Optimal. Leaf size=75 \[ \frac {-a B e-A b e+2 b B d}{2 e^3 (d+e x)^2}-\frac {(b d-a e) (B d-A e)}{3 e^3 (d+e x)^3}-\frac {b B}{e^3 (d+e x)} \]
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Rubi [A] time = 0.05, antiderivative size = 75, 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} \begin {gather*} \frac {-a B e-A b e+2 b B d}{2 e^3 (d+e x)^2}-\frac {(b d-a e) (B d-A e)}{3 e^3 (d+e x)^3}-\frac {b B}{e^3 (d+e x)} \end {gather*}
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)^4} \, dx &=\int \left (\frac {(-b d+a e) (-B d+A e)}{e^2 (d+e x)^4}+\frac {-2 b B d+A b e+a B e}{e^2 (d+e x)^3}+\frac {b B}{e^2 (d+e x)^2}\right ) \, dx\\ &=-\frac {(b d-a e) (B d-A e)}{3 e^3 (d+e x)^3}+\frac {2 b B d-A b e-a B e}{2 e^3 (d+e x)^2}-\frac {b B}{e^3 (d+e x)}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 63, normalized size = 0.84 \begin {gather*} -\frac {a e (2 A e+B (d+3 e x))+b \left (A e (d+3 e x)+2 B \left (d^2+3 d e x+3 e^2 x^2\right )\right )}{6 e^3 (d+e x)^3} \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) (A+B x)}{(d+e x)^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.32, size = 93, normalized size = 1.24 \begin {gather*} -\frac {6 \, B b e^{2} x^{2} + 2 \, B b d^{2} + 2 \, A a e^{2} + {\left (B a + A b\right )} d e + 3 \, {\left (2 \, B b d e + {\left (B a + A b\right )} e^{2}\right )} x}{6 \, {\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.18, size = 69, normalized size = 0.92 \begin {gather*} -\frac {{\left (6 \, B b x^{2} e^{2} + 6 \, B b d x e + 2 \, B b d^{2} + 3 \, B a x e^{2} + 3 \, A b x e^{2} + B a d e + A b d e + 2 \, A a e^{2}\right )} e^{\left (-3\right )}}{6 \, {\left (x e + d\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 79, normalized size = 1.05 \begin {gather*} -\frac {B b}{\left (e x +d \right ) e^{3}}-\frac {A a \,e^{2}-A d b e -B d a e +B b \,d^{2}}{3 \left (e x +d \right )^{3} e^{3}}-\frac {A b e +B a e -2 B b d}{2 \left (e x +d \right )^{2} e^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 93, normalized size = 1.24 \begin {gather*} -\frac {6 \, B b e^{2} x^{2} + 2 \, B b d^{2} + 2 \, A a e^{2} + {\left (B a + A b\right )} d e + 3 \, {\left (2 \, B b d e + {\left (B a + A b\right )} e^{2}\right )} x}{6 \, {\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 91, normalized size = 1.21 \begin {gather*} -\frac {\frac {2\,A\,a\,e^2+2\,B\,b\,d^2+A\,b\,d\,e+B\,a\,d\,e}{6\,e^3}+\frac {x\,\left (A\,b\,e+B\,a\,e+2\,B\,b\,d\right )}{2\,e^2}+\frac {B\,b\,x^2}{e}}{d^3+3\,d^2\,e\,x+3\,d\,e^2\,x^2+e^3\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.63, size = 107, normalized size = 1.43 \begin {gather*} \frac {- 2 A a e^{2} - A b d e - B a d e - 2 B b d^{2} - 6 B b e^{2} x^{2} + x \left (- 3 A b e^{2} - 3 B a e^{2} - 6 B b d e\right )}{6 d^{3} e^{3} + 18 d^{2} e^{4} x + 18 d e^{5} x^{2} + 6 e^{6} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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