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

Optimal. Leaf size=145 \[ \frac{(d+e x)^3 (d g+e f)^2}{5 d e^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{2 (d+e x)^2 (e f-4 d g) (d g+e f)}{15 d^2 e^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{(d+e x) \left (7 d^2 g^2-6 d e f g+2 e^2 f^2\right )}{15 d^3 e^3 \sqrt{d^2-e^2 x^2}} \]

[Out]

((e*f + d*g)^2*(d + e*x)^3)/(5*d*e^3*(d^2 - e^2*x^2)^(5/2)) + (2*(e*f - 4*d*g)*(
e*f + d*g)*(d + e*x)^2)/(15*d^2*e^3*(d^2 - e^2*x^2)^(3/2)) + ((2*e^2*f^2 - 6*d*e
*f*g + 7*d^2*g^2)*(d + e*x))/(15*d^3*e^3*Sqrt[d^2 - e^2*x^2])

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Rubi [A]  time = 0.435677, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097 \[ \frac{(d+e x)^3 (d g+e f)^2}{5 d e^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{2 (d+e x)^2 (e f-4 d g) (d g+e f)}{15 d^2 e^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{(d+e x) \left (7 d^2 g^2-6 d e f g+2 e^2 f^2\right )}{15 d^3 e^3 \sqrt{d^2-e^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

((e*f + d*g)^2*(d + e*x)^3)/(5*d*e^3*(d^2 - e^2*x^2)^(5/2)) + (2*(e*f - 4*d*g)*(
e*f + d*g)*(d + e*x)^2)/(15*d^2*e^3*(d^2 - e^2*x^2)^(3/2)) + ((2*e^2*f^2 - 6*d*e
*f*g + 7*d^2*g^2)*(d + e*x))/(15*d^3*e^3*Sqrt[d^2 - e^2*x^2])

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Rubi in Sympy [A]  time = 59.1491, size = 212, normalized size = 1.46 \[ \frac{g^{2} \sqrt{d^{2} - e^{2} x^{2}}}{d e^{3} \left (d - e x\right )} - \frac{2 g \sqrt{d^{2} - e^{2} x^{2}} \left (d g + e f\right )}{3 d e^{3} \left (d - e x\right )^{2}} + \frac{\sqrt{d^{2} - e^{2} x^{2}} \left (d g + e f\right )^{2}}{5 d e^{3} \left (d - e x\right )^{3}} - \frac{2 g \sqrt{d^{2} - e^{2} x^{2}} \left (d g + e f\right )}{3 d^{2} e^{3} \left (d - e x\right )} + \frac{2 \sqrt{d^{2} - e^{2} x^{2}} \left (d g + e f\right )^{2}}{15 d^{2} e^{3} \left (d - e x\right )^{2}} + \frac{2 \sqrt{d^{2} - e^{2} x^{2}} \left (d g + e f\right )^{2}}{15 d^{3} e^{3} \left (d - e x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.124361, size = 105, normalized size = 0.72 \[ \frac{\sqrt{d^2-e^2 x^2} \left (2 d^4 g^2-6 d^3 e g (f+g x)+d^2 e^2 \left (7 f^2+18 f g x+7 g^2 x^2\right )-6 d e^3 f x (f+g x)+2 e^4 f^2 x^2\right )}{15 d^3 e^3 (d-e x)^3} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[d^2 - e^2*x^2]*(2*d^4*g^2 + 2*e^4*f^2*x^2 - 6*d^3*e*g*(f + g*x) - 6*d*e^3*
f*x*(f + g*x) + d^2*e^2*(7*f^2 + 18*f*g*x + 7*g^2*x^2)))/(15*d^3*e^3*(d - e*x)^3
)

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Maple [A]  time = 0.012, size = 131, normalized size = 0.9 \[{\frac{ \left ( -ex+d \right ) \left ( ex+d \right ) ^{4} \left ( 7\,{d}^{2}{e}^{2}{g}^{2}{x}^{2}-6\,d{e}^{3}fg{x}^{2}+2\,{e}^{4}{f}^{2}{x}^{2}-6\,{d}^{3}e{g}^{2}x+18\,{d}^{2}{e}^{2}fgx-6\,d{e}^{3}{f}^{2}x+2\,{d}^{4}{g}^{2}-6\,{d}^{3}efg+7\,{d}^{2}{e}^{2}{f}^{2} \right ) }{15\,{d}^{3}{e}^{3}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*(-e*x+d)*(e*x+d)^4*(7*d^2*e^2*g^2*x^2-6*d*e^3*f*g*x^2+2*e^4*f^2*x^2-6*d^3*e
*g^2*x+18*d^2*e^2*f*g*x-6*d*e^3*f^2*x+2*d^4*g^2-6*d^3*e*f*g+7*d^2*e^2*f^2)/d^3/e
^3/(-e^2*x^2+d^2)^(7/2)

