Optimal. Leaf size=127 \[ \frac {A e (2 c d-b e)-B \left (3 c d^2-e (2 b d-a e)\right )}{2 e^4 (d+e x)^2}+\frac {(B d-A e) \left (a e^2-b d e+c d^2\right )}{3 e^4 (d+e x)^3}+\frac {-A c e-b B e+3 B c d}{e^4 (d+e x)}+\frac {B c \log (d+e x)}{e^4} \]
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Rubi [A] time = 0.12, antiderivative size = 126, normalized size of antiderivative = 0.99, number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {771} \[ -\frac {-B e (2 b d-a e)-A e (2 c d-b e)+3 B c d^2}{2 e^4 (d+e x)^2}+\frac {(B d-A e) \left (a e^2-b d e+c d^2\right )}{3 e^4 (d+e x)^3}+\frac {-A c e-b B e+3 B c d}{e^4 (d+e x)}+\frac {B c \log (d+e x)}{e^4} \]
Antiderivative was successfully verified.
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Rule 771
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a+b x+c x^2\right )}{(d+e x)^4} \, dx &=\int \left (\frac {(-B d+A e) \left (c d^2-b d e+a e^2\right )}{e^3 (d+e x)^4}+\frac {3 B c d^2-B e (2 b d-a e)-A e (2 c d-b e)}{e^3 (d+e x)^3}+\frac {-3 B c d+b B e+A c e}{e^3 (d+e x)^2}+\frac {B c}{e^3 (d+e x)}\right ) \, dx\\ &=\frac {(B d-A e) \left (c d^2-b d e+a e^2\right )}{3 e^4 (d+e x)^3}-\frac {3 B c d^2-B e (2 b d-a e)-A e (2 c d-b e)}{2 e^4 (d+e x)^2}+\frac {3 B c d-b B e-A c e}{e^4 (d+e x)}+\frac {B c \log (d+e x)}{e^4}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 130, normalized size = 1.02 \[ \frac {-A e \left (e (2 a e+b d+3 b e x)+2 c \left (d^2+3 d e x+3 e^2 x^2\right )\right )+B \left (c d \left (11 d^2+27 d e x+18 e^2 x^2\right )-e \left (a e (d+3 e x)+2 b \left (d^2+3 d e x+3 e^2 x^2\right )\right )\right )+6 B c (d+e x)^3 \log (d+e x)}{6 e^4 (d+e x)^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.10, size = 185, normalized size = 1.46 \[ \frac {11 \, B c d^{3} - 2 \, A a e^{3} - 2 \, {\left (B b + A c\right )} d^{2} e - {\left (B a + A b\right )} d e^{2} + 6 \, {\left (3 \, B c d e^{2} - {\left (B b + A c\right )} e^{3}\right )} x^{2} + 3 \, {\left (9 \, B c d^{2} e - 2 \, {\left (B b + A c\right )} d e^{2} - {\left (B a + A b\right )} e^{3}\right )} x + 6 \, {\left (B c e^{3} x^{3} + 3 \, B c d e^{2} x^{2} + 3 \, B c d^{2} e x + B c d^{3}\right )} \log \left (e x + d\right )}{6 \, {\left (e^{7} x^{3} + 3 \, d e^{6} x^{2} + 3 \, d^{2} e^{5} x + d^{3} e^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 138, normalized size = 1.09 \[ B c e^{\left (-4\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {{\left (6 \, {\left (3 \, B c d e - B b e^{2} - A c e^{2}\right )} x^{2} + 3 \, {\left (9 \, B c d^{2} - 2 \, B b d e - 2 \, A c d e - B a e^{2} - A b e^{2}\right )} x + {\left (11 \, B c d^{3} - 2 \, B b d^{2} e - 2 \, A c d^{2} e - B a d e^{2} - A b d e^{2} - 2 \, A a e^{3}\right )} e^{\left (-1\right )}\right )} e^{\left (-3\right )}}{6 \, {\left (x e + d\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 225, normalized size = 1.77 \[ -\frac {A a}{3 \left (e x +d \right )^{3} e}+\frac {A b d}{3 \left (e x +d \right )^{3} e^{2}}-\frac {A c \,d^{2}}{3 \left (e x +d \right )^{3} e^{3}}+\frac {B a d}{3 \left (e x +d \right )^{3} e^{2}}-\frac {B b \,d^{2}}{3 \left (e x +d \right )^{3} e^{3}}+\frac {B c \,d^{3}}{3 \left (e x +d \right )^{3} e^{4}}-\frac {A b}{2 \left (e x +d \right )^{2} e^{2}}+\frac {A c d}{\left (e x +d \right )^{2} e^{3}}-\frac {B a}{2 \left (e x +d \right )^{2} e^{2}}+\frac {B b d}{\left (e x +d \right )^{2} e^{3}}-\frac {3 B c \,d^{2}}{2 \left (e x +d \right )^{2} e^{4}}-\frac {A c}{\left (e x +d \right ) e^{3}}-\frac {B b}{\left (e x +d \right ) e^{3}}+\frac {3 B c d}{\left (e x +d \right ) e^{4}}+\frac {B c \ln \left (e x +d \right )}{e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 154, normalized size = 1.21 \[ \frac {11 \, B c d^{3} - 2 \, A a e^{3} - 2 \, {\left (B b + A c\right )} d^{2} e - {\left (B a + A b\right )} d e^{2} + 6 \, {\left (3 \, B c d e^{2} - {\left (B b + A c\right )} e^{3}\right )} x^{2} + 3 \, {\left (9 \, B c d^{2} e - 2 \, {\left (B b + A c\right )} d e^{2} - {\left (B a + A b\right )} e^{3}\right )} x}{6 \, {\left (e^{7} x^{3} + 3 \, d e^{6} x^{2} + 3 \, d^{2} e^{5} x + d^{3} e^{4}\right )}} + \frac {B c \log \left (e x + d\right )}{e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.37, size = 154, normalized size = 1.21 \[ \frac {B\,c\,\ln \left (d+e\,x\right )}{e^4}-\frac {\frac {2\,A\,a\,e^3-11\,B\,c\,d^3+A\,b\,d\,e^2+B\,a\,d\,e^2+2\,A\,c\,d^2\,e+2\,B\,b\,d^2\,e}{6\,e^4}+\frac {x^2\,\left (A\,c\,e+B\,b\,e-3\,B\,c\,d\right )}{e^2}+\frac {x\,\left (A\,b\,e^2+B\,a\,e^2-9\,B\,c\,d^2+2\,A\,c\,d\,e+2\,B\,b\,d\,e\right )}{2\,e^3}}{d^3+3\,d^2\,e\,x+3\,d\,e^2\,x^2+e^3\,x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 16.20, size = 184, normalized size = 1.45 \[ \frac {B c \log {\left (d + e x \right )}}{e^{4}} + \frac {- 2 A a e^{3} - A b d e^{2} - 2 A c d^{2} e - B a d e^{2} - 2 B b d^{2} e + 11 B c d^{3} + x^{2} \left (- 6 A c e^{3} - 6 B b e^{3} + 18 B c d e^{2}\right ) + x \left (- 3 A b e^{3} - 6 A c d e^{2} - 3 B a e^{3} - 6 B b d e^{2} + 27 B c d^{2} e\right )}{6 d^{3} e^{4} + 18 d^{2} e^{5} x + 18 d e^{6} x^{2} + 6 e^{7} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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