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

Optimal. Leaf size=58 \[ -\frac{a^2}{2 c^2 x \sqrt{c x^2}}-\frac{2 a b}{c^2 \sqrt{c x^2}}+\frac{b^2 x \log (x)}{c^2 \sqrt{c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0345514, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ -\frac{a^2}{2 c^2 x \sqrt{c x^2}}-\frac{2 a b}{c^2 \sqrt{c x^2}}+\frac{b^2 x \log (x)}{c^2 \sqrt{c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 13.5369, size = 60, normalized size = 1.03 \[ - \frac{a^{2} \sqrt{c x^{2}}}{2 c^{3} x^{3}} - \frac{2 a b \sqrt{c x^{2}}}{c^{3} x^{2}} + \frac{b^{2} \sqrt{c x^{2}} \log{\left (x \right )}}{c^{3} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0167111, size = 36, normalized size = 0.62 \[ \frac{x^3 \left (2 b^2 x^2 \log (x)-a (a+4 b x)\right )}{2 \left (c x^2\right )^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(x^3*(-(a*(a + 4*b*x)) + 2*b^2*x^2*Log[x]))/(2*(c*x^2)^(5/2))

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Maple [A]  time = 0.006, size = 34, normalized size = 0.6 \[{\frac{{x}^{3} \left ( 2\,{b}^{2}\ln \left ( x \right ){x}^{2}-4\,abx-{a}^{2} \right ) }{2} \left ( c{x}^{2} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*x^3*(2*b^2*ln(x)*x^2-4*a*b*x-a^2)/(c*x^2)^(5/2)

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Maxima [A]  time = 1.35256, size = 51, normalized size = 0.88 \[ -\frac{2 \, a b x^{2}}{\left (c x^{2}\right )^{\frac{3}{2}} c} + \frac{b^{2} \log \left (x\right )}{c^{\frac{5}{2}}} - \frac{a^{2}}{2 \, c^{\frac{5}{2}} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a*b*x^2/((c*x^2)^(3/2)*c) + b^2*log(x)/c^(5/2) - 1/2*a^2/(c^(5/2)*x^2)

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Fricas [A]  time = 0.213256, size = 49, normalized size = 0.84 \[ \frac{{\left (2 \, b^{2} x^{2} \log \left (x\right ) - 4 \, a b x - a^{2}\right )} \sqrt{c x^{2}}}{2 \, c^{3} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(2*b^2*x^2*log(x) - 4*a*b*x - a^2)*sqrt(c*x^2)/(c^3*x^3)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.542564, size = 4, normalized size = 0.07 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x