Optimal. Leaf size=142 \[ \frac {a b}{2 \left (a+b x^2\right ) (b c-a d)^3}+\frac {a d+b c}{2 \left (c+d x^2\right ) (b c-a d)^3}+\frac {c}{4 \left (c+d x^2\right )^2 (b c-a d)^2}+\frac {b (2 a d+b c) \log \left (a+b x^2\right )}{2 (b c-a d)^4}-\frac {b (2 a d+b c) \log \left (c+d x^2\right )}{2 (b c-a d)^4} \]
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Rubi [A] time = 0.15, antiderivative size = 142, 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, 77} \[ \frac {a b}{2 \left (a+b x^2\right ) (b c-a d)^3}+\frac {a d+b c}{2 \left (c+d x^2\right ) (b c-a d)^3}+\frac {c}{4 \left (c+d x^2\right )^2 (b c-a d)^2}+\frac {b (2 a d+b c) \log \left (a+b x^2\right )}{2 (b c-a d)^4}-\frac {b (2 a d+b c) \log \left (c+d x^2\right )}{2 (b c-a d)^4} \]
Antiderivative was successfully verified.
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Rule 77
Rule 446
Rubi steps
\begin {align*} \int \frac {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 {x}{(a+b x)^2 (c+d x)^3} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (-\frac {a b^2}{(b c-a d)^3 (a+b x)^2}+\frac {b^2 (b c+2 a d)}{(b c-a d)^4 (a+b x)}-\frac {c d}{(b c-a d)^2 (c+d x)^3}-\frac {d (b c+a d)}{(b c-a d)^3 (c+d x)^2}-\frac {b d (b c+2 a d)}{(b c-a d)^4 (c+d x)}\right ) \, dx,x,x^2\right )\\ &=\frac {a b}{2 (b c-a d)^3 \left (a+b x^2\right )}+\frac {c}{4 (b c-a d)^2 \left (c+d x^2\right )^2}+\frac {b c+a d}{2 (b c-a d)^3 \left (c+d x^2\right )}+\frac {b (b c+2 a d) \log \left (a+b x^2\right )}{2 (b c-a d)^4}-\frac {b (b c+2 a d) \log \left (c+d x^2\right )}{2 (b c-a d)^4}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 121, normalized size = 0.85 \[ \frac {\frac {c (b c-a d)^2}{\left (c+d x^2\right )^2}+\frac {2 a b (b c-a d)}{a+b x^2}+\frac {2 (a d+b c) (b c-a d)}{c+d x^2}+2 b (2 a d+b c) \log \left (a+b x^2\right )-2 b (2 a d+b c) \log \left (c+d x^2\right )}{4 (b c-a d)^4} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.49, size = 598, normalized size = 4.21 \[ \frac {5 \, a b^{2} c^{3} - 4 \, a^{2} b c^{2} d - a^{3} c d^{2} + 2 \, {\left (b^{3} c^{2} d + a b^{2} c d^{2} - 2 \, a^{2} b d^{3}\right )} x^{4} + {\left (3 \, b^{3} c^{3} + 4 \, a b^{2} c^{2} d - 5 \, a^{2} b c d^{2} - 2 \, a^{3} d^{3}\right )} x^{2} + 2 \, {\left ({\left (b^{3} c d^{2} + 2 \, a b^{2} d^{3}\right )} x^{6} + a b^{2} c^{3} + 2 \, a^{2} b c^{2} d + {\left (2 \, b^{3} c^{2} d + 5 \, a b^{2} c d^{2} + 2 \, a^{2} b d^{3}\right )} x^{4} + {\left (b^{3} c^{3} + 4 \, a b^{2} c^{2} d + 4 \, a^{2} b c d^{2}\right )} x^{2}\right )} \log \left (b x^{2} + a\right ) - 2 \, {\left ({\left (b^{3} c d^{2} + 2 \, a b^{2} d^{3}\right )} x^{6} + a b^{2} c^{3} + 2 \, a^{2} b c^{2} d + {\left (2 \, b^{3} c^{2} d + 5 \, a b^{2} c d^{2} + 2 \, a^{2} b d^{3}\right )} x^{4} + {\left (b^{3} c^{3} + 4 \, a b^{2} c^{2} d + 4 \, a^{2} b c d^{2}\right )} x^{2}\right )} \log \left (d x^{2} + c\right )}{4 \, {\left (a b^{4} c^{6} - 4 \, a^{2} b^{3} c^{5} d + 6 \, a^{3} b^{2} c^{4} d^{2} - 4 \, a^{4} b c^{3} d^{3} + a^{5} c^{2} d^{4} + {\left (b^{5} c^{4} d^{2} - 4 \, a