3.270 \(\int \frac{x^3 \left (a+b x^2\right )}{(-c+d x)^{3/2} (c+d x)^{3/2}} \, dx\)

Optimal. Leaf size=115 \[ \frac{2 \sqrt{d x-c} \sqrt{c+d x} \left (3 a d^2+4 b c^2\right )}{3 d^6}-\frac{x^2 \left (3 a d^2+4 b c^2\right )}{3 d^4 \sqrt{d x-c} \sqrt{c+d x}}+\frac{b x^4}{3 d^2 \sqrt{d x-c} \sqrt{c+d x}} \]

[Out]

-((4*b*c^2 + 3*a*d^2)*x^2)/(3*d^4*Sqrt[-c + d*x]*Sqrt[c + d*x]) + (b*x^4)/(3*d^2
*Sqrt[-c + d*x]*Sqrt[c + d*x]) + (2*(4*b*c^2 + 3*a*d^2)*Sqrt[-c + d*x]*Sqrt[c +
d*x])/(3*d^6)

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Rubi [A]  time = 0.316393, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129 \[ \frac{2 \sqrt{d x-c} \sqrt{c+d x} \left (3 a d^2+4 b c^2\right )}{3 d^6}-\frac{x^2 \left (3 a d^2+4 b c^2\right )}{3 d^4 \sqrt{d x-c} \sqrt{c+d x}}+\frac{b x^4}{3 d^2 \sqrt{d x-c} \sqrt{c+d x}} \]

Antiderivative was successfully verified.

[In]  Int[(x^3*(a + b*x^2))/((-c + d*x)^(3/2)*(c + d*x)^(3/2)),x]

[Out]

-((4*b*c^2 + 3*a*d^2)*x^2)/(3*d^4*Sqrt[-c + d*x]*Sqrt[c + d*x]) + (b*x^4)/(3*d^2
*Sqrt[-c + d*x]*Sqrt[c + d*x]) + (2*(4*b*c^2 + 3*a*d^2)*Sqrt[-c + d*x]*Sqrt[c +
d*x])/(3*d^6)

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Rubi in Sympy [A]  time = 18.6131, size = 102, normalized size = 0.89 \[ \frac{b x^{4}}{3 d^{2} \sqrt{- c + d x} \sqrt{c + d x}} - \frac{x^{2} \left (3 a d^{2} + 4 b c^{2}\right )}{3 d^{4} \sqrt{- c + d x} \sqrt{c + d x}} + \frac{2 \sqrt{- c + d x} \sqrt{c + d x} \left (3 a d^{2} + 4 b c^{2}\right )}{3 d^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(b*x**2+a)/(d*x-c)**(3/2)/(d*x+c)**(3/2),x)

[Out]

b*x**4/(3*d**2*sqrt(-c + d*x)*sqrt(c + d*x)) - x**2*(3*a*d**2 + 4*b*c**2)/(3*d**
4*sqrt(-c + d*x)*sqrt(c + d*x)) + 2*sqrt(-c + d*x)*sqrt(c + d*x)*(3*a*d**2 + 4*b
*c**2)/(3*d**6)

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Mathematica [A]  time = 0.0896928, size = 72, normalized size = 0.63 \[ \frac{-6 a c^2 d^2+3 a d^4 x^2-8 b c^4+4 b c^2 d^2 x^2+b d^4 x^4}{3 d^6 \sqrt{d x-c} \sqrt{c+d x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^3*(a + b*x^2))/((-c + d*x)^(3/2)*(c + d*x)^(3/2)),x]

[Out]

(-8*b*c^4 - 6*a*c^2*d^2 + 4*b*c^2*d^2*x^2 + 3*a*d^4*x^2 + b*d^4*x^4)/(3*d^6*Sqrt
[-c + d*x]*Sqrt[c + d*x])

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Maple [A]  time = 0.008, size = 68, normalized size = 0.6 \[ -{\frac{-b{d}^{4}{x}^{4}-3\,a{d}^{4}{x}^{2}-4\,b{c}^{2}{d}^{2}{x}^{2}+6\,a{c}^{2}{d}^{2}+8\,b{c}^{4}}{3\,{d}^{6}}{\frac{1}{\sqrt{dx+c}}}{\frac{1}{\sqrt{dx-c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(b*x^2+a)/(d*x-c)^(3/2)/(d*x+c)^(3/2),x)

