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

Optimal. Leaf size=73 \[ -\frac{2 b \sqrt{c+d x^2} (b c-a d)}{d^3}-\frac{(b c-a d)^2}{d^3 \sqrt{c+d x^2}}+\frac{b^2 \left (c+d x^2\right )^{3/2}}{3 d^3} \]

[Out]

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

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Rubi [A]  time = 0.158262, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{2 b \sqrt{c+d x^2} (b c-a d)}{d^3}-\frac{(b c-a d)^2}{d^3 \sqrt{c+d x^2}}+\frac{b^2 \left (c+d x^2\right )^{3/2}}{3 d^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 23.3565, size = 63, normalized size = 0.86 \[ \frac{b^{2} \left (c + d x^{2}\right )^{\frac{3}{2}}}{3 d^{3}} + \frac{2 b \sqrt{c + d x^{2}} \left (a d - b c\right )}{d^{3}} - \frac{\left (a d - b c\right )^{2}}{d^{3} \sqrt{c + d x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0747515, size = 65, normalized size = 0.89 \[ \frac{-3 a^2 d^2+6 a b d \left (2 c+d x^2\right )+b^2 \left (-8 c^2-4 c d x^2+d^2 x^4\right )}{3 d^3 \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.008, size = 69, normalized size = 1. \[ -{\frac{-{b}^{2}{d}^{2}{x}^{4}-6\,ab{d}^{2}{x}^{2}+4\,{b}^{2}cd{x}^{2}+3\,{a}^{2}{d}^{2}-12\,cabd+8\,{b}^{2}{c}^{2}}{3\,{d}^{3}}{\frac{1}{\sqrt{d{x}^{2}+c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(-b^2*d^2*x^4-6*a*b*d^2*x^2+4*b^2*c*d*x^2+3*a^2*d^2-12*a*b*c*d+8*b^2*c^2)/(
d*x^2+c)^(1/2)/d^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.228579, size = 107, normalized size = 1.47 \[ \frac{{\left (b^{2} d^{2} x^{4} - 8 \, b^{2} c^{2} + 12 \, a b c d - 3 \, a^{2} d^{2} - 2 \,{\left (2 \, b^{2} c d - 3 \, a b d^{2}\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{3 \,{\left (d^{4} x^{2} + c d^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(b^2*d^2*x^4 - 8*b^2*c^2 + 12*a*b*c*d - 3*a^2*d^2 - 2*(2*b^2*c*d - 3*a*b*d^2
)*x^2)*sqrt(d*x^2 + c)/(d^4*x^2 + c*d^3)

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Sympy [A]  time = 3.54453, size = 155, normalized size = 2.12 \[ \begin{cases} - \frac{a^{2}}{d \sqrt{c + d x^{2}}} + \frac{4 a b c}{d^{2} \sqrt{c + d x^{2}}} + \frac{2 a b x^{2}}{d \sqrt{c + d x^{2}}} - \frac{8 b^{2} c^{2}}{3 d^{3} \sqrt{c + d x^{2}}} - \frac{4 b^{2} c x^{2}}{3 d^{2} \sqrt{c + d x^{2}}} + \frac{b^{2} x^{4}}{3 d \sqrt{c + d x^{2}}} & \text{for}\: d \neq 0 \\\frac{\frac{a^{2} x^{2}}{2} + \frac{a b x^{4}}{2} + \frac{b^{2} x^{6}}{6}}{c^{\frac{3}{2}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-a**2/(d*sqrt(c + d*x**2)) + 4*a*b*c/(d**2*sqrt(c + d*x**2)) + 2*a*b*
x**2/(d*sqrt(c + d*x**2)) - 8*b**2*c**2/(3*d**3*sqrt(c + d*x**2)) - 4*b**2*c*x**
2/(3*d**2*sqrt(c + d*x**2)) + b**2*x**4/(3*d*sqrt(c + d*x**2)), Ne(d, 0)), ((a**
2*x**2/2 + a*b*x**4/2 + b**2*x**6/6)/c**(3/2), True))

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GIAC/XCAS [A]  time = 0.235683, size = 108, normalized size = 1.48 \[ \frac{{\left (d x^{2} + c\right )}^{\frac{3}{2}} b^{2} - 6 \, \sqrt{d x^{2} + c} b^{2} c + 6 \, \sqrt{d x^{2} + c} a b d - \frac{3 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}}{\sqrt{d x^{2} + c}}}{3 \, d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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