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3.1461 \(\int \frac{(A+B x) \sqrt{d+e x}}{\left (a-c x^2\right )^3} \, dx\)

Optimal. Leaf size=372 \[ \frac{\left (a B e \left (2 \sqrt{c} d-3 \sqrt{a} e\right )-A \left (-18 \sqrt{a} c d e+5 a \sqrt{c} e^2+12 c^{3/2} d^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{32 a^{5/2} c^{5/4} \left (\sqrt{c} d-\sqrt{a} e\right )^{3/2}}-\frac{\left (a B e \left (3 \sqrt{a} e+2 \sqrt{c} d\right )-A \left (18 \sqrt{a} c d e+5 a \sqrt{c} e^2+12 c^{3/2} d^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{32 a^{5/2} c^{5/4} \left (\sqrt{a} e+\sqrt{c} d\right )^{3/2}}-\frac{\sqrt{d+e x} \left (a e (A c d-a B e)-c x \left (-5 a A e^2-a B d e+6 A c d^2\right )\right )}{16 a^2 c \left (a-c x^2\right ) \left (c d^2-a e^2\right )}+\frac{\sqrt{d+e x} (a B+A c x)}{4 a c \left (a-c x^2\right )^2} \]

[Out]

((a*B + A*c*x)*Sqrt[d + e*x])/(4*a*c*(a - c*x^2)^2) - (Sqrt[d + e*x]*(a*e*(A*c*d
 - a*B*e) - c*(6*A*c*d^2 - a*B*d*e - 5*a*A*e^2)*x))/(16*a^2*c*(c*d^2 - a*e^2)*(a
 - c*x^2)) + ((a*B*e*(2*Sqrt[c]*d - 3*Sqrt[a]*e) - A*(12*c^(3/2)*d^2 - 18*Sqrt[a
]*c*d*e + 5*a*Sqrt[c]*e^2))*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqr
t[a]*e]])/(32*a^(5/2)*c^(5/4)*(Sqrt[c]*d - Sqrt[a]*e)^(3/2)) - ((a*B*e*(2*Sqrt[c
]*d + 3*Sqrt[a]*e) - A*(12*c^(3/2)*d^2 + 18*Sqrt[a]*c*d*e + 5*a*Sqrt[c]*e^2))*Ar
cTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(32*a^(5/2)*c^(5/4)*
(Sqrt[c]*d + Sqrt[a]*e)^(3/2))

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Rubi [A]  time = 1.54019, antiderivative size = 372, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{\left (a B e \left (2 \sqrt{c} d-3 \sqrt{a} e\right )-A \left (-18 \sqrt{a} c d e+5 a \sqrt{c} e^2+12 c^{3/2} d^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{32 a^{5/2} c^{5/4} \left (\sqrt{c} d-\sqrt{a} e\right )^{3/2}}-\frac{\left (a B e \left (3 \sqrt{a} e+2 \sqrt{c} d\right )-A \left (18 \sqrt{a} c d e+5 a \sqrt{c} e^2+12 c^{3/2} d^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{32 a^{5/2} c^{5/4} \left (\sqrt{a} e+\sqrt{c} d\right )^{3/2}}-\frac{\sqrt{d+e x} \left (a e (A c d-a B e)-c x \left (-5 a A e^2-a B d e+6 A c d^2\right )\right )}{16 a^2 c \left (a-c x^2\right ) \left (c d^2-a e^2\right )}+\frac{\sqrt{d+e x} (a B+A c x)}{4 a c \left (a-c x^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

((a*B + A*c*x)*Sqrt[d + e*x])/(4*a*c*(a - c*x^2)^2) - (Sqrt[d + e*x]*(a*e*(A*c*d
 - a*B*e) - c*(6*A*c*d^2 - a*B*d*e - 5*a*A*e^2)*x))/(16*a^2*c*(c*d^2 - a*e^2)*(a
 - c*x^2)) + ((a*B*e*(2*Sqrt[c]*d - 3*Sqrt[a]*e) - A*(12*c^(3/2)*d^2 - 18*Sqrt[a
]*c*d*e + 5*a*Sqrt[c]*e^2))*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqr
t[a]*e]])/(32*a^(5/2)*c^(5/4)*(Sqrt[c]*d - Sqrt[a]*e)^(3/2)) - ((a*B*e*(2*Sqrt[c
]*d + 3*Sqrt[a]*e) - A*(12*c^(3/2)*d^2 + 18*Sqrt[a]*c*d*e + 5*a*Sqrt[c]*e^2))*Ar
cTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(32*a^(5/2)*c^(5/4)*
(Sqrt[c]*d + Sqrt[a]*e)^(3/2))

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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)**(1/2)/(-c*x**2+a)**3,x)

[Out]

