Optimal. Leaf size=86 \[ \frac{(a+b x)^3 (-4 a B e+A b e+3 b B d)}{12 e (d+e x)^3 (b d-a e)^2}-\frac{(a+b x)^3 (B d-A e)}{4 e (d+e x)^4 (b d-a e)} \]
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Rubi [A] time = 0.0345901, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {27, 78, 37} \[ \frac{(a+b x)^3 (-4 a B e+A b e+3 b B d)}{12 e (d+e x)^3 (b d-a e)^2}-\frac{(a+b x)^3 (B d-A e)}{4 e (d+e x)^4 (b d-a e)} \]
Antiderivative was successfully verified.
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Rule 27
Rule 78
Rule 37
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )}{(d+e x)^5} \, dx &=\int \frac{(a+b x)^2 (A+B x)}{(d+e x)^5} \, dx\\ &=-\frac{(B d-A e) (a+b x)^3}{4 e (b d-a e) (d+e x)^4}+\frac{(3 b B d+A b e-4 a B e) \int \frac{(a+b x)^2}{(d+e x)^4} \, dx}{4 e (b d-a e)}\\ &=-\frac{(B d-A e) (a+b x)^3}{4 e (b d-a e) (d+e x)^4}+\frac{(3 b B d+A b e-4 a B e) (a+b x)^3}{12 e (b d-a e)^2 (d+e x)^3}\\ \end{align*}
Mathematica [A] time = 0.0626825, size = 125, normalized size = 1.45 \[ -\frac{a^2 e^2 (3 A e+B (d+4 e x))+2 a b e \left (A e (d+4 e x)+B \left (d^2+4 d e x+6 e^2 x^2\right )\right )+b^2 \left (A e \left (d^2+4 d e x+6 e^2 x^2\right )+3 B \left (4 d^2 e x+d^3+6 d e^2 x^2+4 e^3 x^3\right )\right )}{12 e^4 (d+e x)^4} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.007, size = 166, normalized size = 1.9 \begin{align*} -{\frac{b \left ( Abe+2\,aBe-3\,Bbd \right ) }{2\,{e}^{4} \left ( ex+d \right ) ^{2}}}-{\frac{A{a}^{2}{e}^{3}-2\,Adab{e}^{2}+A{d}^{2}{b}^{2}e-B{a}^{2}d{e}^{2}+2\,B{d}^{2}abe-{b}^{2}B{d}^{3}}{4\,{e}^{4} \left ( ex+d \right ) ^{4}}}-{\frac{{b}^{2}B}{{e}^{4} \left ( ex+d \right ) }}-{\frac{2\,Aab{e}^{2}-2\,Ad{b}^{2}e+{a}^{2}B{e}^{2}-4\,Bdabe+3\,{b}^{2}B{d}^{2}}{3\,{e}^{4} \left ( ex+d \right ) ^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.17545, size = 252, normalized size = 2.93 \begin{align*} -\frac{12 \, B b^{2} e^{3} x^{3} + 3 \, B b^{2} d^{3} + 3 \, A a^{2} e^{3} +{\left (2 \, B a b + A b^{2}\right )} d^{2} e +{\left (B a^{2} + 2 \, A a b\right )} d e^{2} + 6 \,{\left (3 \, B b^{2} d e^{2} +{\left (2 \, B a b + A b^{2}\right )} e^{3}\right )} x^{2} + 4 \,{\left (3 \, B b^{2} d^{2} e +{\left (2 \, B a b + A b^{2}\right )} d e^{2} +{\left (B a^{2} + 2 \, A a b\right )} e^{3}\right )} x}{12 \,{\left (e^{8} x^{4} + 4 \, d e^{7} x^{3} + 6 \, d^{2} e^{6} x^{2} + 4 \, d^{3} e^{5} x + d^{4} e^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.44526, size = 392, normalized size = 4.56 \begin{align*} -\frac{12 \, B b^{2} e^{3} x^{3} + 3 \, B b^{2} d^{3} + 3 \, A a^{2} e^{3} +{\left (2 \, B a b + A b^{2}\right )} d^{2} e +{\left (B a^{2} + 2 \, A a b\right )} d e^{2} + 6 \,{\left (3 \, B b^{2} d e^{2} +{\left (2 \, B a b + A b^{2}\right )} e^{3}\right )} x^{2} + 4 \,{\left (3 \, B b^{2} d^{2} e +{\left (2 \, B a b + A b^{2}\right )} d e^{2} +{\left (B a^{2} + 2 \, A a b\right )} e^{3}\right )} x}{12 \,{\left (e^{8} x^{4} + 4 \, d e^{7} x^{3} + 6 \, d^{2} e^{6} x^{2} + 4 \, d^{3} e^{5} x + d^{4} e^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 18.7575, size = 221, normalized size = 2.57 \begin{align*} - \frac{3 A a^{2} e^{3} + 2 A a b d e^{2} + A b^{2} d^{2} e + B a^{2} d e^{2} + 2 B a b d^{2} e + 3 B b^{2} d^{3} + 12 B b^{2} e^{3} x^{3} + x^{2} \left (6 A b^{2} e^{3} + 12 B a b e^{3} + 18 B b^{2} d e^{2}\right ) + x \left (8 A a b e^{3} + 4 A b^{2} d e^{2} + 4 B a^{2} e^{3} + 8 B a b d e^{2} + 12 B b^{2} d^{2} e\right )}{12 d^{4} e^{4} + 48 d^{3} e^{5} x + 72 d^{2} e^{6} x^{2} + 48 d e^{7} x^{3} + 12 e^{8} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.1479, size = 331, normalized size = 3.85 \begin{align*} -\frac{1}{12} \,{\left (\frac{12 \, B b^{2} e^{\left (-1\right )}}{x e + d} - \frac{18 \, B b^{2} d e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}} + \frac{12 \, B b^{2} d^{2} e^{\left (-1\right )}}{{\left (x e + d\right )}^{3}} - \frac{3 \, B b^{2} d^{3} e^{\left (-1\right )}}{{\left (x e + d\right )}^{4}} + \frac{12 \, B a b}{{\left (x e + d\right )}^{2}} + \frac{6 \, A b^{2}}{{\left (x e + d\right )}^{2}} - \frac{16 \, B a b d}{{\left (x e + d\right )}^{3}} - \frac{8 \, A b^{2} d}{{\left (x e + d\right )}^{3}} + \frac{6 \, B a b d^{2}}{{\left (x e + d\right )}^{4}} + \frac{3 \, A b^{2} d^{2}}{{\left (x e + d\right )}^{4}} + \frac{4 \, B a^{2} e}{{\left (x e + d\right )}^{3}} + \frac{8 \, A a b e}{{\left (x e + d\right )}^{3}} - \frac{3 \, B a^{2} d e}{{\left (x e + d\right )}^{4}} - \frac{6 \, A a b d e}{{\left (x e + d\right )}^{4}} + \frac{3 \, A a^{2} e^{2}}{{\left (x e + d\right )}^{4}}\right )} e^{\left (-3\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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