3.581 \(\int \frac{(d+e x)^3 (f+g x)^3}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx\)

Optimal. Leaf size=183 \[ \frac{(d+e x)^2 (2 e f-13 d g) (d g+e f)^2}{15 d^2 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{(d+e x)^3 (d g+e f)^3}{5 d e^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{g^3 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^4}+\frac{(d+e x) (d g+e f) \left (32 d^2 g^2-11 d e f g+2 e^2 f^2\right )}{15 d^3 e^4 \sqrt{d^2-e^2 x^2}} \]

[Out]

((e*f + d*g)^3*(d + e*x)^3)/(5*d*e^4*(d^2 - e^2*x^2)^(5/2)) + ((2*e*f - 13*d*g)*
(e*f + d*g)^2*(d + e*x)^2)/(15*d^2*e^4*(d^2 - e^2*x^2)^(3/2)) + ((e*f + d*g)*(2*
e^2*f^2 - 11*d*e*f*g + 32*d^2*g^2)*(d + e*x))/(15*d^3*e^4*Sqrt[d^2 - e^2*x^2]) -
(g^3*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/e^4

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Rubi [A]  time = 0.634761, antiderivative size = 183, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129 \[ \frac{(d+e x)^2 (2 e f-13 d g) (d g+e f)^2}{15 d^2 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{(d+e x)^3 (d g+e f)^3}{5 d e^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{g^3 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^4}+\frac{(d+e x) (d g+e f) \left (32 d^2 g^2-11 d e f g+2 e^2 f^2\right )}{15 d^3 e^4 \sqrt{d^2-e^2 x^2}} \]

Antiderivative was successfully verified.

[In]  Int[((d + e*x)^3*(f + g*x)^3)/(d^2 - e^2*x^2)^(7/2),x]

[Out]

((e*f + d*g)^3*(d + e*x)^3)/(5*d*e^4*(d^2 - e^2*x^2)^(5/2)) + ((2*e*f - 13*d*g)*
(e*f + d*g)^2*(d + e*x)^2)/(15*d^2*e^4*(d^2 - e^2*x^2)^(3/2)) + ((e*f + d*g)*(2*
e^2*f^2 - 11*d*e*f*g + 32*d^2*g^2)*(d + e*x))/(15*d^3*e^4*Sqrt[d^2 - e^2*x^2]) -
(g^3*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/e^4

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Rubi in Sympy [A]  time = 77.108, size = 243, normalized size = 1.33 \[ - \frac{g^{3} \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{e^{4}} + \frac{3 g^{2} \sqrt{d^{2} - e^{2} x^{2}} \left (d g + e f\right )}{d e^{4} \left (d - e x\right )} - \frac{g \sqrt{d^{2} - e^{2} x^{2}} \left (d g + e f\right )^{2}}{d e^{4} \left (d - e x\right )^{2}} + \frac{\sqrt{d^{2} - e^{2} x^{2}} \left (d g + e f\right )^{3}}{5 d e^{4} \left (d - e x\right )^{3}} - \frac{g \sqrt{d^{2} - e^{2} x^{2}} \left (d g + e f\right )^{2}}{d^{2} e^{4} \left (d - e x\right )} + \frac{2 \sqrt{d^{2} - e^{2} x^{2}} \left (d g + e f\right )^{3}}{15 d^{2} e^{4} \left (d - e x\right )^{2}} + \frac{2 \sqrt{d^{2} - e^{2} x^{2}} \left (d g + e f\right )^{3}}{15 d^{3} e^{4} \left (d - e x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**3*(g*x+f)**3/(-e**2*x**2+d**2)**(7/2),x)

[Out]

-g**3*atan(e*x/sqrt(d**2 - e**2*x**2))/e**4 + 3*g**2*sqrt(d**2 - e**2*x**2)*(d*g
+ e*f)/(d*e**4*(d - e*x)) - g*sqrt(d**2 - e**2*x**2)*(d*g + e*f)**2/(d*e**4*(d
- e*x)**2) + sqrt(d**2 - e**2*x**2)*(d*g + e*f)**3/(5*d*e**4*(d - e*x)**3) - g*s
qrt(d**2 - e**2*x**2)*(d*g + e*f)**2/(d**2*e**4*(d - e*x)) + 2*sqrt(d**2 - e**2*
x**2)*(d*g + e*f)**3/(15*d**2*e**4*(d - e*x)**2) + 2*sqrt(d**2 - e**2*x**2)*(d*g
+ e*f)**3/(15*d**3*e**4*(d - e*x))

