Integrand size = 22, antiderivative size = 116 \[ \int \frac {x^5}{\left (a+b x^2\right )^3 \left (c+d x^2\right )} \, dx=-\frac {a^2}{4 b^2 (b c-a d) \left (a+b x^2\right )^2}+\frac {a (2 b c-a d)}{2 b^2 (b c-a d)^2 \left (a+b x^2\right )}+\frac {c^2 \log \left (a+b x^2\right )}{2 (b c-a d)^3}-\frac {c^2 \log \left (c+d x^2\right )}{2 (b c-a d)^3} \]
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Time = 0.08 (sec) , antiderivative size = 116, 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} \[ \int \frac {x^5}{\left (a+b x^2\right )^3 \left (c+d x^2\right )} \, dx=-\frac {a^2}{4 b^2 \left (a+b x^2\right )^2 (b c-a d)}+\frac {a (2 b c-a d)}{2 b^2 \left (a+b x^2\right ) (b c-a d)^2}+\frac {c^2 \log \left (a+b x^2\right )}{2 (b c-a d)^3}-\frac {c^2 \log \left (c+d x^2\right )}{2 (b c-a d)^3} \]
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Rule 90
Rule 457
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {x^2}{(a+b x)^3 (c+d x)} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {a^2}{b (b c-a d) (a+b x)^3}+\frac {a (-2 b c+a d)}{b (b c-a d)^2 (a+b x)^2}+\frac {b c^2}{(b c-a d)^3 (a+b x)}-\frac {c^2 d}{(b c-a d)^3 (c+d x)}\right ) \, dx,x,x^2\right ) \\ & = -\frac {a^2}{4 b^2 (b c-a d) \left (a+b x^2\right )^2}+\frac {a (2 b c-a d)}{2 b^2 (b c-a d)^2 \left (a+b x^2\right )}+\frac {c^2 \log \left (a+b x^2\right )}{2 (b c-a d)^3}-\frac {c^2 \log \left (c+d x^2\right )}{2 (b c-a d)^3} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.99 \[ \int \frac {x^5}{\left (a+b x^2\right )^3 \left (c+d x^2\right )} \, dx=\frac {a (-b c+a d) \left (-3 a b c+a^2 d-4 b^2 c x^2+2 a b d x^2\right )+2 b^2 c^2 \left (a+b x^2\right )^2 \log \left (a+b x^2\right )-2 b^2 c^2 \left (a+b x^2\right )^2 \log \left (c+d x^2\right )}{4 b^2 (b c-a d)^3 \left (a+b x^2\right )^2} \]
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Time = 2.68 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.07
method | result | size |
default | \(-\frac {c^{2} \ln \left (b \,x^{2}+a \right )-\frac {a^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{2 b^{2} \left (b \,x^{2}+a \right )^{2}}+\frac {a \left (a^{2} d^{2}-3 a b c d +2 b^{2} c^{2}\right )}{b^{2} \left (b \,x^{2}+a \right )}}{2 \left (a d -b c \right )^{3}}+\frac {c^{2} \ln \left (d \,x^{2}+c \right )}{2 \left (a d -b c \right )^{3}}\) | \(124\) |
norman | \(\frac {\frac {\left (-a d +3 b c \right ) a^{2}}{4 b^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {a \left (-a d +2 b c \right ) x^{2}}{2 b \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{\left (b \,x^{2}+a \right )^{2}}-\frac {c^{2} \ln \left (b \,x^{2}+a \right )}{2 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {c^{2} \ln \left (d \,x^{2}+c \right )}{2 a^{3} d^{3}-6 a^{2} b c \,d^{2}+6 a \,b^{2} c^{2} d -2 b^{3} c^{3}}\) | \(196\) |
risch | \(\frac {-\frac {a \left (a d -2 b c \right ) x^{2}}{2 b \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {a^{2} \left (a d -3 b c \right )}{4 b^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{\left (b \,x^{2}+a \right )^{2}}+\frac {c^{2} \ln \left (d \,x^{2}+c \right )}{2 a^{3} d^{3}-6 a^{2} b c \,d^{2}+6 a \,b^{2} c^{2} d -2 b^{3} c^{3}}-\frac {c^{2} \ln \left (-b \,x^{2}-a \right )}{2 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}\) | \(197\) |
parallelrisch | \(-\frac {2 \ln \left (b \,x^{2}+a \right ) x^{4} b^{4} c^{2}-2 \ln \left (d \,x^{2}+c \right ) x^{4} b^{4} c^{2}+4 \ln \left (b \,x^{2}+a \right ) x^{2} a \,b^{3} c^{2}-4 \ln \left (d \,x^{2}+c \right ) x^{2} a \,b^{3} c^{2}+2 a^{3} b \,d^{2} x^{2}-6 a^{2} b^{2} c d \,x^{2}+4 a \,b^{3} c^{2} x^{2}+2 \ln \left (b \,x^{2}+a \right ) a^{2} b^{2} c^{2}-2 \ln \left (d \,x^{2}+c \right ) a^{2} b^{2} c^{2}+a^{4} d^{2}-4 a^{3} c d b +3 a^{2} b^{2} c^{2}}{4 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (b \,x^{2}+a \right )^{2} b^{2}}\) | \(233\) |
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Leaf count of result is larger than twice the leaf count of optimal. 290 vs. \(2 (108) = 216\).