Timed out

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Mathematica [A]  time = 0.990141, size = 384, normalized size = 1.03 \[ -\frac{\left (A \left (-18 \sqrt{a} c d e+5 a \sqrt{c} e^2+12 c^{3/2} d^2\right )+a B e \left (3 \sqrt{a} e-2 \sqrt{c} d\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-\sqrt{a} \sqrt{c} e}}\right )}{32 a^{5/2} \sqrt{c d-\sqrt{a} \sqrt{c} e} \left (c^{3/2} d-\sqrt{a} c e\right )}+\frac{\left (A \left (18 \sqrt{a} c d e+5 a \sqrt{c} e^2+12 c^{3/2} d^2\right )-a B e \left (3 \sqrt{a} e+2 \sqrt{c} d\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} \sqrt{c} e+c d}}\right )}{32 a^{5/2} \sqrt{\sqrt{a} \sqrt{c} e+c d} \left (\sqrt{a} c e+c^{3/2} d\right )}+\frac{\sqrt{d+e x} \left (\frac{\left (a-c x^2\right ) \left (a^2 B e^2-a c e (A (d+5 e x)+B d x)+6 A c^2 d^2 x\right )}{c d^2-a e^2}+4 a (a B+A c x)\right )}{16 a^2 c \left (a-c x^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[d + e*x]*(4*a*(a*B + A*c*x) + ((a - c*x^2)*(a^2*B*e^2 + 6*A*c^2*d^2*x - a*
c*e*(B*d*x + A*(d + 5*e*x))))/(c*d^2 - a*e^2)))/(16*a^2*c*(a - c*x^2)^2) - ((a*B
*e*(-2*Sqrt[c]*d + 3*Sqrt[a]*e) + A*(12*c^(3/2)*d^2 - 18*Sqrt[a]*c*d*e + 5*a*Sqr
t[c]*e^2))*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - Sqrt[a]*Sqrt[c]*e]])/(32*a
^(5/2)*Sqrt[c*d - Sqrt[a]*Sqrt[c]*e]*(c^(3/2)*d - Sqrt[a]*c*e)) + ((-(a*B*e*(2*S
qrt[c]*d + 3*Sqrt[a]*e)) + A*(12*c^(3/2)*d^2 + 18*Sqrt[a]*c*d*e + 5*a*Sqrt[c]*e^
2))*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d + Sqrt[a]*Sqrt[c]*e]])/(32*a^(5/2)*
Sqrt[c*d + Sqrt[a]*Sqrt[c]*e]*(c^(3/2)*d + Sqrt[a]*c*e))

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Maple [B]  time = 0.256, size = 6481, normalized size = 17.4 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(e*x+d)^(1/2)/(-c*x^2+a)^3,x)

[Out]

