Optimal. Leaf size=120 \[ -\frac{b (d+e x)^8 (-2 a B e-A b e+3 b B d)}{8 e^4}+\frac{(d+e x)^7 (b d-a e) (-a B e-2 A b e+3 b B d)}{7 e^4}-\frac{(d+e x)^6 (b d-a e)^2 (B d-A e)}{6 e^4}+\frac{b^2 B (d+e x)^9}{9 e^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.285558, 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} \[ -\frac{b (d+e x)^8 (-2 a B e-A b e+3 b B d)}{8 e^4}+\frac{(d+e x)^7 (b d-a e) (-a B e-2 A b e+3 b B d)}{7 e^4}-\frac{(d+e x)^6 (b d-a e)^2 (B d-A e)}{6 e^4}+\frac{b^2 B (d+e x)^9}{9 e^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 27
Rule 77
Rubi steps
\begin{align*} \int (A+B x) (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right ) \, dx &=\int (a+b x)^2 (A+B x) (d+e x)^5 \, dx\\ &=\int \left (\frac{(-b d+a e)^2 (-B d+A e) (d+e x)^5}{e^3}+\frac{(-b d+a e) (-3 b B d+2 A b e+a B e) (d+e x)^6}{e^3}+\frac{b (-3 b B d+A b e+2 a B e) (d+e x)^7}{e^3}+\frac{b^2 B (d+e x)^8}{e^3}\right ) \, dx\\ &=-\frac{(b d-a e)^2 (B d-A e) (d+e x)^6}{6 e^4}+\frac{(b d-a e) (3 b B d-2 A b e-a B e) (d+e x)^7}{7 e^4}-\frac{b (3 b B d-A b e-2 a B e) (d+e x)^8}{8 e^4}+\frac{b^2 B (d+e x)^9}{9 e^4}\\ \end{align*}
Mathematica [B] time = 0.115407, size = 330, normalized size = 2.75 \[ \frac{1}{6} e^2 x^6 \left (a^2 e^2 (A e+5 B d)+10 a b d e (A e+2 B d)+10 b^2 d^2 (A e+B d)\right )+d e x^5 \left (a^2 e^2 (A e+2 B d)+4 a b d e (A e+B d)+b^2 d^2 (2 A e+B d)\right )+\frac{1}{4} d^2 x^4 \left (10 a^2 e^2 (A e+B d)+10 a b d e (2 A e+B d)+b^2 d^2 (5 A e+B d)\right )+\frac{1}{3} d^3 x^3 \left (A \left (10 a^2 e^2+10 a b d e+b^2 d^2\right )+a B d (5 a e+2 b d)\right )+\frac{1}{7} e^3 x^7 \left (a^2 B e^2+2 a b e (A e+5 B d)+5 b^2 d (A e+2 B d)\right )+a^2 A d^5 x+\frac{1}{2} a d^4 x^2 (5 a A e+a B d+2 A b d)+\frac{1}{8} b e^4 x^8 (2 a B e+A b e+5 b B d)+\frac{1}{9} b^2 B e^5 x^9 \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.003, size = 394, normalized size = 3.3 \begin{align*}{\frac{B{e}^{5}{b}^{2}{x}^{9}}{9}}+{\frac{ \left ( \left ( A{e}^{5}+5\,Bd{e}^{4} \right ){b}^{2}+2\,B{e}^{5}ab \right ){x}^{8}}{8}}+{\frac{ \left ( \left ( 5\,Ad{e}^{4}+10\,B{d}^{2}{e}^{3} \right ){b}^{2}+2\, \left ( A{e}^{5}+5\,Bd{e}^{4} \right ) ab+B{e}^{5}{a}^{2} \right ){x}^{7}}{7}}+{\frac{ \left ( \left ( 10\,A{d}^{2}{e}^{3}+10\,B{d}^{3}{e}^{2} \right ){b}^{2}+2\, \left ( 5\,Ad{e}^{4}+10\,B{d}^{2}{e}^{3} \right ) ab+ \left ( A{e}^{5}+5\,Bd{e}^{4} \right ){a}^{2} \right ){x}^{6}}{6}}+{\frac{ \left ( \left ( 10\,A{d}^{3}{e}^{2}+5\,B{d}^{4}e \right ){b}^{2}+2\, \left ( 10\,A{d}^{2}{e}^{3}+10\,B{d}^{3}{e}^{2} \right ) ab+ \left ( 5\,Ad{e}^{4}+10\,B{d}^{2}{e}^{3} \right ){a}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ( \left ( 5\,A{d}^{4}e+B{d}^{5} \right ){b}^{2}+2\, \left ( 10\,A{d}^{3}{e}^{2}+5\,B{d}^{4}e \right ) ab+ \left ( 