Optimal. Leaf size=65 \[ a^5 \log (x)+\frac {5}{2} a^4 b x^2+\frac {5}{2} a^3 b^2 x^4+\frac {5}{3} a^2 b^3 x^6+\frac {5}{8} a b^4 x^8+\frac {b^5 x^{10}}{10} \]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 65, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {5}{3} a^2 b^3 x^6+\frac {5}{2} a^3 b^2 x^4+\frac {5}{2} a^4 b x^2+a^5 \log (x)+\frac {5}{8} a b^4 x^8+\frac {b^5 x^{10}}{10} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^5}{x} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^5}{x} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (5 a^4 b+\frac {a^5}{x}+10 a^3 b^2 x+10 a^2 b^3 x^2+5 a b^4 x^3+b^5 x^4\right ) \, dx,x,x^2\right )\\ &=\frac {5}{2} a^4 b x^2+\frac {5}{2} a^3 b^2 x^4+\frac {5}{3} a^2 b^3 x^6+\frac {5}{8} a b^4 x^8+\frac {b^5 x^{10}}{10}+a^5 \log (x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.00, size = 65, normalized size = 1.00 \begin {gather*} a^5 \log (x)+\frac {5}{2} a^4 b x^2+\frac {5}{2} a^3 b^2 x^4+\frac {5}{3} a^2 b^3 x^6+\frac {5}{8} a b^4 x^8+\frac {b^5 x^{10}}{10} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+b x^2\right )^5}{x} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.90, size = 55, normalized size = 0.85 \begin {gather*} \frac {1}{10} \, b^{5} x^{10} + \frac {5}{8} \, a b^{4} x^{8} + \frac {5}{3} \, a^{2} b^{3} x^{6} + \frac {5}{2} \, a^{3} b^{2} x^{4} + \frac {5}{2} \, a^{4} b x^{2} + a^{5} \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.92, size = 58, normalized size = 0.89 \begin {gather*} \frac {1}{10} \, b^{5} x^{10} + \frac {5}{8} \, a b^{4} x^{8} + \frac {5}{3} \, a^{2} b^{3} x^{6} + \frac {5}{2} \, a^{3} b^{2} x^{4} + \frac {5}{2} \, a^{4} b x^{2} + \frac {1}{2} \, a^{5} \log \left (x^{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 56, normalized size = 0.86 \begin {gather*} \frac {b^{5} x^{10}}{10}+\frac {5 a \,b^{4} x^{8}}{8}+\frac {5 a^{2} b^{3} x^{6}}{3}+\frac {5 a^{3} b^{2} x^{4}}{2}+\frac {5 a^{4} b \,x^{2}}{2}+a^{5} \ln \relax (x ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.37, size = 58, normalized size = 0.89 \begin {gather*} \frac {1}{10} \, b^{5} x^{10} + \frac {5}{8} \, a b^{4} x^{8} + \frac {5}{3} \, a^{2} b^{3} x^{6} + \frac {5}{2} \, a^{3} b^{2} x^{4} + \frac {5}{2} \, a^{4} b x^{2} + \frac {1}{2} \, a^{5} \log \left (x^{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.03, size = 55, normalized size = 0.85 \begin {gather*} a^5\,\ln \relax (x)+\frac {b^5\,x^{10}}{10}+\frac {5\,a^4\,b\,x^2}{2}+\frac {5\,a\,b^4\,x^8}{8}+\frac {5\,a^3\,b^2\,x^4}{2}+\frac {5\,a^2\,b^3\,x^6}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.14, size = 65, normalized size = 1.00 \begin {gather*} a^{5} \log {\relax (x )} + \frac {5 a^{4} b x^{2}}{2} + \frac {5 a^{3} b^{2} x^{4}}{2} + \frac {5 a^{2} b^{3} x^{6}}{3} + \frac {5 a b^{4} x^{8}}{8} + \frac {b^{5} x^{10}}{10} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________