result too large to display

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 65.3802, size = 11884, normalized size = 31.95 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/64*((a^4*c^2*d^2 - a^5*c*e^2 + (a^2*c^4*d^2 - a^3*c^3*e^2)*x^4 - 2*(a^3*c^3*d^
2 - a^4*c^2*e^2)*x^2)*sqrt((144*A^2*c^4*d^7 - 48*A*B*a*c^3*d^6*e + 160*A*B*a^2*c
^2*d^4*e^3 - 150*A*B*a^3*c*d^2*e^5 + 30*A*B*a^4*e^7 + 4*(B^2*a^2*c^2 - 105*A^2*a
*c^3)*d^5*e^2 - 5*(3*B^2*a^3*c - 77*A^2*a^2*c^2)*d^3*e^4 + 15*(B^2*a^4 - 7*A^2*a
^3*c)*d*e^6 + (a^5*c^5*d^6 - 3*a^6*c^4*d^4*e^2 + 3*a^7*c^3*d^2*e^4 - a^8*c^2*e^6
)*sqrt((900*A^2*B^2*c^4*d^6*e^8 - 60*(5*A*B^3*a*c^3 + 21*A^3*B*c^4)*d^5*e^9 + (2
5*B^4*a^2*c^2 - 2070*A^2*B^2*a*c^3 + 441*A^4*c^4)*d^4*e^10 + 8*(115*A*B^3*a^2*c^
2 + 387*A^3*B*a*c^3)*d^3*e^11 - 6*(15*B^4*a^3*c - 136*A^2*B^2*a^2*c^2 + 175*A^4*
a*c^3)*d^2*e^12 - 76*(9*A*B^3*a^3*c + 25*A^3*B*a^2*c^2)*d*e^13 + (81*B^4*a^4 + 4
50*A^2*B^2*a^3*c + 625*A^4*a^2*c^2)*e^14)/(a^5*c^11*d^12 - 6*a^6*c^10*d^10*e^2 +
 15*a^7*c^9*d^8*e^4 - 20*a^8*c^8*d^6*e^6 + 15*a^9*c^7*d^4*e^8 - 6*a^10*c^6*d^2*e
^10 + a^11*c^5*e^12)))/(a^5*c^5*d^6 - 3*a^6*c^4*d^4*e^2 + 3*a^7*c^3*d^2*e^4 - a^
8*c^2*e^6))*log((4320*A^3*B*c^5*d^7*e^4 - 432*(5*A^2*B^2*a*c^4 + 7*A^4*c^5)*d^6*
e^5 + 72*(5*A*B^3*a^2*c^3 - 147*A^3*B*a*c^4)*d^5*e^6 - 4*(5*B^4*a^3*c^2 - 1674*A
^2*B^2*a^2*c^3 - 1971*A^4*a*c^4)*d^4*e^7 - 2*(647*A*B^3*a^3*c^2 - 2727*A^3*B*a^2
*c^3)*d^3*e^8 + 3*(27*B^4*a^4*c - 1672*A^2*B^2*a^3*c^2 - 1875*A^4*a^2*c^3)*d^2*e
^9 + 2*(567*A*B^3*a^4*c + 625*A^3*B*a^3*c^2)*d*e^10 - (81*B^4*a^5 - 625*A^4*a^3*
c^2)*e^11)*sqrt(e*x + d) + (180*A^2*B*a^3*c^5*d^6*e^5 - 6*(10*A*B^2*a^4*c^4 + 21
*A^3*a^3*c^5)*d^5*e^6 + (5*B^3*a^5*c^3 - 447*A^2*B*a^4*c^4)*d^4*e^7 + 6*(37*A*B^
2*a^5*c^3 + 53*A^3*a^4*c^4)*d^3*e^8 - 24*(B^3*a^6*c^2 - 9*A^2*B*a^5*c^3)*d^2*e^9
 - 2*(93*A*B^2*a^6*c^2 + 100*A^3*a^5*c^3)*d*e^10 + 3*(9*B^3*a^7*c + 25*A^2*B*a^6
*c^2)*e^11 - (12*A*a^5*c^9*d^10 - 2*B*a^6*c^8*d^9*e - 55*A*a^6*c^8*d^8*e^2 + 10*
B*a^7*c^7*d^7*e^3 + 98*A*a^7*c^7*d^6*e^4 - 18*B*a^8*c^6*d^5*e^5 - 84*A*a^8*c^6*d
^4*e^6 + 14*B*a^9*c^5*d^3*e^7 + 34*A*a^9*c^5*d^2*e^8 - 4*B*a^10*c^4*d*e^9 - 5*A*
a^10*c^4*e^10)*sqrt((900*A^2*B^2*c^4*d^6*e^8 - 60*(5*A*B^3*a*c^3 + 21*A^3*B*c^4)
*d^5*e^9 + (25*B^4*a^2*c^2 - 2070*A^2*B^2*a*c^3 + 441*A^4*c^4)*d^4*e^10 + 8*(115
*A*B^3*a^2*c^2 + 387*A^3*B*a*c^3)*d^3*e^11 - 6*(15*B^4*a^3*c - 136*A^2*B^2*a^2*c