Time = 0.25 (sec) , antiderivative size = 290, normalized size of antiderivative = 2.50 \[ \int \frac {x^5}{\left (a+b x^2\right )^3 \left (c+d x^2\right )} \, dx=\frac {3 \, a^{2} b^{2} c^{2} - 4 \, a^{3} b c d + a^{4} d^{2} + 2 \, {\left (2 \, a b^{3} c^{2} - 3 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{2} + 2 \, {\left (b^{4} c^{2} x^{4} + 2 \, a b^{3} c^{2} x^{2} + a^{2} b^{2} c^{2}\right )} \log \left (b x^{2} + a\right ) - 2 \, {\left (b^{4} c^{2} x^{4} + 2 \, a b^{3} c^{2} x^{2} + a^{2} b^{2} c^{2}\right )} \log \left (d x^{2} + c\right )}{4 \, {\left (a^{2} b^{5} c^{3} - 3 \, a^{3} b^{4} c^{2} d + 3 \, a^{4} b^{3} c d^{2} - a^{5} b^{2} d^{3} + {\left (b^{7} c^{3} - 3 \, a b^{6} c^{2} d + 3 \, a^{2} b^{5} c d^{2} - a^{3} b^{4} d^{3}\right )} x^{4} + 2 \, {\left (a b^{6} c^{3} - 3 \, a^{2} b^{5} c^{2} d + 3 \, a^{3} b^{4} c d^{2} - a^{4} b^{3} d^{3}\right )} x^{2}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 418 vs. \(2 (97) = 194\).
Time = 3.17 (sec) , antiderivative size = 418, normalized size of antiderivative = 3.60 \[ \int \frac {x^5}{\left (a+b x^2\right )^3 \left (c+d x^2\right )} \, dx=\frac {c^{2} \log {\left (x^{2} + \frac {- \frac {a^{4} c^{2} d^{4}}{\left (a d - b c\right )^{3}} + \frac {4 a^{3} b c^{3} d^{3}}{\left (a d - b c\right )^{3}} - \frac {6 a^{2} b^{2} c^{4} d^{2}}{\left (a d - b c\right )^{3}} + \frac {4 a b^{3} c^{5} d}{\left (a d - b c\right )^{3}} + a c^{2} d - \frac {b^{4} c^{6}}{\left (a d - b c\right )^{3}} + b c^{3}}{2 b c^{2} d} \right )}}{2 \left (a d - b c\right )^{3}} - \frac {c^{2} \log {\left (x^{2} + \frac {\frac {a^{4} c^{2} d^{4}}{\left (a d - b c\right )^{3}} - \frac {4 a^{3} b c^{3} d^{3}}{\left (a d - b c\right )^{3}} + \frac {6 a^{2} b^{2} c^{4} d^{2}}{\left (a d - b c\right )^{3}} - \frac {4 a b^{3} c^{5} d}{\left (a d - b c\right )^{3}} + a c^{2} d + \frac {b^{4} c^{6}}{\left (a d - b c\right )^{3}} + b c^{3}}{2 b c^{2} d} \right )}}{2 \left (a d - b c\right )^{3}} + \frac {- a^{3} d + 3 a^{2} b c + x^{2} \left (- 2 a^{2} b d + 4 a b^{2} c\right )}{4 a^{4} b^{2} d^{2} - 8 a^{3} b^{3} c d + 4 a^{2} b^{4} c^{2} + x^{4} \cdot \left (4 a^{2} b^{4} d^{2} - 8 a b^{5} c d + 4 b^{6} c^{2}\right ) + x^{2} \cdot \left (8 a^{3} b^{3} d^{2} - 16 a^{2} b^{4} c d + 8 a b^{5} c^{2}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 236 vs. \(2 (108) = 216\).
