Integrand size = 22, antiderivative size = 88 \[ \int \frac {x^3 \left (c+d x^2\right )^2}{\left (a+b x^2\right )^2} \, dx=\frac {d (b c-a d) x^2}{b^3}+\frac {d^2 x^4}{4 b^2}+\frac {a (b c-a d)^2}{2 b^4 \left (a+b x^2\right )}+\frac {(b c-3 a d) (b c-a d) \log \left (a+b x^2\right )}{2 b^4} \]
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Time = 0.10 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {457, 78} \[ \int \frac {x^3 \left (c+d x^2\right )^2}{\left (a+b x^2\right )^2} \, dx=\frac {a (b c-a d)^2}{2 b^4 \left (a+b x^2\right )}+\frac {(b c-3 a d) (b c-a d) \log \left (a+b x^2\right )}{2 b^4}+\frac {d x^2 (b c-a d)}{b^3}+\frac {d^2 x^4}{4 b^2} \]
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Rule 78
Rule 457
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {x (c+d x)^2}{(a+b x)^2} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {2 d (b c-a d)}{b^3}+\frac {d^2 x}{b^2}-\frac {a (-b c+a d)^2}{b^3 (a+b x)^2}+\frac {(b c-3 a d) (b c-a d)}{b^3 (a+b x)}\right ) \, dx,x,x^2\right ) \\ & = \frac {d (b c-a d) x^2}{b^3}+\frac {d^2 x^4}{4 b^2}+\frac {a (b c-a d)^2}{2 b^4 \left (a+b x^2\right )}+\frac {(b c-3 a d) (b c-a d) \log \left (a+b x^2\right )}{2 b^4} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.99 \[ \int \frac {x^3 \left (c+d x^2\right )^2}{\left (a+b x^2\right )^2} \, dx=\frac {4 b d (b c-a d) x^2+b^2 d^2 x^4+\frac {2 a (b c-a d)^2}{a+b x^2}+2 \left (b^2 c^2-4 a b c d+3 a^2 d^2\right ) \log \left (a+b x^2\right )}{4 b^4} \]
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Time = 2.74 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.93
method | result | size |
default | \(\frac {\left (-b d \,x^{2}+2 a d -2 b c \right )^{2}}{4 b^{4}}+\frac {\left (a d -b c \right ) \left (\frac {\left (3 a d -b c \right ) \ln \left (b \,x^{2}+a \right )}{b}+\frac {\left (a d -b c \right ) a}{b \left (b \,x^{2}+a \right )}\right )}{2 b^{3}}\) | \(82\) |
norman | \(\frac {\frac {d^{2} x^{6}}{4 b}-\frac {d \left (3 a d -4 b c \right ) x^{4}}{4 b^{2}}-\frac {\left (3 a^{3} d^{2}-4 a^{2} b c d +b^{2} c^{2} a \right ) x^{2}}{2 a \,b^{3}}}{b \,x^{2}+a}+\frac {\left (3 a^{2} d^{2}-4 a b c d +b^{2} c^{2}\right ) \ln \left (b \,x^{2}+a \right )}{2 b^{4}}\) | \(113\) |
risch | \(\frac {d^{2} x^{4}}{4 b^{2}}-\frac {x^{2} a \,d^{2}}{b^{3}}+\frac {x^{2} c d}{b^{2}}+\frac {a^{2} d^{2}}{b^{4}}-\frac {2 a c d}{b^{3}}+\frac {c^{2}}{b^{2}}+\frac {a^{3} d^{2}}{2 b^{4} \left (b \,x^{2}+a \right )}-\frac {a^{2} c d}{b^{3} \left (b \,x^{2}+a \right )}+\frac {a \,c^{2}}{2 b^{2} \left (b \,x^{2}+a \right )}+\frac {3 \ln \left (b \,x^{2}+a \right ) a^{2} d^{2}}{2 b^{4}}-\frac {2 \ln \left (b \,x^{2}+a \right ) a c d}{b^{3}}+\frac {\ln \left (b \,x^{2}+a \right ) c^{2}}{2 b^{2}}\) | \(167\) |
parallelrisch | \(\frac {b^{3} d^{2} x^{6}-3 x^{4} a \,b^{2} d^{2}+4 x^{4} b^{3} c d +6 \ln \left (b \,x^{2}+a \right ) x^{2} a^{2} b \,d^{2}-8 \ln \left (b \,x^{2}+a \right ) x^{2} a \,b^{2} c d +2 \ln \left (b \,x^{2}+a \right ) x^{2} b^{3} c^{2}+6 \ln \left (b \,x^{2}+a \right ) a^{3} d^{2}-8 \ln \left (b \,x^{2}+a \right ) a^{2} b c d +2 \ln \left (b \,x^{2}+a \right ) a \,b^{2} c^{2}+6 a^{3} d^{2}-8 a^{2} b c d +2 b^{2} c^{2} a}{4 b^{4} \left (b \,x^{2}+a \right )}\) | \(180\) |
