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

Optimal. Leaf size=23 \[ \frac{x \log (a+b x)}{b c \sqrt{c x^2}} \]

[Out]

(x*Log[a + b*x])/(b*c*Sqrt[c*x^2])

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Rubi [A]  time = 0.0129196, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{x \log (a+b x)}{b c \sqrt{c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(x*Log[a + b*x])/(b*c*Sqrt[c*x^2])

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Rubi in Sympy [A]  time = 12.1337, size = 20, normalized size = 0.87 \[ \frac{\sqrt{c x^{2}} \log{\left (a + b x \right )}}{b c^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(c*x**2)*log(a + b*x)/(b*c**2*x)

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Mathematica [A]  time = 0.00547395, size = 22, normalized size = 0.96 \[ \frac{x^3 \log (a+b x)}{b \left (c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(x^3*Log[a + b*x])/(b*(c*x^2)^(3/2))

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Maple [A]  time = 0.004, size = 21, normalized size = 0.9 \[{\frac{{x}^{3}\ln \left ( bx+a \right ) }{b} \left ( c{x}^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.210982, size = 31, normalized size = 1.35 \[ \frac{\sqrt{c x^{2}} \log \left (b x + a\right )}{b c^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(c*x^2)*log(b*x + a)/(b*c^2*x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.211095, size = 47, normalized size = 2.04 \[ -\frac{{\rm ln}\left ({\left | -{\left (\sqrt{c} x - \sqrt{c x^{2}}\right )} b - 2 \, a \sqrt{c} \right |}\right )}{b c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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