^2 + 175*A^4*a*c^3)*d^2*e^12 - 76*(9*A*B^3*a^3*c + 25*A^3*B*a^2*c^2)*d*e^13 + (8
1*B^4*a^4 + 450*A^2*B^2*a^3*c + 625*A^4*a^2*c^2)*e^14)/(a^5*c^11*d^12 - 6*a^6*c^
10*d^10*e^2 + 15*a^7*c^9*d^8*e^4 - 20*a^8*c^8*d^6*e^6 + 15*a^9*c^7*d^4*e^8 - 6*a
^10*c^6*d^2*e^10 + a^11*c^5*e^12)))*sqrt((144*A^2*c^4*d^7 - 48*A*B*a*c^3*d^6*e +
 160*A*B*a^2*c^2*d^4*e^3 - 150*A*B*a^3*c*d^2*e^5 + 30*A*B*a^4*e^7 + 4*(B^2*a^2*c
^2 - 105*A^2*a*c^3)*d^5*e^2 - 5*(3*B^2*a^3*c - 77*A^2*a^2*c^2)*d^3*e^4 + 15*(B^2
*a^4 - 7*A^2*a^3*c)*d*e^6 + (a^5*c^5*d^6 - 3*a^6*c^4*d^4*e^2 + 3*a^7*c^3*d^2*e^4
 - a^8*c^2*e^6)*sqrt((900*A^2*B^2*c^4*d^6*e^8 - 60*(5*A*B^3*a*c^3 + 21*A^3*B*c^4
)*d^5*e^9 + (25*B^4*a^2*c^2 - 2070*A^2*B^2*a*c^3 + 441*A^4*c^4)*d^4*e^10 + 8*(11
5*A*B^3*a^2*c^2 + 387*A^3*B*a*c^3)*d^3*e^11 - 6*(15*B^4*a^3*c - 136*A^2*B^2*a^2*
c^2 + 175*A^4*a*c^3)*d^2*e^12 - 76*(9*A*B^3*a^3*c + 25*A^3*B*a^2*c^2)*d*e^13 + (
81*B^4*a^4 + 450*A^2*B^2*a^3*c + 625*A^4*a^2*c^2)*e^14)/(a^5*c^11*d^12 - 6*a^6*c
^10*d^10*e^2 + 15*a^7*c^9*d^8*e^4 - 20*a^8*c^8*d^6*e^6 + 15*a^9*c^7*d^4*e^8 - 6*
a^10*c^6*d^2*e^10 + a^11*c^5*e^12)))/(a^5*c^5*d^6 - 3*a^6*c^4*d^4*e^2 + 3*a^7*c^
3*d^2*e^4 - a^8*c^2*e^6))) - (a^4*c^2*d^2 - a^5*c*e^2 + (a^2*c^4*d^2 - a^3*c^3*e
^2)*x^4 - 2*(a^3*c^3*d^2 - a^4*c^2*e^2)*x^2)*sqrt((144*A^2*c^4*d^7 - 48*A*B*a*c^
3*d^6*e + 160*A*B*a^2*c^2*d^4*e^3 - 150*A*B*a^3*c*d^2*e^5 + 30*A*B*a^4*e^7 + 4*(
B^2*a^2*c^2 - 105*A^2*a*c^3)*d^5*e^2 - 5*(3*B^2*a^3*c - 77*A^2*a^2*c^2)*d^3*e^4
+ 15*(B^2*a^4 - 7*A^2*a^3*c)*d*e^6 + (a^5*c^5*d^6 - 3*a^6*c^4*d^4*e^2 + 3*a^7*c^
3*d^2*e^4 - a^8*c^2*e^6)*sqrt((900*A^2*B^2*c^4*d^6*e^8 - 60*(5*A*B^3*a*c^3 + 21*
A^3*B*c^4)*d^5*e^9 + (25*B^4*a^2*c^2 - 2070*A^2*B^2*a*c^3 + 441*A^4*c^4)*d^4*e^1
0 + 8*(115*A*B^3*a^2*c^2 + 387*A^3*B*a*c^3)*d^3*e^11 - 6*(15*B^4*a^3*c - 136*A^2
*B^2*a^2*c^2 + 175*A^4*a*c^3)*d^2*e^12 - 76*(9*A*B^3*a^3*c + 25*A^3*B*a^2*c^2)*d
*e^13 + (81*B^4*a^4 + 450*A^2*B^2*a^3*c + 625*A^4*a^2*c^2)*e^14)/(a^5*c^11*d^12
- 6*a^6*c^10*d^10*e^2 + 15*a^7*c^9*d^8*e^4 - 20*a^8*c^8*d^6*e^6 + 15*a^9*c^7*d^4
*e^8 - 6*a^10*c^6*d^2*e^10 + a^11*c^5*e^12)))/(a^5*c^5*d^6 - 3*a^6*c^4*d^4*e^2 +
 3*a^7*c^3*d^2*e^4 - a^8*c^2*e^6))*log((4320*A^3*B*c^5*d^7*e^4 - 432*(5*A^2*B^2*
a*c^4 + 7*A^4*c^5)*d^6*e^5 + 72*(5*A*B^3*a^2*c^3 - 147*A^3*B*a*c^4)*d^5*e^6 - 4*
(5*B^4*a^3*c^2 - 1674*A^2*B^2*a^2*c^3 - 1971*A^4*a*c^4)*d^4*e^7 - 2*(647*A*B^3*a