b^{4} c^{3} d^{3} + 6 \, a^{2} b^{3} c^{2} d^{4} - 4 \, a^{3} b^{2} c d^{5} + a^{4} b d^{6}\right )} x^{6} + {\left (2 \, b^{5} c^{5} d - 7 \, a b^{4} c^{4} d^{2} + 8 \, a^{2} b^{3} c^{3} d^{3} - 2 \, a^{3} b^{2} c^{2} d^{4} - 2 \, a^{4} b c d^{5} + a^{5} d^{6}\right )} x^{4} + {\left (b^{5} c^{6} - 2 \, a b^{4} c^{5} d - 2 \, a^{2} b^{3} c^{4} d^{2} + 8 \, a^{3} b^{2} c^{3} d^{3} - 7 \, a^{4} b c^{2} d^{4} + 2 \, a^{5} c d^{5}\right )} x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.49, size = 267, normalized size = 1.88 \[ \frac {\frac {2 \, a b^{5}}{{\left (b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}\right )} {\left (b x^{2} + a\right )}} - \frac {2 \, {\left (b^{4} c + 2 \, a b^{3} d\right )} \log \left ({\left | \frac {b c}{b x^{2} + a} - \frac {a d}{b x^{2} + a} + d \right |}\right )}{b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}} - \frac {3 \, b^{3} c d^{2} + 2 \, a b^{2} d^{3} + \frac {2 \, {\left (2 \, b^{5} c^{2} d - a b^{4} c d^{2} - a^{2} b^{3} d^{3}\right )}}{{\left (b x^{2} + a\right )} b}}{{\left (b c - a d\right )}^{4} {\left (\frac {b c}{b x^{2} + a} - \frac {a d}{b x^{2} + a} + d\right )}^{2}}}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 283, normalized size = 1.99 \[ \frac {a^{2} c \,d^{2}}{4 \left (a d -b c \right )^{4} \left (d \,x^{2}+c \right )^{2}}-\frac {a b \,c^{2} d}{2 \left (a d -b c \right )^{4} \left (d \,x^{2}+c \right )^{2}}+\frac {b^{2} c^{3}}{4 \left (a d -b c \right )^{4} \left (d \,x^{2}+c \right )^{2}}-\frac {a^{2} b d}{2 \left (a d -b c \right )^{4} \left (b \,x^{2}+a \right )}-\frac {a^{2} d^{2}}{2 \left (a d -b c \right )^{4} \left (d \,x^{2}+c \right )}+\frac {a \,b^{2} c}{2 \left (a d -b c \right )^{4} \left (b \,x^{2}+a \right )}+\frac {a b d \ln \left (b \,x^{2}+a \right )}{\left (a d -b c \right )^{4}}-\frac {a b d \ln \left (d \,x^{2}+c \right )}{\left (a d -b c \right )^{4}}+\frac {b^{2} c^{2}}{2 \left (a d -b c \right )^{4} \left (d \,x^{2}+c \right )}+\frac {b^{2} c \ln \left (b \,x^{2}+a \right )}{2 \left (a d -b c \right )^{4}}-\frac {b^{2} c \ln \left (d \,x^{2}+c \right )}{2 \left (a d -b c \right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.23, size = 415, normalized size = 2.92 \[ \frac {{\left (b^{2} c + 2 \, a b d\right )} \log \left (b x^{2} + a\right )}{2 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )}} - \frac {{\left (b^{2} c + 2 \, a b d\right )} \log \left (d x^{2} + c\right )}{2 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )}} + \frac {2 \, {\left (b^{2} c d + 2 \, a b d^{2}\right )} x^{4} + 5 \, a b c^{2} + a^{2} c d + {\left (3 \, b^{2} c^{2} + 7 \, a b c d + 2 \, a^{2} d^{2}\right )} x^{2}}{4 \, {\left (a b^{3} c^{5} - 3 \, a^{2} b^{2} c^{4} d + 3 \, a^{3} b c^{3} d^{2} - a^{4} c^{2} d^{3} + {\left (b^{4} c^{3} d^{2} - 3 \, a b^{3} c^{2} d^{3} + 3 \, a^{2} b^{2} c d^{4} - a^{3} b d^{5}\right )} x^{6} + {\left (2 \, b^{4} c^{4} d - 5 \, a b^{3} c^{3} d^{2} + 3 \, a^{2} b^{2} c^{2} d^{3} + a^{3} b c d^{4} - a^{4} d^{5}\right )} x^{4} + {\left (b^{4} c^{5} - a b^{3} c^{4} d - 3 \, a^{2} b^{2} c^{3} d^{2} + 5 \, a^{3} b c^{2} d^{3} - 2 \, a^{4} c d^{4}\right )} x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.70, size = 926, normalized size = 6.52 \[ \frac {5\,a\,b^2\,c^3-a^3\,c\,d^2-2\,a^3\,d^3\,x^2+3\,b^3\,c^3\,x^2-4\,a^2\,b\,d^3\,x^4+2\,b^3\,c^2\,d\,x^4+a\,b^2\,c^3\,\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^2\,b\,c^2\,d+b^3\,c^3\,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 )\,4{}\mathrm {i}+a^2\,b\,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 )\,8{}\mathrm {i}+a\,b^2\,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 )\,8{}\mathrm {i}+b^3\,c^2\,d\,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 )\,8{}\mathrm {i}+b^3\,c\,d^2\,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 )\,4{}\mathrm {i}+4\,a\,b^2\,c^2\,d\,x^2-5\,a^2\,b\,c\,d^2\,x^2+2\,a\,b^2\,c\,d^2\,x^4+a^2\,b\,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 )\,8{}\mathrm {i}+a\,b^2\,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 )\,16{}\mathrm {i}+a^2\,b\,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 )\,16{}\mathrm {i}+a\,b^2\,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 )\,20{}\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} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 7.95, size = 784, normalized size = 5.52 \[ - \frac {b \left (2 a d + b c\right ) \log {\left (x^{2} + \frac {- \frac {a^{5} b d^{5} \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} + \frac {5 a^{4} b^{2} c d^{4} \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} - \frac {10 a^{3} b^{3} c^{2} d^{3} \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} + \frac {10 a^{2} b^{4} c^{3} d^{2} \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} + 2 a^{2} b d^{2} - \frac {5 a b^{5} c^{4} d \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} + 3 a b^{2} c d + \frac {b^{6} c^{5} \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} + b^{3} c^{2}}{4 a b^{2} d^{2} + 2 b^{3} c d} \right )}}{2 \left (a d - b c\right )^{4}} + \frac {b \left (2 a d + b c\right ) \log {\left (x^{2} + \frac {\frac {a^{5} b d^{5} \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} - \frac {5 a^{4} b^{2} c d^{4} \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} + \frac {10 a^{3} b^{3} c^{2} d^{3} \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} - \frac {10 a^{2} b^{4} c^{3} d^{2} \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} + 2 a^{2} b d^{2} + \frac {5 a b^{5} c^{4} d \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} + 3 a b^{2} c d - \frac {b^{6} c^{5} \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} + b^{3} c^{2}}{4 a b^{2} d^{2} + 2 b^{3} c d} \right )}}{2 \left (a d - b c\right )^{4}} + \frac {- a^{2} c d - 5 a b c^{2} + x^{4} \left (- 4 a b d^{2} - 2 b^{2} c d\right ) + x^{2} \left (- 2 a^{2} d^{2} - 7 a b c d - 3 b^{2} c^{2}\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 )} \]
Verification of antiderivative is not currently implemented for this CAS.
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