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

Optimal. Leaf size=56 \[ -\frac{a^2}{c \sqrt{c x^2}}+\frac{2 a b x \log (x)}{c \sqrt{c x^2}}+\frac{b^2 x^2}{c \sqrt{c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0344171, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ -\frac{a^2}{c \sqrt{c x^2}}+\frac{2 a b x \log (x)}{c \sqrt{c x^2}}+\frac{b^2 x^2}{c \sqrt{c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0122496, size = 33, normalized size = 0.59 \[ \frac{x^2 \left (-a^2+2 a b x \log (x)+b^2 x^2\right )}{\left (c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.007, size = 32, normalized size = 0.6 \[{{x}^{2} \left ( 2\,ab\ln \left ( x \right ) x+{b}^{2}{x}^{2}-{a}^{2} \right ) \left ( c{x}^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.35351, size = 57, normalized size = 1.02 \[ \frac{b^{2} x^{2}}{\sqrt{c x^{2}} c} + \frac{2 \, a b \log \left (x\right )}{c^{\frac{3}{2}}} - \frac{a^{2}}{\sqrt{c x^{2}} c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.219729, size = 46, normalized size = 0.82 \[ \frac{{\left (b^{2} x^{2} + 2 \, a b x \log \left (x\right ) - a^{2}\right )} \sqrt{c x^{2}}}{c^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.222928, size = 93, normalized size = 1.66 \[ \frac{\frac{\sqrt{c x^{2}} b^{2}}{c} - \frac{2 \,{\left (a b{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2}} \right |}\right ) - \frac{a^{2} \sqrt{c}}{\sqrt{c} x - \sqrt{c x^{2}}}\right )}}{\sqrt{c}}}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(sqrt(c*x^2)*b^2/c - 2*(a*b*ln(abs(-sqrt(c)*x + sqrt(c*x^2))) - a^2*sqrt(c)/(sqr
t(c)*x - sqrt(c*x^2)))/sqrt(c))/c