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

Optimal. Leaf size=40 \[ \frac{c \sqrt{c x^2}}{b}-\frac{a c \sqrt{c x^2} \log (a+b x)}{b^2 x} \]

[Out]

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

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Rubi [A]  time = 0.0298071, antiderivative size = 40, 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 \[ \frac{c \sqrt{c x^2}}{b}-\frac{a c \sqrt{c x^2} \log (a+b x)}{b^2 x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a c \sqrt{c x^{2}} \log{\left (a + b x \right )}}{b^{2} x} + \frac{c \sqrt{c x^{2}} \int \frac{1}{b}\, dx}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a*c*sqrt(c*x**2)*log(a + b*x)/(b**2*x) + c*sqrt(c*x**2)*Integral(1/b, x)/x

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Mathematica [A]  time = 0.0103214, size = 30, normalized size = 0.75 \[ \frac{c^2 x (b x-a \log (a+b x))}{b^2 \sqrt{c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.005, size = 29, normalized size = 0.7 \[ -{\frac{a\ln \left ( bx+a \right ) -bx}{{b}^{2}{x}^{3}} \left ( c{x}^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.234928, size = 39, normalized size = 0.98 \[ \frac{{\left (b c x - a c \log \left (b x + a\right )\right )} \sqrt{c x^{2}}}{b^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.204419, size = 50, normalized size = 1.25 \[ c^{\frac{3}{2}}{\left (\frac{x{\rm sign}\left (x\right )}{b} - \frac{a{\rm ln}\left ({\left | b x + a \right |}\right ){\rm sign}\left (x\right )}{b^{2}} + \frac{a{\rm ln}\left ({\left | a \right |}\right ){\rm sign}\left (x\right )}{b^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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