3.1458 \(\int \frac{(A+B x) (d+e x)^{7/2}}{\left (a-c x^2\right )^3} \, dx\)

Optimal. Leaf size=396 \[ \frac{\left (\sqrt{c} d-\sqrt{a} e\right )^{3/2} \left (7 a B e \left (3 \sqrt{a} e+2 \sqrt{c} d\right )-A \left (18 \sqrt{a} c d e+5 a \sqrt{c} e^2+12 c^{3/2} d^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{32 a^{5/2} c^{11/4}}-\frac{\left (\sqrt{a} e+\sqrt{c} d\right )^{3/2} \left (7 a B e \left (2 \sqrt{c} d-3 \sqrt{a} e\right )-A \left (-18 \sqrt{a} c d e+5 a \sqrt{c} e^2+12 c^{3/2} d^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{32 a^{5/2} c^{11/4}}+\frac{\sqrt{d+e x} \left (x \left (2 A c d \left (3 c d^2-2 a e^2\right )-7 a B e \left (a e^2+c d^2\right )\right )+a e \left (-5 a A e^2-14 a B d e+7 A c d^2\right )\right )}{16 a^2 c^2 \left (a-c x^2\right )}+\frac{(d+e x)^{5/2} (x (a B e+A c d)+a (A e+B d))}{4 a c \left (a-c x^2\right )^2} \]

[Out]

((d + e*x)^(5/2)*(a*(B*d + A*e) + (A*c*d + a*B*e)*x))/(4*a*c*(a - c*x^2)^2) + (S
qrt[d + e*x]*(a*e*(7*A*c*d^2 - 14*a*B*d*e - 5*a*A*e^2) + (2*A*c*d*(3*c*d^2 - 2*a
*e^2) - 7*a*B*e*(c*d^2 + a*e^2))*x))/(16*a^2*c^2*(a - c*x^2)) + ((Sqrt[c]*d - Sq
rt[a]*e)^(3/2)*(7*a*B*e*(2*Sqrt[c]*d + 3*Sqrt[a]*e) - A*(12*c^(3/2)*d^2 + 18*Sqr
t[a]*c*d*e + 5*a*Sqrt[c]*e^2))*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d -
Sqrt[a]*e]])/(32*a^(5/2)*c^(11/4)) - ((Sqrt[c]*d + Sqrt[a]*e)^(3/2)*(7*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 + Sqrt[a]*e]])/(32*a^(5/2)*c^
(11/4))

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Rubi [A]  time = 1.65113, antiderivative size = 396, 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 \[ \frac{\left (\sqrt{c} d-\sqrt{a} e\right )^{3/2} \left (7 a B e \left (3 \sqrt{a} e+2 \sqrt{c} d\right )-A \left (18 \sqrt{a} c d e+5 a \sqrt{c} e^2+12 c^{3/2} d^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{32 a^{5/2} c^{11/4}}-\frac{\left (\sqrt{a} e+\sqrt{c} d\right )^{3/2} \left (7 a B e \left (2 \sqrt{c} d-3 \sqrt{a} e\right )-A \left (-18 \sqrt{a} c d e+5 a \sqrt{c} e^2+12 c^{3/2} d^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{32 a^{5/2} c^{11/4}}+\frac{\sqrt{d+e x} \left (x \left (2 A c d \left (3 c d^2-2 a e^2\right )-7 a B e \left (a e^2+c d^2\right )\right )+a e \left (-5 a A e^2-14 a B d e+7 A c d^2\right )\right )}{16 a^2 c^2 \left (a-c x^2\right )}+\frac{(d+e x)^{5/2} (x (a B e+A c d)+a (A e+B d))}{4 a c \left (a-c x^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

((d + e*x)^(5/2)*(a*(B*d + A*e) + (A*c*d + a*B*e)*x))/(4*a*c*(a - c*x^2)^2) + (S
qrt[d + e*x]*(a*e*(7*A*c*d^2 - 14*a*B*d*e - 5*a*A*e^2) + (2*A*c*d*(3*c*d^2 - 2*a
*e^2) - 7*a*B*e*(c*d^2 + a*e^2))*x))/(16*a^2*c^2*(a - c*x^2)) + ((Sqrt[c]*d - Sq
rt[a]*e)^(3/2)*(7*a*B*e*(2*Sqrt[c]*d + 3*Sqrt[a]*e) - A*(12*c^(3/2)*d^2 + 18*Sqr
t[a]*c*d*e + 5*a*Sqrt[c]*e^2))*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d -
Sqrt[a]*e]])/(32*a^(5/2)*c^(11/4)) - ((Sqrt[c]*d + Sqrt[a]*e)^(3/2)*(7*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 + Sqrt[a]*e]])/(32*a^(5/2)*c^
(11/4))

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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)**(7/2)/(-c*x**2+a)**3,x)

[Out]