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Mathematica [A]  time = 0.307271, size = 145, normalized size = 0.79 \[ \frac{\frac{\sqrt{d^2-e^2 x^2} (d g+e f) \left (22 d^4 g^2-d^3 e g (16 f+51 g x)+d^2 e^2 \left (7 f^2+33 f g x+32 g^2 x^2\right )-d e^3 f x (6 f+11 g x)+2 e^4 f^2 x^2\right )}{d^3 (d-e x)^3}-15 g^3 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{15 e^4} \]

Antiderivative was successfully verified.

[In]  Integrate[((d + e*x)^3*(f + g*x)^3)/(d^2 - e^2*x^2)^(7/2),x]

[Out]

(((e*f + d*g)*Sqrt[d^2 - e^2*x^2]*(22*d^4*g^2 + 2*e^4*f^2*x^2 - d*e^3*f*x*(6*f +
11*g*x) - d^3*e*g*(16*f + 51*g*x) + d^2*e^2*(7*f^2 + 33*f*g*x + 32*g^2*x^2)))/(
d^3*(d - e*x)^3) - 15*g^3*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(15*e^4)

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Maple [B]  time = 0.016, size = 713, normalized size = 3.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^3*(g*x+f)^3/(-e^2*x^2+d^2)^(7/2),x)

[Out]

1/3*x^2*e/(-e^2*x^2+d^2)^(5/2)*f^3+3*x^4/(-e^2*x^2+d^2)^(5/2)*d*g^3+22/15*d^5/e^
4/(-e^2*x^2+d^2)^(5/2)*g^3+7/15/e/(-e^2*x^2+d^2)^(5/2)*d^2*f^3+4/5*d*f^3*x/(-e^2
*x^2+d^2)^(5/2)+1/15/d*f^3*x/(-e^2*x^2+d^2)^(3/2)+2/15/d^3*f^3*x/(-e^2*x^2+d^2)^
(1/2)+1/5*e*g^3*x^5/(-e^2*x^2+d^2)^(5/2)-1/3/e*g^3*x^3/(-e^2*x^2+d^2)^(3/2)+8/5/
e^3*g^3*x/(-e^2*x^2+d^2)^(1/2)-1/e^3*g^3/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*
x^2+d^2)^(1/2))-d^2/e*x^2/(-e^2*x^2+d^2)^(5/2)*f*g^2-21/10*d^3/e^2*x/(-e^2*x^2+d
^2)^(5/2)*f*g^2+9/10*d^2/e*x/(-e^2*x^2+d^2)^(5/2)*f^2*g+7/10/e^2*x/(-e^2*x^2+d^2
)^(3/2)*d*f*g^2+7/5/d/e^2*x/(-e^2*x^2+d^2)^(1/2)*f*g^2-3/5/d^2/e*x/(-e^2*x^2+d^2
)^(1/2)*f^2*g+3*x^4*e/(-e^2*x^2+d^2)^(5/2)*f*g^2-11/3*d^3/e^2*x^2/(-e^2*x^2+d^2)
^(5/2)*g^3+2/5*d^4/e^3/(-e^2*x^2+d^2)^(5/2)*f*g^2+3/10/e^3*x/(-e^2*x^2+d^2)^(3/2
)*d^2*g^3-3/10/e*x/(-e^2*x^2+d^2)^(3/2)*f^2*g-3/5/e^2/(-e^2*x^2+d^2)^(5/2)*d^3*f
^2*g+9/2*x^3/(-e^2*x^2+d^2)^(5/2)*d*f*g^2+3/2*x^3*e/(-e^2*x^2+d^2)^(5/2)*f^2*g-9
/10*d^4/e^3*x/(-e^2*x^2+d^2)^(5/2)*g^3+3*x^2/(-e^2*x^2+d^2)^(5/2)*d*f^2*g+3/2*x^
3/e/(-e^2*x^2+d^2)^(5/2)*d^2*g^3

