Optimal. Leaf size=126 \[ -\frac {b^2}{2 \left (a+b x^2\right ) (b c-a d)^3}-\frac {3 b^2 d \log \left (a+b x^2\right )}{2 (b c-a d)^4}+\frac {3 b^2 d \log \left (c+d x^2\right )}{2 (b c-a d)^4}-\frac {b d}{\left (c+d x^2\right ) (b c-a d)^3}-\frac {d}{4 \left (c+d x^2\right )^2 (b c-a d)^2} \]
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Rubi [A] time = 0.11, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {444, 44} \begin {gather*} -\frac {b^2}{2 \left (a+b x^2\right ) (b c-a d)^3}-\frac {3 b^2 d \log \left (a+b x^2\right )}{2 (b c-a d)^4}+\frac {3 b^2 d \log \left (c+d x^2\right )}{2 (b c-a d)^4}-\frac {b d}{\left (c+d x^2\right ) (b c-a d)^3}-\frac {d}{4 \left (c+d x^2\right )^2 (b c-a d)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 44
Rule 444
Rubi steps
\begin {align*} \int \frac {x}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{(a+b x)^2 (c+d x)^3} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {b^3}{(b c-a d)^3 (a+b x)^2}-\frac {3 b^3 d}{(b c-a d)^4 (a+b x)}+\frac {d^2}{(b c-a d)^2 (c+d x)^3}+\frac {2 b d^2}{(b c-a d)^3 (c+d x)^2}+\frac {3 b^2 d^2}{(b c-a d)^4 (c+d x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {b^2}{2 (b c-a d)^3 \left (a+b x^2\right )}-\frac {d}{4 (b c-a d)^2 \left (c+d x^2\right )^2}-\frac {b d}{(b c-a d)^3 \left (c+d x^2\right )}-\frac {3 b^2 d \log \left (a+b x^2\right )}{2 (b c-a d)^4}+\frac {3 b^2 d \log \left (c+d x^2\right )}{2 (b c-a d)^4}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 107, normalized size = 0.85 \begin {gather*} -\frac {\frac {2 b^2 (b c-a d)}{a+b x^2}+6 b^2 d \log \left (a+b x^2\right )+\frac {4 b d (b c-a d)}{c+d x^2}+\frac {d (b c-a d)^2}{\left (c+d x^2\right )^2}-6 b^2 d \log \left (c+d x^2\right )}{4 (b c-a d)^4} \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 {x}{\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 = 0.93, size = 507, normalized size = 4.02 \begin {gather*} -\frac {2 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d - 6 \, a^{2} b c d^{2} + a^{3} d^{3} + 6 \, {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{4} + 3 \, {\left (3 \, b^{3} c^{2} d - 2 \, a b^{2} c d^{2} - a^{2} b d^{3}\right )} x^{2} + 6 \, {\left (b^{3} d^{3} x^{6} + a b^{2} c^{2} d + {\left (2 \, b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{4} + {\left (b^{3} c^{2} d + 2 \, a b^{2} c d^{2}\right )} x^{2}\right )} \log \left (b x^{2} + a\right ) - 6 \, {\left (b^{3} d^{3} x^{6} + a b^{2} c^{2} d + {\left (2 \, b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{4} + {\left (b^{3} c^{2} d + 2 \, a b^{2} c d^{2}\right )} x^{2}\right )} \log \left (d x^{2} + c\right )}{4 \, {\left (a b^{4} c^{6} - 4 \, a^{2} b^{3} c^{5} d + 6 \, a^{3} b^{2} c^{4} d^{2} - 4 \, a^{4} b c^{3} d^{3} + a^{5} c^{2} d^{4} + {\left (b^{5} c^{4} d^{2} - 4 \, a b^{4} c^{3} d^{3} + 6 \, a^{2} b^{3} c^{2} d^{4} - 4 \, a^{3} b^{2} c d^{5} + a^{4} b d^{6}\right )} x^{6} + {\left (2 \, b^{5} c^{5} d - 7 \, a b^{4} c^{4} d^{2} + 8 \, a^{2} b^{3} c^{3} d^{3} - 2 \, a^{3} b^{2} c^{2} d^{4} - 2 \, a^{4} b c d^{5} + a^{5} d^{6}\right )} x^{4} + {\left (b^{5} c^{6} - 2 \, a b^{4} c^{5} d - 2 \, a^{2} b^{3} c^{4} d^{2} + 8 \, a^{3} b^{2} c^{3} d^{3} - 7 \, a^{4} b c^{2} d^{4} + 2 \, a^{5} c d^{5}\right )} x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.43, size = 229, normalized size = 1.82 \begin {gather*} \frac {3 \, b^{3} d \log \left ({\left | \frac {b c}{b x^{2} + a} - \frac {a d}{b x^{2} + a} + d \right |}\right )}{2 \, {\left (b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}\right )}} - \frac {b^{5}}{2 \, {\left (b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}\right )} {\left (b x^{2} + a\right )}} + \frac {5 \, b^{2} d^{3} + \frac {6 \, {\left (b^{4} c d^{2} - a b^{3} d^{3}\right )}}{{\left (b x^{2} + a\right )} b}}{4 \, {\left (b c - a d\right )}^{4} {\left (\frac {b c}{b x^{2} + a} - \frac {a d}{b x^{2} + a} + d\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 234, normalized size = 1.86 \begin {gather*} -\frac {a^{2} d^{3}}{4 \left (a d -b c \right )^{4} \left (d \,x^{2}+c \right )^{2}}+\frac {a b c \,d^{2}}{2 \left (a d -b c \right )^{4} \left (d \,x^{2}+c \right )^{2}}-\frac {b^{2} c^{2} d}{4 \left (a d -b c \right )^{4} \left (d \,x^{2}+c \right )^{2}}+\frac {a \,b^{2} d}{2 \left (a d -b c \right )^{4} \left (b \,x^{2}+a \right )}+\frac {a b \,d^{2}}{\left (a d -b c \right )^{4} \left (d \,x^{2}+c \right )}-\frac {b^{3} c}{2 \left (a d -b c \right )^{4} \left (b \,x^{2}+a \right )}-\frac {b^{2} c d}{\left (a d -b c \right )^{4} \left (d \,x^{2}+c \right )}-\frac {3 b^{2} d \ln \left (b \,x^{2}+a \right )}{2 \left (a d -b c \right )^{4}}+\frac {3 b^{2} d \ln \left (d \,x^{2}+c \right )}{2 \left (a d -b c \right )^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.26, size = 394, normalized size = 3.13 \begin {gather*} -\frac {3 \, b^{2} d \log \left (b x^{2} + a\right )}{2 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )}} + \frac {3 \, b^{2} d \log \left (d x^{2} + c\right )}{2 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )}} - \frac {6 \, b^{2} d^{2} x^{4} + 2 \, b^{2} c^{2} + 5 \, a b c d - a^{2} d^{2} + 3 \, {\left (3 \, b^{2} c d + a b d^{2}\right )} x^{2}}{4 \, {\left (a b^{3} c^{5} - 3 \, a^{2} b^{2} c^{4} d + 3 \, a^{3} b c^{3} d^{2} - a^{4} c^{2} d^{3} + {\left (b^{4} c^{3} d^{2} - 3 \, a b^{3} c^{2} d^{3} + 3 \, a^{2} b^{2} c d^{4} - a^{3} b d^{5}\right )} x^{6} + {\left (2 \, b^{4} c^{4} d - 5 \, a b^{3} c^{3} d^{2} + 3 \, a^{2} b^{2} c^{2} d^{3} + a^{3} b c d^{4} - a^{4} d^{5}\right )} x^{4} + {\left (b^{4} c^{5} - a b^{3} c^{4} d - 3 \, a^{2} b^{2} c^{3} d^{2} + 5 \, a^{3} b c^{2} d^{3} - 2 \, a^{4} c d^{4}\right )} x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.59, size = 707, normalized size = 5.61 \begin {gather*} -\frac {a^3\,d^3+2\,b^3\,c^3-3\,a^2\,b\,d^3\,x^2-6\,a\,b^2\,d^3\,x^4+9\,b^3\,c^2\,d\,x^2+6\,b^3\,c\,d^2\,x^4+3\,a\,b^2\,c^2\,d-6\,a^2\,b\,c\,d^2+b^3\,d^3\,x^6\,\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 )\,12{}\mathrm {i}+a\,b^2\,d^3\,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 )\,12{}\mathrm {i}+b^3\,c^2\,d\,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 )\,12{}\mathrm {i}+b^3\,c\,d^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 )\,24{}\mathrm {i}-6\,a\,b^2\,c\,d^2\,x^2+a\,b^2\,c^2\,d\,\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 )\,12{}\mathrm {i}+a\,b^2\,c\,d^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 )\,24{}\mathrm {i}}{4\,a^5\,c^2\,d^4+8\,a^5\,c\,d^5\,x^2+4\,a^5\,d^6\,x^4-16\,a^4\,b\,c^3\,d^3-28\,a^4\,b\,c^2\,d^4\,x^2-8\,a^4\,b\,c\,d^5\,x^4+4\,a^4\,b\,d^6\,x^6+24\,a^3\,b^2\,c^4\,d^2+32\,a^3\,b^2\,c^3\,d^3\,x^2-8\,a^3\,b^2\,c^2\,d^4\,x^4-16\,a^3\,b^2\,c\,d^5\,x^6-16\,a^2\,b^3\,c^5\,d-8\,a^2\,b^3\,c^4\,d^2\,x^2+32\,a^2\,b^3\,c^3\,d^3\,x^4+24\,a^2\,b^3\,c^2\,d^4\,x^6+4\,a\,b^4\,c^6-8\,a\,b^4\,c^5\,d\,x^2-28\,a\,b^4\,c^4\,d^2\,x^4-16\,a\,b^4\,c^3\,d^3\,x^6+4\,b^5\,c^6\,x^2+8\,b^5\,c^5\,d\,x^4+4\,b^5\,c^4\,d^2\,x^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 10.06, size = 643, normalized size = 5.10 \begin {gather*} \frac {3 b^{2} d \log {\left (x^{2} + \frac {- \frac {3 a^{5} b^{2} d^{6}}{\left (a d - b c\right )^{4}} + \frac {15 a^{4} b^{3} c d^{5}}{\left (a d - b c\right )^{4}} - \frac {30 a^{3} b^{4} c^{2} d^{4}}{\left (a d - b c\right )^{4}} + \frac {30 a^{2} b^{5} c^{3} d^{3}}{\left (a d - b c\right )^{4}} - \frac {15 a b^{6} c^{4} d^{2}}{\left (a d - b c\right )^{4}} + 3 a b^{2} d^{2} + \frac {3 b^{7} c^{5} d}{\left (a d - b c\right )^{4}} + 3 b^{3} c d}{6 b^{3} d^{2}} \right )}}{2 \left (a d - b c\right )^{4}} - \frac {3 b^{2} d \log {\left (x^{2} + \frac {\frac {3 a^{5} b^{2} d^{6}}{\left (a d - b c\right )^{4}} - \frac {15 a^{4} b^{3} c d^{5}}{\left (a d - b c\right )^{4}} + \frac {30 a^{3} b^{4} c^{2} d^{4}}{\left (a d - b c\right )^{4}} - \frac {30 a^{2} b^{5} c^{3} d^{3}}{\left (a d - b c\right )^{4}} + \frac {15 a b^{6} c^{4} d^{2}}{\left (a d - b c\right )^{4}} + 3 a b^{2} d^{2} - \frac {3 b^{7} c^{5} d}{\left (a d - b c\right )^{4}} + 3 b^{3} c d}{6 b^{3} d^{2}} \right )}}{2 \left (a d - b c\right )^{4}} + \frac {- a^{2} d^{2} + 5 a b c d + 2 b^{2} c^{2} + 6 b^{2} d^{2} x^{4} + x^{2} \left (3 a b d^{2} + 9 b^{2} c d\right )}{4 a^{4} c^{2} d^{3} - 12 a^{3} b c^{3} d^{2} + 12 a^{2} b^{2} c^{4} d - 4 a b^{3} c^{5} + x^{6} \left (4 a^{3} b d^{5} - 12 a^{2} b^{2} c d^{4} + 12 a b^{3} c^{2} d^{3} - 4 b^{4} c^{3} d^{2}\right ) + x^{4} \left (4 a^{4} d^{5} - 4 a^{3} b c d^{4} - 12 a^{2} b^{2} c^{2} d^{3} + 20 a b^{3} c^{3} d^{2} - 8 b^{4} c^{4} d\right ) + x^{2} \left (8 a^{4} c d^{4} - 20 a^{3} b c^{2} d^{3} + 12 a^{2} b^{2} c^{3} d^{2} + 4 a b^{3} c^{4} d - 4 b^{4} c^{5}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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