Optimal. Leaf size=120 \[ \frac{b (-2 a B e-A b e+3 b B d)}{4 e^4 (d+e x)^4}-\frac{(b d-a e) (-a B e-2 A b e+3 b B d)}{5 e^4 (d+e x)^5}+\frac{(b d-a e)^2 (B d-A e)}{6 e^4 (d+e x)^6}-\frac{b^2 B}{3 e^4 (d+e x)^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0761238, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {77} \[ \frac{b (-2 a B e-A b e+3 b B d)}{4 e^4 (d+e x)^4}-\frac{(b d-a e) (-a B e-2 A b e+3 b B d)}{5 e^4 (d+e x)^5}+\frac{(b d-a e)^2 (B d-A e)}{6 e^4 (d+e x)^6}-\frac{b^2 B}{3 e^4 (d+e x)^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 77
Rubi steps
\begin{align*} \int \frac{(a+b x)^2 (A+B x)}{(d+e x)^7} \, dx &=\int \left (\frac{(-b d+a e)^2 (-B d+A e)}{e^3 (d+e x)^7}+\frac{(-b d+a e) (-3 b B d+2 A b e+a B e)}{e^3 (d+e x)^6}+\frac{b (-3 b B d+A b e+2 a B e)}{e^3 (d+e x)^5}+\frac{b^2 B}{e^3 (d+e x)^4}\right ) \, dx\\ &=\frac{(b d-a e)^2 (B d-A e)}{6 e^4 (d+e x)^6}-\frac{(b d-a e) (3 b B d-2 A b e-a B e)}{5 e^4 (d+e x)^5}+\frac{b (3 b B d-A b e-2 a B e)}{4 e^4 (d+e x)^4}-\frac{b^2 B}{3 e^4 (d+e x)^3}\\ \end{align*}
Mathematica [A] time = 0.0608497, size = 126, normalized size = 1.05 \[ -\frac{2 a^2 e^2 (5 A e+B (d+6 e x))+2 a b e \left (2 A e (d+6 e x)+B \left (d^2+6 d e x+15 e^2 x^2\right )\right )+b^2 \left (A e \left (d^2+6 d e x+15 e^2 x^2\right )+B \left (6 d^2 e x+d^3+15 d e^2 x^2+20 e^3 x^3\right )\right )}{60 e^4 (d+e x)^6} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.006, size = 166, normalized size = 1.4 \begin{align*} -{\frac{{a}^{2}A{e}^{3}-2\,Adab{e}^{2}+A{d}^{2}{b}^{2}e-Bd{a}^{2}{e}^{2}+2\,B{d}^{2}abe-{b}^{2}B{d}^{3}}{6\,{e}^{4} \left ( ex+d \right ) ^{6}}}-{\frac{B{b}^{2}}{3\,{e}^{4} \left ( ex+d \right ) ^{3}}}-{\frac{2\,Aba{e}^{2}-2\,Ad{b}^{2}e+B{a}^{2}{e}^{2}-4\,Bdabe+3\,{b}^{2}B{d}^{2}}{5\,{e}^{4} \left ( ex+d \right ) ^{5}}}-{\frac{b \left ( Abe+2\,Bae-3\,Bbd \right ) }{4\,{e}^{4} \left ( ex+d \right ) ^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.52274, size = 281, normalized size = 2.34 \begin{align*} -\frac{20 \, B b^{2} e^{3} x^{3} + B b^{2} d^{3} + 10 \, A a^{2} e^{3} +{\left (2 \, B a b + A b^{2}\right )} d^{2} e + 2 \,{\left (B a^{2} + 2 \, A a b\right )} d e^{2} + 15 \,{\left (B b^{2} d e^{2} +{\left (2 \, B a b + A b^{2}\right )} e^{3}\right )} x^{2} + 6 \,{\left (B b^{2} d^{2} e +{\left (2 \, B a b + A b^{2}\right )} d e^{2} + 2 \,{\left (B a^{2} + 2 \, A a b\right )} e^{3}\right )} x}{60 \,{\left (e^{10} x^{6} + 6 \, d e^{9} x^{5} + 15 \, d^{2} e^{8} x^{4} + 20 \, d^{3} e^{7} x^{3} + 15 \, d^{4} e^{6} x^{2} + 6 \, d^{5} e^{5} x + d^{6} e^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.7932, size = 440, normalized size = 3.67 \begin{align*} -\frac{20 \, B b^{2} e^{3} x^{3} + B b^{2} d^{3} + 10 \, A a^{2} e^{3} +{\left (2 \, B a b + A b^{2}\right )} d^{2} e + 2 \,{\left (B a^{2} + 2 \, A a b\right )} d e^{2} + 15 \,{\left (B b^{2} d e^{2} +{\left (2 \, B a b + A b^{2}\right )} e^{3}\right )} x^{2} + 6 \,{\left (B b^{2} d^{2} e +{\left (2 \, B a b + A b^{2}\right )} d e^{2} + 2 \,{\left (B a^{2} + 2 \, A a b\right )} e^{3}\right )} x}{60 \,{\left (e^{10} x^{6} + 6 \, d e^{9} x^{5} + 15 \, d^{2} e^{8} x^{4} + 20 \, d^{3} e^{7} x^{3} + 15 \, d^{4} e^{6} x^{2} + 6 \, d^{5} e^{5} x + d^{6} e^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 90.8951, size = 245, normalized size = 2.04 \begin{align*} - \frac{10 A a^{2} e^{3} + 4 A a b d e^{2} + A b^{2} d^{2} e + 2 B a^{2} d e^{2} + 2 B a b d^{2} e + B b^{2} d^{3} + 20 B b^{2} e^{3} x^{3} + x^{2} \left (15 A b^{2} e^{3} + 30 B a b e^{3} + 15 B b^{2} d e^{2}\right ) + x \left (24 A a b e^{3} + 6 A b^{2} d e^{2} + 12 B a^{2} e^{3} + 12 B a b d e^{2} + 6 B b^{2} d^{2} e\right )}{60 d^{6} e^{4} + 360 d^{5} e^{5} x + 900 d^{4} e^{6} x^{2} + 1200 d^{3} e^{7} x^{3} + 900 d^{2} e^{8} x^{4} + 360 d e^{9} x^{5} + 60 e^{10} x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 2.99092, size = 213, normalized size = 1.78 \begin{align*} -\frac{{\left (20 \, B b^{2} x^{3} e^{3} + 15 \, B b^{2} d x^{2} e^{2} + 6 \, B b^{2} d^{2} x e + B b^{2} d^{3} + 30 \, B a b x^{2} e^{3} + 15 \, A b^{2} x^{2} e^{3} + 12 \, B a b d x e^{2} + 6 \, A b^{2} d x e^{2} + 2 \, B a b d^{2} e + A b^{2} d^{2} e + 12 \, B a^{2} x e^{3} + 24 \, A a b x e^{3} + 2 \, B a^{2} d e^{2} + 4 \, A a b d e^{2} + 10 \, A a^{2} e^{3}\right )} e^{\left (-4\right )}}{60 \,{\left (x e + d\right )}^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]