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

Optimal. Leaf size=61 \[ \frac{a^2 x \log (x)}{c \sqrt{c x^2}}+\frac{2 a b x^2}{c \sqrt{c x^2}}+\frac{b^2 x^3}{2 c \sqrt{c x^2}} \]

[Out]

(2*a*b*x^2)/(c*Sqrt[c*x^2]) + (b^2*x^3)/(2*c*Sqrt[c*x^2]) + (a^2*x*Log[x])/(c*Sq
rt[c*x^2])

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Rubi [A]  time = 0.0318025, antiderivative size = 61, 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 \[ \frac{a^2 x \log (x)}{c \sqrt{c x^2}}+\frac{2 a b x^2}{c \sqrt{c x^2}}+\frac{b^2 x^3}{2 c \sqrt{c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*a*b*x^2)/(c*Sqrt[c*x^2]) + (b^2*x^3)/(2*c*Sqrt[c*x^2]) + (a^2*x*Log[x])/(c*Sq
rt[c*x^2])

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0138377, size = 34, normalized size = 0.56 \[ \frac{x^3 \left (2 a^2 \log (x)+b x (4 a+b x)\right )}{2 \left (c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.008, size = 33, normalized size = 0.5 \[{\frac{{x}^{3} \left ({b}^{2}{x}^{2}+2\,{a}^{2}\ln \left ( x \right ) +4\,abx \right ) }{2} \left ( c{x}^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.34873, size = 61, normalized size = 1. \[ \frac{b^{2} x^{3}}{2 \, \sqrt{c x^{2}} c} + \frac{2 \, a b x^{2}}{\sqrt{c x^{2}} c} + \frac{a^{2} \log \left (x\right )}{c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.213048, size = 47, normalized size = 0.77 \[ \frac{{\left (b^{2} x^{2} + 4 \, a b x + 2 \, a^{2} \log \left (x\right )\right )} \sqrt{c x^{2}}}{2 \, c^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2} \left (a + b x\right )^{2}}{\left (c x^{2}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x**2*(a + b*x)**2/(c*x**2)**(3/2), x)

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GIAC/XCAS [A]  time = 0.215587, size = 74, normalized size = 1.21 \[ -\frac{\frac{2 \, a^{2}{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2}} \right |}\right )}{\sqrt{c}} - \sqrt{c x^{2}}{\left (\frac{b^{2} x}{c} + \frac{4 \, a b}{c}\right )}}{2 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(2*a^2*ln(abs(-sqrt(c)*x + sqrt(c*x^2)))/sqrt(c) - sqrt(c*x^2)*(b^2*x/c + 4
*a*b/c))/c