Integrand size = 22, antiderivative size = 98 \[ \int \frac {\left (c+d x^2\right )^3}{x^3 \left (a+b x^2\right )^2} \, dx=-\frac {c^3}{2 a^2 x^2}-\frac {(b c-a d)^3}{2 a^2 b^2 \left (a+b x^2\right )}-\frac {c^2 (2 b c-3 a d) \log (x)}{a^3}+\frac {(b c-a d)^2 (2 b c+a d) \log \left (a+b x^2\right )}{2 a^3 b^2} \]
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Time = 0.10 (sec) , antiderivative size = 98, 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, 90} \[ \int \frac {\left (c+d x^2\right )^3}{x^3 \left (a+b x^2\right )^2} \, dx=\frac {(b c-a d)^2 (a d+2 b c) \log \left (a+b x^2\right )}{2 a^3 b^2}-\frac {c^2 \log (x) (2 b c-3 a d)}{a^3}-\frac {(b c-a d)^3}{2 a^2 b^2 \left (a+b x^2\right )}-\frac {c^3}{2 a^2 x^2} \]
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Rule 90
Rule 457
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {(c+d x)^3}{x^2 (a+b x)^2} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {c^3}{a^2 x^2}+\frac {c^2 (-2 b c+3 a d)}{a^3 x}-\frac {(-b c+a d)^3}{a^2 b (a+b x)^2}+\frac {(-b c+a d)^2 (2 b c+a d)}{a^3 b (a+b x)}\right ) \, dx,x,x^2\right ) \\ & = -\frac {c^3}{2 a^2 x^2}-\frac {(b c-a d)^3}{2 a^2 b^2 \left (a+b x^2\right )}-\frac {c^2 (2 b c-3 a d) \log (x)}{a^3}+\frac {(b c-a d)^2 (2 b c+a d) \log \left (a+b x^2\right )}{2 a^3 b^2} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.89 \[ \int \frac {\left (c+d x^2\right )^3}{x^3 \left (a+b x^2\right )^2} \, dx=\frac {-\frac {a c^3}{x^2}+\frac {a (-b c+a d)^3}{b^2 \left (a+b x^2\right )}+2 c^2 (-2 b c+3 a d) \log (x)+\frac {(b c-a d)^2 (2 b c+a d) \log \left (a+b x^2\right )}{b^2}}{2 a^3} \]
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Time = 2.73 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.02
method | result | size |
default | \(-\frac {c^{3}}{2 a^{2} x^{2}}+\frac {c^{2} \left (3 a d -2 b c \right ) \ln \left (x \right )}{a^{3}}+\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (\frac {\left (a d +2 b c \right ) \ln \left (b \,x^{2}+a \right )}{b^{2}}+\frac {\left (a d -b c \right ) a}{b^{2} \left (b \,x^{2}+a \right )}\right )}{2 a^{3}}\) | \(100\) |
norman | \(\frac {-\frac {c^{3}}{2 a}+\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -2 b^{3} c^{3}\right ) x^{2}}{2 a^{2} b^{2}}}{x^{2} \left (b \,x^{2}+a \right )}+\frac {c^{2} \left (3 a d -2 b c \right ) \ln \left (x \right )}{a^{3}}+\frac {\left (a^{3} d^{3}-3 a \,b^{2} c^{2} d +2 b^{3} c^{3}\right ) \ln \left (b \,x^{2}+a \right )}{2 a^{3} b^{2}}\) | \(131\) |
risch | \(\frac {-\frac {c^{3}}{2 a}+\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -2 b^{3} c^{3}\right ) x^{2}}{2 a^{2} b^{2}}}{x^{2} \left (b \,x^{2}+a \right )}+\frac {3 c^{2} \ln \left (x \right ) d}{a^{2}}-\frac {2 c^{3} \ln \left (x \right ) b}{a^{3}}+\frac {\ln \left (-b \,x^{2}-a \right ) d^{3}}{2 b^{2}}-\frac {3 \ln \left (-b \,x^{2}-a \right ) c^{2} d}{2 a^{2}}+\frac {b \ln \left (-b \,x^{2}-a \right ) c^{3}}{a^{3}}\) | \(151\) |
parallelrisch | \(\frac {6 \ln \left (x \right ) x^{4} a \,b^{3} c^{2} d -4 \ln \left (x \right ) x^{4} b^{4} c^{3}+\ln \left (b \,x^{2}+a \right ) x^{4} a^{3} b \,d^{3}-3 \ln \left (b \,x^{2}+a \right ) x^{4} a \,b^{3} c^{2} d +2 \ln \left (b \,x^{2}+a \right ) x^{4} b^{4} c^{3}+6 \ln \left (x \right ) x^{2} a^{2} b^{2} c^{2} d -4 \ln \left (x \right ) x^{2} a \,b^{3} c^{3}+\ln \left (b \,x^{2}+a \right ) x^{2} a^{4} d^{3}-3 \ln \left (b \,x^{2}+a \right ) x^{2} a^{2} b^{2} c^{2} d +2 \ln \left (b \,x^{2}+a \right ) x^{2} a \,b^{3} c^{3}+a^{4} d^{3} x^{2}-3 a^{3} b c \,d^{2} x^{2}+3 a^{2} b^{2} c^{2} d \,x^{2}-2 b^{3} c^{3} a \,x^{2}-a^{2} b^{2} c^{3}}{2 a^{3} b^{2} x^{2} \left (b \,x^{2}+a \right )}\) | \(262\) |
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Leaf count of result is larger than twice the leaf count of optimal. 209 vs. \(2 (92) = 184\).
