Optimal. Leaf size=443 \[ -\frac {e \left (2 a B d e \left (c d^2-11 a e^2\right )-3 A \left (-5 a^2 e^4+4 a c d^2 e^2+c^2 d^4\right )\right )}{8 a^2 (d+e x) \left (a e^2+c d^2\right )^3}-\frac {x \left (2 a B e \left (c d^2-2 a e^2\right )-3 A c d \left (3 a e^2+c d^2\right )\right )+a e \left (-5 a A e^2+6 a B d e+A c d^2\right )}{8 a^2 \left (a+c x^2\right ) (d+e x) \left (a e^2+c d^2\right )^2}-\frac {\sqrt {c} \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \left (2 a B d e \left (-15 a^2 e^4+10 a c d^2 e^2+c^2 d^4\right )-3 A \left (-5 a^3 e^6+15 a^2 c d^2 e^4+5 a c^2 d^4 e^2+c^3 d^6\right )\right )}{8 a^{5/2} \left (a e^2+c d^2\right )^4}-\frac {a (B d-A e)-x (a B e+A c d)}{4 a \left (a+c x^2\right )^2 (d+e x) \left (a e^2+c d^2\right )}+\frac {e^4 \log \left (a+c x^2\right ) \left (-a B e^2-6 A c d e+5 B c d^2\right )}{2 \left (a e^2+c d^2\right )^4}-\frac {e^4 \log (d+e x) \left (-a B e^2-6 A c d e+5 B c d^2\right )}{\left (a e^2+c d^2\right )^4} \]
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Rubi [A] time = 0.72, antiderivative size = 443, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {823, 801, 635, 205, 260} \begin {gather*} -\frac {e \left (2 a B d e \left (c d^2-11 a e^2\right )-3 A \left (-5 a^2 e^4+4 a c d^2 e^2+c^2 d^4\right )\right )}{8 a^2 (d+e x) \left (a e^2+c d^2\right )^3}-\frac {\sqrt {c} \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \left (2 a B d e \left (-15 a^2 e^4+10 a c d^2 e^2+c^2 d^4\right )-3 A \left (15 a^2 c d^2 e^4-5 a^3 e^6+5 a c^2 d^4 e^2+c^3 d^6\right )\right )}{8 a^{5/2} \left (a e^2+c d^2\right )^4}-\frac {x \left (2 a B e \left (c d^2-2 a e^2\right )-3 A c d \left (3 a e^2+c d^2\right )\right )+a e \left (-5 a A e^2+6 a B d e+A c d^2\right )}{8 a^2 \left (a+c x^2\right ) (d+e x) \left (a e^2+c d^2\right )^2}-\frac {a (B d-A e)-x (a B e+A c d)}{4 a \left (a+c x^2\right )^2 (d+e x) \left (a e^2+c d^2\right )}+\frac {e^4 \log \left (a+c x^2\right ) \left (-a B e^2-6 A c d e+5 B c d^2\right )}{2 \left (a e^2+c d^2\right )^4}-\frac {e^4 \log (d+e x) \left (-a B e^2-6 A c d e+5 B c d^2\right )}{\left (a e^2+c d^2\right )^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 260
Rule 635
Rule 801
Rule 823
Rubi steps
\begin {align*} \int \frac {A+B x}{(d+e x)^2 \left (a+c x^2\right )^3} \, dx &=-\frac {a (B d-A e)-(A c d+a B e) x}{4 a \left (c d^2+a e^2\right ) (d+e x) \left (a+c x^2\right )^2}-\frac {\int \frac {-c \left (3 A c d^2-2 a B d e+5 a A e^2\right )-4 c e (A c d+a B e) x}{(d+e x)^2 \left (a+c x^2\right )^2} \, dx}{4 a c \left (c d^2+a e^2\right )}\\ &=-\frac {a (B d-A e)-(A c d+a B e) x}{4 a \left (c d^2+a e^2\right ) (d+e x) \left (a+c x^2\right )^2}-\frac {a e \left (A c d^2+6 a B d e-5 a A e^2\right )+\left (2 a B e \left (c d^2-2 a e^2\right )-3 A c d \left (c d^2+3 a e^2\right )\right ) x}{8 a^2 \left (c d^2+a e^2\right )^2 (d+e x) \left (a+c x^2\right )}+\frac {\int \frac {-c^2 \left (2 a B d e \left (c d^2+7 a e^2\right )-3 A \left (c^2 d^4+2 a c d^2 e^2+5 a^2 e^4\right )\right )-2 c^2 e \left (2 a B e \left (c d^2-2 a e^2\right )-3 A c d \left (c d^2+3 a e^2\right )\right ) x}{(d+e x)^2 \left (a+c x^2\right )} \, dx}{8 a^2 c^2 \left (c d^2+a e^2\right )^2}\\ &=-\frac {a (B d-A e)-(A c d+a B e) x}{4 a \left (c d^2+a e^2\right ) (d+e x) \left (a+c x^2\right )^2}-\frac {a e \left (A c d^2+6 a B d e-5 a A e^2\right )+\left (2 a B e \left (c d^2-2 a e^2\right )-3 A c d \left (c d^2+3 a e^2\right )\right ) x}{8 a^2 \left (c d^2+a e^2\right )^2 (d+e x) \left (a+c x^2\right )}+\frac {\int \left (\frac {c^2 e^2 \left (2 a B d e \left (c d^2-11 a e^2\right )-3 A \left (c^2 d^4+4 a c d^2 e^2-5 a^2 e^4\right )\right )}{\left (c d^2+a e^2\right ) (d+e x)^2}+\frac {8 a^2 c^2 e^5 \left (-5 B c d^2+6 A c d e+a B e^2\right )}{\left (c d^2+a e^2\right )^2 (d+e x)}+\frac {c^3 \left (-2 a B d e \left (c^2 d^4+10 a c d^2 e^2-15 a^2 e^4\right )+3 A \left (c^3 d^6+5 a c^2 d^4 e^2+15 a^2 c d^2 e^4-5 a^3 e^6\right )+8 a^2 e^4 \left (5 B c d^2-6 A c d e-a B e^2\right ) x\right )}{\left (c d^2+a e^2\right )^2 \left (a+c x^2\right )}\right ) \, dx}{8 a^2 c^2 \left (c d^2+a e^2\right )^2}\\ &=-\frac {e \left (2 a B d e \left (c d^2-11 a e^2\right )-3 A \left (c^2 d^4+4 a c d^2 e^2-5 a^2 e^4\right )\right )}{8 a^2 \left (c d^2+a e^2\right )^3 (d+e x)}-\frac {a (B d-A e)-(A c d+a B e) x}{4 a \left (c d^2+a e^2\right ) (d+e x) \left (a+c x^2\right )^2}-\frac {a e \left (A c d^2+6 a B d e-5 a A e^2\right )+\left (2 a B e \left (c d^2-2 a e^2\right )-3 A c d \left (c d^2+3 a e^2\right )\right ) x}{8 a^2 \left (c d^2+a e^2\right )^2 (d+e x) \left (a+c x^2\right )}-\frac {e^4 \left (5 B c d^2-6 A c d e-a B e^2\right ) \log (d+e x)}{\left (c d^2+a e^2\right )^4}+\frac {c \int \frac {-2 a B d e \left (c^2 d^4+10 a c d^2 e^2-15 a^2 e^4\right )+3 A \left (c^3 d^6+5 a c^2 d^4 e^2+15 a^2 c d^2 e^4-5 a^3 e^6\right )+8 a^2 e^4 \left (5 B c d^2-6 A c d e-a B e^2\right ) x}{a+c x^2} \, dx}{8 a^2 \left (c d^2+a e^2\right )^4}\\ &=-\frac {e \left (2 a B d e \left (c d^2-11 a e^2\right )-3 A \left (c^2 d^4+4 a c d^2 e^2-5 a^2 e^4\right )\right )}{8 a^2 \left (c d^2+a e^2\right )^3 (d+e x)}-\frac {a (B d-A e)-(A c d+a B e) x}{4 a \left (c d^2+a e^2\right ) (d+e x) \left (a+c x^2\right )^2}-\frac {a e \left (A c d^2+6 a B d e-5 a A e^2\right )+\left (2 a B e \left (c d^2-2 a e^2\right )-3 A c d \left (c d^2+3 