Timed out

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Mathematica [A]  time = 1.94751, size = 444, normalized size = 1.12 \[ \frac{\frac{2 \sqrt{a} \sqrt{c} \sqrt{d+e x} \left (a^3 \left (-e^2\right ) (5 A e+7 B (2 d+e x))+a^2 c \left (A e \left (11 d^2+4 d e x+9 e^2 x^2\right )+B \left (4 d^3+5 d^2 e x+26 d e^2 x^2+11 e^3 x^3\right )\right )+a c^2 d x \left (A \left (10 d^2+d e x+8 e^2 x^2\right )+7 B d e x^2\right )-6 A c^3 d^3 x^3\right )}{\left (a-c x^2\right )^2}-\frac{\left (\sqrt{c} d-\sqrt{a} e\right )^2 \left (A \left (18 \sqrt{a} c d e+5 a \sqrt{c} e^2+12 c^{3/2} d^2\right )-7 a B e \left (3 \sqrt{a} e+2 \sqrt{c} d\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-\sqrt{a} \sqrt{c} e}}\right )}{\sqrt{c d-\sqrt{a} \sqrt{c} e}}+\frac{\left (\sqrt{a} e+\sqrt{c} d\right )^2 \left (A \left (-18 \sqrt{a} c d e+5 a \sqrt{c} e^2+12 c^{3/2} d^2\right )+7 a B e \left (3 \sqrt{a} e-2 \sqrt{c} d\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} \sqrt{c} e+c d}}\right )}{\sqrt{\sqrt{a} \sqrt{c} e+c d}}}{32 a^{5/2} c^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(d + e*x)^(7/2))/(a - c*x^2)^3,x]

[Out]

((2*Sqrt[a]*Sqrt[c]*Sqrt[d + e*x]*(-6*A*c^3*d^3*x^3 - a^3*e^2*(5*A*e + 7*B*(2*d
+ e*x)) + a*c^2*d*x*(7*B*d*e*x^2 + A*(10*d^2 + d*e*x + 8*e^2*x^2)) + a^2*c*(A*e*
(11*d^2 + 4*d*e*x + 9*e^2*x^2) + B*(4*d^3 + 5*d^2*e*x + 26*d*e^2*x^2 + 11*e^3*x^
3))))/(a - c*x^2)^2 - ((Sqrt[c]*d - Sqrt[a]*e)^2*(-7*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[(Sqrt[
c]*Sqrt[d + e*x])/Sqrt[c*d - Sqrt[a]*Sqrt[c]*e]])/Sqrt[c*d - Sqrt[a]*Sqrt[c]*e]
+ ((Sqrt[c]*d + Sqrt[a]*e)^2*(7*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[(Sqrt[c]*Sqrt[d + e*x])/Sq
rt[c*d + Sqrt[a]*Sqrt[c]*e]])/Sqrt[c*d + Sqrt[a]*Sqrt[c]*e])/(32*a^(5/2)*c^(5/2)
)

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Maple [B]  time = 0.14, size = 2456, normalized size = 6.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(e*x+d)^(7/2)/(-c*x^2+a)^3,x)

[Out]

