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

Optimal. Leaf size=106 \[ -\frac{a^4 \sqrt{c x^2}}{b^5 x (a+b x)}+\frac{3 a^2 \sqrt{c x^2}}{b^4}-\frac{4 a^3 \sqrt{c x^2} \log (a+b x)}{b^5 x}-\frac{a x \sqrt{c x^2}}{b^3}+\frac{x^2 \sqrt{c x^2}}{3 b^2} \]

[Out]

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

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Rubi [A]  time = 0.039058, antiderivative size = 106, 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, Rules used = {15, 43} \[ -\frac{a^4 \sqrt{c x^2}}{b^5 x (a+b x)}+\frac{3 a^2 \sqrt{c x^2}}{b^4}-\frac{4 a^3 \sqrt{c x^2} \log (a+b x)}{b^5 x}-\frac{a x \sqrt{c x^2}}{b^3}+\frac{x^2 \sqrt{c x^2}}{3 b^2} \]

Antiderivative was successfully verified.

[In]

Int[(x^3*Sqrt[c*x^2])/(a + b*x)^2,x]

[Out]

(3*a^2*Sqrt[c*x^2])/b^4 - (a*x*Sqrt[c*x^2])/b^3 + (x^2*Sqrt[c*x^2])/(3*b^2) - (a^4*Sqrt[c*x^2])/(b^5*x*(a + b*
x)) - (4*a^3*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^3 \sqrt{c x^2}}{(a+b x)^2} \, dx &=\frac{\sqrt{c x^2} \int \frac{x^4}{(a+b x)^2} \, dx}{x}\\ &=\frac{\sqrt{c x^2} \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 \sqrt{c x^2}}{b^4}-\frac{a x \sqrt{c x^2}}{b^3}+\frac{x^2 \sqrt{c x^2}}{3 b^2}-\frac{a^4 \sqrt{c x^2}}{b^5 x (a+b x)}-\frac{4 a^3 \sqrt{c x^2} \log (a+b x)}{b^5 x}\\ \end{align*}

Mathematica [A]  time = 0.0275, size = 81, normalized size = 0.76 \[ \frac{c x \left (6 a^2 b^2 x^2+9 a^3 b x-12 a^3 (a+b x) \log (a+b x)-3 a^4-2 a b^3 x^3+b^4 x^4\right )}{3 b^5 \sqrt{c x^2} (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c*x*(-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*Sqrt
[c*x^2]*(a + b*x))

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Maple [A]  time = 0.009, 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 \left ( bx+a \right ) }\sqrt{c{x}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(c*x^2)^(1/2)/(b*x+a)^2,x)

[Out]

-1/3*(c*x^2)^(1/2)*(-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/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^3*(c*x^2)^(1/2)/(b*x+a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.65858, size = 177, normalized size = 1.67 \begin{align*} \frac{{\left (b^{4} x^{4} - 2 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} + 9 \, a^{3} b x - 3 \, a^{4} - 12 \,{\left (a^{3} b x + a^{4}\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^3*(c*x^2)^(1/2)/(b*x+a)^2,x, algorithm="fricas")

[Out]

1/3*(b^4*x^4 - 2*a*b^3*x^3 + 6*a^2*b^2*x^2 + 9*a^3*b*x - 3*a^4 - 12*(a^3*b*x + a^4)*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^{3} \sqrt{c x^{2}}}{\left (a + b x\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(c*x**2)**(1/2)/(b*x+a)**2,x)

[Out]

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

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Giac [A]  time = 1.05304, size = 130, normalized size = 1.23 \begin{align*} -\frac{1}{3} \, \sqrt{c}{\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^3*(c*x^2)^(1/2)/(b*x+a)^2,x, algorithm="giac")

[Out]

-1/3*sqrt(c)*(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)