Integrand size = 22, antiderivative size = 117 \[ \int \frac {x^3 \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=\frac {3 d (b c-a d)^2 x^2}{2 b^4}+\frac {d^2 (3 b c-2 a d) x^4}{4 b^3}+\frac {d^3 x^6}{6 b^2}+\frac {a (b c-a d)^3}{2 b^5 \left (a+b x^2\right )}+\frac {(b c-4 a d) (b c-a d)^2 \log \left (a+b x^2\right )}{2 b^5} \]
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Time = 0.12 (sec) , antiderivative size = 117, 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} \[ \int \frac {x^3 \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=\frac {a (b c-a d)^3}{2 b^5 \left (a+b x^2\right )}+\frac {(b c-4 a d) (b c-a d)^2 \log \left (a+b x^2\right )}{2 b^5}+\frac {3 d x^2 (b c-a d)^2}{2 b^4}+\frac {d^2 x^4 (3 b c-2 a d)}{4 b^3}+\frac {d^3 x^6}{6 b^2} \]
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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)^3}{(a+b x)^2} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {3 d (b c-a d)^2}{b^4}+\frac {d^2 (3 b c-2 a d) x}{b^3}+\frac {d^3 x^2}{b^2}+\frac {a (-b c+a d)^3}{b^4 (a+b x)^2}+\frac {(b c-4 a d) (b c-a d)^2}{b^4 (a+b x)}\right ) \, dx,x,x^2\right ) \\ & = \frac {3 d (b c-a d)^2 x^2}{2 b^4}+\frac {d^2 (3 b c-2 a d) x^4}{4 b^3}+\frac {d^3 x^6}{6 b^2}+\frac {a (b c-a d)^3}{2 b^5 \left (a+b x^2\right )}+\frac {(b c-4 a d) (b c-a d)^2 \log \left (a+b x^2\right )}{2 b^5} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.91 \[ \int \frac {x^3 \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=\frac {18 b d (b c-a d)^2 x^2+3 b^2 d^2 (3 b c-2 a d) x^4+2 b^3 d^3 x^6-\frac {6 a (-b c+a d)^3}{a+b x^2}+6 (b c-4 a d) (b c-a d)^2 \log \left (a+b x^2\right )}{12 b^5} \]
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Time = 2.61 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.17
method | result | size |
default | \(\frac {d \left (\frac {b^{2} d^{2} x^{6}}{6}+\frac {\left (-2 a b \,d^{2}+3 b^{2} c d \right ) x^{4}}{4}+\frac {\left (3 a^{2} d^{2}-6 a b c d +3 b^{2} c^{2}\right ) x^{2}}{2}\right )}{b^{4}}-\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (\frac {\left (4 a d -b c \right ) \ln \left (b \,x^{2}+a \right )}{b}+\frac {\left (a d -b c \right ) a}{b \left (b \,x^{2}+a \right )}\right )}{2 b^{4}}\) | \(137\) |
norman | \(\frac {\frac {d^{3} x^{8}}{6 b}+\frac {d \left (4 a^{2} d^{2}-9 a b c d +6 b^{2} c^{2}\right ) x^{4}}{4 b^{3}}-\frac {d^{2} \left (4 a d -9 b c \right ) x^{6}}{12 b^{2}}+\frac {\left (4 a^{4} d^{3}-9 a^{3} b c \,d^{2}+6 a^{2} b^{2} c^{2} d -a \,b^{3} c^{3}\right ) x^{2}}{2 a \,b^{4}}}{b \,x^{2}+a}-\frac {\left (4 a^{3} d^{3}-9 a^{2} b c \,d^{2}+6 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (b \,x^{2}+a \right )}{2 b^{5}}\) | \(177\) |
