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

Optimal. Leaf size=35 \[ \frac{a x \log (x)}{c \sqrt{c x^2}}+\frac{b x^2}{c \sqrt{c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0179818, antiderivative size = 35, 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 \[ \frac{a x \log (x)}{c \sqrt{c x^2}}+\frac{b x^2}{c \sqrt{c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00607968, size = 21, normalized size = 0.6 \[ \frac{x^3 (a \log (x)+b x)}{\left (c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(x^3*(b*x + a*Log[x]))/(c*x^2)^(3/2)

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Maple [A]  time = 0.004, size = 20, normalized size = 0.6 \[{{x}^{3} \left ( bx+a\ln \left ( x \right ) \right ) \left ( c{x}^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.3457, size = 31, normalized size = 0.89 \[ \frac{b x^{2}}{\sqrt{c x^{2}} c} + \frac{a \log \left (x\right )}{c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.210445, size = 30, normalized size = 0.86 \[ \frac{\sqrt{c x^{2}}{\left (b x + a \log \left (x\right )\right )}}{c^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(c*x^2)*(b*x + a*log(x))/(c^2*x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.212585, size = 54, normalized size = 1.54 \[ -\frac{\frac{a{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2}} \right |}\right )}{\sqrt{c}} - \frac{\sqrt{c x^{2}} b}{c}}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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