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Maxima [A]  time = 0.781927, size = 1220, normalized size = 6.67 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^3*(g*x + f)^3/(-e^2*x^2 + d^2)^(7/2),x, algorithm="maxima")

[Out]

1/15*e^3*g^3*x*(15*x^4/((-e^2*x^2 + d^2)^(5/2)*e^2) - 20*d^2*x^2/((-e^2*x^2 + d^
2)^(5/2)*e^4) + 8*d^4/((-e^2*x^2 + d^2)^(5/2)*e^6)) - 1/3*e*g^3*x*(3*x^2/((-e^2*
x^2 + d^2)^(3/2)*e^2) - 2*d^2/((-e^2*x^2 + d^2)^(3/2)*e^4)) + 1/5*d*f^3*x/(-e^2*
x^2 + d^2)^(5/2) + 3/5*d^2*f^3/((-e^2*x^2 + d^2)^(5/2)*e) + 3/5*d^3*f^2*g/((-e^2
*x^2 + d^2)^(5/2)*e^2) + 4/15*f^3*x/((-e^2*x^2 + d^2)^(3/2)*d) + 4/15*d^2*g^3*x/
((-e^2*x^2 + d^2)^(3/2)*e^3) + 8/15*f^3*x/(sqrt(-e^2*x^2 + d^2)*d^3) - 7/15*g^3*
x/(sqrt(-e^2*x^2 + d^2)*e^3) + 3*(e^3*f*g^2 + d*e^2*g^3)*x^4/((-e^2*x^2 + d^2)^(
5/2)*e^2) - g^3*arcsin(e^2*x/sqrt(d^2*e^2))/(sqrt(e^2)*e^3) + 3/2*(e^3*f^2*g + 3
*d*e^2*f*g^2 + d^2*e*g^3)*x^3/((-e^2*x^2 + d^2)^(5/2)*e^2) - 4*(e^3*f*g^2 + d*e^
2*g^3)*d^2*x^2/((-e^2*x^2 + d^2)^(5/2)*e^4) + 1/3*(e^3*f^3 + 9*d*e^2*f^2*g + 9*d
^2*e*f*g^2 + d^3*g^3)*x^2/((-e^2*x^2 + d^2)^(5/2)*e^2) - 9/10*(e^3*f^2*g + 3*d*e
^2*f*g^2 + d^2*e*g^3)*d^2*x/((-e^2*x^2 + d^2)^(5/2)*e^4) + 3/5*(d*e^2*f^3 + 3*d^
2*e*f^2*g + d^3*f*g^2)*x/((-e^2*x^2 + d^2)^(5/2)*e^2) + 8/5*(e^3*f*g^2 + d*e^2*g
^3)*d^4/((-e^2*x^2 + d^2)^(5/2)*e^6) - 2/15*(e^3*f^3 + 9*d*e^2*f^2*g + 9*d^2*e*f
*g^2 + d^3*g^3)*d^2/((-e^2*x^2 + d^2)^(5/2)*e^4) + 3/10*(e^3*f^2*g + 3*d*e^2*f*g
^2 + d^2*e*g^3)*x/((-e^2*x^2 + d^2)^(3/2)*e^4) - 1/5*(d*e^2*f^3 + 3*d^2*e*f^2*g
+ d^3*f*g^2)*x/((-e^2*x^2 + d^2)^(3/2)*d^2*e^2) + 3/5*(e^3*f^2*g + 3*d*e^2*f*g^2
+ d^2*e*g^3)*x/(sqrt(-e^2*x^2 + d^2)*d^2*e^4) - 2/5*(d*e^2*f^3 + 3*d^2*e*f^2*g
+ d^3*f*g^2)*x/(sqrt(-e^2*x^2 + d^2)*d^4*e^2)