risch | \(\frac {d^{3} x^{6}}{6 b^{2}}-\frac {d^{3} x^{4} a}{2 b^{3}}+\frac {3 d^{2} x^{4} c}{4 b^{2}}+\frac {3 d^{3} a^{2} x^{2}}{2 b^{4}}-\frac {3 d^{2} a c \,x^{2}}{b^{3}}+\frac {3 d \,c^{2} x^{2}}{2 b^{2}}-\frac {a^{4} d^{3}}{2 b^{5} \left (b \,x^{2}+a \right )}+\frac {3 a^{3} c \,d^{2}}{2 b^{4} \left (b \,x^{2}+a \right )}-\frac {3 a^{2} c^{2} d}{2 b^{3} \left (b \,x^{2}+a \right )}+\frac {a \,c^{3}}{2 b^{2} \left (b \,x^{2}+a \right )}-\frac {2 \ln \left (b \,x^{2}+a \right ) a^{3} d^{3}}{b^{5}}+\frac {9 \ln \left (b \,x^{2}+a \right ) a^{2} c \,d^{2}}{2 b^{4}}-\frac {3 \ln \left (b \,x^{2}+a \right ) a \,c^{2} d}{b^{3}}+\frac {\ln \left (b \,x^{2}+a \right ) c^{3}}{2 b^{2}}\) | \(229\) |
parallelrisch | \(-\frac {-2 b^{4} d^{3} x^{8}+4 x^{6} a \,b^{3} d^{3}-9 x^{6} b^{4} c \,d^{2}-12 x^{4} a^{2} b^{2} d^{3}+27 x^{4} a \,b^{3} c \,d^{2}-18 x^{4} b^{4} c^{2} d +24 \ln \left (b \,x^{2}+a \right ) x^{2} a^{3} b \,d^{3}-54 \ln \left (b \,x^{2}+a \right ) x^{2} a^{2} b^{2} c \,d^{2}+36 \ln \left (b \,x^{2}+a \right ) x^{2} a \,b^{3} c^{2} d -6 \ln \left (b \,x^{2}+a \right ) x^{2} b^{4} c^{3}+24 \ln \left (b \,x^{2}+a \right ) a^{4} d^{3}-54 \ln \left (b \,x^{2}+a \right ) a^{3} b c \,d^{2}+36 \ln \left (b \,x^{2}+a \right ) a^{2} b^{2} c^{2} d -6 \ln \left (b \,x^{2}+a \right ) a \,b^{3} c^{3}+24 a^{4} d^{3}-54 a^{3} b c \,d^{2}+36 a^{2} b^{2} c^{2} d -6 a \,b^{3} c^{3}}{12 b^{5} \left (b \,x^{2}+a \right )}\) | \(283\) |
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Leaf count of result is larger than twice the leaf count of optimal. 254 vs. \(2 (107) = 214\).
Time = 0.24 (sec) , antiderivative size = 254, normalized size of antiderivative = 2.17 \[ \int \frac {x^3 \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=\frac {2 \, b^{4} d^{3} x^{8} + 6 \, a b^{3} c^{3} - 18 \, a^{2} b^{2} c^{2} d + 18 \, a^{3} b c d^{2} - 6 \, a^{4} d^{3} + {\left (9 \, b^{4} c d^{2} - 4 \, a b^{3} d^{3}\right )} x^{6} + 3 \, {\left (6 \, b^{4} c^{2} d - 9 \, a b^{3} c d^{2} + 4 \, a^{2} b^{2} d^{3}\right )} x^{4} + 18 \, {\left (a b^{3} c^{2} d - 2 \, a^{2} b^{2} c d^{2} + a^{3} b d^{3}\right )} x^{2} + 6 \, {\left (a b^{3} c^{3} - 6 \, a^{2} b^{2} c^{2} d + 9 \, a^{3} b c d^{2} - 4 \, a^{4} d^{3} + {\left (b^{4} c^{3} - 6 \, a b^{3} c^{2} d + 9 \, a^{2} b^{2} c d^{2} - 4 \, a^{3} b d^{3}\right )} x^{2}\right )} \log \left (b x^{2} + a\right )}{12 \, {\left (b^{6} x^{2} + a b^{5}\right )}} \]
