Integrand size = 20, antiderivative size = 87 \[ \int \frac {x \left (c+d x^2\right )^3}{a+b x^2} \, dx=\frac {d (b c-a d)^2 x^2}{2 b^3}+\frac {(b c-a d) \left (c+d x^2\right )^2}{4 b^2}+\frac {\left (c+d x^2\right )^3}{6 b}+\frac {(b c-a d)^3 \log \left (a+b x^2\right )}{2 b^4} \]
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Time = 0.06 (sec) , antiderivative size = 87, 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 = {455, 45} \[ \int \frac {x \left (c+d x^2\right )^3}{a+b x^2} \, dx=\frac {(b c-a d)^3 \log \left (a+b x^2\right )}{2 b^4}+\frac {d x^2 (b c-a d)^2}{2 b^3}+\frac {\left (c+d x^2\right )^2 (b c-a d)}{4 b^2}+\frac {\left (c+d x^2\right )^3}{6 b} \]
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Rule 45
Rule 455
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {(c+d x)^3}{a+b x} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {d (b c-a d)^2}{b^3}+\frac {(b c-a d)^3}{b^3 (a+b x)}+\frac {d (b c-a d) (c+d x)}{b^2}+\frac {d (c+d x)^2}{b}\right ) \, dx,x,x^2\right ) \\ & = \frac {d (b c-a d)^2 x^2}{2 b^3}+\frac {(b c-a d) \left (c+d x^2\right )^2}{4 b^2}+\frac {\left (c+d x^2\right )^3}{6 b}+\frac {(b c-a d)^3 \log \left (a+b x^2\right )}{2 b^4} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.94 \[ \int \frac {x \left (c+d x^2\right )^3}{a+b x^2} \, dx=\frac {b d x^2 \left (6 a^2 d^2-3 a b d \left (6 c+d x^2\right )+b^2 \left (18 c^2+9 c d x^2+2 d^2 x^4\right )\right )+6 (b c-a d)^3 \log \left (a+b x^2\right )}{12 b^4} \]
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Time = 2.66 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.29
method | result | size |
norman | \(\frac {d^{3} x^{6}}{6 b}+\frac {d \left (a^{2} d^{2}-3 a b c d +3 b^{2} c^{2}\right ) x^{2}}{2 b^{3}}-\frac {d^{2} \left (a d -3 b c \right ) x^{4}}{4 b^{2}}-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (b \,x^{2}+a \right )}{2 b^{4}}\) | \(112\) |
default | \(\frac {d \left (\frac {1}{3} b^{2} d^{2} x^{6}-\frac {1}{2} x^{4} a b \,d^{2}+\frac {3}{2} x^{4} b^{2} c d +a^{2} d^{2} x^{2}-3 a b c d \,x^{2}+3 b^{2} c^{2} x^{2}\right )}{2 b^{3}}+\frac {\left (-a^{3} d^{3}+3 a^{2} b c \,d^{2}-3 a \,b^{2} c^{2} d +b^{3} c^{3}\right ) \ln \left (b \,x^{2}+a \right )}{2 b^{4}}\) | \(119\) |
parallelrisch | \(-\frac {-2 b^{3} d^{3} x^{6}+3 a \,b^{2} d^{3} x^{4}-9 x^{4} b^{3} c \,d^{2}-6 x^{2} a^{2} b \,d^{3}+18 x^{2} a \,b^{2} c \,d^{2}-18 x^{2} b^{3} c^{2} d +6 \ln \left (b \,x^{2}+a \right ) a^{3} d^{3}-18 \ln \left (b \,x^{2}+a \right ) a^{2} b c \,d^{2}+18 \ln \left (b \,x^{2}+a \right ) a \,b^{2} c^{2} d -6 \ln \left (b \,x^{2}+a \right ) b^{3} c^{3}}{12 b^{4}}\) | \(147\) |
risch | \(\frac {d^{3} x^{6}}{6 b}-\frac {d^{3} x^{4} a}{4 b^{2}}+\frac {3 d^{2} x^{4} c}{4 b}+\frac {d^{3} a^{2} x^{2}}{2 b^{3}}-\frac {3 d^{2} a c \,x^{2}}{2 b^{2}}+\frac {3 d \,c^{2} x^{2}}{2 b}-\frac {\ln \left (b \,x^{2}+a \right ) a^{3} d^{3}}{2 b^{4}}+\frac {3 \ln \left (b \,x^{2}+a \right ) a^{2} c \,d^{2}}{2 b^{3}}-\frac {3 \ln \left (b \,x^{2}+a \right ) a \,c^{2} d}{2 b^{2}}+\frac {\ln \left (b \,x^{2}+a \right ) c^{3}}{2 b}\) | \(149\) |
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Time = 0.23 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.38 \[ \int \frac {x \left (c+d x^2\right )^3}{a+b x^2} \, dx=\frac {2 \, b^{3} d^{3} x^{6} + 3 \, {\left (3 \, b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{4} + 6 \, {\left (3 \, b^{3} c^{2} d - 3 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{2} + 6 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (b x^{2} + a\right )}{12 \, b^{4}} \]
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Time = 0.28 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.08 \[ \int \frac {x \left (c+d x^2\right )^3}{a+b x^2} \, dx=x^{4} \left (- \frac {a d^{3}}{4 b^{2}} + \frac {3 c d^{2}}{4 b}\right ) + x^{2} \left (\frac {a^{2} d^{3}}{2 b^{3}} - \frac {3 a c d^{2}}{2 b^{2}} + \frac {3 c^{2} d}{2 b}\right ) + \frac {d^{3} x^{6}}{6 b} - \frac {\left (a d - b c\right )^{3} \log {\left (a + b x^{2} \right )}}{2 b^{4}} \]
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Time = 0.19 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.37 \[ \int \frac {x \left (c+d x^2\right )^3}{a+b x^2} \, dx=\frac {2 \, b^{2} d^{3} x^{6} + 3 \, {\left (3 \, b^{2} c d^{2} - a b d^{3}\right )} x^{4} + 6 \, {\left (3 \, b^{2} c^{2} d - 3 \, a b c d^{2} + a^{2} d^{3}\right )} x^{2}}{12 \, b^{3}} + \frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (b x^{2} + a\right )}{2 \, b^{4}} \]
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Time = 0.28 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.43 \[ \int \frac {x \left (c+d x^2\right )^3}{a+b x^2} \, dx=\frac {2 \, b^{2} d^{3} x^{6} + 9 \, b^{2} c d^{2} x^{4} - 3 \, a b d^{3} x^{4} + 18 \, b^{2} c^{2} d x^{2} - 18 \, a b c d^{2} x^{2} + 6 \, a^{2} d^{3} x^{2}}{12 \, b^{3}} + \frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{4}} \]
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Time = 0.07 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.41 \[ \int \frac {x \left (c+d x^2\right )^3}{a+b x^2} \, dx=x^2\,\left (\frac {3\,c^2\,d}{2\,b}+\frac {a\,\left (\frac {a\,d^3}{b^2}-\frac {3\,c\,d^2}{b}\right )}{2\,b}\right )-x^4\,\left (\frac {a\,d^3}{4\,b^2}-\frac {3\,c\,d^2}{4\,b}\right )-\frac {\ln \left (b\,x^2+a\right )\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{2\,b^4}+\frac {d^3\,x^6}{6\,b} \]
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