Optimal. Leaf size=134 \[ \frac {A e (2 c d-b e)-B \left (3 c d^2-e (2 b d-a e)\right )}{4 e^4 (d+e x)^4}+\frac {(B d-A e) \left (a e^2-b d e+c d^2\right )}{5 e^4 (d+e x)^5}+\frac {-A c e-b B e+3 B c d}{3 e^4 (d+e x)^3}-\frac {B c}{2 e^4 (d+e x)^2} \]
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Rubi [A] time = 0.11, 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}{4 e^4 (d+e x)^4}+\frac {(B d-A e) \left (a e^2-b d e+c d^2\right )}{5 e^4 (d+e x)^5}+\frac {-A c e-b B e+3 B c d}{3 e^4 (d+e x)^3}-\frac {B c}{2 e^4 (d+e x)^2} \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)^6} \, dx &=\int \left (\frac {(-B d+A e) \left (c d^2-b d e+a e^2\right )}{e^3 (d+e x)^6}+\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)^5}+\frac {-3 B c d+b B e+A c e}{e^3 (d+e x)^4}+\frac {B c}{e^3 (d+e x)^3}\right ) \, dx\\ &=\frac {(B d-A e) \left (c d^2-b d e+a e^2\right )}{5 e^4 (d+e x)^5}-\frac {3 B c d^2-B e (2 b d-a e)-A e (2 c d-b e)}{4 e^4 (d+e x)^4}+\frac {3 B c d-b B e-A c e}{3 e^4 (d+e x)^3}-\frac {B c}{2 e^4 (d+e x)^2}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 122, normalized size = 0.91 \begin {gather*} -\frac {A e \left (3 e (4 a e+b d+5 b e x)+2 c \left (d^2+5 d e x+10 e^2 x^2\right )\right )+B \left (e \left (3 a e (d+5 e x)+2 b \left (d^2+5 d e x+10 e^2 x^2\right )\right )+3 c \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )\right )}{60 e^4 (d+e x)^5} \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)^6} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.37, size = 173, normalized size = 1.29 \begin {gather*} -\frac {30 \, B c e^{3} x^{3} + 3 \, B c d^{3} + 12 \, A a e^{3} + 2 \, {\left (B b + A c\right )} d^{2} e + 3 \, {\left (B a + A b\right )} d e^{2} + 10 \, {\left (3 \, B c d e^{2} + 2 \, {\left (B b + A c\right )} e^{3}\right )} x^{2} + 5 \, {\left (3 \, B c d^{2} e + 2 \, {\left (B b + A c\right )} d e^{2} + 3 \, {\left (B a + A b\right )} e^{3}\right )} x}{60 \, {\left (e^{9} x^{5} + 5 \, d e^{8} x^{4} + 10 \, d^{2} e^{7} x^{3} + 10 \, d^{3} e^{6} x^{2} + 5 \, d^{4} e^{5} x + d^{5} e^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 135, normalized size = 1.01 \begin {gather*} -\frac {{\left (30 \, B c x^{3} e^{3} + 30 \, B c d x^{2} e^{2} + 15 \, B c d^{2} x e + 3 \, B c d^{3} + 20 \, B b x^{2} e^{3} + 20 \, A c x^{2} e^{3} + 10 \, B b d x e^{2} + 10 \, A c d x e^{2} + 2 \, B b d^{2} e + 2 \, A c d^{2} e + 15 \, B a x e^{3} + 15 \, A b x e^{3} + 3 \, B a d e^{2} + 3 \, A b d e^{2} + 12 \, A a e^{3}\right )} e^{\left (-4\right )}}{60 \, {\left (x e + d\right )}^{5}} \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}{2 \left (e x +d \right )^{2} 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}}{4 \left (e x +d \right )^{4} 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}}{5 \left (e x +d \right )^{5} e^{4}}-\frac {A c e +B b e -3 B c d}{3 \left (e x +d \right )^{3} e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.57, size = 173, normalized size = 1.29 \begin {gather*} -\frac {30 \, B c e^{3} x^{3} + 3 \, B c d^{3} + 12 \, A a e^{3} + 2 \, {\left (B b + A c\right )} d^{2} e + 3 \, {\left (B a + A b\right )} d e^{2} + 10 \, {\left (3 \, B c d e^{2} + 2 \, {\left (B b + A c\right )} e^{3}\right )} x^{2} + 5 \, {\left (3 \, B c d^{2} e + 2 \, {\left (B b + A c\right )} d e^{2} + 3 \, {\left (B a + A b\right )} e^{3}\right )} x}{60 \, {\left (e^{9} x^{5} + 5 \, d e^{8} x^{4} + 10 \, d^{2} e^{7} x^{3} + 10 \, d^{3} e^{6} x^{2} + 5 \, d^{4} e^{5} x + d^{5} e^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 180, normalized size = 1.34 \begin {gather*} -\frac {\frac {12\,A\,a\,e^3+3\,B\,c\,d^3+3\,A\,b\,d\,e^2+3\,B\,a\,d\,e^2+2\,A\,c\,d^2\,e+2\,B\,b\,d^2\,e}{60\,e^4}+\frac {x^2\,\left (2\,A\,c\,e+2\,B\,b\,e+3\,B\,c\,d\right )}{6\,e^2}+\frac {x\,\left (3\,A\,b\,e^2+3\,B\,a\,e^2+3\,B\,c\,d^2+2\,A\,c\,d\,e+2\,B\,b\,d\,e\right )}{12\,e^3}+\frac {B\,c\,x^3}{2\,e}}{d^5+5\,d^4\,e\,x+10\,d^3\,e^2\,x^2+10\,d^2\,e^3\,x^3+5\,d\,e^4\,x^4+e^5\,x^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 114.50, size = 212, normalized size = 1.58 \begin {gather*} \frac {- 12 A a e^{3} - 3 A b d e^{2} - 2 A c d^{2} e - 3 B a d e^{2} - 2 B b d^{2} e - 3 B c d^{3} - 30 B c e^{3} x^{3} + x^{2} \left (- 20 A c e^{3} - 20 B b e^{3} - 30 B c d e^{2}\right ) + x \left (- 15 A b e^{3} - 10 A c d e^{2} - 15 B a e^{3} - 10 B b d e^{2} - 15 B c d^{2} e\right )}{60 d^{5} e^{4} + 300 d^{4} e^{5} x + 600 d^{3} e^{6} x^{2} + 600 d^{2} e^{7} x^{3} + 300 d e^{8} x^{4} + 60 e^{9} x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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