Integrand size = 20, antiderivative size = 75 \[ \int \frac {x^5 \left (c+d x^2\right )}{a+b x^2} \, dx=-\frac {a (b c-a d) x^2}{2 b^3}+\frac {(b c-a d) x^4}{4 b^2}+\frac {d x^6}{6 b}+\frac {a^2 (b c-a d) \log \left (a+b x^2\right )}{2 b^4} \]
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Time = 0.07 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {457, 78} \[ \int \frac {x^5 \left (c+d x^2\right )}{a+b x^2} \, dx=\frac {a^2 (b c-a d) \log \left (a+b x^2\right )}{2 b^4}-\frac {a x^2 (b c-a d)}{2 b^3}+\frac {x^4 (b c-a d)}{4 b^2}+\frac {d x^6}{6 b} \]
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Rule 78
Rule 457
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {x^2 (c+d x)}{a+b x} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {a (-b c+a d)}{b^3}+\frac {(b c-a d) x}{b^2}+\frac {d x^2}{b}-\frac {a^2 (-b c+a d)}{b^3 (a+b x)}\right ) \, dx,x,x^2\right ) \\ & = -\frac {a (b c-a d) x^2}{2 b^3}+\frac {(b c-a d) x^4}{4 b^2}+\frac {d x^6}{6 b}+\frac {a^2 (b c-a d) \log \left (a+b x^2\right )}{2 b^4} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.95 \[ \int \frac {x^5 \left (c+d x^2\right )}{a+b x^2} \, dx=\frac {b x^2 \left (6 a^2 d-3 a b \left (2 c+d x^2\right )+b^2 x^2 \left (3 c+2 d x^2\right )\right )+6 a^2 (b c-a d) \log \left (a+b x^2\right )}{12 b^4} \]
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Time = 2.60 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.91
method | result | size |
norman | \(\frac {d \,x^{6}}{6 b}-\frac {\left (a d -b c \right ) x^{4}}{4 b^{2}}+\frac {\left (a d -b c \right ) a \,x^{2}}{2 b^{3}}-\frac {a^{2} \left (a d -b c \right ) \ln \left (b \,x^{2}+a \right )}{2 b^{4}}\) | \(68\) |
default | \(\frac {\frac {1}{3} b^{2} d \,x^{6}-\frac {1}{2} a b d \,x^{4}+\frac {1}{2} b^{2} c \,x^{4}+a^{2} d \,x^{2}-a b c \,x^{2}}{2 b^{3}}-\frac {a^{2} \left (a d -b c \right ) \ln \left (b \,x^{2}+a \right )}{2 b^{4}}\) | \(74\) |
parallelrisch | \(-\frac {-2 x^{6} d \,b^{3}+3 x^{4} a \,b^{2} d -3 x^{4} b^{3} c -6 a^{2} b d \,x^{2}+6 a \,b^{2} c \,x^{2}+6 \ln \left (b \,x^{2}+a \right ) a^{3} d -6 \ln \left (b \,x^{2}+a \right ) a^{2} b c}{12 b^{4}}\) | \(84\) |
risch | \(\frac {d \,x^{6}}{6 b}-\frac {a d \,x^{4}}{4 b^{2}}+\frac {c \,x^{4}}{4 b}+\frac {a^{2} d \,x^{2}}{2 b^{3}}-\frac {a c \,x^{2}}{2 b^{2}}-\frac {a^{3} \ln \left (b \,x^{2}+a \right ) d}{2 b^{4}}+\frac {a^{2} \ln \left (b \,x^{2}+a \right ) c}{2 b^{3}}\) | \(86\) |
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Time = 0.26 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00 \[ \int \frac {x^5 \left (c+d x^2\right )}{a+b x^2} \, dx=\frac {2 \, b^{3} d x^{6} + 3 \, {\left (b^{3} c - a b^{2} d\right )} x^{4} - 6 \, {\left (a b^{2} c - a^{2} b d\right )} x^{2} + 6 \, {\left (a^{2} b c - a^{3} d\right )} \log \left (b x^{2} + a\right )}{12 \, b^{4}} \]
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Time = 0.18 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.93 \[ \int \frac {x^5 \left (c+d x^2\right )}{a+b x^2} \, dx=- \frac {a^{2} \left (a d - b c\right ) \log {\left (a + b x^{2} \right )}}{2 b^{4}} + x^{4} \left (- \frac {a d}{4 b^{2}} + \frac {c}{4 b}\right ) + x^{2} \left (\frac {a^{2} d}{2 b^{3}} - \frac {a c}{2 b^{2}}\right ) + \frac {d x^{6}}{6 b} \]
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Time = 0.21 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.99 \[ \int \frac {x^5 \left (c+d x^2\right )}{a+b x^2} \, dx=\frac {2 \, b^{2} d x^{6} + 3 \, {\left (b^{2} c - a b d\right )} x^{4} - 6 \, {\left (a b c - a^{2} d\right )} x^{2}}{12 \, b^{3}} + \frac {{\left (a^{2} b c - a^{3} d\right )} \log \left (b x^{2} + a\right )}{2 \, b^{4}} \]
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Time = 0.29 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.03 \[ \int \frac {x^5 \left (c+d x^2\right )}{a+b x^2} \, dx=\frac {2 \, b^{2} d x^{6} + 3 \, b^{2} c x^{4} - 3 \, a b d x^{4} - 6 \, a b c x^{2} + 6 \, a^{2} d x^{2}}{12 \, b^{3}} + \frac {{\left (a^{2} b c - a^{3} d\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{4}} \]
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Time = 0.07 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.01 \[ \int \frac {x^5 \left (c+d x^2\right )}{a+b x^2} \, dx=x^4\,\left (\frac {c}{4\,b}-\frac {a\,d}{4\,b^2}\right )+\frac {d\,x^6}{6\,b}-\frac {\ln \left (b\,x^2+a\right )\,\left (a^3\,d-a^2\,b\,c\right )}{2\,b^4}-\frac {a\,x^2\,\left (\frac {c}{b}-\frac {a\,d}{b^2}\right )}{2\,b} \]
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