Optimal. Leaf size=64 \[ -\frac{5 a^3 b^2}{x^2}+10 a^2 b^3 \log (x)-\frac{5 a^4 b}{4 x^4}-\frac{a^5}{6 x^6}+\frac{5}{2} a b^4 x^2+\frac{b^5 x^4}{4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0340468, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {266, 43} \[ -\frac{5 a^3 b^2}{x^2}+10 a^2 b^3 \log (x)-\frac{5 a^4 b}{4 x^4}-\frac{a^5}{6 x^6}+\frac{5}{2} a b^4 x^2+\frac{b^5 x^4}{4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^5}{x^7} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^5}{x^4} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (5 a b^4+\frac{a^5}{x^4}+\frac{5 a^4 b}{x^3}+\frac{10 a^3 b^2}{x^2}+\frac{10 a^2 b^3}{x}+b^5 x\right ) \, dx,x,x^2\right )\\ &=-\frac{a^5}{6 x^6}-\frac{5 a^4 b}{4 x^4}-\frac{5 a^3 b^2}{x^2}+\frac{5}{2} a b^4 x^2+\frac{b^5 x^4}{4}+10 a^2 b^3 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0043543, size = 64, normalized size = 1. \[ -\frac{5 a^3 b^2}{x^2}+10 a^2 b^3 \log (x)-\frac{5 a^4 b}{4 x^4}-\frac{a^5}{6 x^6}+\frac{5}{2} a b^4 x^2+\frac{b^5 x^4}{4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.005, size = 57, normalized size = 0.9 \begin{align*} -{\frac{{a}^{5}}{6\,{x}^{6}}}-{\frac{5\,{a}^{4}b}{4\,{x}^{4}}}-5\,{\frac{{a}^{3}{b}^{2}}{{x}^{2}}}+{\frac{5\,a{b}^{4}{x}^{2}}{2}}+{\frac{{b}^{5}{x}^{4}}{4}}+10\,{a}^{2}{b}^{3}\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 2.04804, size = 82, normalized size = 1.28 \begin{align*} \frac{1}{4} \, b^{5} x^{4} + \frac{5}{2} \, a b^{4} x^{2} + 5 \, a^{2} b^{3} \log \left (x^{2}\right ) - \frac{60 \, a^{3} b^{2} x^{4} + 15 \, a^{4} b x^{2} + 2 \, a^{5}}{12 \, x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.07728, size = 139, normalized size = 2.17 \begin{align*} \frac{3 \, b^{5} x^{10} + 30 \, a b^{4} x^{8} + 120 \, a^{2} b^{3} x^{6} \log \left (x\right ) - 60 \, a^{3} b^{2} x^{4} - 15 \, a^{4} b x^{2} - 2 \, a^{5}}{12 \, x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.485417, size = 63, normalized size = 0.98 \begin{align*} 10 a^{2} b^{3} \log{\left (x \right )} + \frac{5 a b^{4} x^{2}}{2} + \frac{b^{5} x^{4}}{4} - \frac{2 a^{5} + 15 a^{4} b x^{2} + 60 a^{3} b^{2} x^{4}}{12 x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 2.66172, size = 97, normalized size = 1.52 \begin{align*} \frac{1}{4} \, b^{5} x^{4} + \frac{5}{2} \, a b^{4} x^{2} + 5 \, a^{2} b^{3} \log \left (x^{2}\right ) - \frac{110 \, a^{2} b^{3} x^{6} + 60 \, a^{3} b^{2} x^{4} + 15 \, a^{4} b x^{2} + 2 \, a^{5}}{12 \, x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]