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

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

[Out]

(c*Sqrt[c*x^2]*Log[a + b*x])/(b*x)

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Rubi [A]  time = 0.0114327, 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 \[ \frac{c \sqrt{c x^2} \log (a+b x)}{b x} \]

Antiderivative was successfully verified.

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

[Out]

(c*Sqrt[c*x^2]*Log[a + b*x])/(b*x)

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Rubi in Sympy [A]  time = 13.0937, size = 19, normalized size = 0.83 \[ \frac{c \sqrt{c x^{2}} \log{\left (a + b x \right )}}{b x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c*sqrt(c*x**2)*log(a + b*x)/(b*x)

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Mathematica [A]  time = 0.00525604, size = 22, normalized size = 0.96 \[ \frac{\left (c x^2\right )^{3/2} \log (a+b x)}{b x^3} \]

Antiderivative was successfully verified.

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

[Out]

((c*x^2)^(3/2)*Log[a + b*x])/(b*x^3)

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Maple [A]  time = 0.005, size = 21, normalized size = 0.9 \[{\frac{\ln \left ( bx+a \right ) }{b{x}^{3}} \left ( c{x}^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.35739, size = 18, normalized size = 0.78 \[ \frac{c^{\frac{3}{2}} \log \left (b x + a\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c^(3/2)*log(b*x + a)/b

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Fricas [A]  time = 0.211369, size = 28, normalized size = 1.22 \[ \frac{\sqrt{c x^{2}} c \log \left (b x + a\right )}{b x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(c*x^2)*c*log(b*x + a)/(b*x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.206746, size = 38, normalized size = 1.65 \[ c^{\frac{3}{2}}{\left (\frac{{\rm ln}\left ({\left | b x + a \right |}\right ){\rm sign}\left (x\right )}{b} - \frac{{\rm ln}\left ({\left | a \right |}\right ){\rm sign}\left (x\right )}{b}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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