3.909 \(\int \frac{x^5}{\sqrt{c x^2} (a+b x)^2} \, dx\)

Optimal. Leaf size=107 \[ -\frac{a^4 x}{b^5 \sqrt{c x^2} (a+b x)}+\frac{3 a^2 x^2}{b^4 \sqrt{c x^2}}-\frac{4 a^3 x \log (a+b x)}{b^5 \sqrt{c x^2}}-\frac{a x^3}{b^3 \sqrt{c x^2}}+\frac{x^4}{3 b^2 \sqrt{c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.032868, antiderivative size = 107, 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} \[ -\frac{a^4 x}{b^5 \sqrt{c x^2} (a+b x)}+\frac{3 a^2 x^2}{b^4 \sqrt{c x^2}}-\frac{4 a^3 x \log (a+b x)}{b^5 \sqrt{c x^2}}-\frac{a x^3}{b^3 \sqrt{c x^2}}+\frac{x^4}{3 b^2 \sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]

Int[x^5/(Sqrt[c*x^2]*(a + b*x)^2),x]

[Out]

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

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

Mathematica [A]  time = 0.0149218, size = 80, normalized size = 0.75 \[ \frac{x \left (6 a^2 b^2 x^2+9 a^3 b x-12 a^3 (a+b x) \log (a+b x)-3 a^4-2 a b^3 x^3+b^4 x^4\right )}{3 b^5 \sqrt{c x^2} (a+b x)} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5/(Sqrt[c*x^2]*(a + b*x)^2),x]

[Out]

(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 = 0.004, size = 86, normalized size = 0.8 \begin{align*} -{\frac{x \left ( -{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} \right ) }{3\,{b}^{5} \left ( bx+a \right ) }{\frac{1}{\sqrt{c{x}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(b*x+a)^2/(c*x^2)^(1/2),x)

[Out]

-1/3*x*(-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)/(c*x^2)^(1/2
)/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^5/(b*x+a)^2/(c*x^2)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.25569, size = 182, normalized size = 1.7 \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} c x^{2} + a b^{5} c x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x+a)^2/(c*x^2)^(1/2),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*c*x^2 + a*b^5*c*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{5}}{\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**5/(b*x+a)**2/(c*x**2)**(1/2),x)

[Out]

Integral(x**5/(sqrt(c*x**2)*(a + b*x)**2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{5}}{\sqrt{c x^{2}}{\left (b x + a\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x+a)^2/(c*x^2)^(1/2),x, algorithm="giac")

[Out]

integrate(x^5/(sqrt(c*x^2)*(b*x + a)^2), x)