Optimal. Leaf size=62 \[ 20 a^2 b^3 \sqrt{x}+10 a^3 b^2 \log (x)-\frac{10 a^4 b}{\sqrt{x}}-\frac{a^5}{x}+5 a b^4 x+\frac{2}{3} b^5 x^{3/2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.033022, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ 20 a^2 b^3 \sqrt{x}+10 a^3 b^2 \log (x)-\frac{10 a^4 b}{\sqrt{x}}-\frac{a^5}{x}+5 a b^4 x+\frac{2}{3} b^5 x^{3/2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (a+b \sqrt{x}\right )^5}{x^2} \, dx &=2 \operatorname{Subst}\left (\int \frac{(a+b x)^5}{x^3} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (10 a^2 b^3+\frac{a^5}{x^3}+\frac{5 a^4 b}{x^2}+\frac{10 a^3 b^2}{x}+5 a b^4 x+b^5 x^2\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{a^5}{x}-\frac{10 a^4 b}{\sqrt{x}}+20 a^2 b^3 \sqrt{x}+5 a b^4 x+\frac{2}{3} b^5 x^{3/2}+10 a^3 b^2 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0283227, size = 62, normalized size = 1. \[ 20 a^2 b^3 \sqrt{x}+10 a^3 b^2 \log (x)-\frac{10 a^4 b}{\sqrt{x}}-\frac{a^5}{x}+5 a b^4 x+\frac{2}{3} b^5 x^{3/2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.003, size = 55, normalized size = 0.9 \begin{align*} -{\frac{{a}^{5}}{x}}+5\,a{b}^{4}x+{\frac{2\,{b}^{5}}{3}{x}^{{\frac{3}{2}}}}+10\,{a}^{3}{b}^{2}\ln \left ( x \right ) -10\,{\frac{{a}^{4}b}{\sqrt{x}}}+20\,{a}^{2}{b}^{3}\sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.976525, size = 74, normalized size = 1.19 \begin{align*} \frac{2}{3} \, b^{5} x^{\frac{3}{2}} + 5 \, a b^{4} x + 10 \, a^{3} b^{2} \log \left (x\right ) + 20 \, a^{2} b^{3} \sqrt{x} - \frac{10 \, a^{4} b \sqrt{x} + a^{5}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.46863, size = 143, normalized size = 2.31 \begin{align*} \frac{15 \, a b^{4} x^{2} + 60 \, a^{3} b^{2} x \log \left (\sqrt{x}\right ) - 3 \, a^{5} + 2 \,{\left (b^{5} x^{2} + 30 \, a^{2} b^{3} x - 15 \, a^{4} b\right )} \sqrt{x}}{3 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.756582, size = 61, normalized size = 0.98 \begin{align*} - \frac{a^{5}}{x} - \frac{10 a^{4} b}{\sqrt{x}} + 10 a^{3} b^{2} \log{\left (x \right )} + 20 a^{2} b^{3} \sqrt{x} + 5 a b^{4} x + \frac{2 b^{5} x^{\frac{3}{2}}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.11656, size = 76, normalized size = 1.23 \begin{align*} \frac{2}{3} \, b^{5} x^{\frac{3}{2}} + 5 \, a b^{4} x + 10 \, a^{3} b^{2} \log \left ({\left | x \right |}\right ) + 20 \, a^{2} b^{3} \sqrt{x} - \frac{10 \, a^{4} b \sqrt{x} + a^{5}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]