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

Optimal. Leaf size=76 \[ \frac{\sqrt{d x-c} \sqrt{c+d x} \left (a d^2+2 b c^2\right )}{c^2 d^4}-\frac{x^2 \left (\frac{a}{c^2}+\frac{b}{d^2}\right )}{\sqrt{d x-c} \sqrt{c+d x}} \]

[Out]

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

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Rubi [A]  time = 0.181657, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ \frac{\sqrt{d x-c} \sqrt{c+d x} \left (a d^2+2 b c^2\right )}{c^2 d^4}-\frac{x^2 \left (\frac{a}{c^2}+\frac{b}{d^2}\right )}{\sqrt{d x-c} \sqrt{c+d x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 11.3348, size = 65, normalized size = 0.86 \[ - \frac{x^{2} \left (\frac{a}{c^{2}} + \frac{b}{d^{2}}\right )}{\sqrt{- c + d x} \sqrt{c + d x}} + \frac{\sqrt{- c + d x} \sqrt{c + d x} \left (a d^{2} + 2 b c^{2}\right )}{c^{2} d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.07601, size = 45, normalized size = 0.59 \[ \frac{-a d^2-2 b c^2+b d^2 x^2}{d^4 \sqrt{d x-c} \sqrt{c+d x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.005, size = 43, normalized size = 0.6 \[ -{\frac{-b{d}^{2}{x}^{2}+a{d}^{2}+2\,b{c}^{2}}{{d}^{4}}{\frac{1}{\sqrt{dx+c}}}{\frac{1}{\sqrt{dx-c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.38135, size = 93, normalized size = 1.22 \[ \frac{b x^{2}}{\sqrt{d^{2} x^{2} - c^{2}} d^{2}} - \frac{2 \, b c^{2}}{\sqrt{d^{2} x^{2} - c^{2}} d^{4}} - \frac{a}{\sqrt{d^{2} x^{2} - c^{2}} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b*x^2/(sqrt(d^2*x^2 - c^2)*d^2) - 2*b*c^2/(sqrt(d^2*x^2 - c^2)*d^4) - a/(sqrt(d^
2*x^2 - c^2)*d^2)

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Fricas [A]  time = 0.247176, size = 196, normalized size = 2.58 \[ -\frac{2 \, b d^{4} x^{4} + 2 \, b c^{4} + a c^{2} d^{2} -{\left (5 \, b c^{2} d^{2} + 2 \, a d^{4}\right )} x^{2} - 2 \,{\left (b d^{3} x^{3} -{\left (2 \, b c^{2} d + a d^{3}\right )} x\right )} \sqrt{d x + c} \sqrt{d x - c}}{2 \, d^{7} x^{3} - 2 \, c^{2} d^{5} x -{\left (2 \, d^{6} x^{2} - c^{2} d^{4}\right )} \sqrt{d x + c} \sqrt{d x - c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(2*b*d^4*x^4 + 2*b*c^4 + a*c^2*d^2 - (5*b*c^2*d^2 + 2*a*d^4)*x^2 - 2*(b*d^3*x^3
- (2*b*c^2*d + a*d^3)*x)*sqrt(d*x + c)*sqrt(d*x - c))/(2*d^7*x^3 - 2*c^2*d^5*x
- (2*d^6*x^2 - c^2*d^4)*sqrt(d*x + c)*sqrt(d*x - c))

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Sympy [A]  time = 71.8296, size = 201, normalized size = 2.64 \[ a \left (- \frac{{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{1}{4}, \frac{3}{4}, 1 & 0, 1, \frac{3}{2} \\\frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1, \frac{3}{2} & 0 \end{matrix} \middle |{\frac{c^{2}}{d^{2} x^{2}}} \right )}}{2 \pi ^{\frac{3}{2}} c d^{2}} - \frac{i{G_{6, 6}^{2, 6}\left (\begin{matrix} -1, - \frac{1}{2}, - \frac{1}{4}, 0, \frac{1}{4}, 1 & \\- \frac{1}{4}, \frac{1}{4} & -1, - \frac{1}{2}, \frac{1}{2}, 0 \end{matrix} \middle |{\frac{c^{2} e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{2 \pi ^{\frac{3}{2}} c d^{2}}\right ) + b \left (\frac{c{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{3}{4}, - \frac{1}{4} & -1, 0, \frac{1}{2}, 1 \\- \frac{3}{4}, - \frac{1}{2}, - \frac{1}{4}, 0, \frac{1}{2}, 0 & \end{matrix} \middle |{\frac{c^{2}}{d^{2} x^{2}}} \right )}}{2 \pi ^{\frac{3}{2}} d^{4}} - \frac{i c{G_{6, 6}^{2, 6}\left (\begin{matrix} -2, - \frac{3}{2}, - \frac{5}{4}, -1, - \frac{3}{4}, 1 & \\- \frac{5}{4}, - \frac{3}{4} & -2, - \frac{3}{2}, - \frac{1}{2}, 0 \end{matrix} \middle |{\frac{c^{2} e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{2 \pi ^{\frac{3}{2}} d^{4}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.22925, size = 127, normalized size = 1.67 \[ \frac{{\left (2 \,{\left (d x + c\right )} b d^{8} - \frac{5 \, b c^{2} d^{8} + a d^{10}}{c}\right )} \sqrt{d x + c}}{32 \, \sqrt{d x - c}} + \frac{2 \,{\left (b c^{2} + a d^{2}\right )}}{{\left ({\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{2} + 2 \, c\right )} d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/32*(2*(d*x + c)*b*d^8 - (5*b*c^2*d^8 + a*d^10)/c)*sqrt(d*x + c)/sqrt(d*x - c)
+ 2*(b*c^2 + a*d^2)/(((sqrt(d*x + c) - sqrt(d*x - c))^2 + 2*c)*d^4)