Optimal. Leaf size=77 \[ \frac {-a B e-A b e+2 b B d}{3 e^3 (d+e x)^3}-\frac {(b d-a e) (B d-A e)}{4 e^3 (d+e x)^4}-\frac {b B}{2 e^3 (d+e x)^2} \]
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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} \[ \frac {-a B e-A b e+2 b B d}{3 e^3 (d+e x)^3}-\frac {(b d-a e) (B d-A e)}{4 e^3 (d+e x)^4}-\frac {b B}{2 e^3 (d+e x)^2} \]
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)^5} \, dx &=\int \left (\frac {(-b d+a e) (-B d+A e)}{e^2 (d+e x)^5}+\frac {-2 b B d+A b e+a B e}{e^2 (d+e x)^4}+\frac {b B}{e^2 (d+e x)^3}\right ) \, dx\\ &=-\frac {(b d-a e) (B d-A e)}{4 e^3 (d+e x)^4}+\frac {2 b B d-A b e-a B e}{3 e^3 (d+e x)^3}-\frac {b B}{2 e^3 (d+e x)^2}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 62, normalized size = 0.81 \[ -\frac {a e (3 A e+B (d+4 e x))+b \left (A e (d+4 e x)+B \left (d^2+4 d e x+6 e^2 x^2\right )\right )}{12 e^3 (d+e x)^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.15, size = 102, normalized size = 1.32 \[ -\frac {6 \, B b e^{2} x^{2} + B b d^{2} + 3 \, A a e^{2} + {\left (B a + A b\right )} d e + 4 \, {\left (B b d e + {\left (B a + A b\right )} e^{2}\right )} x}{12 \, {\left (e^{7} x^{4} + 4 \, d e^{6} x^{3} + 6 \, d^{2} e^{5} x^{2} + 4 \, d^{3} e^{4} x + d^{4} e^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.18, size = 121, normalized size = 1.57 \[ -\frac {1}{12} \, {\left (\frac {6 \, B b e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}} - \frac {8 \, B b d e^{\left (-2\right )}}{{\left (x e + d\right )}^{3}} + \frac {3 \, B b d^{2} e^{\left (-2\right )}}{{\left (x e + d\right )}^{4}} + \frac {4 \, B a e^{\left (-1\right )}}{{\left (x e + d\right )}^{3}} + \frac {4 \, A b e^{\left (-1\right )}}{{\left (x e + d\right )}^{3}} - \frac {3 \, B a d e^{\left (-1\right )}}{{\left (x e + d\right )}^{4}} - \frac {3 \, A b d e^{\left (-1\right )}}{{\left (x e + d\right )}^{4}} + \frac {3 \, A a}{{\left (x e + d\right )}^{4}}\right )} e^{\left (-1\right )} \]
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 \[ -\frac {B b}{2 \left (e x +d \right )^{2} e^{3}}-\frac {A b e +B a e -2 B b d}{3 \left (e x +d \right )^{3} e^{3}}-\frac {A a \,e^{2}-A d b e -B d a e +B b \,d^{2}}{4 \left (e x +d \right )^{4} e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 102, normalized size = 1.32 \[ -\frac {6 \, B b e^{2} x^{2} + B b d^{2} + 3 \, A a e^{2} + {\left (B a + A b\right )} d e + 4 \, {\left (B b d e + {\left (B a + A b\right )} e^{2}\right )} x}{12 \, {\left (e^{7} x^{4} + 4 \, d e^{6} x^{3} + 6 \, d^{2} e^{5} x^{2} + 4 \, d^{3} e^{4} x + d^{4} e^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.06, size = 101, normalized size = 1.31 \[ -\frac {\frac {3\,A\,a\,e^2+B\,b\,d^2+A\,b\,d\,e+B\,a\,d\,e}{12\,e^3}+\frac {x\,\left (A\,b\,e+B\,a\,e+B\,b\,d\right )}{3\,e^2}+\frac {B\,b\,x^2}{2\,e}}{d^4+4\,d^3\,e\,x+6\,d^2\,e^2\,x^2+4\,d\,e^3\,x^3+e^4\,x^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.96, size = 117, normalized size = 1.52 \[ \frac {- 3 A a e^{2} - A b d e - B a d e - B b d^{2} - 6 B b e^{2} x^{2} + x \left (- 4 A b e^{2} - 4 B a e^{2} - 4 B b d e\right )}{12 d^{4} e^{3} + 48 d^{3} e^{4} x + 72 d^{2} e^{5} x^{2} + 48 d e^{6} x^{3} + 12 e^{7} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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