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

Optimal. Leaf size=89 \[ \frac{a^3 c \sqrt{c x^2}}{b^4 x (a+b x)}+\frac{3 a^2 c \sqrt{c x^2} \log (a+b x)}{b^4 x}-\frac{2 a c \sqrt{c x^2}}{b^3}+\frac{c x \sqrt{c x^2}}{2 b^2} \]

[Out]

(-2*a*c*Sqrt[c*x^2])/b^3 + (c*x*Sqrt[c*x^2])/(2*b^2) + (a^3*c*Sqrt[c*x^2])/(b^4*
x*(a + b*x)) + (3*a^2*c*Sqrt[c*x^2]*Log[a + b*x])/(b^4*x)

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Rubi [A]  time = 0.0759953, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{a^3 c \sqrt{c x^2}}{b^4 x (a+b x)}+\frac{3 a^2 c \sqrt{c x^2} \log (a+b x)}{b^4 x}-\frac{2 a c \sqrt{c x^2}}{b^3}+\frac{c x \sqrt{c x^2}}{2 b^2} \]

Antiderivative was successfully verified.

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

[Out]

(-2*a*c*Sqrt[c*x^2])/b^3 + (c*x*Sqrt[c*x^2])/(2*b^2) + (a^3*c*Sqrt[c*x^2])/(b^4*
x*(a + b*x)) + (3*a^2*c*Sqrt[c*x^2]*Log[a + b*x])/(b^4*x)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{a^{3} c \sqrt{c x^{2}}}{b^{4} x \left (a + b x\right )} + \frac{3 a^{2} c \sqrt{c x^{2}} \log{\left (a + b x \right )}}{b^{4} x} - \frac{2 a c \sqrt{c x^{2}}}{b^{3}} + \frac{c \sqrt{c x^{2}} \int x\, dx}{b^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0326959, size = 71, normalized size = 0.8 \[ \frac{\left (c x^2\right )^{3/2} \left (2 a^3-4 a^2 b x+6 a^2 (a+b x) \log (a+b x)-3 a b^2 x^2+b^3 x^3\right )}{2 b^4 x^3 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

((c*x^2)^(3/2)*(2*a^3 - 4*a^2*b*x - 3*a*b^2*x^2 + b^3*x^3 + 6*a^2*(a + b*x)*Log[
a + b*x]))/(2*b^4*x^3*(a + b*x))

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Maple [A]  time = 0.007, size = 76, normalized size = 0.9 \[{\frac{{b}^{3}{x}^{3}+6\,\ln \left ( bx+a \right ) x{a}^{2}b-3\,a{b}^{2}{x}^{2}+6\,{a}^{3}\ln \left ( bx+a \right ) -4\,{a}^{2}bx+2\,{a}^{3}}{2\,{x}^{3} \left ( bx+a \right ){b}^{4}} \left ( c{x}^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2)^(3/2)/(b*x+a)^2,x)

[Out]

1/2*(c*x^2)^(3/2)*(b^3*x^3+6*ln(b*x+a)*x*a^2*b-3*a*b^2*x^2+6*a^3*ln(b*x+a)-4*a^2
*b*x+2*a^3)/x^3/(b*x+a)/b^4

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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, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.213576, size = 107, normalized size = 1.2 \[ \frac{{\left (b^{3} c x^{3} - 3 \, a b^{2} c x^{2} - 4 \, a^{2} b c x + 2 \, a^{3} c + 6 \,{\left (a^{2} b c x + a^{3} c\right )} \log \left (b x + a\right )\right )} \sqrt{c x^{2}}}{2 \,{\left (b^{5} x^{2} + a b^{4} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(b^3*c*x^3 - 3*a*b^2*c*x^2 - 4*a^2*b*c*x + 2*a^3*c + 6*(a^2*b*c*x + a^3*c)*l
og(b*x + a))*sqrt(c*x^2)/(b^5*x^2 + a*b^4*x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.213992, size = 108, normalized size = 1.21 \[ \frac{1}{2} \, c^{\frac{3}{2}}{\left (\frac{6 \, a^{2}{\rm ln}\left ({\left | b x + a \right |}\right ){\rm sign}\left (x\right )}{b^{4}} + \frac{2 \, a^{3}{\rm sign}\left (x\right )}{{\left (b x + a\right )} b^{4}} - \frac{2 \,{\left (3 \, a^{2}{\rm ln}\left ({\left | a \right |}\right ) + a^{2}\right )}{\rm sign}\left (x\right )}{b^{4}} + \frac{b^{2} x^{2}{\rm sign}\left (x\right ) - 4 \, a b x{\rm sign}\left (x\right )}{b^{4}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*c^(3/2)*(6*a^2*ln(abs(b*x + a))*sign(x)/b^4 + 2*a^3*sign(x)/((b*x + a)*b^4)
- 2*(3*a^2*ln(abs(a)) + a^2)*sign(x)/b^4 + (b^2*x^2*sign(x) - 4*a*b*x*sign(x))/b
^4)