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

Optimal. Leaf size=117 \[ \frac{(d+e x)^3 (d g+e f)}{5 d e^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{2 (d+e x) (2 e f-3 d g)}{15 d e^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{x (2 e f-3 d g)}{15 d^3 e \sqrt{d^2-e^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.18492, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103 \[ \frac{(d+e x)^3 (d g+e f)}{5 d e^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{2 (d+e x) (2 e f-3 d g)}{15 d e^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{x (2 e f-3 d g)}{15 d^3 e \sqrt{d^2-e^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 15.6559, size = 102, normalized size = 0.87 \[ \frac{\left (d + e x\right )^{3} \left (d g + e f\right )}{5 d e^{2} \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}} - \frac{2 \left (d + e x\right ) \left (3 d g - 2 e f\right )}{15 d e^{2} \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}} - \frac{x \left (3 d g - 2 e f\right )}{15 d^{3} e \sqrt{d^{2} - e^{2} x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0851478, size = 77, normalized size = 0.66 \[ \frac{\sqrt{d^2-e^2 x^2} \left (-3 d^3 g+d^2 e (7 f+9 g x)-3 d e^2 x (2 f+g x)+2 e^3 f x^2\right )}{15 d^3 e^2 (d-e x)^3} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[d^2 - e^2*x^2]*(-3*d^3*g + 2*e^3*f*x^2 - 3*d*e^2*x*(2*f + g*x) + d^2*e*(7*
f + 9*g*x)))/(15*d^3*e^2*(d - e*x)^3)

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Maple [A]  time = 0.011, size = 85, normalized size = 0.7 \[ -{\frac{ \left ( -ex+d \right ) \left ( ex+d \right ) ^{4} \left ( 3\,d{e}^{2}g{x}^{2}-2\,{e}^{3}f{x}^{2}-9\,{d}^{2}egx+6\,d{e}^{2}fx+3\,{d}^{3}g-7\,{d}^{2}ef \right ) }{15\,{d}^{3}{e}^{2}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/15*(-e*x+d)*(e*x+d)^4*(3*d*e^2*g*x^2-2*e^3*f*x^2-9*d^2*e*g*x+6*d*e^2*f*x+3*d^
3*g-7*d^2*e*f)/d^3/e^2/(-e^2*x^2+d^2)^(7/2)

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Maxima [A]  time = 0.69434, size = 504, normalized size = 4.31 \[ \frac{e g x^{3}}{2 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} + \frac{d f x}{5 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} - \frac{3 \, d^{2} g x}{10 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e} + \frac{3 \, d^{2} f}{5 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e} + \frac{d^{3} g}{5 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{2}} + \frac{4 \, f x}{15 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} d} + \frac{g x}{10 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} e} + \frac{8 \, f x}{15 \, \sqrt{-e^{2} x^{2} + d^{2}} d^{3}} + \frac{g x}{5 \, \sqrt{-e^{2} x^{2} + d^{2}} d^{2} e} + \frac{{\left (e^{3} f + 3 \, d e^{2} g\right )} x^{2}}{3 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{2}} + \frac{3 \,{\left (d e^{2} f + d^{2} e g\right )} x}{5 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{2}} - \frac{2 \,{\left (e^{3} f + 3 \, d e^{2} g\right )} d^{2}}{15 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{4}} - \frac{{\left (d e^{2} f + d^{2} e g\right )} x}{5 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} d^{2} e^{2}} - \frac{2 \,{\left (d e^{2} f + d^{2} e g\right )} x}{5 \, \sqrt{-e^{2} x^{2} + d^{2}} d^{4} e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.283317, size = 363, normalized size = 3.1 \[ -\frac{60 \, d^{4} f x - 3 \,{\left (3 \, e^{4} f - 2 \, d e^{3} g\right )} x^{5} + 5 \,{\left (7 \, d e^{3} f - 3 \, d^{2} e^{2} g\right )} x^{4} - 5 \,{\left (4 \, d^{2} e^{2} f + 3 \, d^{3} e g\right )} x^{3} - 30 \,{\left (2 \, d^{3} e f - d^{4} g\right )} x^{2} - 5 \,{\left (e^{3} f x^{4} + 12 \, d^{3} f x +{\left (2 \, d e^{2} f - 3 \, d^{2} e g\right )} x^{3} - 6 \,{\left (2 \, d^{2} e f - d^{3} g\right )} x^{2}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{15 \,{\left (d^{3} e^{5} x^{5} - 5 \, d^{4} e^{4} x^{4} + 5 \, d^{5} e^{3} x^{3} + 5 \, d^{6} e^{2} x^{2} - 10 \, d^{7} e x + 4 \, d^{8} +{\left (d^{3} e^{4} x^{4} - 7 \, d^{5} e^{2} x^{2} + 10 \, d^{6} e x - 4 \, d^{7}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/15*(60*d^4*f*x - 3*(3*e^4*f - 2*d*e^3*g)*x^5 + 5*(7*d*e^3*f - 3*d^2*e^2*g)*x^
4 - 5*(4*d^2*e^2*f + 3*d^3*e*g)*x^3 - 30*(2*d^3*e*f - d^4*g)*x^2 - 5*(e^3*f*x^4
+ 12*d^3*f*x + (2*d*e^2*f - 3*d^2*e*g)*x^3 - 6*(2*d^2*e*f - d^3*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. \[ \int \frac{\left (d + e x\right )^{3} \left (f + g x\right )}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{7}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.293895, size = 188, normalized size = 1.61 \[ -\frac{\sqrt{-x^{2} e^{2} + d^{2}}{\left ({\left (15 \, d f -{\left (x{\left (\frac{{\left (3 \, d^{2} g e^{7} - 2 \, d f e^{8}\right )} x^{2} e^{\left (-4\right )}}{d^{4}} - \frac{5 \,{\left (3 \, d^{4} g e^{5} - d^{3} f e^{6}\right )} e^{\left (-4\right )}}{d^{4}}\right )} - \frac{5 \,{\left (3 \, d^{5} g e^{4} + d^{4} f e^{5}\right )} e^{\left (-4\right )}}{d^{4}}\right )} x\right )} x - \frac{{\left (3 \, d^{7} g e^{2} - 7 \, d^{6} f e^{3}\right )} e^{\left (-4\right )}}{d^{4}}\right )}}{15 \,{\left (x^{2} e^{2} - d^{2}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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