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Fricas [A]  time = 0.309537, size = 887, normalized size = 4.85 \[ \frac{9 \,{\left (e^{8} f^{3} - 2 \, d e^{7} f^{2} g + 3 \, d^{2} e^{6} f g^{2} + 6 \, d^{3} e^{5} g^{3}\right )} x^{5} - 5 \,{\left (7 \, d e^{7} f^{3} - 9 \, d^{2} e^{6} f^{2} g - 3 \, d^{3} e^{5} f g^{2} + 13 \, d^{4} e^{4} g^{3}\right )} x^{4} + 5 \,{\left (4 \, d^{2} e^{6} f^{3} + 9 \, d^{3} e^{5} f^{2} g - 12 \, d^{4} e^{4} f g^{2} - 17 \, d^{5} e^{3} g^{3}\right )} x^{3} + 30 \,{\left (2 \, d^{3} e^{5} f^{3} - 3 \, d^{4} e^{4} f^{2} g + 5 \, d^{6} e^{2} g^{3}\right )} x^{2} - 60 \,{\left (d^{4} e^{4} f^{3} + d^{7} e g^{3}\right )} x + 30 \,{\left (d^{3} e^{5} g^{3} x^{5} - 5 \, d^{4} e^{4} g^{3} x^{4} + 5 \, d^{5} e^{3} g^{3} x^{3} + 5 \, d^{6} e^{2} g^{3} x^{2} - 10 \, d^{7} e g^{3} x + 4 \, d^{8} g^{3} +{\left (d^{3} e^{4} g^{3} x^{4} - 7 \, d^{5} e^{2} g^{3} x^{2} + 10 \, d^{6} e g^{3} x - 4 \, d^{7} g^{3}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) + 5 \,{\left ({\left (e^{7} f^{3} - 3 \, d^{2} e^{5} f g^{2} - 2 \, d^{3} e^{4} g^{3}\right )} x^{4} +{\left (2 \, d e^{6} f^{3} - 9 \, d^{2} e^{5} f^{2} g + 12 \, d^{3} e^{4} f g^{2} + 23 \, d^{4} e^{3} g^{3}\right )} x^{3} - 6 \,{\left (2 \, d^{2} e^{5} f^{3} - 3 \, d^{3} e^{4} f^{2} g + 5 \, d^{5} e^{2} g^{3}\right )} x^{2} + 12 \,{\left (d^{3} e^{4} f^{3} + d^{6} e g^{3}\right )} x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{15 \,{\left (d^{3} e^{9} x^{5} - 5 \, d^{4} e^{8} x^{4} + 5 \, d^{5} e^{7} x^{3} + 5 \, d^{6} e^{6} x^{2} - 10 \, d^{7} e^{5} x + 4 \, d^{8} e^{4} +{\left (d^{3} e^{8} x^{4} - 7 \, d^{5} e^{6} x^{2} + 10 \, d^{6} e^{5} x - 4 \, d^{7} e^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^3*(g*x + f)^3/(-e^2*x^2 + d^2)^(7/2),x, algorithm="fricas")

[Out]

1/15*(9*(e^8*f^3 - 2*d*e^7*f^2*g + 3*d^2*e^6*f*g^2 + 6*d^3*e^5*g^3)*x^5 - 5*(7*d
*e^7*f^3 - 9*d^2*e^6*f^2*g - 3*d^3*e^5*f*g^2 + 13*d^4*e^4*g^3)*x^4 + 5*(4*d^2*e^
6*f^3 + 9*d^3*e^5*f^2*g - 12*d^4*e^4*f*g^2 - 17*d^5*e^3*g^3)*x^3 + 30*(2*d^3*e^5
*f^3 - 3*d^4*e^4*f^2*g + 5*d^6*e^2*g^3)*x^2 - 60*(d^4*e^4*f^3 + d^7*e*g^3)*x + 3
0*(d^3*e^5*g^3*x^5 - 5*d^4*e^4*g^3*x^4 + 5*d^5*e^3*g^3*x^3 + 5*d^6*e^2*g^3*x^2 -
10*d^7*e*g^3*x + 4*d^8*g^3 + (d^3*e^4*g^3*x^4 - 7*d^5*e^2*g^3*x^2 + 10*d^6*e*g^
3*x - 4*d^7*g^3)*sqrt(-e^2*x^2 + d^2))*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x))
+ 5*((e^7*f^3 - 3*d^2*e^5*f*g^2 - 2*d^3*e^4*g^3)*x^4 + (2*d*e^6*f^3 - 9*d^2*e^5
*f^2*g + 12*d^3*e^4*f*g^2 + 23*d^4*e^3*g^3)*x^3 - 6*(2*d^2*e^5*f^3 - 3*d^3*e^4*f
^2*g + 5*d^5*e^2*g^3)*x^2 + 12*(d^3*e^4*f^3 + d^6*e*g^3)*x)*sqrt(-e^2*x^2 + d^2)
)/(d^3*e^9*x^5 - 5*d^4*e^8*x^4 + 5*d^5*e^7*x^3 + 5*d^6*e^6*x^2 - 10*d^7*e^5*x +
4*d^8*e^4 + (d^3*e^8*x^4 - 7*d^5*e^6*x^2 + 10*d^6*e^5*x - 4*d^7*e^4)*sqrt(-e^2*x
^2 + d^2))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d + e x\right )^{3} \left (f + g x\right )^{3}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{7}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**3*(g*x+f)**3/(-e**2*x**2+d**2)**(7/2),x)

