Optimal. Leaf size=143 \[ -\frac{5 b^4 (d+e x)^{11} (b d-a e)}{11 e^6}+\frac{b^3 (d+e x)^{10} (b d-a e)^2}{e^6}-\frac{10 b^2 (d+e x)^9 (b d-a e)^3}{9 e^6}+\frac{5 b (d+e x)^8 (b d-a e)^4}{8 e^6}-\frac{(d+e x)^7 (b d-a e)^5}{7 e^6}+\frac{b^5 (d+e x)^{12}}{12 e^6} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.31053, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {27, 43} \[ -\frac{5 b^4 (d+e x)^{11} (b d-a e)}{11 e^6}+\frac{b^3 (d+e x)^{10} (b d-a e)^2}{e^6}-\frac{10 b^2 (d+e x)^9 (b d-a e)^3}{9 e^6}+\frac{5 b (d+e x)^8 (b d-a e)^4}{8 e^6}-\frac{(d+e x)^7 (b d-a e)^5}{7 e^6}+\frac{b^5 (d+e x)^{12}}{12 e^6} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 27
Rule 43
Rubi steps
\begin{align*} \int (a+b x) (d+e x)^6 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx &=\int (a+b x)^5 (d+e x)^6 \, dx\\ &=\int \left (\frac{(-b d+a e)^5 (d+e x)^6}{e^5}+\frac{5 b (b d-a e)^4 (d+e x)^7}{e^5}-\frac{10 b^2 (b d-a e)^3 (d+e x)^8}{e^5}+\frac{10 b^3 (b d-a e)^2 (d+e x)^9}{e^5}-\frac{5 b^4 (b d-a e) (d+e x)^{10}}{e^5}+\frac{b^5 (d+e x)^{11}}{e^5}\right ) \, dx\\ &=-\frac{(b d-a e)^5 (d+e x)^7}{7 e^6}+\frac{5 b (b d-a e)^4 (d+e x)^8}{8 e^6}-\frac{10 b^2 (b d-a e)^3 (d+e x)^9}{9 e^6}+\frac{b^3 (b d-a e)^2 (d+e x)^{10}}{e^6}-\frac{5 b^4 (b d-a e) (d+e x)^{11}}{11 e^6}+\frac{b^5 (d+e x)^{12}}{12 e^6}\\ \end{align*}
Mathematica [B] time = 0.072454, size = 501, normalized size = 3.5 \[ \frac{1}{2} b^3 e^4 x^{10} \left (2 a^2 e^2+6 a b d e+3 b^2 d^2\right )+\frac{5}{9} b^2 e^3 x^9 \left (12 a^2 b d e^2+2 a^3 e^3+15 a b^2 d^2 e+4 b^3 d^3\right )+\frac{5}{8} b e^2 x^8 \left (30 a^2 b^2 d^2 e^2+12 a^3 b d e^3+a^4 e^4+20 a b^3 d^3 e+3 b^4 d^4\right )+\frac{1}{7} e x^7 \left (200 a^2 b^3 d^3 e^2+150 a^3 b^2 d^2 e^3+30 a^4 b d e^4+a^5 e^5+75 a b^4 d^4 e+6 b^5 d^5\right )+\frac{1}{6} d x^6 \left (150 a^2 b^3 d^3 e^2+200 a^3 b^2 d^2 e^3+75 a^4 b d e^4+6 a^5 e^5+30 a b^4 d^4 e+b^5 d^5\right )+a d^2 x^5 \left (30 a^2 b^2 d^2 e^2+20 a^3 b d e^3+3 a^4 e^4+12 a b^3 d^3 e+b^4 d^4\right )+\frac{5}{4} a^2 d^3 x^4 \left (15 a^2 b d e^2+4 a^3 e^3+12 a b^2 d^2 e+2 b^3 d^3\right )+\frac{5}{3} a^3 d^4 x^3 \left (3 a^2 e^2+6 a b d e+2 b^2 d^2\right )+\frac{1}{2} a^4 