^3*c^2 - 2727*A^3*B*a^2*c^3)*d^3*e^8 + 3*(27*B^4*a^4*c - 1672*A^2*B^2*a^3*c^2 -
1875*A^4*a^2*c^3)*d^2*e^9 + 2*(567*A*B^3*a^4*c + 625*A^3*B*a^3*c^2)*d*e^10 - (81
*B^4*a^5 - 625*A^4*a^3*c^2)*e^11)*sqrt(e*x + d) - (180*A^2*B*a^3*c^5*d^6*e^5 - 6
*(10*A*B^2*a^4*c^4 + 21*A^3*a^3*c^5)*d^5*e^6 + (5*B^3*a^5*c^3 - 447*A^2*B*a^4*c^
4)*d^4*e^7 + 6*(37*A*B^2*a^5*c^3 + 53*A^3*a^4*c^4)*d^3*e^8 - 24*(B^3*a^6*c^2 - 9
*A^2*B*a^5*c^3)*d^2*e^9 - 2*(93*A*B^2*a^6*c^2 + 100*A^3*a^5*c^3)*d*e^10 + 3*(9*B
^3*a^7*c + 25*A^2*B*a^6*c^2)*e^11 - (12*A*a^5*c^9*d^10 - 2*B*a^6*c^8*d^9*e - 55*
A*a^6*c^8*d^8*e^2 + 10*B*a^7*c^7*d^7*e^3 + 98*A*a^7*c^7*d^6*e^4 - 18*B*a^8*c^6*d
^5*e^5 - 84*A*a^8*c^6*d^4*e^6 + 14*B*a^9*c^5*d^3*e^7 + 34*A*a^9*c^5*d^2*e^8 - 4*
B*a^10*c^4*d*e^9 - 5*A*a^10*c^4*e^10)*sqrt((900*A^2*B^2*c^4*d^6*e^8 - 60*(5*A*B^
3*a*c^3 + 21*A^3*B*c^4)*d^5*e^9 + (25*B^4*a^2*c^2 - 2070*A^2*B^2*a*c^3 + 441*A^4
*c^4)*d^4*e^10 + 8*(115*A*B^3*a^2*c^2 + 387*A^3*B*a*c^3)*d^3*e^11 - 6*(15*B^4*a^
3*c - 136*A^2*B^2*a^2*c^2 + 175*A^4*a*c^3)*d^2*e^12 - 76*(9*A*B^3*a^3*c + 25*A^3
*B*a^2*c^2)*d*e^13 + (81*B^4*a^4 + 450*A^2*B^2*a^3*c + 625*A^4*a^2*c^2)*e^14)/(a
^5*c^11*d^12 - 6*a^6*c^10*d^10*e^2 + 15*a^7*c^9*d^8*e^4 - 20*a^8*c^8*d^6*e^6 + 1
5*a^9*c^7*d^4*e^8 - 6*a^10*c^6*d^2*e^10 + a^11*c^5*e^12)))*sqrt((144*A^2*c^4*d^7
 - 48*A*B*a*c^3*d^6*e + 160*A*B*a^2*c^2*d^4*e^3 - 150*A*B*a^3*c*d^2*e^5 + 30*A*B
*a^4*e^7 + 4*(B^2*a^2*c^2 - 105*A^2*a*c^3)*d^5*e^2 - 5*(3*B^2*a^3*c - 77*A^2*a^2
*c^2)*d^3*e^4 + 15*(B^2*a^4 - 7*A^2*a^3*c)*d*e^6 + (a^5*c^5*d^6 - 3*a^6*c^4*d^4*
e^2 + 3*a^7*c^3*d^2*e^4 - a^8*c^2*e^6)*sqrt((900*A^2*B^2*c^4*d^6*e^8 - 60*(5*A*B
^3*a*c^3 + 21*A^3*B*c^4)*d^5*e^9 + (25*B^4*a^2*c^2 - 2070*A^2*B^2*a*c^3 + 441*A^
4*c^4)*d^4*e^10 + 8*(115*A*B^3*a^2*c^2 + 387*A^3*B*a*c^3)*d^3*e^11 - 6*(15*B^4*a
^3*c - 136*A^2*B^2*a^2*c^2 + 175*A^4*a*c^3)*d^2*e^12 - 76*(9*A*B^3*a^3*c + 25*A^
3*B*a^2*c^2)*d*e^13 + (81*B^4*a^4 + 450*A^2*B^2*a^3*c + 625*A^4*a^2*c^2)*e^14)/(
a^5*c^11*d^12 - 6*a^6*c^10*d^10*e^2 + 15*a^7*c^9*d^8*e^4 - 20*a^8*c^8*d^6*e^6 +
15*a^9*c^7*d^4*e^8 - 6*a^10*c^6*d^2*e^10 + a^11*c^5*e^12)))/(a^5*c^5*d^6 - 3*a^6
*c^4*d^4*e^2 + 3*a^7*c^3*d^2*e^4 - a^8*c^2*e^6))) + (a^4*c^2*d^2 - a^5*c*e^2 + (
a^2*c^4*d^2 - a^3*c^3*e^2)*x^4 - 2*(a^3*c^3*d^2 - a^4*c^2*e^2)*x^2)*sqrt((144*A^
2*c^4*d^7 - 48*A*B*a*c^3*d^6*e + 160*A*B*a^2*c^2*d^4*e^3 - 150*A*B*a^3*c*d^2*e^5
