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]
________________________________________________________________________________________
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]
[Out]
Rule 15
Rule 43
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]