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

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

[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 = 0.0373899, 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, Rules used = {15, 43} \[ -\frac{a^4 c \sqrt{c x^2}}{b^5 x (a+b x)}+\frac{3 a^2 c \sqrt{c x^2}}{b^4}-\frac{4 a^3 c \sqrt{c x^2} \log (a+b x)}{b^5 x}-\frac{a c x \sqrt{c x^2}}{b^3}+\frac{c x^2 \sqrt{c x^2}}{3 b^2} \]

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)

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[(a^IntPart[m]*(a*x^n)^FracPart[m])/x^(n*FracPart[m]), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{x \left (c x^2\right )^{3/2}}{(a+b x)^2} \, dx &=\frac{\left (c \sqrt{c x^2}\right ) \int \frac{x^4}{(a+b x)^2} \, dx}{x}\\ &=\frac{\left (c \sqrt{c x^2}\right ) \int \left (\frac{3 a^2}{b^4}-\frac{2 a x}{b^3}+\frac{x^2}{b^2}+\frac{a^4}{b^4 (a+b x)^2}-\frac{4 a^3}{b^4 (a+b x)}\right ) \, dx}{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}-\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}\\ \end{align*}

Mathematica [A]  time = 0.0196553, size = 82, normalized size = 0.74 \[ \frac{\left (c x^2\right )^{3/2} \left (6 a^2 b^2 x^2+9 a^3 b x-12 a^3 (a+b x) \log (a+b x)-3 a^4-2 a b^3 x^3+b^4 x^4\right )}{3 b^5 x^3 (a+b x)} \]

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 = 0.004, size = 88, normalized size = 0.8 \begin{align*} -{\frac{-{b}^{4}{x}^{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\,bx{a}^{3}+3\,{a}^{4}}{3\,{b}^{5}{x}^{3} \left ( bx+a \right ) } \left ( c{x}^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}

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)*(-b^4*x^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*b*x*a^3+3*a^4)/
x^3/b^5/(b*x+a)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.37743, size = 196, normalized size = 1.77 \begin{align*} \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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(c*x^2)^(3/2)/(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. \begin{align*} \int \frac{x \left (c x^{2}\right )^{\frac{3}{2}}}{\left (a + b x\right )^{2}}\, dx \end{align*}

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 [A]  time = 1.0597, size = 130, normalized size = 1.17 \begin{align*} -\frac{1}{3} \, c^{\frac{3}{2}}{\left (\frac{12 \, a^{3} \log \left ({\left | b x + a \right |}\right ) \mathrm{sgn}\left (x\right )}{b^{5}} + \frac{3 \, a^{4} \mathrm{sgn}\left (x\right )}{{\left (b x + a\right )} b^{5}} - \frac{3 \,{\left (4 \, a^{3} \log \left ({\left | a \right |}\right ) + a^{3}\right )} \mathrm{sgn}\left (x\right )}{b^{5}} - \frac{b^{4} x^{3} \mathrm{sgn}\left (x\right ) - 3 \, a b^{3} x^{2} \mathrm{sgn}\left (x\right ) + 9 \, a^{2} b^{2} x \mathrm{sgn}\left (x\right )}{b^{6}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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