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

Optimal. Leaf size=111 \[ -\frac{a^4 c \sqrt{c x^2}}{b^5 x (a+b x)}-\frac{4 a^3 c \sqrt{c x^2} \log (a+b x)}{b^5 x}+\frac{3 a^2 c \sqrt{c x^2}}{b^4}-\frac{a c x \sqrt{c x^2}}{b^3}+\frac{c x^2 \sqrt{c x^2}}{3 b^2} \]

[Out]

(3*a^2*c*Sqrt[c*x^2])/b^4 - (a*c*x*Sqrt[c*x^2])/b^3 + (c*x^2*Sqrt[c*x^2])/(3*b^2
) - (a^4*c*Sqrt[c*x^2])/(b^5*x*(a + b*x)) - (4*a^3*c*Sqrt[c*x^2]*Log[a + b*x])/(
b^5*x)

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Rubi [A]  time = 0.0928012, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ -\frac{a^4 c \sqrt{c x^2}}{b^5 x (a+b x)}-\frac{4 a^3 c \sqrt{c x^2} \log (a+b x)}{b^5 x}+\frac{3 a^2 c \sqrt{c x^2}}{b^4}-\frac{a c x \sqrt{c x^2}}{b^3}+\frac{c x^2 \sqrt{c x^2}}{3 b^2} \]

Antiderivative was successfully verified.

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

[Out]

(3*a^2*c*Sqrt[c*x^2])/b^4 - (a*c*x*Sqrt[c*x^2])/b^3 + (c*x^2*Sqrt[c*x^2])/(3*b^2
) - (a^4*c*Sqrt[c*x^2])/(b^5*x*(a + b*x)) - (4*a^3*c*Sqrt[c*x^2]*Log[a + b*x])/(
b^5*x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0388168, size = 82, normalized size = 0.74 \[ \frac{\left (c x^2\right )^{3/2} \left (-3 a^4+9 a^3 b x-12 a^3 (a+b x) \log (a+b x)+6 a^2 b^2 x^2-2 a b^3 x^3+b^4 x^4\right )}{3 b^5 x^3 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

((c*x^2)^(3/2)*(-3*a^4 + 9*a^3*b*x + 6*a^2*b^2*x^2 - 2*a*b^3*x^3 + b^4*x^4 - 12*
a^3*(a + b*x)*Log[a + b*x]))/(3*b^5*x^3*(a + b*x))

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Maple [A]  time = 0.007, size = 88, normalized size = 0.8 \[ -{\frac{-{x}^{4}{b}^{4}+2\,{x}^{3}a{b}^{3}+12\,\ln \left ( bx+a \right ) x{a}^{3}b-6\,{x}^{2}{a}^{2}{b}^{2}+12\,{a}^{4}\ln \left ( bx+a \right ) -9\,x{a}^{3}b+3\,{a}^{4}}{3\,{x}^{3} \left ( bx+a \right ){b}^{5}} \left ( c{x}^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(c*x^2)^(3/2)*(-x^4*b^4+2*x^3*a*b^3+12*ln(b*x+a)*x*a^3*b-6*x^2*a^2*b^2+12*a
^4*ln(b*x+a)-9*x*a^3*b+3*a^4)/x^3/(b*x+a)/b^5

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.215826, size = 123, normalized size = 1.11 \[ \frac{{\left (b^{4} c x^{4} - 2 \, a b^{3} c x^{3} + 6 \, a^{2} b^{2} c x^{2} + 9 \, a^{3} b c x - 3 \, a^{4} c - 12 \,{\left (a^{3} b c x + a^{4} c\right )} \log \left (b x + a\right )\right )} \sqrt{c x^{2}}}{3 \,{\left (b^{6} x^{2} + a b^{5} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(b^4*c*x^4 - 2*a*b^3*c*x^3 + 6*a^2*b^2*c*x^2 + 9*a^3*b*c*x - 3*a^4*c - 12*(a
^3*b*c*x + a^4*c)*log(b*x + a))*sqrt(c*x^2)/(b^6*x^2 + a*b^5*x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.206547, size = 130, normalized size = 1.17 \[ -\frac{1}{3} \, c^{\frac{3}{2}}{\left (\frac{12 \, a^{3}{\rm ln}\left ({\left | b x + a \right |}\right ){\rm sign}\left (x\right )}{b^{5}} + \frac{3 \, a^{4}{\rm sign}\left (x\right )}{{\left (b x + a\right )} b^{5}} - \frac{3 \,{\left (4 \, a^{3}{\rm ln}\left ({\left | a \right |}\right ) + a^{3}\right )}{\rm sign}\left (x\right )}{b^{5}} - \frac{b^{4} x^{3}{\rm sign}\left (x\right ) - 3 \, a b^{3} x^{2}{\rm sign}\left (x\right ) + 9 \, a^{2} b^{2} x{\rm sign}\left (x\right )}{b^{6}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*c^(3/2)*(12*a^3*ln(abs(b*x + a))*sign(x)/b^5 + 3*a^4*sign(x)/((b*x + a)*b^5
) - 3*(4*a^3*ln(abs(a)) + a^3)*sign(x)/b^5 - (b^4*x^3*sign(x) - 3*a*b^3*x^2*sign
(x) + 9*a^2*b^2*x*sign(x))/b^6)