[Out]

Integral((d + e*x)**3*(f + g*x)**3/(-(-d + e*x)*(d + e*x))**(7/2), x)

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GIAC/XCAS [A]  time = 0.294936, size = 417, normalized size = 2.28 \[ -g^{3} \arcsin \left (\frac{x e}{d}\right ) e^{\left (-4\right )}{\rm sign}\left (d\right ) - \frac{\sqrt{-x^{2} e^{2} + d^{2}}{\left ({\left ({\left ({\left (x{\left (\frac{{\left (32 \, d^{4} g^{3} e^{8} + 21 \, d^{3} f g^{2} e^{9} - 9 \, d^{2} f^{2} g e^{10} + 2 \, d f^{3} e^{11}\right )} x e^{\left (-7\right )}}{d^{4}} + \frac{45 \,{\left (d^{5} g^{3} e^{7} + d^{4} f g^{2} e^{8}\right )} e^{\left (-7\right )}}{d^{4}}\right )} - \frac{5 \,{\left (7 \, d^{6} g^{3} e^{6} - 3 \, d^{5} f g^{2} e^{7} - 9 \, d^{4} f^{2} g e^{8} + d^{3} f^{3} e^{9}\right )} e^{\left (-7\right )}}{d^{4}}\right )} x - \frac{5 \,{\left (11 \, d^{7} g^{3} e^{5} + 3 \, d^{6} f g^{2} e^{6} - 9 \, d^{5} f^{2} g e^{7} - d^{4} f^{3} e^{8}\right )} e^{\left (-7\right )}}{d^{4}}\right )} x + \frac{15 \,{\left (d^{8} g^{3} e^{4} + d^{5} f^{3} e^{7}\right )} e^{\left (-7\right )}}{d^{4}}\right )} x + \frac{{\left (22 \, d^{9} g^{3} e^{3} + 6 \, d^{8} f g^{2} e^{4} - 9 \, d^{7} f^{2} g e^{5} + 7 \, d^{6} f^{3} e^{6}\right )} e^{\left (-7\right )}}{d^{4}}\right )}}{15 \,{\left (x^{2} e^{2} - d^{2}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^3*(g*x + f)^3/(-e^2*x^2 + d^2)^(7/2),x, algorithm="giac")

[Out]

-g^3*arcsin(x*e/d)*e^(-4)*sign(d) - 1/15*sqrt(-x^2*e^2 + d^2)*((((x*((32*d^4*g^3
*e^8 + 21*d^3*f*g^2*e^9 - 9*d^2*f^2*g*e^10 + 2*d*f^3*e^11)*x*e^(-7)/d^4 + 45*(d^
5*g^3*e^7 + d^4*f*g^2*e^8)*e^(-7)/d^4) - 5*(7*d^6*g^3*e^6 - 3*d^5*f*g^2*e^7 - 9*
d^4*f^2*g*e^8 + d^3*f^3*e^9)*e^(-7)/d^4)*x - 5*(11*d^7*g^3*e^5 + 3*d^6*f*g^2*e^6
- 9*d^5*f^2*g*e^7 - d^4*f^3*e^8)*e^(-7)/d^4)*x + 15*(d^8*g^3*e^4 + d^5*f^3*e^7)
*e^(-7)/d^4)*x + (22*d^9*g^3*e^3 + 6*d^8*f*g^2*e^4 - 9*d^7*f^2*g*e^5 + 7*d^6*f^3
*e^6)*e^(-7)/d^4)/(x^2*e^2 - d^2)^3