Optimal. Leaf size=178 \[ -\frac {1}{2 a c^3 x^2}+\frac {d^2}{4 c^2 (b c-a d) \left (c+d x^2\right )^2}+\frac {d^2 (3 b c-2 a d)}{2 c^3 (b c-a d)^2 \left (c+d x^2\right )}-\frac {(b c+3 a d) \log (x)}{a^2 c^4}+\frac {b^4 \log \left (a+b x^2\right )}{2 a^2 (b c-a d)^3}-\frac {d^2 \left (6 b^2 c^2-8 a b c d+3 a^2 d^2\right ) \log \left (c+d x^2\right )}{2 c^4 (b c-a d)^3} \]
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Rubi [A]
time = 0.15, antiderivative size = 178, 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 = {457, 90}
\begin {gather*} \frac {b^4 \log \left (a+b x^2\right )}{2 a^2 (b c-a d)^3}-\frac {d^2 \left (3 a^2 d^2-8 a b c d+6 b^2 c^2\right ) \log \left (c+d x^2\right )}{2 c^4 (b c-a d)^3}-\frac {\log (x) (3 a d+b c)}{a^2 c^4}+\frac {d^2 (3 b c-2 a d)}{2 c^3 \left (c+d x^2\right ) (b c-a d)^2}+\frac {d^2}{4 c^2 \left (c+d x^2\right )^2 (b c-a d)}-\frac {1}{2 a c^3 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 90
Rule 457
Rubi steps
\begin {align*} \int \frac {1}{x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{x^2 (a+b x) (c+d x)^3} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {1}{a c^3 x^2}+\frac {-b c-3 a d}{a^2 c^4 x}-\frac {b^5}{a^2 (-b c+a d)^3 (a+b x)}-\frac {d^3}{c^2 (b c-a d) (c+d x)^3}-\frac {d^3 (3 b c-2 a d)}{c^3 (b c-a d)^2 (c+d x)^2}-\frac {d^3 \left (6 b^2 c^2-8 a b c d+3 a^2 d^2\right )}{c^4 (b c-a d)^3 (c+d x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {1}{2 a c^3 x^2}+\frac {d^2}{4 c^2 (b c-a d) \left (c+d x^2\right )^2}+\frac {d^2 (3 b c-2 a d)}{2 c^3 (b c-a d)^2 \left (c+d x^2\right )}-\frac {(b c+3 a d) \log (x)}{a^2 c^4}+\frac {b^4 \log \left (a+b x^2\right )}{2 a^2 (b c-a d)^3}-\frac {d^2 \left (6 b^2 c^2-8 a b c d+3 a^2 d^2\right ) \log \left (c+d x^2\right )}{2 c^4 (b c-a d)^3}\\ \end {align*}
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Mathematica [A]
time = 0.32, size = 171, normalized size = 0.96 \begin {gather*} \frac {1}{4} \left (-\frac {2}{a c^3 x^2}+\frac {d^2}{c^2 (b c-a d) \left (c+d x^2\right )^2}+\frac {2 d^2 (3 b c-2 a d)}{c^3 (b c-a d)^2 \left (c+d x^2\right )}-\frac {4 (b c+3 a d) \log (x)}{a^2 c^4}-\frac {2 b^4 \log \left (a+b x^2\right )}{a^2 (-b c+a d)^3}-\frac {2 d^2 \left (6 b^2 c^2-8 a b c d+3 a^2 d^2\right ) \log \left (c+d x^2\right )}{c^4 (b c-a d)^3}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.17, size = 187, normalized size = 1.05
method | result | size |
default | \(-\frac {b^{4} \ln \left (b \,x^{2}+a \right )}{2 a^{2} \left (a d -b c \right )^{3}}+\frac {d^{3} \left (-\frac {c \left (2 a^{2} d^{2}-5 a b c d +3 b^{2} c^{2}\right )}{d \left (d \,x^{2}+c \right )}-\frac {c^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{2 d \left (d \,x^{2}+c \right )^{2}}+\frac {\left (3 a^{2} d^{2}-8 a b c d +6 b^{2} c^{2}\right ) \ln \left (d \,x^{2}+c \right )}{d}\right )}{2 c^{4} \left (a d -b c \right )^{3}}-\frac {1}{2 a \,c^{3} x^{2}}+\frac {\left (-3 a d -b c \right ) \ln \left (x \right )}{a^{2} c^{4}}\) | \(187\) |