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Time = 0.23 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.82 \[ \int \frac {x^3 \left (c+d x^2\right )^2}{\left (a+b x^2\right )^2} \, dx=\frac {b^{3} d^{2} x^{6} + 2 \, a b^{2} c^{2} - 4 \, a^{2} b c d + 2 \, a^{3} d^{2} + {\left (4 \, b^{3} c d - 3 \, a b^{2} d^{2}\right )} x^{4} + 4 \, {\left (a b^{2} c d - a^{2} b d^{2}\right )} x^{2} + 2 \, {\left (a b^{2} c^{2} - 4 \, a^{2} b c d + 3 \, a^{3} d^{2} + {\left (b^{3} c^{2} - 4 \, a b^{2} c d + 3 \, a^{2} b d^{2}\right )} x^{2}\right )} \log \left (b x^{2} + a\right )}{4 \, {\left (b^{5} x^{2} + a b^{4}\right )}} \]
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Time = 0.52 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.12 \[ \int \frac {x^3 \left (c+d x^2\right )^2}{\left (a+b x^2\right )^2} \, dx=x^{2} \left (- \frac {a d^{2}}{b^{3}} + \frac {c d}{b^{2}}\right ) + \frac {a^{3} d^{2} - 2 a^{2} b c d + a b^{2} c^{2}}{2 a b^{4} + 2 b^{5} x^{2}} + \frac {d^{2} x^{4}}{4 b^{2}} + \frac {\left (a d - b c\right ) \left (3 a d - b c\right ) \log {\left (a + b x^{2} \right )}}{2 b^{4}} \]
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Time = 0.23 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.22 \[ \int \frac {x^3 \left (c+d x^2\right )^2}{\left (a+b x^2\right )^2} \, dx=\frac {a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}}{2 \, {\left (b^{5} x^{2} + a b^{4}\right )}} + \frac {b d^{2} x^{4} + 4 \, {\left (b c d - a d^{2}\right )} x^{2}}{4 \, b^{3}} + \frac {{\left (b^{2} c^{2} - 4 \, a b c d + 3 \, a^{2} d^{2}\right )} \log \left (b x^{2} + a\right )}{2 \, b^{4}} \]
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Time = 0.30 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.85 \[ \int \frac {x^3 \left (c+d x^2\right )^2}{\left (a+b x^2\right )^2} \, dx=\frac {\frac {{\left (b x^{2} + a\right )}^{2} {\left (d^{2} + \frac {2 \, {\left (2 \, b^{2} c d - 3 \, a b d^{2}\right )}}{{\left (b x^{2} + a\right )} b}\right )}}{b^{3}} - \frac {2 \, {\left (b^{2} c^{2} - 4 \, a b c d + 3 \, a^{2} d^{2}\right )} \log \left (\frac {{\left | b x^{2} + a \right |}}{{\left (b x^{2} + a\right )}^{2} {\left | b \right |}}\right )}{b^{3}} + \frac {2 \, {\left (\frac {a b^{4} c^{2}}{b x^{2} + a} - \frac {2 \, a^{2} b^{3} c d}{b x^{2} + a} + \frac {a^{3} b^{2} d^{2}}{b x^{2} + a}\right )}}{b^{5}}}{4 \, b} \]
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Time = 0.08 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.27 \[ \int \frac {x^3 \left (c+d x^2\right )^2}{\left (a+b x^2\right )^2} \, dx=\frac {a^3\,d^2-2\,a^2\,b\,c\,d+a\,b^2\,c^2}{2\,b\,\left (b^4\,x^2+a\,b^3\right )}-x^2\,\left (\frac {a\,d^2}{b^3}-\frac {c\,d}{b^2}\right )+\frac {d^2\,x^4}{4\,b^2}+\frac {\ln \left (b\,x^2+a\right )\,\left (3\,a^2\,d^2-4\,a\,b\,c\,d+b^2\,c^2\right )}{2\,b^4} \]
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