Optimal. Leaf size=226 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \left (A c d \left (3 a e^2+c d^2\right )+a \left (c d^2-a e^2\right ) (C d-B e)\right )}{2 a^{3/2} \sqrt {c} \left (a e^2+c d^2\right )^2}-\frac {a (a C e-A c e+B c d)-c x (a B e-a C d+A c d)}{2 a c \left (a+c x^2\right ) \left (a e^2+c d^2\right )}-\frac {e \log \left (a+c x^2\right ) \left (A e^2-B d e+C d^2\right )}{2 \left (a e^2+c d^2\right )^2}+\frac {e \log (d+e x) \left (A e^2-B d e+C d^2\right )}{\left (a e^2+c d^2\right )^2} \]
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Rubi [A] time = 0.43, antiderivative size = 226, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {1647, 801, 635, 205, 260} \[ \frac {\tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \left (A c d \left (3 a e^2+c d^2\right )+a \left (c d^2-a e^2\right ) (C d-B e)\right )}{2 a^{3/2} \sqrt {c} \left (a e^2+c d^2\right )^2}-\frac {a (a C e-A c e+B c d)-c x (a B e-a C d+A c d)}{2 a c \left (a+c x^2\right ) \left (a e^2+c d^2\right )}-\frac {e \log \left (a+c x^2\right ) \left (A e^2-B d e+C d^2\right )}{2 \left (a e^2+c d^2\right )^2}+\frac {e \log (d+e x) \left (A e^2-B d e+C d^2\right )}{\left (a e^2+c d^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 205
Rule 260
Rule 635
Rule 801
Rule 1647
Rubi steps
\begin {align*} \int \frac {A+B x+C x^2}{(d+e x) \left (a+c x^2\right )^2} \, dx &=-\frac {a (B c d-A c e+a C e)-c (A c d-a C d+a B e) x}{2 a c \left (c d^2+a e^2\right ) \left (a+c x^2\right )}-\frac {\int \frac {-\frac {c \left (a d (C d-B e)+A \left (c d^2+2 a e^2\right )\right )}{c d^2+a e^2}-\frac {c e (A c d-a C d+a B e) x}{c d^2+a e^2}}{(d+e x) \left (a+c x^2\right )} \, dx}{2 a c}\\ &=-\frac {a (B c d-A c e+a C e)-c (A c d-a C d+a B e) x}{2 a c \left (c d^2+a e^2\right ) \left (a+c x^2\right )}-\frac {\int \left (-\frac {2 a c e^2 \left (C d^2-B d e+A e^2\right )}{\left (c d^2+a e^2\right )^2 (d+e x)}+\frac {c \left (-a (C d-B e) \left (c d^2-a e^2\right )-A c d \left (c d^2+3 a e^2\right )+2 a c e \left (C d^2-B d e+A e^2\right ) x\right )}{\left (c d^2+a e^2\right )^2 \left (a+c x^2\right )}\right ) \, dx}{2 a c}\\ &=-\frac {a (B c d-A c e+a C e)-c (A c d-a C d+a B e) x}{2 a c \left (c d^2+a e^2\right ) \left (a+c x^2\right )}+\frac {e \left (C d^2-B d e+A e^2\right ) \log (d+e x)}{\left (c d^2+a e^2\right )^2}-\frac {\int \frac {-a (C d-B e) \left (c d^2-a e^2\right )-A c d \left (c d^2+3 a e^2\right )+2 a c e \left (C d^2-B d e+A e^2\right ) x}{a+c x^2} \, dx}{2 a \left (c d^2+a e^2\right )^2}\\ &=-\frac {a (B c d-A c e+a C e)-c (A c d-a C d+a B e) x}{2 a c \left (c d^2+a e^2\right ) \left (a+c x^2\right )}+\frac {e \left (C d^2-B d e+A e^2\right ) \log (d+e x)}{\left (c d^2+a e^2\right )^2}-\frac {\left (c e \left (C d^2-B d e+A e^2\right )\right ) \int \frac {x}{a+c x^2} \, dx}{\left (c d^2+a e^2\right )^2}+\frac {\left (a (C d-B e) \left (c d^2-a e^2\right )+A c d \left (c d^2+3 a e^2\right )\right ) \int \frac {1}{a+c x^2} \, dx}{2 a \left (c d^2+a e^2\right )^2}\\ &=-\frac {a (B c d-A c e+a C e)-c (A c d-a C d+a B e) x}{2 a c \left (c d^2+a e^2\right ) \left (a+c x^2\right )}+\frac {\left (a (C d-B e) \left (c d^2-a e^2\right )+A c d \left (c d^2+3 a e^2\right )\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {c} \left (c d^2+a e^2\right )^2}+\frac {e \left (C d^2-B d e+A e^2\right ) \log (d+e x)}{\left (c d^2+a e^2\right )^2}-\frac {e \left (C d^2-B d e+A e^2\right ) \log \left (a+c x^2\right )}{2 \left (c d^2+a e^2\right )^2}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 195, normalized size = 0.86 \[ \frac {\frac {\tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \left (A c d \left (3 a e^2+c d^2\right )+a \left (c d^2-a e^2\right ) (C d-B e)\right )}{a^{3/2} \sqrt {c}}+\frac {\left (a e^2+c d^2\right ) \left (a^2 (-C) e+a c (A e-B d+B e x-C d x)+A c^2 d x\right )}{a c \left (a+c x^2\right )}-e \log \left (a+c x^2\right ) \left (e (A e-B d)+C d^2\right )+2 e \log (d+e x) \left (e (A e-B d)+C d^2\right )}{2 \left (a e^2+c d^2\right )^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 64.28, size = 1024, normalized size = 4.53 \[ \left [-\frac {2 \, B a^{2} c^{2} d^{3} + 2 \, B a^{3} c d e^{2} + 2 \, {\left (C a^{3} c - A a^{2} c^{2}\right )} d^{2} e + 2 \, {\left (C a^{4} - A a^{3} c\right )} e^{3} - {\left (B a^{2} c d^{2} e - B a^{3} e^{3} - {\left (C a^{2} c + A a c^{2}\right )} d^{3} + {\left (C a^{3} - 3 \, A a^{2} c\right )} d e^{2} + {\left (B a c^{2} d^{2} e - B a^{2} c e^{3} - {\left (C a c^{2} + A c^{3}\right )} d^{3} + {\left (C a^{2} c - 3 \, A a c^{2}\right )} d e^{2}\right )} x^{2}\right )} \sqrt {-a c} \log \left (\frac {c x^{2} - 2 \, \sqrt {-a c} x - a}{c x^{2} + a}\right ) - 2 \, {\left (B a^{2} c^{2} d^{2} e + B a^{3} c e^{3} - {\left (C a^{2} c^{2} - A a c^{3}\right )} d^{3} - {\left (C a^{3} c - A a^{2} c^{2}\right )} d e^{2}\right )} x + 2 \, {\left (C a^{3} c d^{2} e - B a^{3} c d e^{2} + A a^{3} c e^{3} + {\left (C a^{2} c^{2} d^{2} e - B a^{2} c^{2} d e^{2} + A a^{2} c^{2} e^{3}\right )} x^{2}\right )} \log \left (c x^{2} + a\right ) - 4 \, {\left (C a^{3} c d^{2} e - B a^{3} c d e^{2} + A a^{3} c e^{3} + {\left (C a^{2} c^{2} d^{2} e - B a^{2} c^{2} d e^{2} + A a^{2} c^{2} e^{3}\right )} x^{2}\right )} \log \left (e x + d\right )}{4 \, {\left (a^{3} c^{3} d^{4} + 2 \, a^{4} c^{2} d^{2} e^{2} + a^{5} c e^{4} + {\left (a^{2} c^{4} d^{4} + 2 \, a^{3} c^{3} d^{2} e^{2} + a^{4} c^{2} e^{4}\right )} x^{2}\right )}}, -\frac {B a^{2} c^{2} d^{3} + B a^{3} c d e^{2} + {\left (C a^{3} c - A a^{2} c^{2}\right )} d^{2} e + {\left (C a^{4} - A a^{3} c\right )} e^{3} + {\left (B a^{2} c d^{2} e - B a^{3} e^{3} - {\left (C a^{2} c + A a c^{2}\right )} d^{3} + {\left (C a^{3} - 3 \, A a^{2} c\right )} d e^{2} + {\left (B a c^{2} d^{2} e - B a^{2} c e^{3} - {\left (C a c^{2} + A c^{3}\right )} d^{3} + {\left (C a^{2} c - 3 \, A a c^{2}\right )} d e^{2}\right )} x^{2}\right )} \sqrt {a c} \arctan \left (\frac {\sqrt {a c} x}{a}\right ) - {\left (B a^{2} c^{2} d^{2} e + B a^{3} c e^{3} - {\left (C a^{2} c^{2} - A a c^{3}\right )} d^{3} - {\left (C a^{3} c - A a^{2} c^{2}\right )} d e^{2}\right )} x + {\left (C a^{3} c d^{2} e - B a^{3} c d e^{2} + A a^{3} c e^{3} + {\left (C a^{2} c^{2} d^{2} e - B a^{2} c^{2} d e^{2} + A a^{2} c^{2} e^{3}\right )} x^{2}\right )} \log \left (c x^{2} + a\right ) - 2 \, {\left (C a^{3} c d^{2} e - B a^{3} c d e^{2} + A a^{3} c e^{3} + {\left (C a^{2} c^{2} d^{2} e - B a^{2} c^{2} d e^{2} + A a^{2} c^{2} e^{3}\right )} x^{2}\right )} \log \left (e x + d\right )}{2 \, {\left (a^{3} c^{3} d^{4} + 2 \, a^{4} c^{2} d^{2} e^{2} + a^{5} c e^{4} + {\left (a^{2} c^{4} d^{4} + 2 \, a^{3} c^{3} d^{2} e^{2} + a^{4} c^{2} e^{4}\right )} x^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 350, normalized size = 1.55 \[ -\frac {{\left (C d^{2} e - B d e^{2} + A e^{3}\right )} \log \left (c x^{2} + a\right )}{2 \, {\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )}} + \frac {{\left (C d^{2} e^{2} - B d e^{3} + A e^{4}\right )} \log \left ({\left | x e + d \right |}\right )}{c^{2} d^{4} e + 2 \, a c d^{2} e^{3} + a^{2} e^{5}} + \frac {{\left (C a c d^{3} + A c^{2} d^{3} - B a c d^{2} e - C a^{2} d e^{2} + 3 \, A a c d e^{2} + B a^{2} e^{3}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \, {\left (a c^{2} d^{4} + 2 \, a^{2} c d^{2} e^{2} + a^{3} e^{4}\right )} \sqrt {a c}} - \frac {B a c^{2} d^{3} + C a^{2} c d^{2} e - A a c^{2} d^{2} e + B a^{2} c d e^{2} + C a^{3} e^{3} - A a^{2} c e^{3} + {\left (C a c^{2} d^{3} - A c^{3} d^{3} - B a c^{2} d^{2} e + C a^{2} c d e^{2} - A a c^{2} d e^{2} - B a^{2} c e^{3}\right )} x}{2 \, {\left (c d^{2} + a e^{2}\right )}^{2} {\left (c x^{2} + a\right )} a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 742, normalized size = 3.28 \[ \frac {A \,c^{2} d^{3} x}{2 \left (a \,e^{2}+c \,d^{2}\right )^{2} \left (c \,x^{2}+a \right ) a}+\frac {A \,c^{2} d^{3} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \left (a \,e^{2}+c \,d^{2}\right )^{2} \sqrt {a c}\, a}+\frac {A c d \,e^{2} x}{2 \left (a \,e^{2}+c \,d^{2}\right )^{2} \left (c \,x^{2}+a \right )}+\frac {3 A c d \,e^{2} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \left (a \,e^{2}+c \,d^{2}\right )^{2} \sqrt {a c}}+\frac {B a \,e^{3} x}{2 \left (a \,e^{2}+c \,d^{2}\right )^{2} \left (c \,x^{2}+a \right )}+\frac {B a \,e^{3} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \left (a \,e^{2}+c \,d^{2}\right )^{2} \sqrt {a c}}+\frac {B c \,d^{2} e x}{2 \left (a \,e^{2}+c \,d^{2}\right )^{2} \left (c \,x^{2}+a \right )}-\frac {B c \,d^{2} e \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \left (a \,e^{2}+c \,d^{2}\right )^{2} \sqrt {a c}}-\frac {C a d \,e^{2} x}{2 \left (a \,e^{2}+c \,d^{2}\right )^{2} \left (c \,x^{2}+a \right )}-\frac {C a d \,e^{2} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \left (a \,e^{2}+c \,d^{2}\right )^{2} \sqrt {a c}}-\frac {C c \,d^{3} x}{2 \left (a \,e^{2}+c \,d^{2}\right )^{2} \left (c \,x^{2}+a \right )}+\frac {C c \,d^{3} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \left (a \,e^{2}+c \,d^{2}\right )^{2} \sqrt {a c}}+\frac {A a \,e^{3}}{2 \left (a \,e^{2}+c \,d^{2}\right )^{2} \left (c \,x^{2}+a \right )}+\frac {A c \,d^{2} e}{2 \left (a \,e^{2}+c \,d^{2}\right )^{2} \left (c \,x^{2}+a \right )}-\frac {A \,e^{3} \ln \left (c \,x^{2}+a \right )}{2 \left (a \,e^{2}+c \,d^{2}\right )^{2}}+\frac {A \,e^{3} \ln \left (e x +d \right )}{\left (a \,e^{2}+c \,d^{2}\right )^{2}}-\frac {B a d \,e^{2}}{2 \left (a \,e^{2}+c \,d^{2}\right )^{2} \left (c \,x^{2}+a \right )}-\frac {B c \,d^{3}}{2 \left (a \,e^{2}+c \,d^{2}\right )^{2} \left (c \,x^{2}+a \right )}+\frac {B d \,e^{2} \ln \left (c \,x^{2}+a \right )}{2 \left (a \,e^{2}+c \,d^{2}\right )^{2}}-\frac {B d \,e^{2} \ln \left (e x +d \right )}{\left (a \,e^{2}+c \,d^{2}\right )^{2}}-\frac {C \,a^{2} e^{3}}{2 \left (a \,e^{2}+c \,d^{2}\right )^{2} \left (c \,x^{2}+a \right ) c}-\frac {C a \,d^{2} e}{2 \left (a \,e^{2}+c \,d^{2}\right )^{2} \left (c \,x^{2}+a \right )}-\frac {C \,d^{2} e \ln \left (c \,x^{2}+a \right )}{2 \left (a \,e^{2}+c \,d^{2}\right )^{2}}+\frac {C \,d^{2} e \ln \left (e x +d \right )}{\left (a \,e^{2}+c \,d^{2}\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.00, size = 293, normalized size = 1.30 \[ -\frac {{\left (C d^{2} e - B d e^{2} + A e^{3}\right )} \log \left (c x^{2} + a\right )}{2 \, {\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )}} + \frac {{\left (C d^{2} e - B d e^{2} + A e^{3}\right )} \log \left (e x + d\right )}{c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}} - \frac {{\left (B a c d^{2} e - B a^{2} e^{3} - {\left (C a c + A c^{2}\right )} d^{3} + {\left (C a^{2} - 3 \, A a c\right )} d e^{2}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \, {\left (a c^{2} d^{4} + 2 \, a^{2} c d^{2} e^{2} + a^{3} e^{4}\right )} \sqrt {a c}} - \frac {B a c d + {\left (C a^{2} - A a c\right )} e - {\left (B a c e - {\left (C a c - A c^{2}\right )} d\right )} x}{2 \, {\left (a^{2} c^{2} d^{2} + a^{3} c e^{2} + {\left (a c^{3} d^{2} + a^{2} c^{2} e^{2}\right )} x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 7.68, size = 1493, normalized size = 6.61 \[ \text {result too large to display} \]
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 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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