norman | \(\frac {-\frac {1}{2 a c}+\frac {\left (6 a^{2} d^{3}-10 a b c \,d^{2}+3 b^{2} c^{2} d \right ) d \,x^{4}}{2 a \,c^{3} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {\left (9 a^{2} d^{3}-15 a b c \,d^{2}+4 b^{2} c^{2} d \right ) d^{2} x^{6}}{4 c^{4} a \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{x^{2} \left (d \,x^{2}+c \right )^{2}}-\frac {b^{4} \ln \left (b \,x^{2}+a \right )}{2 a^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {\left (3 a d +b c \right ) \ln \left (x \right )}{a^{2} c^{4}}+\frac {d^{2} \left (3 a^{2} d^{2}-8 a b c d +6 b^{2} c^{2}\right ) \ln \left (d \,x^{2}+c \right )}{2 c^{4} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}\) | \(297\) |
risch | \(\frac {-\frac {d^{2} \left (3 a^{2} d^{2}-5 a b c d +b^{2} c^{2}\right ) x^{4}}{2 c^{3} a \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {d \left (9 a^{2} d^{2}-15 a b c d +4 b^{2} c^{2}\right ) x^{2}}{4 c^{2} a \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {1}{2 a c}}{x^{2} \left (d \,x^{2}+c \right )^{2}}-\frac {3 \ln \left (x \right ) d}{a \,c^{4}}-\frac {\ln \left (x \right ) b}{a^{2} c^{3}}+\frac {3 d^{4} \ln \left (-d \,x^{2}-c \right ) a^{2}}{2 c^{4} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {4 d^{3} \ln \left (-d \,x^{2}-c \right ) a b}{c^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {3 d^{2} \ln \left (-d \,x^{2}-c \right ) b^{2}}{c^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {b^{4} \ln \left (b \,x^{2}+a \right )}{2 a^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}\) | \(396\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 364 vs.
\(2 (168) = 336\).
time = 0.30, size = 364, normalized size = 2.04 \begin {gather*} \frac {b^{4} \log \left (b x^{2} + a\right )}{2 \, {\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}\right )}} - \frac {{\left (6 \, b^{2} c^{2} d^{2} - 8 \, a b c d^{3} + 3 \, a^{2} d^{4}\right )} \log \left (d x^{2} + c\right )}{2 \, {\left (b^{3} c^{7} - 3 \, a b^{2} c^{6} d + 3 \, a^{2} b c^{5} d^{2} - a^{3} c^{4} d^{3}\right )}} - \frac {2 \, b^{2} c^{4} - 4 \, a b c^{3} d + 2 \, a^{2} c^{2} d^{2} + 2 \, {\left (b^{2} c^{2} d^{2} - 5 \, a b c d^{3} + 3 \, a^{2} d^{4}\right )} x^{4} + {\left (4 \, b^{2} c^{3} d - 15 \, a b c^{2} d^{2} + 9 \, a^{2} c d^{3}\right )} x^{2}}{4 \, {\left ({\left (a b^{2} c^{5} d^{2} - 2 \, a^{2} b c^{4} d^{3} + a^{3} c^{3} d^{4}\right )} x^{6} + 2 \, {\left (a b^{2} c^{6} d - 2 \, a^{2} b c^{5} d^{2} + a^{3} c^{4} d^{3}\right )} x^{4} + {\left (a b^{2} c^{7} - 2 \, a^{2} b c^{6} d + a^{3} c^{5} d^{2}\right )} x^{2}\right )}} - \frac {{\left (b c + 3 \, a d\right )} \log \left (x^{2}\right )}{2 \, a^{2} c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 640 vs.
\(2 (168) = 336\).
time = 9.49, size = 640, normalized size = 3.60 \begin {gather*} -\frac {2 \, a b^{3} c^{6} - 6 \, a^{2} b^{2} c^{5} d + 6 \, a^{3} b c^{4} d^{2} - 2 \, a^{4} c^{3} d^{3} + 2 \, {\left (a b^{3} c^{4} d^{2} - 6 \, a^{2} b^{2} c^{3} d^{3} + 8 \, a^{3} b c^{2} d^{4} - 3 \, a^{4} c d^{5}\right )} x^{4} + {\left (4 \, a b^{3} c^{5} d - 19 \, a^{2} b^{2} c^{4} d^{2} + 24 \, a^{3} b c^{3} d^{3} - 9 \, a^{4} c^{2} d^{4}\right )} x^{2} - 2 \, {\left (b^{4} c^{4} d^{2} x^{6} + 2 \, b^{4} c^{5} d x^{4} + b^{4} c^{6} x^{2}\right )} \log \left (b x^{2} + a\right ) + 