Optimal. Leaf size=215 \[ \frac {b^4 (2 b c-5 a d) \log \left (a+b x^2\right )}{2 a^3 (b c-a d)^4}-\frac {\log (x) (3 a d+2 b c)}{a^3 c^4}-\frac {b^4}{2 a^2 \left (a+b x^2\right ) (b c-a d)^3}+\frac {d^3 \left (3 a^2 d^2-10 a b c d+10 b^2 c^2\right ) \log \left (c+d x^2\right )}{2 c^4 (b c-a d)^4}-\frac {1}{2 a^2 c^3 x^2}-\frac {d^3 (2 b c-a d)}{c^3 \left (c+d x^2\right ) (b c-a d)^3}-\frac {d^3}{4 c^2 \left (c+d x^2\right )^2 (b c-a d)^2} \]
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Rubi [A] time = 0.29, antiderivative size = 215, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {446, 88} \begin {gather*} \frac {d^3 \left (3 a^2 d^2-10 a b c d+10 b^2 c^2\right ) \log \left (c+d x^2\right )}{2 c^4 (b c-a d)^4}-\frac {b^4}{2 a^2 \left (a+b x^2\right ) (b c-a d)^3}+\frac {b^4 (2 b c-5 a d) \log \left (a+b x^2\right )}{2 a^3 (b c-a d)^4}-\frac {\log (x) (3 a d+2 b c)}{a^3 c^4}-\frac {1}{2 a^2 c^3 x^2}-\frac {d^3 (2 b c-a d)}{c^3 \left (c+d x^2\right ) (b c-a d)^3}-\frac {d^3}{4 c^2 \left (c+d x^2\right )^2 (b c-a d)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 88
Rule 446
Rubi steps
\begin {align*} \int \frac {1}{x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x^2 (a+b x)^2 (c+d x)^3} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {1}{a^2 c^3 x^2}+\frac {-2 b c-3 a d}{a^3 c^4 x}-\frac {b^5}{a^2 (-b c+a d)^3 (a+b x)^2}-\frac {b^5 (-2 b c+5 a d)}{a^3 (-b c+a d)^4 (a+b x)}+\frac {d^4}{c^2 (b c-a d)^2 (c+d x)^3}+\frac {2 d^4 (2 b c-a d)}{c^3 (b c-a d)^3 (c+d x)^2}+\frac {d^4 \left (10 b^2 c^2-10 a b c d+3 a^2 d^2\right )}{c^4 (b c-a d)^4 (c+d x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {1}{2 a^2 c^3 x^2}-\frac {b^4}{2 a^2 (b c-a d)^3 \left (a+b x^2\right )}-\frac {d^3}{4 c^2 (b c-a d)^2 \left (c+d x^2\right )^2}-\frac {d^3 (2 b c-a d)}{c^3 (b c-a d)^3 \left (c+d x^2\right )}-\frac {(2 b c+3 a d) \log (x)}{a^3 c^4}+\frac {b^4 (2 b c-5 a d) \log \left (a+b x^2\right )}{2 a^3 (b c-a d)^4}+\frac {d^3 \left (10 b^2 c^2-10 a b c d+3 a^2 d^2\right ) \log \left (c+d x^2\right )}{2 c^4 (b c-a d)^4}\\ \end {align*}
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Mathematica [A] time = 0.32, size = 208, normalized size = 0.97 \begin {gather*} \frac {1}{4} \left (\frac {2 b^4 (2 b c-5 a d) \log \left (a+b x^2\right )}{a^3 (b c-a d)^4}-\frac {4 \log (x) (3 a d+2 b c)}{a^3 c^4}+\frac {2 b^4}{a^2 \left (a+b x^2\right ) (a d-b c)^3}+\frac {2 d^3 \left (3 a^2 d^2-10 a b c d+10 b^2 c^2\right ) \log \left (c+d x^2\right )}{c^4 (b c-a d)^4}-\frac {2}{a^2 c^3 x^2}+\frac {4 d^3 (a d-2 b c)}{c^3 \left (c+d x^2\right ) (b c-a d)^3}-\frac {d^3}{c^2 \left (c+d x^2\right )^2 (b c-a d)^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 72.33, size = 1227, normalized size = 5.71