-7/16*e^7*a^2*c^3/(a^5*c^5*e^8)^(1/2)/((-a^2*c^3*e^3*d+(a^5*c^5*e^8)^(1/2))*c*e)
^(1/2)*arctan(a*c^2*e^2*(e*x+d)^(1/2)/((-a^2*c^3*e^3*d+(a^5*c^5*e^8)^(1/2))*c*e)
^(1/2))*B*d^3-19/32*e^8*a^2*c^3/(a^5*c^5*e^8)^(1/2)/((a^2*c^3*e^3*d+(a^5*c^5*e^8
)^(1/2))*c*e)^(1/2)*arctanh(a*c^2*e^2*(e*x+d)^(1/2)/((a^2*c^3*e^3*d+(a^5*c^5*e^8
)^(1/2))*c*e)^(1/2))*A*d^2+3/8*e^6*a*c^4/(a^5*c^5*e^8)^(1/2)/((-a^2*c^3*e^3*d+(a
^5*c^5*e^8)^(1/2))*c*e)^(1/2)*arctan(a*c^2*e^2*(e*x+d)^(1/2)/((-a^2*c^3*e^3*d+(a
^5*c^5*e^8)^(1/2))*c*e)^(1/2))*A*d^4+3/8*e^6*a*c^4/(a^5*c^5*e^8)^(1/2)/((a^2*c^3
*e^3*d+(a^5*c^5*e^8)^(1/2))*c*e)^(1/2)*arctanh(a*c^2*e^2*(e*x+d)^(1/2)/((a^2*c^3
*e^3*d+(a^5*c^5*e^8)^(1/2))*c*e)^(1/2))*A*d^4+7/8*e^9*a^3*c^2/(a^5*c^5*e^8)^(1/2
)/((a^2*c^3*e^3*d+(a^5*c^5*e^8)^(1/2))*c*e)^(1/2)*arctanh(a*c^2*e^2*(e*x+d)^(1/2
)/((a^2*c^3*e^3*d+(a^5*c^5*e^8)^(1/2))*c*e)^(1/2))*B*d-19/32*e^8*a^2*c^3/(a^5*c^
5*e^8)^(1/2)/((-a^2*c^3*e^3*d+(a^5*c^5*e^8)^(1/2))*c*e)^(1/2)*arctan(a*c^2*e^2*(
e*x+d)^(1/2)/((-a^2*c^3*e^3*d+(a^5*c^5*e^8)^(1/2))*c*e)^(1/2))*A*d^2-17/16*e^3/(
c*e^2*x^2-a*e^2)^2/a*(e*x+d)^(1/2)*A*d^4-7/16*e^2/(c*e^2*x^2-a*e^2)^2/a*(e*x+d)^
(1/2)*B*d^5-21/32*e^6*a/c/((-a^2*c^3*e^3*d+(a^5*c^5*e^8)^(1/2))*c*e)^(1/2)*arcta
n(a*c^2*e^2*(e*x+d)^(1/2)/((-a^2*c^3*e^3*d+(a^5*c^5*e^8)^(1/2))*c*e)^(1/2))*B+21
/32*e^6*a/c/((a^2*c^3*e^3*d+(a^5*c^5*e^8)^(1/2))*c*e)^(1/2)*arctanh(a*c^2*e^2*(e
*x+d)^(1/2)/((a^2*c^3*e^3*d+(a^5*c^5*e^8)^(1/2))*c*e)^(1/2))*B-7/16*e^4/(c*e^2*x
^2-a*e^2)^2/c*(e*x+d)^(5/2)*B*d-7/8*e^4/(c*e^2*x^2-a*e^2)^2/c*(e*x+d)^(3/2)*B*d^
2+7/8*e^4/(c*e^2*x^2-a*e^2)^2/c*(e*x+d)^(1/2)*B*d^3+1/2*e^3/(c*e^2*x^2-a*e^2)^2/
a*(e*x+d)^(7/2)*A*d+7/16*e^2/(c*e^2*x^2-a*e^2)^2/a*(e*x+d)^(7/2)*B*d^2-23/16*e^3
/(c*e^2*x^2-a*e^2)^2/a*(e*x+d)^(5/2)*A*d^2-21/16*e^2/(c*e^2*x^2-a*e^2)^2/a*(e*x+
d)^(5/2)*B*d^3-7/8*e^5/(c*e^2*x^2-a*e^2)^2/c*(e*x+d)^(3/2)*A*d+2*e^3/(c*e^2*x^2-
a*e^2)^2/a*(e*x+d)^(3/2)*A*d^3-7/16*e^6/(c*e^2*x^2-a*e^2)^2/c^2*a*(e*x+d)^(3/2)*
B+21/16*e^2/(c*e^2*x^2-a*e^2)^2/a*(e*x+d)^(3/2)*B*d^4-5/16*e^7/(c*e^2*x^2-a*e^2)
^2*a/c^2*(e*x+d)^(1/2)*A+e^5/(c*e^2*x^2-a*e^2)^2/c*(e*x+d)^(1/2)*A*d^2+5/32*e^10