d^5 x^2 (6 a e+5 b d)+a^5 d^6 x+\frac{1}{11} b^4 e^5 x^{11} (5 a e+6 b d)+\frac{1}{12} b^5 e^6 x^{12} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.003, size = 817, normalized size = 5.7 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.03383, size = 698, normalized size = 4.88 \begin{align*} \frac{1}{12} \, b^{5} e^{6} x^{12} + a^{5} d^{6} x + \frac{1}{11} \,{\left (6 \, b^{5} d e^{5} + 5 \, a b^{4} e^{6}\right )} x^{11} + \frac{1}{2} \,{\left (3 \, b^{5} d^{2} e^{4} + 6 \, a b^{4} d e^{5} + 2 \, a^{2} b^{3} e^{6}\right )} x^{10} + \frac{5}{9} \,{\left (4 \, b^{5} d^{3} e^{3} + 15 \, a b^{4} d^{2} e^{4} + 12 \, a^{2} b^{3} d e^{5} + 2 \, a^{3} b^{2} e^{6}\right )} x^{9} + \frac{5}{8} \,{\left (3 \, b^{5} d^{4} e^{2} + 20 \, a b^{4} d^{3} e^{3} + 30 \, a^{2} b^{3} d^{2} e^{4} + 12 \, a^{3} b^{2} d e^{5} + a^{4} b e^{6}\right )} x^{8} + \frac{1}{7} \,{\left (6 \, b^{5} d^{5} e + 75 \, a b^{4} d^{4} e^{2} + 200 \, a^{2} b^{3} d^{3} e^{3} + 150 \, a^{3} b^{2} d^{2} e^{4} + 30 \, a^{4} b d e^{5} + a^{5} e^{6}\right )} x^{7} + \frac{1}{6} \,{\left (b^{5} d^{6} + 30 \, a b^{4} d^{5} e + 150 \, a^{2} b^{3} d^{4} e^{2} + 200 \, a^{3} b^{2} d^{3} e^{3} + 75 \, a^{4} b d^{2} e^{4} + 6 \, a^{5} d e^{5}\right )} x^{6} +{\left (a b^{4} d^{6} + 12 \, a^{2} b^{3} d^{5} e + 30 \, a^{3} b^{2} d^{4} e^{2} + 20 \, a^{4} b d^{3} e^{3} + 3 \, a^{5} d^{2} e^{4}\right )} x^{5} + \frac{5}{4} \,{\left (2 \, a^{2} b^{3} d^{6} + 12 \, a^{3} b^{2} d^{5} e + 15 \, a^{4} b d^{4} e^{2} + 4 \, a^{5} d^{3} e^{3}\right )} x^{4} + \frac{5}{3} \,{\left (2 \, a^{3} b^{2} d^{6} + 6 \, a^{4} b d^{5} e + 3 \, a^{5} d^{4} e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (5 \, a^{4} b d^{6} + 6 \, a^{5} d^{5} e\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.31551, size = 1251, normalized size = 8.75 \begin{align*} \frac{1}{12} x^{12} e^{6} b^{5} + \frac{6}{11} x^{11} e^{5} d b^{5} + \frac{5}{11} x^{11} e^{6} b^{4} a + \frac{3}{2} x^{10} e^{4} d^{2} b^{5} + 3 x^{10} e^{5} d b^{4} a + x^{10} e^{6} b^{3} a^{2} + \frac{20}{9} x^{9} e^{3} d^{3} b^{5} + \frac{25}{3} x^{9} e^{4} d^{2} b^{4} a + \frac{20}{3} x^{9} e^{5} d