2 \, {\left ({\left (6 \, a^{2} b^{2} c^{2} d^{4} - 8 \, a^{3} b c d^{5} + 3 \, a^{4} d^{6}\right )} x^{6} + 2 \, {\left (6 \, a^{2} b^{2} c^{3} d^{3} - 8 \, a^{3} b c^{2} d^{4} + 3 \, a^{4} c d^{5}\right )} x^{4} + {\left (6 \, a^{2} b^{2} c^{4} d^{2} - 8 \, a^{3} b c^{3} d^{3} + 3 \, a^{4} c^{2} d^{4}\right )} x^{2}\right )} \log \left (d x^{2} + c\right ) + 4 \, {\left ({\left (b^{4} c^{4} d^{2} - 6 \, a^{2} b^{2} c^{2} d^{4} + 8 \, a^{3} b c d^{5} - 3 \, a^{4} d^{6}\right )} x^{6} + 2 \, {\left (b^{4} c^{5} d - 6 \, a^{2} b^{2} c^{3} d^{3} + 8 \, a^{3} b c^{2} d^{4} - 3 \, a^{4} c d^{5}\right )} x^{4} + {\left (b^{4} c^{6} - 6 \, a^{2} b^{2} c^{4} d^{2} + 8 \, a^{3} b c^{3} d^{3} - 3 \, a^{4} c^{2} d^{4}\right )} x^{2}\right )} \log \left (x\right )}{4 \, {\left ({\left (a^{2} b^{3} c^{7} d^{2} - 3 \, a^{3} b^{2} c^{6} d^{3} + 3 \, a^{4} b c^{5} d^{4} - a^{5} c^{4} d^{5}\right )} x^{6} + 2 \, {\left (a^{2} b^{3} c^{8} d - 3 \, a^{3} b^{2} c^{7} d^{2} + 3 \, a^{4} b c^{6} d^{3} - a^{5} c^{5} d^{4}\right )} x^{4} + {\left (a^{2} b^{3} c^{9} - 3 \, a^{3} b^{2} c^{8} d + 3 \, a^{4} b c^{7} d^{2} - a^{5} c^{6} d^{3}\right )} x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 357 vs.
\(2 (168) = 336\).
time = 1.50, size = 357, normalized size = 2.01 \begin {gather*} \frac {b^{5} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, {\left (a^{2} b^{4} c^{3} - 3 \, a^{3} b^{3} c^{2} d + 3 \, a^{4} b^{2} c d^{2} - a^{5} b d^{3}\right )}} - \frac {{\left (6 \, b^{2} c^{2} d^{3} - 8 \, a b c d^{4} + 3 \, a^{2} d^{5}\right )} \log \left ({\left | d x^{2} + c \right |}\right )}{2 \, {\left (b^{3} c^{7} d - 3 \, a b^{2} c^{6} d^{2} + 3 \, a^{2} b c^{5} d^{3} - a^{3} c^{4} d^{4}\right )}} + \frac {18 \, b^{2} c^{2} d^{4} x^{4} - 24 \, a b c d^{5} x^{4} + 9 \, a^{2} d^{6} x^{4} + 42 \, b^{2} c^{3} d^{3} x^{2} - 58 \, a b c^{2} d^{4} x^{2} + 22 \, a^{2} c d^{5} x^{2} + 25 \, b^{2} c^{4} d^{2} - 36 \, a b c^{3} d^{3} + 14 \, a^{2} c^{2} d^{4}}{4 \, {\left (b^{3} c^{7} - 3 \, a b^{2} c^{6} d + 3 \, a^{2} b c^{5} d^{2} - a^{3} c^{4} d^{3}\right )} {\left (d x^{2} + c\right )}^{2}} - \frac {{\left (b c + 3 \, a d\right )} \log \left (x^{2}\right )}{2 \, a^{2} c^{4}} + \frac {b c x^{2} + 3 \, a d x^{2} - a c}{2 \, a^{2} c^{4} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.85, size = 314, normalized size = 1.76 \begin {gather*} -\frac {\frac {1}{2\,a\,c}+\frac {x^4\,\left (3\,a^2\,d^4-5\,a\,b\,c\,d^3+b^2\,c^2\,d^2\right )}{2\,a\,c^3\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {x^2\,\left (9\,a^2\,d^3-15\,a\,b\,c\,d^2+4\,b^2\,c^2\,d\right )}{4\,a\,c^2\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}}{c^2\,x^2+2\,c\,d\,x^4+d^2\,x^6}-\frac {\ln \left (d\,x^2+c\right )\,\left (3\,a^2\,d^4-8\,a\,b\,c\,d^3+6\,b^2\,c^2\,d^2\right )}{-2\,a^3\,c^4\,d^3+6\,a^2\,b\,c^5\,d^2-6\,a\,b^2\,c^6\,d+2\,b^3\,c^7}-\frac {b^4\,\ln \left (b\,x^2+a\right )}{2\,\left (a^5\,d^3-3\,a^4\,b\,c\,d^2+3\,a^3\,b^2\,c^2\,d-a^2\,b^3\,c^3\right )}-\frac {\ln \left (x\right )\,\left (3\,a\,d+b\,c\right )}{a^2\,c^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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