*a^3*c^2/(a^5*c^5*e^8)^(1/2)/((a^2*c^3*e^3*d+(a^5*c^5*e^8)^(1/2))*c*e)^(1/2)*arc
tanh(a*c^2*e^2*(e*x+d)^(1/2)/((a^2*c^3*e^3*d+(a^5*c^5*e^8)^(1/2))*c*e)^(1/2))*A-
3/16*e^3/a*c/((-a^2*c^3*e^3*d+(a^5*c^5*e^8)^(1/2))*c*e)^(1/2)*arctan(a*c^2*e^2*(
e*x+d)^(1/2)/((-a^2*c^3*e^3*d+(a^5*c^5*e^8)^(1/2))*c*e)^(1/2))*A*d^3+5/32*e^10*a
^3*c^2/(a^5*c^5*e^8)^(1/2)/((-a^2*c^3*e^3*d+(a^5*c^5*e^8)^(1/2))*c*e)^(1/2)*arct
an(a*c^2*e^2*(e*x+d)^(1/2)/((-a^2*c^3*e^3*d+(a^5*c^5*e^8)^(1/2))*c*e)^(1/2))*A+3
/16*e^3/a*c/((a^2*c^3*e^3*d+(a^5*c^5*e^8)^(1/2))*c*e)^(1/2)*arctanh(a*c^2*e^2*(e
*x+d)^(1/2)/((a^2*c^3*e^3*d+(a^5*c^5*e^8)^(1/2))*c*e)^(1/2))*A*d^3-3/8*e/(c*e^2*
x^2-a*e^2)^2/a^2*c*(e*x+d)^(7/2)*A*d^3+9/8*e/(c*e^2*x^2-a*e^2)^2*c/a^2*(e*x+d)^(
5/2)*A*d^4-9/8*e/(c*e^2*x^2-a*e^2)^2*c/a^2*(e*x+d)^(3/2)*A*d^5+3/8*e/(c*e^2*x^2-
a*e^2)^2/a^2*c*(e*x+d)^(1/2)*A*d^6-7/16*e^6/(c*e^2*x^2-a*e^2)^2*a/c^2*(e*x+d)^(1
/2)*B*d+7/8*e^9*a^3*c^2/(a^5*c^5*e^8)^(1/2)/((-a^2*c^3*e^3*d+(a^5*c^5*e^8)^(1/2)
)*c*e)^(1/2)*arctan(a*c^2*e^2*(e*x+d)^(1/2)/((-a^2*c^3*e^3*d+(a^5*c^5*e^8)^(1/2)
)*c*e)^(1/2))*B*d-7/16*e^7*a^2*c^3/(a^5*c^5*e^8)^(1/2)/((a^2*c^3*e^3*d+(a^5*c^5*
e^8)^(1/2))*c*e)^(1/2)*arctanh(a*c^2*e^2*(e*x+d)^(1/2)/((a^2*c^3*e^3*d+(a^5*c^5*
e^8)^(1/2))*c*e)^(1/2))*B*d^3+7/32*e^4/((-a^2*c^3*e^3*d+(a^5*c^5*e^8)^(1/2))*c*e
)^(1/2)*arctan(a*c^2*e^2*(e*x+d)^(1/2)/((-a^2*c^3*e^3*d+(a^5*c^5*e^8)^(1/2))*c*e
)^(1/2))*B*d^2+1/4*e^5/((-a^2*c^3*e^3*d+(a^5*c^5*e^8)^(1/2))*c*e)^(1/2)*arctan(a
*c^2*e^2*(e*x+d)^(1/2)/((-a^2*c^3*e^3*d+(a^5*c^5*e^8)^(1/2))*c*e)^(1/2))*A*d+11/
16*e^4/(c*e^2*x^2-a*e^2)^2/c*(e*x+d)^(7/2)*B+9/16*e^5/(c*e^2*x^2-a*e^2)^2/c*(e*x
+d)^(5/2)*A-1/4*e^5/((a^2*c^3*e^3*d+(a^5*c^5*e^8)^(1/2))*c*e)^(1/2)*arctanh(a*c^
2*e^2*(e*x+d)^(1/2)/((a^2*c^3*e^3*d+(a^5*c^5*e^8)^(1/2))*c*e)^(1/2))*A*d-7/32*e^
4/((a^2*c^3*e^3*d+(a^5*c^5*e^8)^(1/2))*c*e)^(1/2)*arctanh(a*c^2*e^2*(e*x+d)^(1/2
)/((a^2*c^3*e^3*d+(a^5*c^5*e^8)^(1/2))*c*e)^(1/2))*B*d^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\int \frac{{\left (B x + A\right )}{\left (e x + d\right )}^{\frac{7}{2}}}{{\left (c x^{2} - a\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(B*x + A)*(e*x + d)^(7/2)/(c*x^2 - a)^3,x, algorithm="maxima")

