Optimal. Leaf size=134 \[ \frac {A e (2 c d-b e)-B \left (3 c d^2-e (2 b d-a e)\right )}{5 e^4 (d+e x)^5}+\frac {(B d-A e) \left (a e^2-b d e+c d^2\right )}{6 e^4 (d+e x)^6}+\frac {-A c e-b B e+3 B c d}{4 e^4 (d+e x)^4}-\frac {B c}{3 e^4 (d+e x)^3} \]
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Rubi [A] time = 0.10, antiderivative size = 133, 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} \begin {gather*} -\frac {-B e (2 b d-a e)-A e (2 c d-b e)+3 B c d^2}{5 e^4 (d+e x)^5}+\frac {(B d-A e) \left (a e^2-b d e+c d^2\right )}{6 e^4 (d+e x)^6}+\frac {-A c e-b B e+3 B c d}{4 e^4 (d+e x)^4}-\frac {B c}{3 e^4 (d+e x)^3} \end {gather*}
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)^7} \, dx &=\int \left (\frac {(-B d+A e) \left (c d^2-b d e+a e^2\right )}{e^3 (d+e x)^7}+\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)^6}+\frac {-3 B c d+b B e+A c e}{e^3 (d+e x)^5}+\frac {B c}{e^3 (d+e x)^4}\right ) \, dx\\ &=\frac {(B d-A e) \left (c d^2-b d e+a e^2\right )}{6 e^4 (d+e x)^6}-\frac {3 B c d^2-B e (2 b d-a e)-A e (2 c d-b e)}{5 e^4 (d+e x)^5}+\frac {3 B c d-b B e-A c e}{4 e^4 (d+e x)^4}-\frac {B c}{3 e^4 (d+e x)^3}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 119, normalized size = 0.89 \begin {gather*} -\frac {A e \left (2 e (5 a e+b d+6 b e x)+c \left (d^2+6 d e x+15 e^2 x^2\right )\right )+B \left (e \left (2 a e (d+6 e x)+b \left (d^2+6 d e x+15 e^2 x^2\right )\right )+c \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )\right )}{60 e^4 (d+e x)^6} \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+b x+c x^2\right )}{(d+e x)^7} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.36, size = 178, normalized size = 1.33 \begin {gather*} -\frac {20 \, B c e^{3} x^{3} + B c d^{3} + 10 \, A a e^{3} + {\left (B b + A c\right )} d^{2} e + 2 \, {\left (B a + A b\right )} d e^{2} + 15 \, {\left (B c d e^{2} + {\left (B b + A c\right )} e^{3}\right )} x^{2} + 6 \, {\left (B c d^{2} e + {\left (B b + A c\right )} d e^{2} + 2 \, {\left (B a + A b\right )} e^{3}\right )} x}{60 \, {\left (e^{10} x^{6} + 6 \, d e^{9} x^{5} + 15 \, d^{2} e^{8} x^{4} + 20 \, d^{3} e^{7} x^{3} + 15 \, d^{4} e^{6} x^{2} + 6 \, d^{5} e^{5} x + d^{6} e^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 132, normalized size = 0.99 \begin {gather*} -\frac {{\left (20 \, B c x^{3} e^{3} + 15 \, B c d x^{2} e^{2} + 6 \, B c d^{2} x e + B c d^{3} + 15 \, B b x^{2} e^{3} + 15 \, A c x^{2} e^{3} + 6 \, B b d x e^{2} + 6 \, A c d x e^{2} + B b d^{2} e + A c d^{2} e + 12 \, B a x e^{3} + 12 \, A b x e^{3} + 2 \, B a d e^{2} + 2 \, A b d e^{2} + 10 \, A a e^{3}\right )} e^{\left (-4\right )}}{60 \, {\left (x e + d\right )}^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 142, normalized size = 1.06 \begin {gather*} -\frac {B c}{3 \left (e x +d \right )^{3} e^{4}}-\frac {a A \,e^{3}-A b d \,e^{2}+A c \,d^{2} e -a B d \,e^{2}+B \,d^{2} b e -B c \,d^{3}}{6 \left (e x +d \right )^{6} e^{4}}-\frac {A c e +B b e -3 B c d}{4 \left (e x +d \right )^{4} e^{4}}-\frac {A b \,e^{2}-2 A c d e +B a \,e^{2}-2 B b d e +3 B c \,d^{2}}{5 \left (e x +d \right )^{5} e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.52, size = 178, normalized size = 1.33 \begin {gather*} -\frac {20 \, B c e^{3} x^{3} + B c d^{3} + 10 \, A a e^{3} + {\left (B b + A c\right )} d^{2} e + 2 \, {\left (B a + A b\right )} d e^{2} + 15 \, {\left (B c d e^{2} + {\left (B b + A c\right )} e^{3}\right )} x^{2} + 6 \, {\left (B c d^{2} e + {\left (B b + A c\right )} d e^{2} + 2 \, {\left (B a + A b\right )} e^{3}\right )} x}{60 \, {\left (e^{10} x^{6} + 6 \, d e^{9} x^{5} + 15 \, d^{2} e^{8} x^{4} + 20 \, d^{3} e^{7} x^{3} + 15 \, d^{4} e^{6} x^{2} + 6 \, d^{5} e^{5} x + d^{6} e^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.36, size = 182, normalized size = 1.36 \begin {gather*} -\frac {\frac {10\,A\,a\,e^3+B\,c\,d^3+2\,A\,b\,d\,e^2+2\,B\,a\,d\,e^2+A\,c\,d^2\,e+B\,b\,d^2\,e}{60\,e^4}+\frac {x^2\,\left (A\,c\,e+B\,b\,e+B\,c\,d\right )}{4\,e^2}+\frac {x\,\left (2\,A\,b\,e^2+2\,B\,a\,e^2+B\,c\,d^2+A\,c\,d\,e+B\,b\,d\,e\right )}{10\,e^3}+\frac {B\,c\,x^3}{3\,e}}{d^6+6\,d^5\,e\,x+15\,d^4\,e^2\,x^2+20\,d^3\,e^3\,x^3+15\,d^2\,e^4\,x^4+6\,d\,e^5\,x^5+e^6\,x^6} \end {gather*}
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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