 + 30*A*B*a^4*e^7 + 4*(B^2*a^2*c^2 - 105*A^2*a*c^3)*d^5*e^2 - 5*(3*B^2*a^3*c - 7
7*A^2*a^2*c^2)*d^3*e^4 + 15*(B^2*a^4 - 7*A^2*a^3*c)*d*e^6 - (a^5*c^5*d^6 - 3*a^6
*c^4*d^4*e^2 + 3*a^7*c^3*d^2*e^4 - a^8*c^2*e^6)*sqrt((900*A^2*B^2*c^4*d^6*e^8 -
60*(5*A*B^3*a*c^3 + 21*A^3*B*c^4)*d^5*e^9 + (25*B^4*a^2*c^2 - 2070*A^2*B^2*a*c^3
 + 441*A^4*c^4)*d^4*e^10 + 8*(115*A*B^3*a^2*c^2 + 387*A^3*B*a*c^3)*d^3*e^11 - 6*
(15*B^4*a^3*c - 136*A^2*B^2*a^2*c^2 + 175*A^4*a*c^3)*d^2*e^12 - 76*(9*A*B^3*a^3*
c + 25*A^3*B*a^2*c^2)*d*e^13 + (81*B^4*a^4 + 450*A^2*B^2*a^3*c + 625*A^4*a^2*c^2
)*e^14)/(a^5*c^11*d^12 - 6*a^6*c^10*d^10*e^2 + 15*a^7*c^9*d^8*e^4 - 20*a^8*c^8*d
^6*e^6 + 15*a^9*c^7*d^4*e^8 - 6*a^10*c^6*d^2*e^10 + a^11*c^5*e^12)))/(a^5*c^5*d^
6 - 3*a^6*c^4*d^4*e^2 + 3*a^7*c^3*d^2*e^4 - a^8*c^2*e^6))*log((4320*A^3*B*c^5*d^
7*e^4 - 432*(5*A^2*B^2*a*c^4 + 7*A^4*c^5)*d^6*e^5 + 72*(5*A*B^3*a^2*c^3 - 147*A^
3*B*a*c^4)*d^5*e^6 - 4*(5*B^4*a^3*c^2 - 1674*A^2*B^2*a^2*c^3 - 1971*A^4*a*c^4)*d
^4*e^7 - 2*(647*A*B^3*a^3*c^2 - 2727*A^3*B*a^2*c^3)*d^3*e^8 + 3*(27*B^4*a^4*c -
1672*A^2*B^2*a^3*c^2 - 1875*A^4*a^2*c^3)*d^2*e^9 + 2*(567*A*B^3*a^4*c + 625*A^3*
B*a^3*c^2)*d*e^10 - (81*B^4*a^5 - 625*A^4*a^3*c^2)*e^11)*sqrt(e*x + d) + (180*A^
2*B*a^3*c^5*d^6*e^5 - 6*(10*A*B^2*a^4*c^4 + 21*A^3*a^3*c^5)*d^5*e^6 + (5*B^3*a^5
*c^3 - 447*A^2*B*a^4*c^4)*d^4*e^7 + 6*(37*A*B^2*a^5*c^3 + 53*A^3*a^4*c^4)*d^3*e^
8 - 24*(B^3*a^6*c^2 - 9*A^2*B*a^5*c^3)*d^2*e^9 - 2*(93*A*B^2*a^6*c^2 + 100*A^3*a
^5*c^3)*d*e^10 + 3*(9*B^3*a^7*c + 25*A^2*B*a^6*c^2)*e^11 + (12*A*a^5*c^9*d^10 -
2*B*a^6*c^8*d^9*e - 55*A*a^6*c^8*d^8*e^2 + 10*B*a^7*c^7*d^7*e^3 + 98*A*a^7*c^7*d
^6*e^4 - 18*B*a^8*c^6*d^5*e^5 - 84*A*a^8*c^6*d^4*e^6 + 14*B*a^9*c^5*d^3*e^7 + 34
*A*a^9*c^5*d^2*e^8 - 4*B*a^10*c^4*d*e^9 - 5*A*a^10*c^4*e^10)*sqrt((900*A^2*B^2*c
^4*d^6*e^8 - 60*(5*A*B^3*a*c^3 + 21*A^3*B*c^4)*d^5*e^9 + (25*B^4*a^2*c^2 - 2070*
A^2*B^2*a*c^3 + 441*A^4*c^4)*d^4*e^10 + 8*(115*A*B^3*a^2*c^2 + 387*A^3*B*a*c^3)*
d^3*e^11 - 6*(15*B^4*a^3*c - 136*A^2*B^2*a^2*c^2 + 175*A^4*a*c^3)*d^2*e^12 - 76*
(9*A*B^3*a^3*c + 25*A^3*B*a^2*c^2)*d*e^13 + (81*B^4*a^4 + 450*A^2*B^2*a^3*c + 62
5*A^4*a^2*c^2)*e^14)/(a^5*c^11*d^12 - 6*a^6*c^10*d^10*e^2 + 15*a^7*c^9*d^8*e^4 -
 20*a^8*c^8*d^6*e^6 + 15*a^9*c^7*d^4*e^8 - 6*a^10*c^6*d^2*e^10 + a^11*c^5*e^12))
)*sqrt((144*A^2*c^4*d^7 - 48*A*B*a*c^3*d^6*e + 160*A*B*a^2*c^2*d^4*e^3 - 150*A*B