[Out]

-integrate((B*x + A)*(e*x + d)^(7/2)/(c*x^2 - a)^3, x)

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Fricas [A]  time = 11.5273, size = 9003, normalized size = 22.73 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(B*x + A)*(e*x + d)^(7/2)/(c*x^2 - a)^3,x, algorithm="fricas")

[Out]

-1/64*((a^2*c^4*x^4 - 2*a^3*c^3*x^2 + a^4*c^2)*sqrt((144*A^2*c^4*d^7 - 336*A*B*a
*c^3*d^6*e + 1120*A*B*a^2*c^2*d^4*e^3 - 1050*A*B*a^3*c*d^2*e^5 + 210*A*B*a^4*e^7
+ a^5*c^5*sqrt((44100*A^2*B^2*c^4*d^6*e^8 - 2940*(35*A*B^3*a*c^3 + 3*A^3*B*c^4)
*d^5*e^9 + 49*(1225*B^4*a^2*c^2 - 2070*A^2*B^2*a*c^3 + 9*A^4*c^4)*d^4*e^10 + 56*
(5635*A*B^3*a^2*c^2 + 387*A^3*B*a*c^3)*d^3*e^11 - 42*(5145*B^4*a^3*c - 952*A^2*B
^2*a^2*c^2 + 25*A^4*a*c^3)*d^2*e^12 - 532*(441*A*B^3*a^3*c + 25*A^3*B*a^2*c^2)*d
*e^13 + (194481*B^4*a^4 + 22050*A^2*B^2*a^3*c + 625*A^4*a^2*c^2)*e^14)/(a^5*c^11
)) + 28*(7*B^2*a^2*c^2 - 15*A^2*a*c^3)*d^5*e^2 - 35*(21*B^2*a^3*c - 11*A^2*a^2*c
^2)*d^3*e^4 + 105*(7*B^2*a^4 - A^2*a^3*c)*d*e^6)/(a^5*c^5))*log(-(30240*A^3*B*c^
6*d^9*e^4 - 3024*(35*A^2*B^2*a*c^5 + A^4*c^6)*d^8*e^5 + 504*(245*A*B^3*a^2*c^4 -
207*A^3*B*a*c^5)*d^7*e^6 - 4*(12005*B^4*a^3*c^3 - 108486*A^2*B^2*a^2*c^4 - 2727
*A^4*a*c^5)*d^6*e^7 - 14*(40523*A*B^3*a^3*c^3 - 8019*A^3*B*a^2*c^4)*d^5*e^8 + (2
42501*B^4*a^4*c^2 - 573888*A^2*B^2*a^3*c^3 - 13509*A^4*a^2*c^4)*d^4*e^9 + 28*(29
743*A*B^3*a^4*c^2 - 1051*A^3*B*a^3*c^3)*d^3*e^10 - 2*(194481*B^4*a^5*c - 122892*
A^2*B^2*a^4*c^2 - 3125*A^4*a^3*c^3)*d^2*e^11 - 14*(27783*A*B^3*a^5*c + 625*A^3*B
*a^4*c^2)*d*e^12 + (194481*B^4*a^6 - 625*A^4*a^4*c^2)*e^13)*sqrt(e*x + d) + (126
0*A^2*B*a^3*c^6*d^5*e^5 - 42*(70*A*B^2*a^4*c^5 + 3*A^3*a^3*c^6)*d^4*e^6 + 49*(35
*B^3*a^5*c^4 - 51*A^2*B*a^4*c^5)*d^3*e^7 + 3*(1911*A*B^2*a^5*c^4 + 85*A^3*a^4*c^
5)*d^2*e^8 - 21*(147*B^3*a^6*c^3 - 55*A^2*B*a^5*c^4)*d*e^9 - 5*(441*A*B^2*a^6*c^
3 + 25*A^3*a^5*c^4)*e^10 + (12*A*a^5*c^10*d^3 - 14*B*a^6*c^9*d^2*e - 13*A*a^6*c^
9*d*e^2 + 21*B*a^7*c^8*e^3)*sqrt((44100*A^2*B^2*c^4*d^6*e^8 - 2940*(35*A*B^3*a*c
^3 + 3*A^3*B*c^4)*d^5*e^9 + 49*(1225*B^4*a^2*c^2 - 2070*A^2*B^2*a*c^3 + 9*A^4*c^
4)*d^4*e^10 + 56*(5635*A*B^3*a^2*c^2 + 387*A^3*B*a*c^3)*d^3*e^11 - 42*(5145*B^4*
a^3*c - 952*A^2*B^2*a^2*c^2 + 25*A^4*a*c^3)*d^2*e^12 - 532*(441*A*B^3*a^3*c + 25
*A^3*B*a^2*c^2)*d*e^13 + (194481*B^4*a^4 + 22050*A^2*B^2*a^3*c + 625*A^4*a^2*c^2
)*e^14)/(a^5*c^11)))*sqrt((144*A^2*c^4*d^7 - 336*A*B*a*c^3*d^6*e + 1120*A*B*a^2*
c^2*d^4*e^3 - 1050*A*B*a^3*c*d^2*e^5 + 210*A*B*a^4*e^7 + a^5*c^5*sqrt((44100*A^2
*B^2*c^4*d^6*e^8 - 2940*(35*A*B^3*a*c^3 + 3*A^3*B*c^4)*d^5*e^9 + 49*(1225*B^4*a^
2*c^2 - 2070*A^2*B^2*a*c^3 + 9*A^4*c^4)*d^4*e^10 + 56*(5635*A*B^3*a^2*c^2 + 387*
A^3*B*a*c^3)*d^3*e^11 - 42*(5145*B^4*a^3*c - 952*A^2*B^2*a^2*c^2 + 25*A^4*a*c^3)
*d^2*e^12 - 532*(441*A*B^3*a^3*c + 25*A^3*B*a^2*c^2)*d*e^13 + (194481*B^4*a^4 +
22050*A^2*B^2*a^3*c + 625*A^4*a^2*c^2)*e^14)/(a^5*c^11)) + 28*(7*B^2*a^2*c^2 - 1