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Maxima [A]  time = 0.701287, size = 787, normalized size = 5.43 \[ \frac{e g^{2} x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} - \frac{4 \, d^{2} g^{2} x^{2}}{3 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e} + \frac{d f^{2} x}{5 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} + \frac{3 \, d^{2} f^{2}}{5 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e} + \frac{2 \, d^{3} f g}{5 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{2}} + \frac{8 \, d^{4} g^{2}}{15 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{3}} + \frac{4 \, f^{2} x}{15 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} d} + \frac{8 \, f^{2} x}{15 \, \sqrt{-e^{2} x^{2} + d^{2}} d^{3}} + \frac{{\left (2 \, e^{3} f g + 3 \, d e^{2} g^{2}\right )} x^{3}}{2 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{2}} + \frac{{\left (e^{3} f^{2} + 6 \, d e^{2} f g + 3 \, d^{2} e g^{2}\right )} x^{2}}{3 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{2}} - \frac{3 \,{\left (2 \, e^{3} f g + 3 \, d e^{2} g^{2}\right )} d^{2} x}{10 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{4}} + \frac{{\left (3 \, d e^{2} f^{2} + 6 \, d^{2} e f g + d^{3} g^{2}\right )} x}{5 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{2}} - \frac{2 \,{\left (e^{3} f^{2} + 6 \, d e^{2} f g + 3 \, d^{2} e g^{2}\right )} d^{2}}{15 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{4}} + \frac{{\left (2 \, e^{3} f g + 3 \, d e^{2} g^{2}\right )} x}{10 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} e^{4}} - \frac{{\left (3 \, d e^{2} f^{2} + 6 \, d^{2} e f g + d^{3} g^{2}\right )} x}{15 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} d^{2} e^{2}} + \frac{{\left (2 \, e^{3} f g + 3 \, d e^{2} g^{2}\right )} x}{5 \, \sqrt{-e^{2} x^{2} + d^{2}} d^{2} e^{4}} - \frac{2 \,{\left (3 \, d e^{2} f^{2} + 6 \, d^{2} e f g + d^{3} g^{2}\right )} x}{15 \, \sqrt{-e^{2} x^{2} + d^{2}} d^{4} e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.286283, size = 456, normalized size = 3.14 \[ -\frac{60 \, d^{4} f^{2} x - 3 \,{\left (3 \, e^{4} f^{2} - 4 \, d e^{3} f g + 3 \, d^{2} e^{2} g^{2}\right )} x^{5} + 5 \,{\left (7 \, d e^{3} f^{2} - 6 \, d^{2} e^{2} f g - d^{3} e g^{2}\right )} x^{4} - 10 \,{\left (2 \, d^{2} e^{2} f^{2} + 3 \, d^{3} e f g - 2 \, d^{4} g^{2}\right )} x^{3} - 60 \,{\left (d^{3} e f^{2} - d^{4} f g\right )} x^{2} - 5 \,{\left (12 \, d^{3} f^{2} x +{\left (e^{3} f^{2} - d^{2} e g^{2}\right )} x^{4} + 2 \,{\left (d e^{2} f^{2} - 3 \, d^{2} e f g + 2 \, d^{3} g^{2}\right )} x^{3} - 12 \,{\left (d^{2} e f^{2} - d^{3} f g\right )} x^{2}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{15 \,{\left (d^{3} e^{5} x^{5} - 5 \, d^{4} e^{4} x^{4} + 5 \, d^{5} e^{3} x^{3} + 5 \, d^{6} e^{2} x^{2} - 10 \, d^{7} e x + 4 \, d^{8} +{\left (d^{3} e^{4} x^{4} - 7 \, d^{5} e^{2} x^{2} + 10 \, d^{6} e x - 4 \, d^{7}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/15*(60*d^4*f^2*x - 3*(3*e^4*f^2 - 4*d*e^3*f*g + 3*d^2*e^2*g^2)*x^5 + 5*(7*d*e
^3*f^2 - 6*d^2*e^2*f*g - d^3*e*g^2)*x^4 - 10*(2*d^2*e^2*f^2 + 3*d^3*e*f*g - 2*d^
4*g^2)*x^3 - 60*(d^3*e*f^2 - d^4*f*g)*x^2 - 5*(12*d^3*f^2*x + (e^3*f^2 - d^2*e*g
^2)*x^4 + 2*(d*e^2*f^2 - 3*d^2*e*f*g + 2*d^3*g^2)*x^3 - 12*(d^2*e*f^2 - d^3*f*g)
*x^2)*sqrt(-e^2*x^2 + d^2))/(d^3*e^5*x^5 - 5*d^4*e^4*x^4 + 5*d^5*e^3*x^3 + 5*d^6
*e^2*x^2 - 10*d^7*e*x + 4*d^8 + (d^3*e^4*x^4 - 7*d^5*e^2*x^2 + 10*d^6*e*x - 4*d^
7)*sqrt(-e^2*x^2 + d^2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.28964, size = 267, normalized size = 1.84 \[ -\frac{\sqrt{-x^{2} e^{2} + d^{2}}{\left ({\left (15 \, d f^{2} +{\left ({\left ({\left (15 \, g^{2} e + \frac{{\left (7 \, d^{3} g^{2} e^{6} - 6 \, d^{2} f g e^{7} + 2 \, d f^{2} e^{8}\right )} x e^{\left (-4\right )}}{d^{4}}\right )} x + \frac{5 \,{\left (d^{5} g^{2} e^{4} + 6 \, d^{4} f g e^{5} - d^{3} f^{2} e^{6}\right )} e^{\left (-4\right )}}{d^{4}}\right )} x - \frac{5 \,{\left (d^{6} g^{2} e^{3} - 6 \, d^{5} f g e^{4} - d^{4} f^{2} e^{5}\right )} e^{\left (-4\right )}}{d^{4}}\right )} x\right )} x + \frac{{\left (2 \, d^{8} g^{2} e - 6 \, d^{7} f g e^{2} + 7 \, d^{6} f^{2} e^{3}\right )} e^{\left (-4\right )}}{d^{4}}\right )}}{15 \,{\left (x^{2} e^{2} - d^{2}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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