*a^3*c*d^2*e^5 + 30*A*B*a^4*e^7 + 4*(B^2*a^2*c^2 - 105*A^2*a*c^3)*d^5*e^2 - 5*(3
*B^2*a^3*c - 77*A^2*a^2*c^2)*d^3*e^4 + 15*(B^2*a^4 - 7*A^2*a^3*c)*d*e^6 - (a^5*c
^5*d^6 - 3*a^6*c^4*d^4*e^2 + 3*a^7*c^3*d^2*e^4 - a^8*c^2*e^6)*sqrt((900*A^2*B^2*
c^4*d^6*e^8 - 60*(5*A*B^3*a*c^3 + 21*A^3*B*c^4)*d^5*e^9 + (25*B^4*a^2*c^2 - 2070
*A^2*B^2*a*c^3 + 441*A^4*c^4)*d^4*e^10 + 8*(115*A*B^3*a^2*c^2 + 387*A^3*B*a*c^3)
*d^3*e^11 - 6*(15*B^4*a^3*c - 136*A^2*B^2*a^2*c^2 + 175*A^4*a*c^3)*d^2*e^12 - 76
*(9*A*B^3*a^3*c + 25*A^3*B*a^2*c^2)*d*e^13 + (81*B^4*a^4 + 450*A^2*B^2*a^3*c + 6
25*A^4*a^2*c^2)*e^14)/(a^5*c^11*d^12 - 6*a^6*c^10*d^10*e^2 + 15*a^7*c^9*d^8*e^4
- 20*a^8*c^8*d^6*e^6 + 15*a^9*c^7*d^4*e^8 - 6*a^10*c^6*d^2*e^10 + a^11*c^5*e^12)
))/(a^5*c^5*d^6 - 3*a^6*c^4*d^4*e^2 + 3*a^7*c^3*d^2*e^4 - a^8*c^2*e^6))) - (a^4*
c^2*d^2 - a^5*c*e^2 + (a^2*c^4*d^2 - a^3*c^3*e^2)*x^4 - 2*(a^3*c^3*d^2 - a^4*c^2
*e^2)*x^2)*sqrt((144*A^2*c^4*d^7 - 48*A*B*a*c^3*d^6*e + 160*A*B*a^2*c^2*d^4*e^3
- 150*A*B*a^3*c*d^2*e^5 + 30*A*B*a^4*e^7 + 4*(B^2*a^2*c^2 - 105*A^2*a*c^3)*d^5*e
^2 - 5*(3*B^2*a^3*c - 77*A^2*a^2*c^2)*d^3*e^4 + 15*(B^2*a^4 - 7*A^2*a^3*c)*d*e^6
 - (a^5*c^5*d^6 - 3*a^6*c^4*d^4*e^2 + 3*a^7*c^3*d^2*e^4 - a^8*c^2*e^6)*sqrt((900
*A^2*B^2*c^4*d^6*e^8 - 60*(5*A*B^3*a*c^3 + 21*A^3*B*c^4)*d^5*e^9 + (25*B^4*a^2*c
^2 - 2070*A^2*B^2*a*c^3 + 441*A^4*c^4)*d^4*e^10 + 8*(115*A*B^3*a^2*c^2 + 387*A^3
*B*a*c^3)*d^3*e^11 - 6*(15*B^4*a^3*c - 136*A^2*B^2*a^2*c^2 + 175*A^4*a*c^3)*d^2*
e^12 - 76*(9*A*B^3*a^3*c + 25*A^3*B*a^2*c^2)*d*e^13 + (81*B^4*a^4 + 450*A^2*B^2*
a^3*c + 625*A^4*a^2*c^2)*e^14)/(a^5*c^11*d^12 - 6*a^6*c^10*d^10*e^2 + 15*a^7*c^9
*d^8*e^4 - 20*a^8*c^8*d^6*e^6 + 15*a^9*c^7*d^4*e^8 - 6*a^10*c^6*d^2*e^10 + a^11*
c^5*e^12)))/(a^5*c^5*d^6 - 3*a^6*c^4*d^4*e^2 + 3*a^7*c^3*d^2*e^4 - a^8*c^2*e^6))
*log((4320*A^3*B*c^5*d^7*e^4 - 432*(5*A^2*B^2*a*c^4 + 7*A^4*c^5)*d^6*e^5 + 72*(5
*A*B^3*a^2*c^3 - 147*A^3*B*a*c^4)*d^5*e^6 - 4*(5*B^4*a^3*c^2 - 1674*A^2*B^2*a^2*
c^3 - 1971*A^4*a*c^4)*d^4*e^7 - 2*(647*A*B^3*a^3*c^2 - 2727*A^3*B*a^2*c^3)*d^3*e
^8 + 3*(27*B^4*a^4*c - 1672*A^2*B^2*a^3*c^2 - 1875*A^4*a^2*c^3)*d^2*e^9 + 2*(567
*A*B^3*a^4*c + 625*A^3*B*a^3*c^2)*d*e^10 - (81*B^4*a^5 - 625*A^4*a^3*c^2)*e^11)*
sqrt(e*x + d) - (180*A^2*B*a^3*c^5*d^6*e^5 - 6*(10*A*B^2*a^4*c^4 + 21*A^3*a^3*c^
5)*d^5*e^6 + (5*B^3*a^5*c^3 - 447*A^2*B*a^4*c^4)*d^4*e^7 + 6*(37*A*B^2*a^5*c^3 +