5*A^2*a*c^3)*d^5*e^2 - 35*(21*B^2*a^3*c - 11*A^2*a^2*c^2)*d^3*e^4 + 105*(7*B^2*a
^4 - A^2*a^3*c)*d*e^6)/(a^5*c^5))) - (a^2*c^4*x^4 - 2*a^3*c^3*x^2 + a^4*c^2)*sqr
t((144*A^2*c^4*d^7 - 336*A*B*a*c^3*d^6*e + 1120*A*B*a^2*c^2*d^4*e^3 - 1050*A*B*a
^3*c*d^2*e^5 + 210*A*B*a^4*e^7 + a^5*c^5*sqrt((44100*A^2*B^2*c^4*d^6*e^8 - 2940*
(35*A*B^3*a*c^3 + 3*A^3*B*c^4)*d^5*e^9 + 49*(1225*B^4*a^2*c^2 - 2070*A^2*B^2*a*c
^3 + 9*A^4*c^4)*d^4*e^10 + 56*(5635*A*B^3*a^2*c^2 + 387*A^3*B*a*c^3)*d^3*e^11 -
42*(5145*B^4*a^3*c - 952*A^2*B^2*a^2*c^2 + 25*A^4*a*c^3)*d^2*e^12 - 532*(441*A*B
^3*a^3*c + 25*A^3*B*a^2*c^2)*d*e^13 + (194481*B^4*a^4 + 22050*A^2*B^2*a^3*c + 62
5*A^4*a^2*c^2)*e^14)/(a^5*c^11)) + 28*(7*B^2*a^2*c^2 - 15*A^2*a*c^3)*d^5*e^2 - 3
5*(21*B^2*a^3*c - 11*A^2*a^2*c^2)*d^3*e^4 + 105*(7*B^2*a^4 - A^2*a^3*c)*d*e^6)/(
a^5*c^5))*log(-(30240*A^3*B*c^6*d^9*e^4 - 3024*(35*A^2*B^2*a*c^5 + A^4*c^6)*d^8*
e^5 + 504*(245*A*B^3*a^2*c^4 - 207*A^3*B*a*c^5)*d^7*e^6 - 4*(12005*B^4*a^3*c^3 -
108486*A^2*B^2*a^2*c^4 - 2727*A^4*a*c^5)*d^6*e^7 - 14*(40523*A*B^3*a^3*c^3 - 80
19*A^3*B*a^2*c^4)*d^5*e^8 + (242501*B^4*a^4*c^2 - 573888*A^2*B^2*a^3*c^3 - 13509
*A^4*a^2*c^4)*d^4*e^9 + 28*(29743*A*B^3*a^4*c^2 - 1051*A^3*B*a^3*c^3)*d^3*e^10 -
2*(194481*B^4*a^5*c - 122892*A^2*B^2*a^4*c^2 - 3125*A^4*a^3*c^3)*d^2*e^11 - 14*
(27783*A*B^3*a^5*c + 625*A^3*B*a^4*c^2)*d*e^12 + (194481*B^4*a^6 - 625*A^4*a^4*c
^2)*e^13)*sqrt(e*x + d) - (1260*A^2*B*a^3*c^6*d^5*e^5 - 42*(70*A*B^2*a^4*c^5 + 3
*A^3*a^3*c^6)*d^4*e^6 + 49*(35*B^3*a^5*c^4 - 51*A^2*B*a^4*c^5)*d^3*e^7 + 3*(1911
*A*B^2*a^5*c^4 + 85*A^3*a^4*c^5)*d^2*e^8 - 21*(147*B^3*a^6*c^3 - 55*A^2*B*a^5*c^
4)*d*e^9 - 5*(441*A*B^2*a^6*c^3 + 25*A^3*a^5*c^4)*e^10 + (12*A*a^5*c^10*d^3 - 14
*B*a^6*c^9*d^2*e - 13*A*a^6*c^9*d*e^2 + 21*B*a^7*c^8*e^3)*sqrt((44100*A^2*B^2*c^
4*d^6*e^8 - 2940*(35*A*B^3*a*c^3 + 3*A^3*B*c^4)*d^5*e^9 + 49*(1225*B^4*a^2*c^2 -
2070*A^2*B^2*a*c^3 + 9*A^4*c^4)*d^4*e^10 + 56*(5635*A*B^3*a^2*c^2 + 387*A^3*B*a
*c^3)*d^3*e^11 - 42*(5145*B^4*a^3*c - 952*A^2*B^2*a^2*c^2 + 25*A^4*a*c^3)*d^2*e^
12 - 532*(441*A*B^3*a^3*c + 25*A^3*B*a^2*c^2)*d*e^13 + (194481*B^4*a^4 + 22050*A
^2*B^2*a^3*c + 625*A^4*a^2*c^2)*e^14)/(a^5*c^11)))*sqrt((144*A^2*c^4*d^7 - 336*A
*B*a*c^3*d^6*e + 1120*A*B*a^2*c^2*d^4*e^3 - 1050*A*B*a^3*c*d^2*e^5 + 210*A*B*a^4
*e^7 + a^5*c^5*sqrt((44100*A^2*B^2*c^4*d^6*e^8 - 2940*(35*A*B^3*a*c^3 + 3*A^3*B*
c^4)*d^5*e^9 + 49*(1225*B^4*a^2*c^2 - 2070*A^2*B^2*a*c^3 + 9*A^4*c^4)*d^4*e^10 +
56*(5635*A*B^3*a^2*c^2 + 387*A^3*B*a*c^3)*d^3*e^11 - 42*(5145*B^4*a^3*c - 952*A
^2*B^2*a^2*c^2 + 25*A^4*a*c^3)*d^2*e^12 - 532*(441*A*B^3*a^3*c + 25*A^3*B*a^2*c^
2)*d*e^13 + (194481*B^4*a^4 + 22050*A^2*B^2*a^3*c + 625*A^4*a^2*c^2)*e^14)/(a^5*
c^11)) + 28*(7*B^2*a^2*c^2 - 15*A^2*a*c^3)*d^5*e^2 - 35*(21*B^2*a^3*c - 11*A^2*a
^2*c^2)*d^3*e^4 + 105*(7*B^2*a^4 - A^2*a^3*c)*d*e^6)/(a^5*c^5))) + (a^2*c^4*x^4
- 2*a^3*c^3*x^2 + a^4*c^2)*sqrt((144*A^2*c^4*d^7 - 336*A*B*a*c^3*d^6*e + 1120*A*