Time = 0.25 (sec) , antiderivative size = 209, normalized size of antiderivative = 2.13 \[ \int \frac {\left (c+d x^2\right )^3}{x^3 \left (a+b x^2\right )^2} \, dx=-\frac {a^{2} b^{2} c^{3} + {\left (2 \, a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} x^{2} - {\left ({\left (2 \, b^{4} c^{3} - 3 \, a b^{3} c^{2} d + a^{3} b d^{3}\right )} x^{4} + {\left (2 \, a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + a^{4} d^{3}\right )} x^{2}\right )} \log \left (b x^{2} + a\right ) + 2 \, {\left ({\left (2 \, b^{4} c^{3} - 3 \, a b^{3} c^{2} d\right )} x^{4} + {\left (2 \, a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d\right )} x^{2}\right )} \log \left (x\right )}{2 \, {\left (a^{3} b^{3} x^{4} + a^{4} b^{2} x^{2}\right )}} \]
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Time = 4.10 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.31 \[ \int \frac {\left (c+d x^2\right )^3}{x^3 \left (a+b x^2\right )^2} \, dx=\frac {- a b^{2} c^{3} + x^{2} \left (a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - 2 b^{3} c^{3}\right )}{2 a^{3} b^{2} x^{2} + 2 a^{2} b^{3} x^{4}} + \frac {c^{2} \cdot \left (3 a d - 2 b c\right ) \log {\left (x \right )}}{a^{3}} + \frac {\left (a d - b c\right )^{2} \left (a d + 2 b c\right ) \log {\left (\frac {a}{b} + x^{2} \right )}}{2 a^{3} b^{2}} \]
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Time = 0.20 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.44 \[ \int \frac {\left (c+d x^2\right )^3}{x^3 \left (a+b x^2\right )^2} \, dx=-\frac {a b^{2} c^{3} + {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{2}}{2 \, {\left (a^{2} b^{3} x^{4} + a^{3} b^{2} x^{2}\right )}} - \frac {{\left (2 \, b c^{3} - 3 \, a c^{2} d\right )} \log \left (x^{2}\right )}{2 \, a^{3}} + \frac {{\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d + a^{3} d^{3}\right )} \log \left (b x^{2} + a\right )}{2 \, a^{3} b^{2}} \]
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Time = 0.29 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.60 \[ \int \frac {\left (c+d x^2\right )^3}{x^3 \left (a+b x^2\right )^2} \, dx=-\frac {{\left (2 \, b c^{3} - 3 \, a c^{2} d\right )} \log \left (x^{2}\right )}{2 \, a^{3}} + \frac {{\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d + a^{3} d^{3}\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{3} b^{2}} - \frac {a^{2} b d^{3} x^{4} + 4 \, b^{3} c^{3} x^{2} - 6 \, a b^{2} c^{2} d x^{2} + 6 \, a^{2} b c d^{2} x^{2} - a^{3} d^{3} x^{2} + 2 \, a b^{2} c^{3}}{4 \, {\left (b x^{4} + a x^{2}\right )} a^{2} b^{2}} \]
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Time = 5.04 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.38 \[ \int \frac {\left (c+d x^2\right )^3}{x^3 \left (a+b x^2\right )^2} \, dx=\frac {\ln \left (b\,x^2+a\right )\,\left (a^3\,d^3-3\,a\,b^2\,c^2\,d+2\,b^3\,c^3\right )}{2\,a^3\,b^2}-\frac {\ln \left (x\right )\,\left (2\,b\,c^3-3\,a\,c^2\,d\right )}{a^3}-\frac {\frac {c^3}{2\,a}-\frac {x^2\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-2\,b^3\,c^3\right )}{2\,a^2\,b^2}}{b\,x^4+a\,x^2} \]
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