[Out]

-1/3*(-b*d^4*x^4-3*a*d^4*x^2-4*b*c^2*d^2*x^2+6*a*c^2*d^2+8*b*c^4)/(d*x+c)^(1/2)/
d^6/(d*x-c)^(1/2)

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Maxima [A]  time = 1.40378, size = 166, normalized size = 1.44 \[ \frac{b x^{4}}{3 \, \sqrt{d^{2} x^{2} - c^{2}} d^{2}} + \frac{4 \, b c^{2} x^{2}}{3 \, \sqrt{d^{2} x^{2} - c^{2}} d^{4}} + \frac{a x^{2}}{\sqrt{d^{2} x^{2} - c^{2}} d^{2}} - \frac{8 \, b c^{4}}{3 \, \sqrt{d^{2} x^{2} - c^{2}} d^{6}} - \frac{2 \, a c^{2}}{\sqrt{d^{2} x^{2} - c^{2}} d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*b*x^4/(sqrt(d^2*x^2 - c^2)*d^2) + 4/3*b*c^2*x^2/(sqrt(d^2*x^2 - c^2)*d^4) +
a*x^2/(sqrt(d^2*x^2 - c^2)*d^2) - 8/3*b*c^4/(sqrt(d^2*x^2 - c^2)*d^6) - 2*a*c^2/
(sqrt(d^2*x^2 - c^2)*d^4)

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Fricas [A]  time = 0.254866, size = 354, normalized size = 3.08 \[ -\frac{8 \, b d^{8} x^{8} - 8 \, b c^{8} - 6 \, a c^{6} d^{2} + 24 \,{\left (b c^{2} d^{6} + a d^{8}\right )} x^{6} -{\left (95 \, b c^{4} d^{4} + 72 \, a c^{2} d^{6}\right )} x^{4} + 17 \,{\left (4 \, b c^{6} d^{2} + 3 \, a c^{4} d^{4}\right )} x^{2} - 4 \,{\left (2 \, b d^{7} x^{7} +{\left (7 \, b c^{2} d^{5} + 6 \, a d^{7}\right )} x^{5} - 5 \,{\left (4 \, b c^{4} d^{3} + 3 \, a c^{2} d^{5}\right )} x^{3} + 2 \,{\left (4 \, b c^{6} d + 3 \, a c^{4} d^{3}\right )} x\right )} \sqrt{d x + c} \sqrt{d x - c}}{3 \,{\left (8 \, d^{11} x^{5} - 12 \, c^{2} d^{9} x^{3} + 4 \, c^{4} d^{7} x -{\left (8 \, d^{10} x^{4} - 8 \, c^{2} d^{8} x^{2} + c^{4} d^{6}\right )} \sqrt{d x + c} \sqrt{d x - c}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(8*b*d^8*x^8 - 8*b*c^8 - 6*a*c^6*d^2 + 24*(b*c^2*d^6 + a*d^8)*x^6 - (95*b*c
^4*d^4 + 72*a*c^2*d^6)*x^4 + 17*(4*b*c^6*d^2 + 3*a*c^4*d^4)*x^2 - 4*(2*b*d^7*x^7
+ (7*b*c^2*d^5 + 6*a*d^7)*x^5 - 5*(4*b*c^4*d^3 + 3*a*c^2*d^5)*x^3 + 2*(4*b*c^6*
d + 3*a*c^4*d^3)*x)*sqrt(d*x + c)*sqrt(d*x - c))/(8*d^11*x^5 - 12*c^2*d^9*x^3 +
4*c^4*d^7*x - (8*d^10*x^4 - 8*c^2*d^8*x^2 + c^4*d^6)*sqrt(d*x + c)*sqrt(d*x - c)
)

