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

Optimal. Leaf size=68 \[ -\frac{a^2 c \sqrt{c x^2}}{b^3 x (a+b x)}-\frac{2 a c \sqrt{c x^2} \log (a+b x)}{b^3 x}+\frac{c \sqrt{c x^2}}{b^2} \]

[Out]

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

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Rubi [A]  time = 0.0527329, antiderivative size = 68, 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 \[ -\frac{a^2 c \sqrt{c x^2}}{b^3 x (a+b x)}-\frac{2 a c \sqrt{c x^2} \log (a+b x)}{b^3 x}+\frac{c \sqrt{c x^2}}{b^2} \]

Antiderivative was successfully verified.

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

[Out]

(c*Sqrt[c*x^2])/b^2 - (a^2*c*Sqrt[c*x^2])/(b^3*x*(a + b*x)) - (2*a*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. \[ - \frac{a^{2} c \sqrt{c x^{2}}}{b^{3} x \left (a + b x\right )} - \frac{2 a c \sqrt{c x^{2}} \log{\left (a + b x \right )}}{b^{3} x} + \frac{c \sqrt{c x^{2}} \int \frac{1}{b^{2}}\, dx}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0105255, size = 55, normalized size = 0.81 \[ \frac{c^2 x \left (-a^2+a b x-2 a (a+b x) \log (a+b x)+b^2 x^2\right )}{b^3 \sqrt{c x^2} (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.007, size = 62, normalized size = 0.9 \[ -{\frac{2\,\ln \left ( bx+a \right ) xab-{b}^{2}{x}^{2}+2\,{a}^{2}\ln \left ( bx+a \right ) -abx+{a}^{2}}{{x}^{3} \left ( bx+a \right ){b}^{3}} \left ( c{x}^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.209069, size = 85, normalized size = 1.25 \[ \frac{{\left (b^{2} c x^{2} + a b c x - a^{2} c - 2 \,{\left (a b c x + a^{2} c\right )} \log \left (b x + a\right )\right )} \sqrt{c x^{2}}}{b^{4} x^{2} + a b^{3} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.206268, size = 78, normalized size = 1.15 \[ c^{\frac{3}{2}}{\left (\frac{x{\rm sign}\left (x\right )}{b^{2}} - \frac{2 \, a{\rm ln}\left ({\left | b x + a \right |}\right ){\rm sign}\left (x\right )}{b^{3}} + \frac{{\left (2 \, a{\rm ln}\left ({\left | a \right |}\right ) + a\right )}{\rm sign}\left (x\right )}{b^{3}} - \frac{a^{2}{\rm sign}\left (x\right )}{{\left (b x + a\right )} b^{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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