(B^4*a^
5*c + 7*A^2*B^2*a^4*c^2 + A^4*a^3*c^3)*d^2*e^8 + 20*(A*B^3*a^5*c + A^3*B*a^4*c^2
)*d*e^9 + (B^4*a^6 + 2*A^2*B^2*a^5*c + A^4*a^4*c^2)*e^10)/(a*c^9)))/(a*c^4))*log
(-(2*(A*B^3*a*c^5 - A^3*B*c^6)*d^9 + 5*(B^4*a^2*c^4 - A^4*c^6)*d^8*e + 16*(A*B^3
*a^2*c^4 - A^3*B*a*c^5)*d^7*e^2 - 28*(A*B^3*a^3*c^3 - A^3*B*a^2*c^4)*d^5*e^4 - 1
4*(B^4*a^4*c^2 - A^4*a^2*c^4)*d^4*e^5 + 8*(B^4*a^5*c - A^4*a^3*c^3)*d^2*e^7 + 10
*(A*B^3*a^5*c - A^3*B*a^4*c^2)*d*e^8 + (B^4*a^6 - A^4*a^4*c^2)*e^9)*sqrt(e*x + d
) - (2*A*B^2*a*c^6*d^7 + (5*B^3*a^2*c^5 + 9*A^2*B*a*c^6)*d^6*e + 2*(16*A*B^2*a^2
*c^5 + 5*A^3*a*c^6)*d^5*e^2 + 5*(3*B^3*a^3*c^4 + 11*A^2*B*a^2*c^5)*d^4*e^3 + 10*
(5*A*B^2*a^3*c^4 + 2*A^3*a^2*c^5)*d^3*e^4 + (11*B^3*a^4*c^3 + 31*A^2*B*a^3*c^4)*
d^2*e^5 + 2*(6*A*B^2*a^4*c^3 + A^3*a^3*c^4)*d*e^6 + (B^3*a^5*c^2 + A^2*B*a^4*c^3
)*e^7 + (A*a*c^8*d^2 + 2*B*a^2*c^7*d*e + A*a^2*c^7*e^2)*sqrt((4*A^2*B^2*c^6*d^10
+ 20*(A*B^3*a*c^5 + A^3*B*c^6)*d^9*e + 5*(5*B^4*a^2*c^4 + 26*A^2*B^2*a*c^5 + 5*
A^4*c^6)*d^8*e^2 + 240*(A*B^3*a^2*c^4 + A^3*B*a*c^5)*d^7*e^3 + 20*(5*B^4*a^3*c^3
+ 32*A^2*B^2*a^2*c^4 + 5*A^4*a*c^5)*d^6*e^4 + 504*(A*B^3*a^3*c^3 + A^3*B*a^2*c^
4)*d^5*e^5 + 10*(11*B^4*a^4*c^2 + 62*A^2*B^2*a^3*c^3 + 11*A^4*a^2*c^4)*d^4*e^6 +
240*(A*B^3*a^4*c^2 + A^3*B*a^3*c^3)*d^3*e^7 + 20*(B^4*a^5*c + 7*A^2*B^2*a^4*c^2
+ A^4*a^3*c^3)*d^2*e^8 + 20*(A*B^3*a^5*c + A^3*B*a^4*c^2)*d*e^9 + (B^4*a^6 + 2*
A^2*B^2*a^5*c + A^4*a^4*c^2)*e^10)/(a*c^9)))*sqrt((10*A*B*a*c^2*d^4*e + 20*A*B*a
^2*c*d^2*e^3 + 2*A*B*a^3*e^5 + (B^2*a*c^2 + A^2*c^3)*d^5 + 10*(B^2*a^2*c + A^2*a
*c^2)*d^3*e^2 + 5*(B^2*a^3 + A^2*a^2*c)*d*e^4 - a*c^4*sqrt((4*A^2*B^2*c^6*d^10 +
20*(A*B^3*a*c^5 + A^3*B*c^6)*d^9*e + 5*(5*B^4*a^2*c^4 + 26*A^2*B^2*a*c^5 + 5*A^
4*c^6)*d^8*e^2 + 240*(A*B^3*a^2*c^4 + A^3*B*a*c^5)*d^7*e^3 + 20*(5*B^4*a^3*c^3 +
32*A^2*B^2*a^2*c^4 + 5*A^4*a*c^5)*d^6*e^4 + 504*(A*B^3*a^3*c^3 + A^3*B*a^2*c^4)
*d^5*e^5 + 10*(11*B^4*a^4*c^2 + 62*A^2*B^2*a^3*c^3 + 11*A^4*a^2*c^4)*d^4*e^6 + 2
40*(A*B^3*a^4*c^2 + A^3*B*a^3*c^3)*d^3*e^7 + 20*(B^4*a^5*c + 7*A^2*B^2*a^4*c^2 +
A^4*a^3*c^3)*d^2*e^8 + 20*(A*B^3*a^5*c + A^3*B*a^4*c^2)*d*e^9 + (B^4*a^6 + 2*A^
2*B^2*a^5*c + A^4*a^4*c^2)*e^10)/(a*c^9)))/(a*c^4))) + 4*(3*B*c*e^2*x^2 + 23*B*c
*d^2 + 35*A*c*d*e + 15*B*a*e^2 + (11*B*c*d*e + 5*A*c*e^2)*x)*sqrt(e*x + d))/c^2

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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)**(5/2)/(-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)^(5/2)/(c*x^2 - a),x, algorithm="giac")

[Out]

Timed out