Optimal. Leaf size=77 \[ \frac {-a B e-A b e+2 b B d}{4 e^3 (d+e x)^4}-\frac {(b d-a e) (B d-A e)}{5 e^3 (d+e x)^5}-\frac {b B}{3 e^3 (d+e x)^3} \]
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Rubi [A] time = 0.05, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {77} \[ \frac {-a B e-A b e+2 b B d}{4 e^3 (d+e x)^4}-\frac {(b d-a e) (B d-A e)}{5 e^3 (d+e x)^5}-\frac {b B}{3 e^3 (d+e x)^3} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin {align*} \int \frac {(a+b x) (A+B x)}{(d+e x)^6} \, dx &=\int \left (\frac {(-b d+a e) (-B d+A e)}{e^2 (d+e x)^6}+\frac {-2 b B d+A b e+a B e}{e^2 (d+e x)^5}+\frac {b B}{e^2 (d+e x)^4}\right ) \, dx\\ &=-\frac {(b d-a e) (B d-A e)}{5 e^3 (d+e x)^5}+\frac {2 b B d-A b e-a B e}{4 e^3 (d+e x)^4}-\frac {b B}{3 e^3 (d+e x)^3}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 65, normalized size = 0.84 \[ -\frac {3 a e (4 A e+B (d+5 e x))+b \left (3 A e (d+5 e x)+2 B \left (d^2+5 d e x+10 e^2 x^2\right )\right )}{60 e^3 (d+e x)^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 117, normalized size = 1.52 \[ -\frac {20 \, B b e^{2} x^{2} + 2 \, B b d^{2} + 12 \, A a e^{2} + 3 \, {\left (B a + A b\right )} d e + 5 \, {\left (2 \, B b d e + 3 \, {\left (B a + A b\right )} e^{2}\right )} x}{60 \, {\left (e^{8} x^{5} + 5 \, d e^{7} x^{4} + 10 \, d^{2} e^{6} x^{3} + 10 \, d^{3} e^{5} x^{2} + 5 \, d^{4} e^{4} x + d^{5} e^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.21, size = 71, normalized size = 0.92 \[ -\frac {{\left (20 \, B b x^{2} e^{2} + 10 \, B b d x e + 2 \, B b d^{2} + 15 \, B a x e^{2} + 15 \, A b x e^{2} + 3 \, B a d e + 3 \, A b d e + 12 \, A a e^{2}\right )} e^{\left (-3\right )}}{60 \, {\left (x e + d\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 79, normalized size = 1.03 \[ -\frac {B b}{3 \left (e x +d \right )^{3} e^{3}}-\frac {A b e +B a e -2 B b d}{4 \left (e x +d \right )^{4} e^{3}}-\frac {A a \,e^{2}-A d b e -B d a e +B b \,d^{2}}{5 \left (e x +d \right )^{5} e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.55, size = 117, normalized size = 1.52 \[ -\frac {20 \, B b e^{2} x^{2} + 2 \, B b d^{2} + 12 \, A a e^{2} + 3 \, {\left (B a + A b\right )} d e + 5 \, {\left (2 \, B b d e + 3 \, {\left (B a + A b\right )} e^{2}\right )} x}{60 \, {\left (e^{8} x^{5} + 5 \, d e^{7} x^{4} + 10 \, d^{2} e^{6} x^{3} + 10 \, d^{3} e^{5} x^{2} + 5 \, d^{4} e^{4} x + d^{5} e^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.07, size = 118, normalized size = 1.53 \[ -\frac {\frac {12\,A\,a\,e^2+2\,B\,b\,d^2+3\,A\,b\,d\,e+3\,B\,a\,d\,e}{60\,e^3}+\frac {x\,\left (3\,A\,b\,e+3\,B\,a\,e+2\,B\,b\,d\right )}{12\,e^2}+\frac {B\,b\,x^2}{3\,e}}{d^5+5\,d^4\,e\,x+10\,d^3\,e^2\,x^2+10\,d^2\,e^3\,x^3+5\,d\,e^4\,x^4+e^5\,x^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.74, size = 134, normalized size = 1.74 \[ \frac {- 12 A a e^{2} - 3 A b d e - 3 B a d e - 2 B b d^{2} - 20 B b e^{2} x^{2} + x \left (- 15 A b e^{2} - 15 B a e^{2} - 10 B b d e\right )}{60 d^{5} e^{3} + 300 d^{4} e^{4} x + 600 d^{3} e^{5} x^{2} + 600 d^{2} e^{6} x^{3} + 300 d e^{7} x^{4} + 60 e^{8} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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