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Time = 0.97 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.39 \[ \int \frac {x^3 \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=x^{4} \left (- \frac {a d^{3}}{2 b^{3}} + \frac {3 c d^{2}}{4 b^{2}}\right ) + x^{2} \cdot \left (\frac {3 a^{2} d^{3}}{2 b^{4}} - \frac {3 a c d^{2}}{b^{3}} + \frac {3 c^{2} d}{2 b^{2}}\right ) + \frac {- a^{4} d^{3} + 3 a^{3} b c d^{2} - 3 a^{2} b^{2} c^{2} d + a b^{3} c^{3}}{2 a b^{5} + 2 b^{6} x^{2}} + \frac {d^{3} x^{6}}{6 b^{2}} - \frac {\left (a d - b c\right )^{2} \cdot \left (4 a d - b c\right ) \log {\left (a + b x^{2} \right )}}{2 b^{5}} \]
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Time = 0.20 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.49 \[ \int \frac {x^3 \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=\frac {a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}}{2 \, {\left (b^{6} x^{2} + a b^{5}\right )}} + \frac {2 \, b^{2} d^{3} x^{6} + 3 \, {\left (3 \, b^{2} c d^{2} - 2 \, a b d^{3}\right )} x^{4} + 18 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x^{2}}{12 \, b^{4}} + \frac {{\left (b^{3} c^{3} - 6 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} - 4 \, a^{3} d^{3}\right )} \log \left (b x^{2} + a\right )}{2 \, b^{5}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 249 vs. \(2 (107) = 214\).
Time = 0.30 (sec) , antiderivative size = 249, normalized size of antiderivative = 2.13 \[ \int \frac {x^3 \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=\frac {\frac {{\left (2 \, d^{3} + \frac {3 \, {\left (3 \, b^{2} c d^{2} - 4 \, a b d^{3}\right )}}{{\left (b x^{2} + a\right )} b} + \frac {18 \, {\left (b^{4} c^{2} d - 3 \, a b^{3} c d^{2} + 2 \, a^{2} b^{2} d^{3}\right )}}{{\left (b x^{2} + a\right )}^{2} b^{2}}\right )} {\left (b x^{2} + a\right )}^{3}}{b^{4}} - \frac {6 \, {\left (b^{3} c^{3} - 6 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} - 4 \, a^{3} d^{3}\right )} \log \left (\frac {{\left | b x^{2} + a \right |}}{{\left (b x^{2} + a\right )}^{2} {\left | b \right |}}\right )}{b^{4}} + \frac {6 \, {\left (\frac {a b^{6} c^{3}}{b x^{2} + a} - \frac {3 \, a^{2} b^{5} c^{2} d}{b x^{2} + a} + \frac {3 \, a^{3} b^{4} c d^{2}}{b x^{2} + a} - \frac {a^{4} b^{3} d^{3}}{b x^{2} + a}\right )}}{b^{7}}}{12 \, b} \]
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Time = 4.97 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.66 \[ \int \frac {x^3 \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=x^2\,\left (\frac {3\,c^2\,d}{2\,b^2}+\frac {a\,\left (\frac {2\,a\,d^3}{b^3}-\frac {3\,c\,d^2}{b^2}\right )}{b}-\frac {a^2\,d^3}{2\,b^4}\right )-x^4\,\left (\frac {a\,d^3}{2\,b^3}-\frac {3\,c\,d^2}{4\,b^2}\right )-\frac {\ln \left (b\,x^2+a\right )\,\left (4\,a^3\,d^3-9\,a^2\,b\,c\,d^2+6\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{2\,b^5}-\frac {a^4\,d^3-3\,a^3\,b\,c\,d^2+3\,a^2\,b^2\,c^2\,d-a\,b^3\,c^3}{2\,b\,\left (b^5\,x^2+a\,b^4\right )}+\frac {d^3\,x^6}{6\,b^2} \]
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