Optimal. Leaf size=94 \[ \frac {\left (a e^2+c d^2\right ) (B d-A e)}{2 e^4 (d+e x)^2}-\frac {a B e^2-2 A c d e+3 B c d^2}{e^4 (d+e x)}-\frac {c (3 B d-A e) \log (d+e x)}{e^4}+\frac {B c x}{e^3} \]
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Rubi [A] time = 0.07, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {772} \begin {gather*} \frac {\left (a e^2+c d^2\right ) (B d-A e)}{2 e^4 (d+e x)^2}-\frac {a B e^2-2 A c d e+3 B c d^2}{e^4 (d+e x)}-\frac {c (3 B d-A e) \log (d+e x)}{e^4}+\frac {B c x}{e^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 772
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a+c x^2\right )}{(d+e x)^3} \, dx &=\int \left (\frac {B c}{e^3}+\frac {(-B d+A e) \left (c d^2+a e^2\right )}{e^3 (d+e x)^3}+\frac {3 B c d^2-2 A c d e+a B e^2}{e^3 (d+e x)^2}+\frac {c (-3 B d+A e)}{e^3 (d+e x)}\right ) \, dx\\ &=\frac {B c x}{e^3}+\frac {(B d-A e) \left (c d^2+a e^2\right )}{2 e^4 (d+e x)^2}-\frac {3 B c d^2-2 A c d e+a B e^2}{e^4 (d+e x)}-\frac {c (3 B d-A e) \log (d+e x)}{e^4}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 88, normalized size = 0.94 \begin {gather*} \frac {\frac {\left (a e^2+c d^2\right ) (B d-A e)}{(d+e x)^2}-\frac {2 \left (a B e^2-2 A c d e+3 B c d^2\right )}{d+e x}+2 \log (d+e x) (A c e-3 B c d)+2 B c e x}{2 e^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) \left (a+c x^2\right )}{(d+e x)^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.41, size = 168, normalized size = 1.79 \begin {gather*} \frac {2 \, B c e^{3} x^{3} + 4 \, B c d e^{2} x^{2} - 5 \, B c d^{3} + 3 \, A c d^{2} e - B a d e^{2} - A a e^{3} - 2 \, {\left (2 \, B c d^{2} e - 2 \, A c d e^{2} + B a e^{3}\right )} x - 2 \, {\left (3 \, B c d^{3} - A c d^{2} e + {\left (3 \, B c d e^{2} - A c e^{3}\right )} x^{2} + 2 \, {\left (3 \, B c d^{2} e - A c d e^{2}\right )} x\right )} \log \left (e x + d\right )}{2 \, {\left (e^{6} x^{2} + 2 \, d e^{5} x + d^{2} e^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 96, normalized size = 1.02 \begin {gather*} B c x e^{\left (-3\right )} - {\left (3 \, B c d - A c e\right )} e^{\left (-4\right )} \log \left ({\left | x e + d \right |}\right ) - \frac {{\left (5 \, B c d^{3} - 3 \, A c d^{2} e + B a d e^{2} + A a e^{3} + 2 \, {\left (3 \, B c d^{2} e - 2 \, A c d e^{2} + B a e^{3}\right )} x\right )} e^{\left (-4\right )}}{2 \, {\left (x e + d\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 144, normalized size = 1.53 \begin {gather*} -\frac {A a}{2 \left (e x +d \right )^{2} e}-\frac {A c \,d^{2}}{2 \left (e x +d \right )^{2} e^{3}}+\frac {B a d}{2 \left (e x +d \right )^{2} e^{2}}+\frac {B c \,d^{3}}{2 \left (e x +d \right )^{2} e^{4}}+\frac {2 A c d}{\left (e x +d \right ) e^{3}}+\frac {A c \ln \left (e x +d \right )}{e^{3}}-\frac {B a}{\left (e x +d \right ) e^{2}}-\frac {3 B c \,d^{2}}{\left (e x +d \right ) e^{4}}-\frac {3 B c d \ln \left (e x +d \right )}{e^{4}}+\frac {B c x}{e^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.63, size = 111, normalized size = 1.18 \begin {gather*} -\frac {5 \, B c d^{3} - 3 \, A c d^{2} e + B a d e^{2} + A a e^{3} + 2 \, {\left (3 \, B c d^{2} e - 2 \, A c d e^{2} + B a e^{3}\right )} x}{2 \, {\left (e^{6} x^{2} + 2 \, d e^{5} x + d^{2} e^{4}\right )}} + \frac {B c x}{e^{3}} - \frac {{\left (3 \, B c d - A c e\right )} \log \left (e x + d\right )}{e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.76, size = 111, normalized size = 1.18 \begin {gather*} \frac {\ln \left (d+e\,x\right )\,\left (A\,c\,e-3\,B\,c\,d\right )}{e^4}-\frac {\frac {5\,B\,c\,d^3-3\,A\,c\,d^2\,e+B\,a\,d\,e^2+A\,a\,e^3}{2\,e}+x\,\left (3\,B\,c\,d^2-2\,A\,c\,d\,e+B\,a\,e^2\right )}{d^2\,e^3+2\,d\,e^4\,x+e^5\,x^2}+\frac {B\,c\,x}{e^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.27, size = 117, normalized size = 1.24 \begin {gather*} \frac {B c x}{e^{3}} - \frac {c \left (- A e + 3 B d\right ) \log {\left (d + e x \right )}}{e^{4}} + \frac {- A a e^{3} + 3 A c d^{2} e - B a d e^{2} - 5 B c d^{3} + x \left (4 A c d e^{2} - 2 B a e^{3} - 6 B c d^{2} e\right )}{2 d^{2} e^{4} + 4 d e^{5} x + 2 e^{6} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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