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.45, size = 638, normalized size = 2.97 \begin {gather*} \frac {{\left (2 \, b^{6} c - 5 \, a b^{5} d\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, {\left (a^{3} b^{5} c^{4} - 4 \, a^{4} b^{4} c^{3} d + 6 \, a^{5} b^{3} c^{2} d^{2} - 4 \, a^{6} b^{2} c d^{3} + a^{7} b d^{4}\right )}} + \frac {{\left (10 \, b^{2} c^{2} d^{4} - 10 \, a b c d^{5} + 3 \, a^{2} d^{6}\right )} \log \left ({\left | d x^{2} + c \right |}\right )}{2 \, {\left (b^{4} c^{8} d - 4 \, a b^{3} c^{7} d^{2} + 6 \, a^{2} b^{2} c^{6} d^{3} - 4 \, a^{3} b c^{5} d^{4} + a^{4} c^{4} d^{5}\right )}} + \frac {10 \, a^{2} b^{3} c^{2} d^{3} x^{4} - 10 \, a^{3} b^{2} c d^{4} x^{4} + 3 \, a^{4} b d^{5} x^{4} - 4 \, b^{5} c^{5} x^{2} + 10 \, a b^{4} c^{4} d x^{2} - 12 \, a^{2} b^{3} c^{3} d^{2} x^{2} + 18 \, a^{3} b^{2} c^{2} d^{3} x^{2} - 12 \, a^{4} b c d^{4} x^{2} + 3 \, a^{5} d^{5} x^{2} - 2 \, a b^{4} c^{5} + 8 \, a^{2} b^{3} c^{4} d - 12 \, a^{3} b^{2} c^{3} d^{2} + 8 \, a^{4} b c^{2} d^{3} - 2 \, a^{5} c d^{4}}{4 \, {\left (a^{2} b^{4} c^{8} - 4 \, a^{3} b^{3} c^{7} d + 6 \, a^{4} b^{2} c^{6} d^{2} - 4 \, a^{5} b c^{5} d^{3} + a^{6} c^{4} d^{4}\right )} {\left (b x^{4} + a x^{2}\right )}} - \frac {30 \, b^{2} c^{2} d^{5} x^{4} - 30 \, a b c d^{6} x^{4} + 9 \, a^{2} d^{7} x^{4} + 68 \, b^{2} c^{3} d^{4} x^{2} - 72 \, a b c^{2} d^{5} x^{2} + 22 \, a^{2} c d^{6} x^{2} + 39 \, b^{2} c^{4} d^{3} - 44 \, a b c^{3} d^{4} + 14 \, a^{2} c^{2} d^{5}}{4 \, {\left (b^{4} c^{8} - 4 \, a b^{3} c^{7} d + 6 \, a^{2} b^{2} c^{6} d^{2} - 4 \, a^{3} b c^{5} d^{3} + a^{4} c^{4} d^{4}\right )} {\left (d x^{2} + c\right )}^{2}} - \frac {{\left (2 \, b c + 3 \, a d\right )} \log \left (x^{2}\right )}{2 \, a^{3} c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 405, normalized size = 1.88 \begin {gather*} -\frac {a^{2} d^{5}}{4 \left (a d -b c \right )^{4} \left (d \,x^{2}+c \right )^{2} c^{2}}+\frac {a b \,d^{4}}{2 \left (a d -b c \right )^{4} \left (d \,x^{2}+c \right )^{2} c}-\frac {b^{2} d^{3}}{4 \left (a d -b c \right )^{4} \left (d \,x^{2}+c \right )^{2}}-\frac {a^{2} d^{5}}{\left (a d -b c \right )^{4} \left (d \,x^{2}+c \right ) c^{3}}+\frac {3 a^{2} d^{5} \ln \left (d \,x^{2}+c \right )}{2 \left (a d -b c \right )^{4} c^{4}}+\frac {3 a b \,d^{4}}{\left (a d -b c \right )^{4} \left (d \,x^{2}+c \right ) c^{2}}-\frac {5 a b \,d^{4} \ln \left (d \,x^{2}+c \right )}{\left (a d -b c \right )^{4} c^{3}}+\frac {b^{4} d}{2 \left (a d -b c \right )^{4} \left (b \,x^{2}+a \right ) a}-\frac {b^{5} c}{2 \left (a d -b c \right )^{4} \left (b \,x^{2}+a \right ) a^{2}}-\frac {5 b^{4} d \ln \left (b \,x^{2}+a \right )}{2 \left (a d -b c \right )^{4} a^{2}}+\frac {b^{5} c \ln \left (b \,x^{2}+a \right )}{\left (a d -b c \right )^{4} a^{3}}-\frac {2 b^{2} d^{3}}{\left (a d -b c \right )^{4} \left (d \,x^{2}+c \right ) c}+\frac {5 b^{2} d^{3} \ln \left (d \,x^{2}+c \right )}{\left (a d -b c \right )^{4} c^{2}}-\frac {3 d \ln \relax (x )}{a^{2} c^{4}}-\frac {2 b \ln \relax (x )}{a^{3} c^{3}}-\frac {1}{2 a^{2} c^{3} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.48, size = 651, normalized size = 3.03 \begin {gather*} \frac {{\left (2 \, b^{5} c - 5 \, a b^{4} d\right )} \log \left (b x^{2} + a\right )}{2 \, {\left (a^{3} b^{4} c^{4} - 4 \, a^{4} b^{3} c^{3} d + 6 \, a^{5} b^{2} c^{2} d^{2} - 4 \, a^{6} b c d^{3} + a^{7} d^{4}\right )}} + \frac {{\left (10 \, b^{2} c^{2} d^{3} - 10 \, a b c d^{4} + 3 \, a^{2} d^{5}\right )} \log \left (d x^{2} + c\right )}{2 \, {\left (b^{4} c^{8} - 4 \, a b^{3} c^{7} d + 6 \, a^{2} b^{2} c^{6} d^{2} - 4 \, a^{3} b c^{5} d^{3} + a^{4} c^{4} d^{4}\right )}} - \frac {2 \, a b^{3} c^{5} - 6 \, a^{2} b^{2} c^{4} d + 6 \, a^{3} b c^{3} d^{2} - 2 \, a^{4} c^{2} d^{3} + 2 \, {\left (2 \, b^{4} c^{3} d^{2} - 3 \, a b^{3} c^{2} d^{3} + 7 \, a^{2} b^{2} c d^{4} - 3 \, a^{3} b d^{5}\right )} x^{6} + {\left (8 \, b^{4} c^{4} d - 10 \, a b^{3} c^{3} d^{2} + 15 \, a^{2} b^{2} c^{2} d^{3} + 5 \, a^{3} b c d^{4} - 6 \, a^{4} d^{5}\right )} x^{4} + {\left (4 \, b^{4} c^{5} - 2 \, a b^{3} c^{4} d - 6 \, a^{2} b^{2} c^{3} d^{2} + 19 \, a^{3} b c^{2} d^{3} - 9 \, a^{4} c d^{4}\right )} x^{2}}{4 \, {\left ({\left (a^{2} b^{4} c^{6} d^{2} - 3 \, a^{3} b^{3} c^{5} d^{3} + 3 \, a^{4} b^{2} c^{4} d^{4} - a^{5} b c^{3} d^{5}\right )} x^{8} + {\left (2 \, a^{2} b^{4} c^{7} d - 5 \, a^{3} b^{3} c^{6} d^{2} + 3 \, a^{4} b^{2} c^{5} d^{3} + a^{5} b c^{4} d^{4} - a^{6} c^{3} d^{5}\right )} x^{6} + {\left (a^{2} b^{4} c^{8} - a^{3} b^{3} c^{7} d - 3 \, a^{4} b^{2} c^{6} d^{2} + 5 \, a^{5} b c^{5} d^{3} - 2 \, a^{6} c^{4} d^{4}\right )} x^{4} + {\left (a^{3} b^{3} c^{8} - 3 \, a^{4} b^{2} c^{7} d + 3 \, a^{5} b c^{6} d^{2} - a^{6} c^{5} d^{3}\right )} x^{2}\right )}} - \frac {{\left (2 \, b c + 3 \, a d\right )} \log \left (x^{2}\right )}{2 \, a^{3} c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.42, size = 549, normalized size = 2.55 \begin {gather*} \frac {\ln \left (b\,x^2+a\right )\,\left (2\,b^5\,c-5\,a\,b^4\,d\right )}{2\,a^7\,d^4-8\,a^6\,b\,c\,d^3+12\,a^5\,b^2\,c^2\,d^2-8\,a^4\,b^3\,c^3\,d+2\,a^3\,b^4\,c^4}-\frac {\frac {1}{2\,a\,c}-\frac {x^4\,\left (-6\,a^4\,d^5+5\,a^3\,b\,c\,d^4+15\,a^2\,b^2\,c^2\,d^3-10\,a\,b^3\,c^3\,d^2+8\,b^4\,c^4\,d\right )}{4\,a^2\,c^3\,\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 {x^2\,\left (9\,a^4\,d^4-19\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2+2\,a\,b^3\,c^3\,d-4\,b^4\,c^4\right )}{4\,a^2\,c^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 {b\,d^2\,x^6\,\left (3\,a^3\,d^3-7\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-2\,b^3\,c^3\right )}{2\,a^2\,c^3\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}}{x^4\,\left (b\,c^2+2\,a\,d\,c\right )+x^6\,\left (a\,d^2+2\,b\,c\,d\right )+a\,c^2\,x^2+b\,d^2\,x^8}+\frac {\ln \left (d\,x^2+c\right )\,\left (3\,a^2\,d^5-10\,a\,b\,c\,d^4+10\,b^2\,c^2\,d^3\right )}{2\,a^4\,c^4\,d^4-8\,a^3\,b\,c^5\,d^3+12\,a^2\,b^2\,c^6\,d^2-8\,a\,b^3\,c^7\,d+2\,b^4\,c^8}-\frac {\ln \relax (x)\,\left (3\,a\,d+2\,b\,c\right )}{a^3\,c^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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