a e^2\right )\right ) x}{8 a^2 \left (c d^2+a e^2\right )^2 (d+e x) \left (a+c x^2\right )}-\frac {e^4 \left (5 B c d^2-6 A c d e-a B e^2\right ) \log (d+e x)}{\left (c d^2+a e^2\right )^4}+\frac {\left (c e^4 \left (5 B c d^2-6 A c d e-a B e^2\right )\right ) \int \frac {x}{a+c x^2} \, dx}{\left (c d^2+a e^2\right )^4}-\frac {\left (c \left (2 a B d e \left (c^2 d^4+10 a c d^2 e^2-15 a^2 e^4\right )-3 A \left (c^3 d^6+5 a c^2 d^4 e^2+15 a^2 c d^2 e^4-5 a^3 e^6\right )\right )\right ) \int \frac {1}{a+c x^2} \, dx}{8 a^2 \left (c d^2+a e^2\right )^4}\\ &=-\frac {e \left (2 a B d e \left (c d^2-11 a e^2\right )-3 A \left (c^2 d^4+4 a c d^2 e^2-5 a^2 e^4\right )\right )}{8 a^2 \left (c d^2+a e^2\right )^3 (d+e x)}-\frac {a (B d-A e)-(A c d+a B e) x}{4 a \left (c d^2+a e^2\right ) (d+e x) \left (a+c x^2\right )^2}-\frac {a e \left (A c d^2+6 a B d e-5 a A e^2\right )+\left (2 a B e \left (c d^2-2 a e^2\right )-3 A c d \left (c d^2+3 a e^2\right )\right ) x}{8 a^2 \left (c d^2+a e^2\right )^2 (d+e x) \left (a+c x^2\right )}-\frac {\sqrt {c} \left (2 a B d e \left (c^2 d^4+10 a c d^2 e^2-15 a^2 e^4\right )-3 A \left (c^3 d^6+5 a c^2 d^4 e^2+15 a^2 c d^2 e^4-5 a^3 e^6\right )\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{8 a^{5/2} \left (c d^2+a e^2\right )^4}-\frac {e^4 \left (5 B c d^2-6 A c d e-a B e^2\right ) \log (d+e x)}{\left (c d^2+a e^2\right )^4}+\frac {e^4 \left (5 B c d^2-6 A c d e-a B e^2\right ) \log \left (a+c x^2\right )}{2 \left (c d^2+a e^2\right )^4}\\ \end {align*}
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Mathematica [A] time = 0.49, size = 378, normalized size = 0.85 \begin {gather*} \frac {\frac {2 \left (a e^2+c d^2\right )^2 \left (a^2 B e^2-a c (A e (e x-2 d)+B d (d-2 e x))+A c^2 d^2 x\right )}{a \left (a+c x^2\right )^2}+\frac {\left (a e^2+c d^2\right ) \left (4 a^3 B e^4+a^2 c e^2 (A e (16 d-7 e x)-2 B d (6 d-7 e x))-2 a c^2 d^2 e x (B d-6 A e)+3 A c^3 d^4 x\right )}{a^2 \left (a+c x^2\right )}+\frac {\sqrt {c} \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \left (2 a B d e \left (15 a^2 e^4-10 a c d^2 e^2-c^2 d^4\right )+3 A \left (-5 a^3 e^6+15 a^2 c d^2 e^4+5 a c^2 d^4 e^2+c^3 d^6\right )\right )}{a^{5/2}}-4 e^4 \log \left (a+c x^2\right ) \left (a B e^2+6 A c d e-5 B c d^2\right )-\frac {8 e^4 \left (a e^2+c d^2\right ) (A e-B d)}{d+e x}+8 e^4 \log (d+e x) \left (a B e^2+6 A c d e-5 B c d^2\right )}{8 \left (a e^2+c d^2\right )^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 {A+B x}{(d+e x)^2 \left (a+c x^2\right )^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [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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giac [A] time = 0.22, size = 836, normalized size = 1.89 \begin {gather*} \frac {{\left (3 \, A c^{4} d^{6} e^{2} - 2 \, B a c^{3} d^{5} e^{3} + 15 \, A a c^{3} d^{4} e^{4} - 20 \, B a^{2} c^{2} d^{3} e^{5} + 45 \, A a^{2} c^{2} d^{2} e^{6} + 30 \, B a^{3} c d e^{7} - 15 \, A a^{3} c e^{8}\right )} \arctan \left (\frac {{\left (c d - \frac {c d^{2}}{x e + d} - \frac {a e^{2}}{x e + d}\right )} e^{\left (-1\right )}}{\sqrt {a c}}\right ) e^{\left (-2\right )}}{8 \, {\left (a^{2} c^{4} d^{8} + 4 \, a^{3} c^{3} d^{6} e^{2} + 6 \, a^{4} c^{2} d^{4} e^{4} + 4 \, a^{5} c d^{2} e^{6} + a^{6} e^{8}\right )} \sqrt {a c}} + \frac {{\left (5 \, B c d^{2} e^{4} - 6 \, A c d e^{5} - B a e^{6}\right )} \log \left (c - \frac {2 \, c d}{x e + d} + \frac {c d^{2}}{{\left (x e + d\right )}^{2}} + \frac {a e^{2}}{{\left (x e + d\right )}^{2}}\right )}{2 \, {\left (c^{4} d^{8} + 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} + 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )}} + \frac {\frac {B d e^{10}}{x e + d} - \frac {A e^{11}}{x e + d}}{c^{3} d^{6} e^{6} + 3 \, a c^{2} d^{4} e^{8} + 3 \, a^{2} c d^{2} e^{10} + a^{3} e^{12}} + \frac {3 \, A c^{5} d^{5} e - 2 \, B a c^{4} d^{4} e^{2} + 14 \, A a c^{4} d^{3} e^{3} + 32 \, B a^{2} c^{3} d^{2} e^{4} - 29 \, A a^{2} c^{3} d e^{5} - 6 \, B a^{3} c^{2} e^{6} - \frac {{\left (9 \, A c^{5} d^{6} e^{2} - 6 \, B a c^{4} d^{5} e^{3} + 41 \, A a c^{4} d^{4} e^{4} + 116 \, B a^{2} c^{3} d^{3} e^{5} - 121 \, A a^{2} c^{3} d^{2} e^{6} - 38 \, B a^{3} c^{2} d e^{7} + 7 \, A a^{3} c^{2} e^{8}\right )} e^{\left (-1\right )}}{x e + d} + \frac {{\left (9 \, A c^{5} d^{7} e^{3} - 6 \, B a c^{4} d^{6} e^{4} + 45 \, A a c^{4} d^{5} e^{5} + 140 \, B a^{2} c^{3} d^{4} e^{6} - 145 \, A a^{2} c^{3} d^{3} e^{7} - 22 \, B a^{3} c^{2} d^{2} e^{8} - 21 \, A a^{3} c^{2} d e^{9} - 8 \, B a^{4} c e^{10}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}} - \frac {{\left (3 \, A c^{5} d^{8} e^{4} - 2 \, B a c^{4} d^{7} e^{5} + 18 \, A a c^{4} d^{6} e^{6} + 58 \, B a^{2} c^{3} d^{5} e^{7} - 60 \, A a^{2} c^{3} d^{4} e^{8} + 26 \, B a^{3} c^{2} d^{3} e^{9} - 66 \, A a^{3} c^{2} d^{2} e^{10} - 34 \, B a^{4} c d e^{11} + 9 \, A a^{4} c e^{12}\right )} e^{\left (-3\right )}}{{\left (x e + d\right )}^{3}}}{8 \, {\left (c d^{2} + a e^{2}\right )}^{4} a^{2} {\left (c - \frac {2 \, c d}{x e + d} + \frac {c d^{2}}{{\left (x e + d\right )}^{2}} + \frac {a e^{2}}{{\left (x e + d\right )}^{2}}\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.12, size = 1410, normalized size = 3.18