 53*A^3*a^4*c^4)*d^3*e^8 - 24*(B^3*a^6*c^2 - 9*A^2*B*a^5*c^3)*d^2*e^9 - 2*(93*A*
B^2*a^6*c^2 + 100*A^3*a^5*c^3)*d*e^10 + 3*(9*B^3*a^7*c + 25*A^2*B*a^6*c^2)*e^11
+ (12*A*a^5*c^9*d^10 - 2*B*a^6*c^8*d^9*e - 55*A*a^6*c^8*d^8*e^2 + 10*B*a^7*c^7*d
^7*e^3 + 98*A*a^7*c^7*d^6*e^4 - 18*B*a^8*c^6*d^5*e^5 - 84*A*a^8*c^6*d^4*e^6 + 14
*B*a^9*c^5*d^3*e^7 + 34*A*a^9*c^5*d^2*e^8 - 4*B*a^10*c^4*d*e^9 - 5*A*a^10*c^4*e^
10)*sqrt((900*A^2*B^2*c^4*d^6*e^8 - 60*(5*A*B^3*a*c^3 + 21*A^3*B*c^4)*d^5*e^9 +
(25*B^4*a^2*c^2 - 2070*A^2*B^2*a*c^3 + 441*A^4*c^4)*d^4*e^10 + 8*(115*A*B^3*a^2*
c^2 + 387*A^3*B*a*c^3)*d^3*e^11 - 6*(15*B^4*a^3*c - 136*A^2*B^2*a^2*c^2 + 175*A^
4*a*c^3)*d^2*e^12 - 76*(9*A*B^3*a^3*c + 25*A^3*B*a^2*c^2)*d*e^13 + (81*B^4*a^4 +
 450*A^2*B^2*a^3*c + 625*A^4*a^2*c^2)*e^14)/(a^5*c^11*d^12 - 6*a^6*c^10*d^10*e^2
 + 15*a^7*c^9*d^8*e^4 - 20*a^8*c^8*d^6*e^6 + 15*a^9*c^7*d^4*e^8 - 6*a^10*c^6*d^2
*e^10 + a^11*c^5*e^12)))*sqrt((144*A^2*c^4*d^7 - 48*A*B*a*c^3*d^6*e + 160*A*B*a^
2*c^2*d^4*e^3 - 150*A*B*a^3*c*d^2*e^5 + 30*A*B*a^4*e^7 + 4*(B^2*a^2*c^2 - 105*A^
2*a*c^3)*d^5*e^2 - 5*(3*B^2*a^3*c - 77*A^2*a^2*c^2)*d^3*e^4 + 15*(B^2*a^4 - 7*A^
2*a^3*c)*d*e^6 - (a^5*c^5*d^6 - 3*a^6*c^4*d^4*e^2 + 3*a^7*c^3*d^2*e^4 - a^8*c^2*
e^6)*sqrt((900*A^2*B^2*c^4*d^6*e^8 - 60*(5*A*B^3*a*c^3 + 21*A^3*B*c^4)*d^5*e^9 +
 (25*B^4*a^2*c^2 - 2070*A^2*B^2*a*c^3 + 441*A^4*c^4)*d^4*e^10 + 8*(115*A*B^3*a^2
*c^2 + 387*A^3*B*a*c^3)*d^3*e^11 - 6*(15*B^4*a^3*c - 136*A^2*B^2*a^2*c^2 + 175*A
^4*a*c^3)*d^2*e^12 - 76*(9*A*B^3*a^3*c + 25*A^3*B*a^2*c^2)*d*e^13 + (81*B^4*a^4
+ 450*A^2*B^2*a^3*c + 625*A^4*a^2*c^2)*e^14)/(a^5*c^11*d^12 - 6*a^6*c^10*d^10*e^
2 + 15*a^7*c^9*d^8*e^4 - 20*a^8*c^8*d^6*e^6 + 15*a^9*c^7*d^4*e^8 - 6*a^10*c^6*d^
2*e^10 + a^11*c^5*e^12)))/(a^5*c^5*d^6 - 3*a^6*c^4*d^4*e^2 + 3*a^7*c^3*d^2*e^4 -
 a^8*c^2*e^6))) + 4*(4*B*a^2*c*d^2 - A*a^2*c*d*e - 3*B*a^3*e^2 - (6*A*c^3*d^2 -
B*a*c^2*d*e - 5*A*a*c^2*e^2)*x^3 + (A*a*c^2*d*e - B*a^2*c*e^2)*x^2 + (10*A*a*c^2
*d^2 - B*a^2*c*d*e - 9*A*a^2*c*e^2)*x)*sqrt(e*x + d))/(a^4*c^2*d^2 - a^5*c*e^2 +
 (a^2*c^4*d^2 - a^3*c^3*e^2)*x^4 - 2*(a^3*c^3*d^2 - a^4*c^2*e^2)*x^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)**(1/2)/(-c*x**2+a)**3,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)*sqrt(e*x + d)/(c*x^2 - a)^3,x, algorithm="giac")

[Out]

Timed out

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