Optimal. Leaf size=120 \[ \frac{b (-2 a B e-A b e+3 b B d)}{3 e^4 (d+e x)^3}-\frac{(b d-a e) (-a B e-2 A b e+3 b B d)}{4 e^4 (d+e x)^4}+\frac{(b d-a e)^2 (B d-A e)}{5 e^4 (d+e x)^5}-\frac{b^2 B}{2 e^4 (d+e x)^2} \]
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Rubi [A] time = 0.0973412, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {27, 77} \[ \frac{b (-2 a B e-A b e+3 b B d)}{3 e^4 (d+e x)^3}-\frac{(b d-a e) (-a B e-2 A b e+3 b B d)}{4 e^4 (d+e x)^4}+\frac{(b d-a e)^2 (B d-A e)}{5 e^4 (d+e x)^5}-\frac{b^2 B}{2 e^4 (d+e x)^2} \]
Antiderivative was successfully verified.
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Rule 27
Rule 77
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )}{(d+e x)^6} \, dx &=\int \frac{(a+b x)^2 (A+B x)}{(d+e x)^6} \, dx\\ &=\int \left (\frac{(-b d+a e)^2 (-B d+A e)}{e^3 (d+e x)^6}+\frac{(-b d+a e) (-3 b B d+2 A b e+a B e)}{e^3 (d+e x)^5}+\frac{b (-3 b B d+A b e+2 a B e)}{e^3 (d+e x)^4}+\frac{b^2 B}{e^3 (d+e x)^3}\right ) \, dx\\ &=\frac{(b d-a e)^2 (B d-A e)}{5 e^4 (d+e x)^5}-\frac{(b d-a e) (3 b B d-2 A b e-a B e)}{4 e^4 (d+e x)^4}+\frac{b (3 b B d-A b e-2 a B e)}{3 e^4 (d+e x)^3}-\frac{b^2 B}{2 e^4 (d+e x)^2}\\ \end{align*}
Mathematica [A] time = 0.0651761, size = 129, normalized size = 1.08 \[ -\frac{3 a^2 e^2 (4 A e+B (d+5 e x))+2 a b e \left (3 A e (d+5 e x)+2 B \left (d^2+5 d e x+10 e^2 x^2\right )\right )+b^2 \left (2 A e \left (d^2+5 d e x+10 e^2 x^2\right )+3 B \left (5 d^2 e x+d^3+10 d e^2 x^2+10 e^3 x^3\right )\right )}{60 e^4 (d+e x)^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 166, normalized size = 1.4 \begin{align*} -{\frac{{b}^{2}B}{2\,{e}^{4} \left ( ex+d \right ) ^{2}}}-{\frac{2\,Aab{e}^{2}-2\,Ad{b}^{2}e+{a}^{2}B{e}^{2}-4\,Bdabe+3\,{b}^{2}B{d}^{2}}{4\,{e}^{4} \left ( ex+d \right ) ^{4}}}-{\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}}{5\,{e}^{4} \left ( ex+d \right ) ^{5}}}-{\frac{b \left ( Abe+2\,aBe-3\,Bbd \right ) }{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 [A] time = 1.04558, size = 274, normalized size = 2.28 \begin{align*} -\frac{30 \, B b^{2} e^{3} x^{3} + 3 \, B b^{2} d^{3} + 12 \, A a^{2} e^{3} + 2 \,{\left (2 \, B a b + A b^{2}\right )} d^{2} e + 3 \,{\left (B a^{2} + 2 \, A a b\right )} d e^{2} + 10 \,{\left (3 \, B b^{2} d e^{2} + 2 \,{\left (2 \, B a b + A b^{2}\right )} e^{3}\right )} x^{2} + 5 \,{\left (3 \, B b^{2} d^{2} e + 2 \,{\left (2 \, B a b + A b^{2}\right )} d e^{2} + 3 \,{\left (B a^{2} + 2 \, 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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.55711, size = 432, normalized size = 3.6 \begin{align*} -\frac{30 \, B b^{2} e^{3} x^{3} + 3 \, B b^{2} d^{3} + 12 \, A a^{2} e^{3} + 2 \,{\left (2 \, B a b + A b^{2}\right )} d^{2} e + 3 \,{\left (B a^{2} + 2 \, A a b\right )} d e^{2} + 10 \,{\left (3 \, B b^{2} d e^{2} + 2 \,{\left (2 \, B a b + A b^{2}\right )} e^{3}\right )} x^{2} + 5 \,{\left (3 \, B b^{2} d^{2} e + 2 \,{\left (2 \, B a b + A b^{2}\right )} d e^{2} + 3 \,{\left (B a^{2} + 2 \, 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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 44.4137, size = 236, normalized size = 1.97 \begin{align*} - \frac{12 A a^{2} e^{3} + 6 A a b d e^{2} + 2 A b^{2} d^{2} e + 3 B a^{2} d e^{2} + 4 B a b d^{2} e + 3 B b^{2} d^{3} + 30 B b^{2} e^{3} x^{3} + x^{2} \left (20 A b^{2} e^{3} + 40 B a b e^{3} + 30 B b^{2} d e^{2}\right ) + x \left (30 A a b e^{3} + 10 A b^{2} d e^{2} + 15 B a^{2} e^{3} + 20 B a b d e^{2} + 15 B b^{2} 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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12623, size = 216, normalized size = 1.8 \begin{align*} -\frac{{\left (30 \, B b^{2} x^{3} e^{3} + 30 \, B b^{2} d x^{2} e^{2} + 15 \, B b^{2} d^{2} x e + 3 \, B b^{2} d^{3} + 40 \, B a b x^{2} e^{3} + 20 \, A b^{2} x^{2} e^{3} + 20 \, B a b d x e^{2} + 10 \, A b^{2} d x e^{2} + 4 \, B a b d^{2} e + 2 \, A b^{2} d^{2} e + 15 \, B a^{2} x e^{3} + 30 \, A a b x e^{3} + 3 \, B a^{2} d e^{2} + 6 \, A a b d e^{2} + 12 \, A a^{2} e^{3}\right )} e^{\left (-4\right )}}{60 \,{\left (x e + d\right )}^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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