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

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

[Out]

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

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Rubi [A]  time = 0.0443128, 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 \[ \frac{a^2 c \sqrt{c x^2} \log (a+b x)}{b^3 x}-\frac{a c \sqrt{c x^2}}{b^2}+\frac{c x \sqrt{c x^2}}{2 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0141023, size = 42, normalized size = 0.69 \[ \frac{c^2 x \left (2 a^2 \log (a+b x)+b x (b x-2 a)\right )}{2 b^3 \sqrt{c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.009, size = 40, normalized size = 0.7 \[{\frac{{b}^{2}{x}^{2}+2\,{a}^{2}\ln \left ( bx+a \right ) -2\,abx}{2\,{b}^{3}{x}^{3}} \left ( c{x}^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2)^(3/2)/x/(b*x+a),x)

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.217807, size = 57, normalized size = 0.93 \[ \frac{{\left (b^{2} c x^{2} - 2 \, a b c x + 2 \, a^{2} c \log \left (b x + a\right )\right )} \sqrt{c x^{2}}}{2 \, b^{3} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.209592, size = 73, normalized size = 1.2 \[ \frac{1}{2} \, c^{\frac{3}{2}}{\left (\frac{2 \, a^{2}{\rm ln}\left ({\left | b x + a \right |}\right ){\rm sign}\left (x\right )}{b^{3}} - \frac{2 \, a^{2}{\rm ln}\left ({\left | a \right |}\right ){\rm sign}\left (x\right )}{b^{3}} + \frac{b x^{2}{\rm sign}\left (x\right ) - 2 \, a x{\rm sign}\left (x\right )}{b^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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