B*a^2*c^2*d^4*e^3 - 1050*A*B*a^3*c*d^2*e^5 + 210*A*B*a^4*e^7 - a^5*c^5*sqrt((441
00*A^2*B^2*c^4*d^6*e^8 - 2940*(35*A*B^3*a*c^3 + 3*A^3*B*c^4)*d^5*e^9 + 49*(1225*
B^4*a^2*c^2 - 2070*A^2*B^2*a*c^3 + 9*A^4*c^4)*d^4*e^10 + 56*(5635*A*B^3*a^2*c^2
+ 387*A^3*B*a*c^3)*d^3*e^11 - 42*(5145*B^4*a^3*c - 952*A^2*B^2*a^2*c^2 + 25*A^4*
a*c^3)*d^2*e^12 - 532*(441*A*B^3*a^3*c + 25*A^3*B*a^2*c^2)*d*e^13 + (194481*B^4*
a^4 + 22050*A^2*B^2*a^3*c + 625*A^4*a^2*c^2)*e^14)/(a^5*c^11)) + 28*(7*B^2*a^2*c
^2 - 15*A^2*a*c^3)*d^5*e^2 - 35*(21*B^2*a^3*c - 11*A^2*a^2*c^2)*d^3*e^4 + 105*(7
*B^2*a^4 - A^2*a^3*c)*d*e^6)/(a^5*c^5))*log(-(30240*A^3*B*c^6*d^9*e^4 - 3024*(35
*A^2*B^2*a*c^5 + A^4*c^6)*d^8*e^5 + 504*(245*A*B^3*a^2*c^4 - 207*A^3*B*a*c^5)*d^
7*e^6 - 4*(12005*B^4*a^3*c^3 - 108486*A^2*B^2*a^2*c^4 - 2727*A^4*a*c^5)*d^6*e^7
- 14*(40523*A*B^3*a^3*c^3 - 8019*A^3*B*a^2*c^4)*d^5*e^8 + (242501*B^4*a^4*c^2 -
573888*A^2*B^2*a^3*c^3 - 13509*A^4*a^2*c^4)*d^4*e^9 + 28*(29743*A*B^3*a^4*c^2 -
1051*A^3*B*a^3*c^3)*d^3*e^10 - 2*(194481*B^4*a^5*c - 122892*A^2*B^2*a^4*c^2 - 31
25*A^4*a^3*c^3)*d^2*e^11 - 14*(27783*A*B^3*a^5*c + 625*A^3*B*a^4*c^2)*d*e^12 + (
194481*B^4*a^6 - 625*A^4*a^4*c^2)*e^13)*sqrt(e*x + d) + (1260*A^2*B*a^3*c^6*d^5*
e^5 - 42*(70*A*B^2*a^4*c^5 + 3*A^3*a^3*c^6)*d^4*e^6 + 49*(35*B^3*a^5*c^4 - 51*A^
2*B*a^4*c^5)*d^3*e^7 + 3*(1911*A*B^2*a^5*c^4 + 85*A^3*a^4*c^5)*d^2*e^8 - 21*(147
*B^3*a^6*c^3 - 55*A^2*B*a^5*c^4)*d*e^9 - 5*(441*A*B^2*a^6*c^3 + 25*A^3*a^5*c^4)*
e^10 - (12*A*a^5*c^10*d^3 - 14*B*a^6*c^9*d^2*e - 13*A*a^6*c^9*d*e^2 + 21*B*a^7*c
^8*e^3)*sqrt((44100*A^2*B^2*c^4*d^6*e^8 - 2940*(35*A*B^3*a*c^3 + 3*A^3*B*c^4)*d^
5*e^9 + 49*(1225*B^4*a^2*c^2 - 2070*A^2*B^2*a*c^3 + 9*A^4*c^4)*d^4*e^10 + 56*(56
35*A*B^3*a^2*c^2 + 387*A^3*B*a*c^3)*d^3*e^11 - 42*(5145*B^4*a^3*c - 952*A^2*B^2*
a^2*c^2 + 25*A^4*a*c^3)*d^2*e^12 - 532*(441*A*B^3*a^3*c + 25*A^3*B*a^2*c^2)*d*e^
13 + (194481*B^4*a^4 + 22050*A^2*B^2*a^3*c + 625*A^4*a^2*c^2)*e^14)/(a^5*c^11)))
*sqrt((144*A^2*c^4*d^7 - 336*A*B*a*c^3*d^6*e + 1120*A*B*a^2*c^2*d^4*e^3 - 1050*A
*B*a^3*c*d^2*e^5 + 210*A*B*a^4*e^7 - a^5*c^5*sqrt((44100*A^2*B^2*c^4*d^6*e^8 - 2
940*(35*A*B^3*a*c^3 + 3*A^3*B*c^4)*d^5*e^9 + 49*(1225*B^4*a^2*c^2 - 2070*A^2*B^2
*a*c^3 + 9*A^4*c^4)*d^4*e^10 + 56*(5635*A*B^3*a^2*c^2 + 387*A^3*B*a*c^3)*d^3*e^1
1 - 42*(5145*B^4*a^3*c - 952*A^2*B^2*a^2*c^2 + 25*A^4*a*c^3)*d^2*e^12 - 532*(441
*A*B^3*a^3*c + 25*A^3*B*a^2*c^2)*d*e^13 + (194481*B^4*a^4 + 22050*A^2*B^2*a^3*c
+ 625*A^4*a^2*c^2)*e^14)/(a^5*c^11)) + 28*(7*B^2*a^2*c^2 - 15*A^2*a*c^3)*d^5*e^2
- 35*(21*B^2*a^3*c - 11*A^2*a^2*c^2)*d^3*e^4 + 105*(7*B^2*a^4 - A^2*a^3*c)*d*e^
6)/(a^5*c^5))) - (a^2*c^4*x^4 - 2*a^3*c^3*x^2 + a^4*c^2)*sqrt((144*A^2*c^4*d^7 -
336*A*B*a*c^3*d^6*e + 1120*A*B*a^2*c^2*d^4*e^3 - 1050*A*B*a^3*c*d^2*e^5 + 210*A
*B*a^4*e^7 - a^5*c^5*sqrt((44100*A^2*B^2*c^4*d^6*e^8 - 2940*(35*A*B^3*a*c^3 + 3*
A^3*B*c^4)*d^5*e^9 + 49*(1225*B^4*a^2*c^2 - 2070*A^2*B^2*a*c^3 + 9*A^4*c^4)*d^4*
