3.666 \(\int \frac{\left (a+b x^2\right )^2}{x^2 \left (c+d x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=90 \[ -\frac{a^2}{c x \left (c+d x^2\right )^{3/2}}+\frac{x \left (2 a (b c-2 a d)+b^2 c x^2\right )}{3 c^2 \left (c+d x^2\right )^{3/2}}+\frac{4 a x (b c-2 a d)}{3 c^3 \sqrt{c+d x^2}} \]

[Out]

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

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Rubi [A]  time = 0.146317, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{a^2}{c x \left (c+d x^2\right )^{3/2}}+\frac{x \left (2 a (b c-2 a d)+b^2 c x^2\right )}{3 c^2 \left (c+d x^2\right )^{3/2}}+\frac{4 a x (b c-2 a d)}{3 c^3 \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^2/(x^2*(c + d*x^2)^(5/2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 19.5037, size = 83, normalized size = 0.92 \[ - \frac{a^{2}}{c x \left (c + d x^{2}\right )^{\frac{3}{2}}} - \frac{4 a x \left (2 a d - b c\right )}{3 c^{3} \sqrt{c + d x^{2}}} - \frac{x \left (2 a \left (2 a d - b c\right ) - b^{2} c x^{2}\right )}{3 c^{2} \left (c + d x^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**2/x**2/(d*x**2+c)**(5/2),x)

[Out]

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

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Mathematica [A]  time = 0.0850988, size = 76, normalized size = 0.84 \[ \frac{-a^2 \left (3 c^2+12 c d x^2+8 d^2 x^4\right )+2 a b c x^2 \left (3 c+2 d x^2\right )+b^2 c^2 x^4}{3 c^3 x \left (c+d x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^2/(x^2*(c + d*x^2)^(5/2)),x]

[Out]

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

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Maple [A]  time = 0.009, size = 78, normalized size = 0.9 \[ -{\frac{8\,{x}^{4}{a}^{2}{d}^{2}-4\,{x}^{4}abcd-{x}^{4}{b}^{2}{c}^{2}+12\,{x}^{2}{a}^{2}cd-6\,a{c}^{2}b{x}^{2}+3\,{a}^{2}{c}^{2}}{3\,x{c}^{3}} \left ( d{x}^{2}+c \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^2/x^2/(d*x^2+c)^(5/2),x)

[Out]

-1/3*(8*a^2*d^2*x^4-4*a*b*c*d*x^4-b^2*c^2*x^4+12*a^2*c*d*x^2-6*a*b*c^2*x^2+3*a^2
*c^2)/(d*x^2+c)^(3/2)/x/c^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/((d*x^2 + c)^(5/2)*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.230325, size = 124, normalized size = 1.38 \[ \frac{{\left ({\left (b^{2} c^{2} + 4 \, a b c d - 8 \, a^{2} d^{2}\right )} x^{4} - 3 \, a^{2} c^{2} + 6 \,{\left (a b c^{2} - 2 \, a^{2} c d\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{3 \,{\left (c^{3} d^{2} x^{5} + 2 \, c^{4} d x^{3} + c^{5} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2/((d*x^2 + c)^(5/2)*x^2),x, algorithm="fricas")

[Out]

1/3*((b^2*c^2 + 4*a*b*c*d - 8*a^2*d^2)*x^4 - 3*a^2*c^2 + 6*(a*b*c^2 - 2*a^2*c*d)
*x^2)*sqrt(d*x^2 + c)/(c^3*d^2*x^5 + 2*c^4*d*x^3 + c^5*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x^{2}\right )^{2}}{x^{2} \left (c + d x^{2}\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**2/x**2/(d*x**2+c)**(5/2),x)

[Out]

Integral((a + b*x**2)**2/(x**2*(c + d*x**2)**(5/2)), x)

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GIAC/XCAS [A]  time = 0.239057, size = 158, normalized size = 1.76 \[ \frac{x{\left (\frac{{\left (b^{2} c^{4} d + 4 \, a b c^{3} d^{2} - 5 \, a^{2} c^{2} d^{3}\right )} x^{2}}{c^{5} d} + \frac{6 \,{\left (a b c^{4} d - a^{2} c^{3} d^{2}\right )}}{c^{5} d}\right )}}{3 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}}} + \frac{2 \, a^{2} \sqrt{d}}{{\left ({\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} - c\right )} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2/((d*x^2 + c)^(5/2)*x^2),x, algorithm="giac")

[Out]

1/3*x*((b^2*c^4*d + 4*a*b*c^3*d^2 - 5*a^2*c^2*d^3)*x^2/(c^5*d) + 6*(a*b*c^4*d -
a^2*c^3*d^2)/(c^5*d))/(d*x^2 + c)^(3/2) + 2*a^2*sqrt(d)/(((sqrt(d)*x - sqrt(d*x^
2 + c))^2 - c)*c^2)