Integrand size = 22, antiderivative size = 73 \[ \int \frac {\left (c+d x^2\right )^3}{x^3 \left (a+b x^2\right )} \, dx=-\frac {c^3}{2 a x^2}+\frac {d^3 x^2}{2 b}-\frac {c^2 (b c-3 a d) \log (x)}{a^2}+\frac {(b c-a d)^3 \log \left (a+b x^2\right )}{2 a^2 b^2} \]
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Time = 0.05 (sec) , antiderivative size = 73, 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} \[ \int \frac {\left (c+d x^2\right )^3}{x^3 \left (a+b x^2\right )} \, dx=\frac {(b c-a d)^3 \log \left (a+b x^2\right )}{2 a^2 b^2}-\frac {c^2 \log (x) (b c-3 a d)}{a^2}-\frac {c^3}{2 a x^2}+\frac {d^3 x^2}{2 b} \]
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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)} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {d^3}{b}+\frac {c^3}{a x^2}+\frac {c^2 (-b c+3 a d)}{a^2 x}-\frac {(-b c+a d)^3}{a^2 b (a+b x)}\right ) \, dx,x,x^2\right ) \\ & = -\frac {c^3}{2 a x^2}+\frac {d^3 x^2}{2 b}-\frac {c^2 (b c-3 a d) \log (x)}{a^2}+\frac {(b c-a d)^3 \log \left (a+b x^2\right )}{2 a^2 b^2} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.03 \[ \int \frac {\left (c+d x^2\right )^3}{x^3 \left (a+b x^2\right )} \, dx=\frac {a b \left (-b c^3+a d^3 x^4\right )-2 b^2 c^2 (b c-3 a d) x^2 \log (x)+(b c-a d)^3 x^2 \log \left (a+b x^2\right )}{2 a^2 b^2 x^2} \]
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Time = 2.69 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.29
method | result | size |
default | \(\frac {d^{3} x^{2}}{2 b}-\frac {c^{3}}{2 a \,x^{2}}+\frac {c^{2} \left (3 a d -b c \right ) \ln \left (x \right )}{a^{2}}-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (b \,x^{2}+a \right )}{2 a^{2} b^{2}}\) | \(94\) |
norman | \(\frac {-\frac {c^{3}}{2 a}+\frac {d^{3} x^{4}}{2 b}}{x^{2}}+\frac {c^{2} \left (3 a d -b c \right ) \ln \left (x \right )}{a^{2}}-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (b \,x^{2}+a \right )}{2 a^{2} b^{2}}\) | \(96\) |
risch | \(\frac {d^{3} x^{2}}{2 b}-\frac {c^{3}}{2 a \,x^{2}}+\frac {3 c^{2} \ln \left (x \right ) d}{a}-\frac {c^{3} \ln \left (x \right ) b}{a^{2}}-\frac {a \ln \left (b \,x^{2}+a \right ) d^{3}}{2 b^{2}}+\frac {3 \ln \left (b \,x^{2}+a \right ) c \,d^{2}}{2 b}-\frac {3 \ln \left (b \,x^{2}+a \right ) c^{2} d}{2 a}+\frac {b \ln \left (b \,x^{2}+a \right ) c^{3}}{2 a^{2}}\) | \(114\) |
parallelrisch | \(\frac {x^{4} a^{2} b \,d^{3}+6 \ln \left (x \right ) x^{2} a \,b^{2} c^{2} d -2 \ln \left (x \right ) x^{2} b^{3} c^{3}-\ln \left (b \,x^{2}+a \right ) x^{2} a^{3} d^{3}+3 \ln \left (b \,x^{2}+a \right ) x^{2} a^{2} b c \,d^{2}-3 \ln \left (b \,x^{2}+a \right ) x^{2} a \,b^{2} c^{2} d +\ln \left (b \,x^{2}+a \right ) x^{2} b^{3} c^{3}-b^{2} c^{3} a}{2 a^{2} b^{2} x^{2}}\) | \(140\) |
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Time = 0.24 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.44 \[ \int \frac {\left (c+d x^2\right )^3}{x^3 \left (a+b x^2\right )} \, dx=\frac {a^{2} b d^{3} x^{4} - a b^{2} c^{3} + {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{2} \log \left (b x^{2} + a\right ) - 2 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d\right )} x^{2} \log \left (x\right )}{2 \, a^{2} b^{2} x^{2}} \]
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Time = 1.28 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.86 \[ \int \frac {\left (c+d x^2\right )^3}{x^3 \left (a+b x^2\right )} \, dx=\frac {d^{3} x^{2}}{2 b} - \frac {c^{3}}{2 a x^{2}} + \frac {c^{2} \cdot \left (3 a d - b c\right ) \log {\left (x \right )}}{a^{2}} - \frac {\left (a d - b c\right )^{3} \log {\left (\frac {a}{b} + x^{2} \right )}}{2 a^{2} b^{2}} \]
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Time = 0.21 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.33 \[ \int \frac {\left (c+d x^2\right )^3}{x^3 \left (a+b x^2\right )} \, dx=\frac {d^{3} x^{2}}{2 \, b} - \frac {c^{3}}{2 \, a x^{2}} - \frac {{\left (b c^{3} - 3 \, a c^{2} d\right )} \log \left (x^{2}\right )}{2 \, a^{2}} + \frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (b x^{2} + a\right )}{2 \, a^{2} b^{2}} \]
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Time = 0.28 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.64 \[ \int \frac {\left (c+d x^2\right )^3}{x^3 \left (a+b x^2\right )} \, dx=\frac {d^{3} x^{2}}{2 \, b} - \frac {{\left (b c^{3} - 3 \, a c^{2} d\right )} \log \left (x^{2}\right )}{2 \, a^{2}} + \frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{2} b^{2}} + \frac {b c^{3} x^{2} - 3 \, a c^{2} d x^{2} - a c^{3}}{2 \, a^{2} x^{2}} \]
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Time = 5.38 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.30 \[ \int \frac {\left (c+d x^2\right )^3}{x^3 \left (a+b x^2\right )} \, dx=\frac {d^3\,x^2}{2\,b}-\frac {c^3}{2\,a\,x^2}-\frac {\ln \left (x\right )\,\left (b\,c^3-3\,a\,c^2\,d\right )}{a^2}-\frac {\ln \left (b\,x^2+a\right )\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{2\,a^2\,b^2} \]
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