Optimal. Leaf size=46 \[ \frac {x^5}{5 b^2}-\frac {a^2}{5 b^3 \left (a+b x^5\right )}-\frac {2 a \log \left (a+b x^5\right )}{5 b^3} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 46, 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 = {272, 45}
\begin {gather*} -\frac {a^2}{5 b^3 \left (a+b x^5\right )}-\frac {2 a \log \left (a+b x^5\right )}{5 b^3}+\frac {x^5}{5 b^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rule 272
Rubi steps
\begin {align*} \int \frac {x^{14}}{\left (a+b x^5\right )^2} \, dx &=\frac {1}{5} \text {Subst}\left (\int \frac {x^2}{(a+b x)^2} \, dx,x,x^5\right )\\ &=\frac {1}{5} \text {Subst}\left (\int \left (\frac {1}{b^2}+\frac {a^2}{b^2 (a+b x)^2}-\frac {2 a}{b^2 (a+b x)}\right ) \, dx,x,x^5\right )\\ &=\frac {x^5}{5 b^2}-\frac {a^2}{5 b^3 \left (a+b x^5\right )}-\frac {2 a \log \left (a+b x^5\right )}{5 b^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 38, normalized size = 0.83 \begin {gather*} \frac {b x^5-\frac {a^2}{a+b x^5}-2 a \log \left (a+b x^5\right )}{5 b^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.17, size = 44, normalized size = 0.96
method | result | size |
risch | \(\frac {x^{5}}{5 b^{2}}-\frac {a^{2}}{5 b^{3} \left (b \,x^{5}+a \right )}-\frac {2 a \ln \left (b \,x^{5}+a \right )}{5 b^{3}}\) | \(41\) |
norman | \(\frac {\frac {x^{10}}{5 b}-\frac {2 a^{2}}{5 b^{3}}}{b \,x^{5}+a}-\frac {2 a \ln \left (b \,x^{5}+a \right )}{5 b^{3}}\) | \(43\) |
default | \(\frac {x^{5}}{5 b^{2}}-\frac {a \left (\frac {a}{b \left (b \,x^{5}+a \right )}+\frac {2 \ln \left (b \,x^{5}+a \right )}{b}\right )}{5 b^{2}}\) | \(44\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.29, size = 43, normalized size = 0.93 \begin {gather*} \frac {x^{5}}{5 \, b^{2}} - \frac {a^{2}}{5 \, {\left (b^{4} x^{5} + a b^{3}\right )}} - \frac {2 \, a \log \left (b x^{5} + a\right )}{5 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.38, size = 56, normalized size = 1.22 \begin {gather*} \frac {b^{2} x^{10} + a b x^{5} - a^{2} - 2 \, {\left (a b x^{5} + a^{2}\right )} \log \left (b x^{5} + a\right )}{5 \, {\left (b^{4} x^{5} + a b^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.19, size = 42, normalized size = 0.91 \begin {gather*} - \frac {a^{2}}{5 a b^{3} + 5 b^{4} x^{5}} - \frac {2 a \log {\left (a + b x^{5} \right )}}{5 b^{3}} + \frac {x^{5}}{5 b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.31, size = 49, normalized size = 1.07 \begin {gather*} \frac {x^{5}}{5 \, b^{2}} - \frac {2 \, a \log \left ({\left | b x^{5} + a \right |}\right )}{5 \, b^{3}} + \frac {2 \, a b x^{5} + a^{2}}{5 \, {\left (b x^{5} + a\right )} b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.05, size = 45, normalized size = 0.98 \begin {gather*} \frac {x^5}{5\,b^2}-\frac {a^2}{5\,\left (b^4\,x^5+a\,b^3\right )}-\frac {2\,a\,\ln \left (b\,x^5+a\right )}{5\,b^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________