b^{3} a^{2} + \frac{10}{9} x^{9} e^{6} b^{2} a^{3} + \frac{15}{8} x^{8} e^{2} d^{4} b^{5} + \frac{25}{2} x^{8} e^{3} d^{3} b^{4} a + \frac{75}{4} x^{8} e^{4} d^{2} b^{3} a^{2} + \frac{15}{2} x^{8} e^{5} d b^{2} a^{3} + \frac{5}{8} x^{8} e^{6} b a^{4} + \frac{6}{7} x^{7} e d^{5} b^{5} + \frac{75}{7} x^{7} e^{2} d^{4} b^{4} a + \frac{200}{7} x^{7} e^{3} d^{3} b^{3} a^{2} + \frac{150}{7} x^{7} e^{4} d^{2} b^{2} a^{3} + \frac{30}{7} x^{7} e^{5} d b a^{4} + \frac{1}{7} x^{7} e^{6} a^{5} + \frac{1}{6} x^{6} d^{6} b^{5} + 5 x^{6} e d^{5} b^{4} a + 25 x^{6} e^{2} d^{4} b^{3} a^{2} + \frac{100}{3} x^{6} e^{3} d^{3} b^{2} a^{3} + \frac{25}{2} x^{6} e^{4} d^{2} b a^{4} + x^{6} e^{5} d a^{5} + x^{5} d^{6} b^{4} a + 12 x^{5} e d^{5} b^{3} a^{2} + 30 x^{5} e^{2} d^{4} b^{2} a^{3} + 20 x^{5} e^{3} d^{3} b a^{4} + 3 x^{5} e^{4} d^{2} a^{5} + \frac{5}{2} x^{4} d^{6} b^{3} a^{2} + 15 x^{4} e d^{5} b^{2} a^{3} + \frac{75}{4} x^{4} e^{2} d^{4} b a^{4} + 5 x^{4} e^{3} d^{3} a^{5} + \frac{10}{3} x^{3} d^{6} b^{2} a^{3} + 10 x^{3} e d^{5} b a^{4} + 5 x^{3} e^{2} d^{4} a^{5} + \frac{5}{2} x^{2} d^{6} b a^{4} + 3 x^{2} e d^{5} a^{5} + x d^{6} a^{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 0.140361, size = 580, normalized size = 4.06 \begin{align*} a^{5} d^{6} x + \frac{b^{5} e^{6} x^{12}}{12} + x^{11} \left (\frac{5 a b^{4} e^{6}}{11} + \frac{6 b^{5} d e^{5}}{11}\right ) + x^{10} \left (a^{2} b^{3} e^{6} + 3 a b^{4} d e^{5} + \frac{3 b^{5} d^{2} e^{4}}{2}\right ) + x^{9} \left (\frac{10 a^{3} b^{2} e^{6}}{9} + \frac{20 a^{2} b^{3} d e^{5}}{3} + \frac{25 a b^{4} d^{2} e^{4}}{3} + \frac{20 b^{5} d^{3} e^{3}}{9}\right ) + x^{8} \left (\frac{5 a^{4} b e^{6}}{8} + \frac{15 a^{3} b^{2} d e^{5}}{2} + \frac{75 a^{2} b^{3} d^{2} e^{4}}{4} + \frac{25 a b^{4} d^{3} e^{3}}{2} + \frac{15 b^{5} d^{4} e^{2}}{8}\right ) + x^{7} \left (\frac{a^{5} e^{6}}{7} + \frac{30 a^{4} b d e^{5}}{7} + \frac{150 a^{3} b^{2} d^{2} e^{4}}{7} + \frac{200 a^{2} b^{3} d^{3} e^{3}}{7} + \frac{75 a b^{4} d^{4} e^{2}}{7} + \frac{6 b^{5} d^{5} e}{7}\right ) + x^{6} \left (a^{5} d e^{5} + \frac{25 a^{4} b d^{2} e^{4}}{2} + \frac{100 a^{3} b^{2} d^{3} e^{3}}{3} + 25 a^{2} b^{3} d^{4} e^{2} + 5 a b^{4} d^{5} e + \frac{b^{5} d^{6}}{6}\right ) + x^{5} \left (3 a^{5} d^{2} e^{4} + 20 a^{4} b d^{3} e^{3} + 30 a^{3} b^{2} d^{4} e^{2} + 12 a^{2} b^{3} d^{5} e + a b^{4} d^{6}\right ) + x^{4} \left (5 a^{5} d^{3} e^{3} + \frac{75 a^{4} b d^{4} e^{2}}{4} + 15 a^{3} b^{2} d^{5} e + \frac{5 a^{2} b^{3} d^{6}}{2}\right ) + x^{3} \left (5 a^{5} d^{4} e^{2} + 10 a^{4} b d^{5} e + \frac{10 a^{3} b^{2} d^{6}}{3}\right ) + x^{2} \left (3 a^{5} d^{5} e + \frac{5 a^{4} b d^{6}}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.14558, size = 749, normalized size = 5.24 \begin{align*} \frac{1}{12} \, b^{5} x^{12} e^{6} + \frac{6}{11} \, b^{5} d x^{11} e^{5} + \frac{3}{2} \, b^{5} d^{2} x^{10} e^{4} + \frac{20}{9} \, b^{5} d^{3} x^{9} e^{3} + \frac{15}{8} \, b^{5} d^{4} x^{8} e^{2} + \frac{6}{7} \, b^{5} d^{5} x^{7} e + \frac{1}{6} \, b^{5} d^{6} x^{6} + \frac{5}{11} \, a b^{4} x^{11} e^{6} + 3 \, a b^{4} d x^{10} e^{5} + \frac{25}{3} \, a b^{4} d^{2} x^{9} e^{4} + \frac{25}{2} \, a b^{4} d^{3} x^{8} e^{3} + \frac{75}{7} \, a b^{4} d^{4} x^{7} e^{2} + 5 \, a b^{4} d^{5} x^{6} e + a b^{4} d^{6} x^{5} + a^{2} b^{3} x^{10} e^{6} + \frac{20}{3} \, a^{2} b^{3} d x^{9} e^{5} + \frac{75}{4} \, a^{2} b^{3} d^{2} x^{8} e^{4} + \frac{200}{7} \, a^{2} b^{3} d^{3} x^{7} e^{3} + 25 \, a^{2} b^{3} d^{4} x^{6} e^{2} + 12 \, a^{2} b^{3} d^{5} x^{5} e + \frac{5}{2} \, a^{2} b^{3} d^{6} x^{4} + \frac{10}{9} \, a^{3} b^{2} x^{9} e^{6} + \frac{15}{2} \, a^{3} b^{2} d x^{8} e^{5} + \frac{150}{7} \, a^{3} b^{2} d^{2} x^{7} e^{4} + \frac{100}{3} \, a^{3} b^{2} d^{3} x^{6} e^{3} + 30 \, a^{3} b^{2} d^{4} x^{5} e^{2} + 15 \, a^{3} b^{2} d^{5} x^{4} e + \frac{10}{3} \, a^{3} b^{2} d^{6} x^{3} + \frac{5}{8} \, a^{4} b x^{8} e^{6} + \frac{30}{7} \, a^{4} b d x^{7} e^{5} + \frac{25}{2} \, a^{4} b d^{2} x^{6} e^{4} + 20 \, a^{4} b d^{3} x^{5} e^{3} + \frac{75}{4} \, a^{4} b d^{4} x^{4} e^{2} + 10 \, a^{4} b d^{5} x^{3} e + \frac{5}{2} \, a^{4} b d^{6} x^{2} + \frac{1}{7} \, a^{5} x^{7} e^{6} + a^{5} d x^{6} e^{5} + 3 \, a^{5} d^{2} x^{5} e^{4} + 5 \, a^{5} d^{3} x^{4} e^{3} + 5 \, a^{5} d^{4} x^{3} e^{2} + 3 \, a^{5} d^{5} x^{2} e + a^{5} d^{6} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]