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.39, size = 997, normalized size = 2.25 \begin {gather*} \frac {{\left (5 \, B c d^{2} e^{4} - 6 \, A c d e^{5} - B a e^{6}\right )} \log \left (c x^{2} + a\right )}{2 \, {\left (c^{4} d^{8} + 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} + 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )}} - \frac {{\left (5 \, B c d^{2} e^{4} - 6 \, A c d e^{5} - B a e^{6}\right )} \log \left (e x + d\right )}{c^{4} d^{8} + 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} + 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}} + \frac {{\left (3 \, A c^{4} d^{6} - 2 \, B a c^{3} d^{5} e + 15 \, A a c^{3} d^{4} e^{2} - 20 \, B a^{2} c^{2} d^{3} e^{3} + 45 \, A a^{2} c^{2} d^{2} e^{4} + 30 \, B a^{3} c d e^{5} - 15 \, A a^{3} c e^{6}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{8 \, {\left (a^{2} c^{4} d^{8} + 4 \, a^{3} c^{3} d^{6} e^{2} + 6 \, a^{4} c^{2} d^{4} e^{4} + 4 \, a^{5} c d^{2} e^{6} + a^{6} e^{8}\right )} \sqrt {a c}} - \frac {2 \, B a^{2} c^{2} d^{5} - 4 \, A a^{2} c^{2} d^{4} e + 12 \, B a^{3} c d^{3} e^{2} - 20 \, A a^{3} c d^{2} e^{3} - 14 \, B a^{4} d e^{4} + 8 \, A a^{4} e^{5} - {\left (3 \, A c^{4} d^{4} e - 2 \, B a c^{3} d^{3} e^{2} + 12 \, A a c^{3} d^{2} e^{3} + 22 \, B a^{2} c^{2} d e^{4} - 15 \, A a^{2} c^{2} e^{5}\right )} x^{4} - {\left (3 \, A c^{4} d^{5} - 2 \, B a c^{3} d^{4} e + 12 \, A a c^{3} d^{3} e^{2} + 2 \, B a^{2} c^{2} d^{2} e^{3} + 9 \, A a^{2} c^{2} d e^{4} + 4 \, B a^{3} c e^{5}\right )} x^{3} - {\left (5 \, A a c^{3} d^{4} e - 10 \, B a^{2} c^{2} d^{3} e^{2} + 28 \, A a^{2} c^{2} d^{2} e^{3} + 38 \, B a^{3} c d e^{4} - 25 \, A a^{3} c e^{5}\right )} x^{2} - {\left (5 \, A a c^{3} d^{5} + 16 \, A a^{2} c^{2} d^{3} e^{2} + 6 \, B a^{3} c d^{2} e^{3} + 11 \, A a^{3} c d e^{4} + 6 \, B a^{4} e^{5}\right )} x}{8 \, {\left (a^{4} c^{3} d^{7} + 3 \, a^{5} c^{2} d^{5} e^{2} + 3 \, a^{6} c d^{3} e^{4} + a^{7} d e^{6} + {\left (a^{2} c^{5} d^{6} e + 3 \, a^{3} c^{4} d^{4} e^{3} + 3 \, a^{4} c^{3} d^{2} e^{5} + a^{5} c^{2} e^{7}\right )} x^{5} + {\left (a^{2} c^{5} d^{7} + 3 \, a^{3} c^{4} d^{5} e^{2} + 3 \, a^{4} c^{3} d^{3} e^{4} + a^{5} c^{2} d e^{6}\right )} x^{4} + 2 \, {\left (a^{3} c^{4} d^{6} e + 3 \, a^{4} c^{3} d^{4} e^{3} + 3 \, a^{5} c^{2} d^{2} e^{5} + a^{6} c e^{7}\right )} x^{3} + 2 \, {\left (a^{3} c^{4} d^{7} + 3 \, a^{4} c^{3} d^{5} e^{2} + 3 \, a^{5} c^{2} d^{3} e^{4} + a^{6} c d e^{6}\right )} x^{2} + {\left (a^{4} c^{3} d^{6} e + 3 \, a^{5} c^{2} d^{4} e^{3} + 3 \, a^{6} c d^{2} e^{5} + a^{7} e^{7}\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.51, size = 3015, normalized size = 6.81
result too large to display
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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