10\,A{d}^{2}{e}^{3}+10\,B{d}^{3}{e}^{2} \right ){a}^{2} \right ){x}^{4}}{4}}+{\frac{ \left ( A{d}^{5}{b}^{2}+2\, \left ( 5\,A{d}^{4}e+B{d}^{5} \right ) ab+ \left ( 10\,A{d}^{3}{e}^{2}+5\,B{d}^{4}e \right ){a}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 2\,A{d}^{5}ab+ \left ( 5\,A{d}^{4}e+B{d}^{5} \right ){a}^{2} \right ){x}^{2}}{2}}+A{d}^{5}{a}^{2}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.2457, size = 498, normalized size = 4.15 \begin{align*} \frac{1}{9} \, B b^{2} e^{5} x^{9} + A a^{2} d^{5} x + \frac{1}{8} \,{\left (5 \, B b^{2} d e^{4} +{\left (2 \, B a b + A b^{2}\right )} e^{5}\right )} x^{8} + \frac{1}{7} \,{\left (10 \, B b^{2} d^{2} e^{3} + 5 \,{\left (2 \, B a b + A b^{2}\right )} d e^{4} +{\left (B a^{2} + 2 \, A a b\right )} e^{5}\right )} x^{7} + \frac{1}{6} \,{\left (10 \, B b^{2} d^{3} e^{2} + A a^{2} e^{5} + 10 \,{\left (2 \, B a b + A b^{2}\right )} d^{2} e^{3} + 5 \,{\left (B a^{2} + 2 \, A a b\right )} d e^{4}\right )} x^{6} +{\left (B b^{2} d^{4} e + A a^{2} d e^{4} + 2 \,{\left (2 \, B a b + A b^{2}\right )} d^{3} e^{2} + 2 \,{\left (B a^{2} + 2 \, A a b\right )} d^{2} e^{3}\right )} x^{5} + \frac{1}{4} \,{\left (B b^{2} d^{5} + 10 \, A a^{2} d^{2} e^{3} + 5 \,{\left (2 \, B a b + A b^{2}\right )} d^{4} e + 10 \,{\left (B a^{2} + 2 \, A a b\right )} d^{3} e^{2}\right )} x^{4} + \frac{1}{3} \,{\left (10 \, A a^{2} d^{3} e^{2} +{\left (2 \, B a b + A b^{2}\right )} d^{5} + 5 \,{\left (B a^{2} + 2 \, A a b\right )} d^{4} e\right )} x^{3} + \frac{1}{2} \,{\left (5 \, A a^{2} d^{4} e +{\left (B a^{2} + 2 \, A a b\right )} d^{5}\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.38758, size = 1037, normalized size = 8.64 \begin{align*} \frac{1}{9} x^{9} e^{5} b^{2} B + \frac{5}{8} x^{8} e^{4} d b^{2} B + \frac{1}{4} x^{8} e^{5} b a B + \frac{1}{8} x^{8} e^{5} b^{2} A + \frac{10}{7} x^{7} e^{3} d^{2} b^{2} B + \frac{10}{7} x^{7} e^{4} d b a B + \frac{1}{7} x^{7} e^{5} a^{2} B + \frac{5}{7} x^{7} e^{4} d b^{2} A + \frac{2}{7} x^{7} e^{5} b a A + \frac{5}{3} x^{6} e^{2} d^{3} b^{2} B + \frac{10}{3} x^{6} e^{3} d^{2} b a B + \frac{5}{6} x^{6} e^{4} d a^{2} B + \frac{5}{3} x^{6} e^{3} d^{2} b^{2} A + \frac{5}{3} x^{6} e^{4} d b a A + \frac{1}{6} x^{6} e^{5} a^{2} A + x^{5} e d^{4} b^{2} B + 4 x^{5} e^{2} d^{3} b a B + 2 x^{5} e^{3} d^{2} a^{2} B + 2 x^{5} e^{2} d^{3} b^{2} A + 4 x^{5} e^{3} d^{2} b a A + x^{5} e^{4} d a^{2} A + \frac{1}{4} x^{4} d^{5} b^{2} B + \frac{5}{2} x^{4} e d^{4} b a B + \frac{5}{2} x^{4} e^{2} d^{3} a^{2} B + \frac{5}{4} x^{4} e d^{4} b^{2} A + 5 x^{4} e^{2} d^{3} b a A + \frac{5}{2} x^{4} e^{3} d^{2} a^{2} A + \frac{2}{3} x^{3} d^{5} b a B + \frac{5}{3} x^{3} e d^{4} a^{2} B + \frac{1}{3} x^{3} d^{5} b^{2} A + \frac{10}{3} x^{3} e d^{4} b a A + \frac{10}{3} x^{3} e^{2} d^{3} a^{2} A + \frac{1}{2} x^{2} d^{5} a^{2} B + x^{2} d^{5} b a A + \frac{5}{2} x^{2} e d^{4} a^{2} A + x d^{5} a^{2} A \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 0.127359, size = 481, normalized size = 4.01 \begin{align*} A a^{2} d^{5} x + \frac{B b^{2} e^{5} x^{9}}{9} + x^{8} \left (\frac{A b^{2} e^{5}}{8} + \frac{B a b e^{5}}{4} + \frac{5 B b^{2} d e^{4}}{8}\right ) + x^{7} \left (\frac{2 A a b e^{5}}{7} + \frac{5 A b^{2} d e^{4}}{7} + \frac{B a^{2} e^{5}}{7} + \frac{10 B a b d e^{4}}{7} + \frac{10 B b^{2} d^{2} e^{3}}{7}\right ) + x^{6} \left (\frac{A a^{2} e^{5}}{6} + \frac{5 A a b d e^{4}}{3} + \frac{5 A b^{2} d^{2} e^{3}}{3} + \frac{5 B a^{2} d e^{4}}{6} + \frac{10 B a b d^{2} e^{3}}{3} + \frac{5 B b^{2} d^{3} e^{2}}{3}\right ) + x^{5} \left (A a^{2} d e^{4} + 4 A a b d^{2} e^{3} + 2 A b^{2} d^{3} e^{2} + 2 B a^{2} d^{2} e^{3} + 4 B a b d^{3} e^{2} + B b^{2} d^{4} e\right ) + x^{4} \left (\frac{5 A a^{2} d^{2} e^{3}}{2} + 5 A a b d^{3} e^{2} + \frac{5 A b^{2} d^{4} e}{4} + \frac{5 B a^{2} d^{3} e^{2}}{2} + \frac{5 B a b d^{4} e}{2} + \frac{B b^{2} d^{5}}{4}\right ) + x^{3} \left (\frac{10 A a^{2} d^{3} e^{2}}{3} + \frac{10 A a b d^{4} e}{3} + \frac{A b^{2} d^{5}}{3} + \frac{5 B a^{2} d^{4} e}{3} + \frac{2 B a b d^{5}}{3}\right ) + x^{2} \left (\frac{5 A a^{2} d^{4} e}{2} + A a b d^{5} + \frac{B a^{2} d^{5}}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.15398, size = 601, normalized size = 5.01 \begin{align*} \frac{1}{9} \, B b^{2} x^{9} e^{5} + \frac{5}{8} \, B b^{2} d x^{8} e^{4} + \frac{10}{7} \, B b^{2} d^{2} x^{7} e^{3} + \frac{5}{3} \, B b^{2} d^{3} x^{6} e^{2} + B b^{2} d^{4} x^{5} e + \frac{1}{4} \, B b^{2} d^{5} x^{4} + \frac{1}{4} \, B a b x^{8} e^{5} + \frac{1}{8} \, A b^{2} x^{8} e^{5} + \frac{10}{7} \, B a b d x^{7} e^{4} + \frac{5}{7} \, A b^{2} d x^{7} e^{4} + \frac{10}{3} \, B a b d^{2} x^{6} e^{3} + \frac{5}{3} \, A b^{2} d^{2} x^{6} e^{3} + 4 \, B a b d^{3} x^{5} e^{2} + 2 \, A b^{2} d^{3} x^{5} e^{2} + \frac{5}{2} \, B a b d^{4} x^{4} e + \frac{5}{4} \, A b^{2} d^{4} x^{4} e + \frac{2}{3} \, B a b d^{5} x^{3} + \frac{1}{3} \, A b^{2} d^{5} x^{3} + \frac{1}{7} \, B a^{2} x^{7} e^{5} + \frac{2}{7} \, A a b x^{7} e^{5} + \frac{5}{6} \, B a^{2} d x^{6} e^{4} + \frac{5}{3} \, A a b d x^{6} e^{4} + 2 \, B a^{2} d^{2} x^{5} e^{3} + 4 \, A a b d^{2} x^{5} e^{3} + \frac{5}{2} \, B a^{2} d^{3} x^{4} e^{2} + 5 \, A a b d^{3} x^{4} e^{2} + \frac{5}{3} \, B a^{2} d^{4} x^{3} e + \frac{10}{3} \, A a b d^{4} x^{3} e + \frac{1}{2} \, B a^{2} d^{5} x^{2} + A a b d^{5} x^{2} + \frac{1}{6} \, A a^{2} x^{6} e^{5} + A a^{2} d x^{5} e^{4} + \frac{5}{2} \, A a^{2} d^{2} x^{4} e^{3} + \frac{10}{3} \, A a^{2} d^{3} x^{3} e^{2} + \frac{5}{2} \, A a^{2} d^{4} x^{2} e + A a^{2} d^{5} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]