Time = 0.20 (sec) , antiderivative size = 236, normalized size of antiderivative = 2.03 \[ \int \frac {x^5}{\left (a+b x^2\right )^3 \left (c+d x^2\right )} \, dx=\frac {c^{2} \log \left (b x^{2} + a\right )}{2 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )}} - \frac {c^{2} \log \left (d x^{2} + c\right )}{2 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )}} + \frac {3 \, a^{2} b c - a^{3} d + 2 \, {\left (2 \, a b^{2} c - a^{2} b d\right )} x^{2}}{4 \, {\left (a^{2} b^{4} c^{2} - 2 \, a^{3} b^{3} c d + a^{4} b^{2} d^{2} + {\left (b^{6} c^{2} - 2 \, a b^{5} c d + a^{2} b^{4} d^{2}\right )} x^{4} + 2 \, {\left (a b^{5} c^{2} - 2 \, a^{2} b^{4} c d + a^{3} b^{3} d^{2}\right )} x^{2}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 232 vs. \(2 (108) = 216\).
Time = 0.28 (sec) , antiderivative size = 232, normalized size of antiderivative = 2.00 \[ \int \frac {x^5}{\left (a+b x^2\right )^3 \left (c+d x^2\right )} \, dx=\frac {b c^{2} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, {\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )}} - \frac {c^{2} d \log \left ({\left | d x^{2} + c \right |}\right )}{2 \, {\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} - a^{3} d^{4}\right )}} - \frac {3 \, b^{4} c^{2} x^{4} + 2 \, a b^{3} c^{2} x^{2} + 6 \, a^{2} b^{2} c d x^{2} - 2 \, a^{3} b d^{2} x^{2} + 4 \, a^{3} b c d - a^{4} d^{2}}{4 \, {\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} {\left (b x^{2} + a\right )}^{2}} \]
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Time = 5.46 (sec) , antiderivative size = 370, normalized size of antiderivative = 3.19 \[ \int \frac {x^5}{\left (a+b x^2\right )^3 \left (c+d x^2\right )} \, dx=\frac {b^3\,\left (4\,a\,c^2\,x^2+a\,c^2\,x^2\,\mathrm {atan}\left (\frac {a\,d\,x^2\,1{}\mathrm {i}-b\,c\,x^2\,1{}\mathrm {i}}{2\,a\,c+a\,d\,x^2+b\,c\,x^2}\right )\,8{}\mathrm {i}\right )+b\,\left (2\,a^3\,d^2\,x^2-4\,a^3\,c\,d\right )+a^4\,d^2+b^2\,\left (3\,a^2\,c^2-6\,a^2\,c\,d\,x^2+a^2\,c^2\,\mathrm {atan}\left (\frac {a\,d\,x^2\,1{}\mathrm {i}-b\,c\,x^2\,1{}\mathrm {i}}{2\,a\,c+a\,d\,x^2+b\,c\,x^2}\right )\,4{}\mathrm {i}\right )+b^4\,c^2\,x^4\,\mathrm {atan}\left (\frac {a\,d\,x^2\,1{}\mathrm {i}-b\,c\,x^2\,1{}\mathrm {i}}{2\,a\,c+a\,d\,x^2+b\,c\,x^2}\right )\,4{}\mathrm {i}}{-4\,a^5\,b^2\,d^3+12\,a^4\,b^3\,c\,d^2-8\,a^4\,b^3\,d^3\,x^2-12\,a^3\,b^4\,c^2\,d+24\,a^3\,b^4\,c\,d^2\,x^2-4\,a^3\,b^4\,d^3\,x^4+4\,a^2\,b^5\,c^3-24\,a^2\,b^5\,c^2\,d\,x^2+12\,a^2\,b^5\,c\,d^2\,x^4+8\,a\,b^6\,c^3\,x^2-12\,a\,b^6\,c^2\,d\,x^4+4\,b^7\,c^3\,x^4} \]
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