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Sympy [A]  time = 82.2935, size = 226, normalized size = 1.97 \[ a \left (\frac{c{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{3}{4}, - \frac{1}{4} & -1, 0, \frac{1}{2}, 1 \\- \frac{3}{4}, - \frac{1}{2}, - \frac{1}{4}, 0, \frac{1}{2}, 0 & \end{matrix} \middle |{\frac{c^{2}}{d^{2} x^{2}}} \right )}}{2 \pi ^{\frac{3}{2}} d^{4}} - \frac{i c{G_{6, 6}^{2, 6}\left (\begin{matrix} -2, - \frac{3}{2}, - \frac{5}{4}, -1, - \frac{3}{4}, 1 & \\- \frac{5}{4}, - \frac{3}{4} & -2, - \frac{3}{2}, - \frac{1}{2}, 0 \end{matrix} \middle |{\frac{c^{2} e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{2 \pi ^{\frac{3}{2}} d^{4}}\right ) + b \left (\frac{c^{3}{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{7}{4}, - \frac{5}{4} & -2, -1, - \frac{1}{2}, 1 \\- \frac{7}{4}, - \frac{3}{2}, - \frac{5}{4}, -1, - \frac{1}{2}, 0 & \end{matrix} \middle |{\frac{c^{2}}{d^{2} x^{2}}} \right )}}{2 \pi ^{\frac{3}{2}} d^{6}} - \frac{i c^{3}{G_{6, 6}^{2, 6}\left (\begin{matrix} -3, - \frac{5}{2}, - \frac{9}{4}, -2, - \frac{7}{4}, 1 & \\- \frac{9}{4}, - \frac{7}{4} & -3, - \frac{5}{2}, - \frac{3}{2}, 0 \end{matrix} \middle |{\frac{c^{2} e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{2 \pi ^{\frac{3}{2}} d^{6}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(b*x**2+a)/(d*x-c)**(3/2)/(d*x+c)**(3/2),x)

[Out]

a*(c*meijerg(((-3/4, -1/4), (-1, 0, 1/2, 1)), ((-3/4, -1/2, -1/4, 0, 1/2, 0), ()
), c**2/(d**2*x**2))/(2*pi**(3/2)*d**4) - I*c*meijerg(((-2, -3/2, -5/4, -1, -3/4
, 1), ()), ((-5/4, -3/4), (-2, -3/2, -1/2, 0)), c**2*exp_polar(2*I*pi)/(d**2*x**
2))/(2*pi**(3/2)*d**4)) + b*(c**3*meijerg(((-7/4, -5/4), (-2, -1, -1/2, 1)), ((-
7/4, -3/2, -5/4, -1, -1/2, 0), ()), c**2/(d**2*x**2))/(2*pi**(3/2)*d**6) - I*c**
3*meijerg(((-3, -5/2, -9/4, -2, -7/4, 1), ()), ((-9/4, -7/4), (-3, -5/2, -3/2, 0
)), c**2*exp_polar(2*I*pi)/(d**2*x**2))/(2*pi**(3/2)*d**6))

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GIAC/XCAS [A]  time = 0.236065, size = 194, normalized size = 1.69 \[ -\frac{{\left (2 \,{\left ({\left (4 \, b d^{24} - \frac{{\left (d x + c\right )} b d^{24}}{c}\right )}{\left (d x + c\right )} - \frac{10 \, b c^{2} d^{24} + 3 \, a d^{26}}{c}\right )}{\left (d x + c\right )} + \frac{3 \,{\left (9 \, b c^{3} d^{24} + 5 \, a c d^{26}\right )}}{c}\right )} \sqrt{d x + c}}{23040 \, \sqrt{d x - c}} + \frac{2 \,{\left (b c^{4} + a c^{2} d^{2}\right )}}{{\left ({\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{2} + 2 \, c\right )} d^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)*x^3/((d*x + c)^(3/2)*(d*x - c)^(3/2)),x, algorithm="giac")

[Out]

-1/23040*(2*((4*b*d^24 - (d*x + c)*b*d^24/c)*(d*x + c) - (10*b*c^2*d^24 + 3*a*d^
26)/c)*(d*x + c) + 3*(9*b*c^3*d^24 + 5*a*c*d^26)/c)*sqrt(d*x + c)/sqrt(d*x - c)
+ 2*(b*c^4 + a*c^2*d^2)/(((sqrt(d*x + c) - sqrt(d*x - c))^2 + 2*c)*d^6)