e^10 + 56*(5635*A*B^3*a^2*c^2 + 387*A^3*B*a*c^3)*d^3*e^11 - 42*(5145*B^4*a^3*c -
952*A^2*B^2*a^2*c^2 + 25*A^4*a*c^3)*d^2*e^12 - 532*(441*A*B^3*a^3*c + 25*A^3*B*
a^2*c^2)*d*e^13 + (194481*B^4*a^4 + 22050*A^2*B^2*a^3*c + 625*A^4*a^2*c^2)*e^14)
/(a^5*c^11)) + 28*(7*B^2*a^2*c^2 - 15*A^2*a*c^3)*d^5*e^2 - 35*(21*B^2*a^3*c - 11
*A^2*a^2*c^2)*d^3*e^4 + 105*(7*B^2*a^4 - A^2*a^3*c)*d*e^6)/(a^5*c^5))*log(-(3024
0*A^3*B*c^6*d^9*e^4 - 3024*(35*A^2*B^2*a*c^5 + A^4*c^6)*d^8*e^5 + 504*(245*A*B^3
*a^2*c^4 - 207*A^3*B*a*c^5)*d^7*e^6 - 4*(12005*B^4*a^3*c^3 - 108486*A^2*B^2*a^2*
c^4 - 2727*A^4*a*c^5)*d^6*e^7 - 14*(40523*A*B^3*a^3*c^3 - 8019*A^3*B*a^2*c^4)*d^
5*e^8 + (242501*B^4*a^4*c^2 - 573888*A^2*B^2*a^3*c^3 - 13509*A^4*a^2*c^4)*d^4*e^
9 + 28*(29743*A*B^3*a^4*c^2 - 1051*A^3*B*a^3*c^3)*d^3*e^10 - 2*(194481*B^4*a^5*c
- 122892*A^2*B^2*a^4*c^2 - 3125*A^4*a^3*c^3)*d^2*e^11 - 14*(27783*A*B^3*a^5*c +
625*A^3*B*a^4*c^2)*d*e^12 + (194481*B^4*a^6 - 625*A^4*a^4*c^2)*e^13)*sqrt(e*x +
d) - (1260*A^2*B*a^3*c^6*d^5*e^5 - 42*(70*A*B^2*a^4*c^5 + 3*A^3*a^3*c^6)*d^4*e^
6 + 49*(35*B^3*a^5*c^4 - 51*A^2*B*a^4*c^5)*d^3*e^7 + 3*(1911*A*B^2*a^5*c^4 + 85*
A^3*a^4*c^5)*d^2*e^8 - 21*(147*B^3*a^6*c^3 - 55*A^2*B*a^5*c^4)*d*e^9 - 5*(441*A*
B^2*a^6*c^3 + 25*A^3*a^5*c^4)*e^10 - (12*A*a^5*c^10*d^3 - 14*B*a^6*c^9*d^2*e - 1
3*A*a^6*c^9*d*e^2 + 21*B*a^7*c^8*e^3)*sqrt((44100*A^2*B^2*c^4*d^6*e^8 - 2940*(35
*A*B^3*a*c^3 + 3*A^3*B*c^4)*d^5*e^9 + 49*(1225*B^4*a^2*c^2 - 2070*A^2*B^2*a*c^3
+ 9*A^4*c^4)*d^4*e^10 + 56*(5635*A*B^3*a^2*c^2 + 387*A^3*B*a*c^3)*d^3*e^11 - 42*
(5145*B^4*a^3*c - 952*A^2*B^2*a^2*c^2 + 25*A^4*a*c^3)*d^2*e^12 - 532*(441*A*B^3*
a^3*c + 25*A^3*B*a^2*c^2)*d*e^13 + (194481*B^4*a^4 + 22050*A^2*B^2*a^3*c + 625*A
^4*a^2*c^2)*e^14)/(a^5*c^11)))*sqrt((144*A^2*c^4*d^7 - 336*A*B*a*c^3*d^6*e + 112
0*A*B*a^2*c^2*d^4*e^3 - 1050*A*B*a^3*c*d^2*e^5 + 210*A*B*a^4*e^7 - a^5*c^5*sqrt(
(44100*A^2*B^2*c^4*d^6*e^8 - 2940*(35*A*B^3*a*c^3 + 3*A^3*B*c^4)*d^5*e^9 + 49*(1
225*B^4*a^2*c^2 - 2070*A^2*B^2*a*c^3 + 9*A^4*c^4)*d^4*e^10 + 56*(5635*A*B^3*a^2*
c^2 + 387*A^3*B*a*c^3)*d^3*e^11 - 42*(5145*B^4*a^3*c - 952*A^2*B^2*a^2*c^2 + 25*
A^4*a*c^3)*d^2*e^12 - 532*(441*A*B^3*a^3*c + 25*A^3*B*a^2*c^2)*d*e^13 + (194481*
B^4*a^4 + 22050*A^2*B^2*a^3*c + 625*A^4*a^2*c^2)*e^14)/(a^5*c^11)) + 28*(7*B^2*a
^2*c^2 - 15*A^2*a*c^3)*d^5*e^2 - 35*(21*B^2*a^3*c - 11*A^2*a^2*c^2)*d^3*e^4 + 10
5*(7*B^2*a^4 - A^2*a^3*c)*d*e^6)/(a^5*c^5))) - 4*(4*B*a^2*c*d^3 + 11*A*a^2*c*d^2
*e - 14*B*a^3*d*e^2 - 5*A*a^3*e^3 - (6*A*c^3*d^3 - 7*B*a*c^2*d^2*e - 8*A*a*c^2*d
*e^2 - 11*B*a^2*c*e^3)*x^3 + (A*a*c^2*d^2*e + 26*B*a^2*c*d*e^2 + 9*A*a^2*c*e^3)*
x^2 + (10*A*a*c^2*d^3 + 5*B*a^2*c*d^2*e + 4*A*a^2*c*d*e^2 - 7*B*a^3*e^3)*x)*sqrt
(e*x + d))/(a^2*c^4*x^4 - 2*a^3*c^3*x^2 + a^4*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)**(7/2)/(-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)*(e*x + d)^(7/2)/(c*x^2 - a)^3,x, algorithm="giac")

[Out]

Timed out