Optimal. Leaf size=268 \[ \frac{1}{6} x^6 \left (6 a^2 c^2 e+12 a b^2 c e+12 a b c^2 d+4 b^3 c d+b^4 e\right )+\frac{1}{5} x^5 \left (12 a^2 b c e+6 a^2 c^2 d+12 a b^2 c d+4 a b^3 e+b^4 d\right )+\frac{1}{2} a x^4 \left (2 a^2 c e+3 a b^2 e+6 a b c d+2 b^3 d\right )+\frac{2}{3} a^2 x^3 \left (2 a b e+2 a c d+3 b^2 d\right )+\frac{1}{2} a^3 x^2 (a e+4 b d)+a^4 d x+\frac{1}{4} c^2 x^8 \left (2 a c e+3 b^2 e+2 b c d\right )+\frac{2}{7} c x^7 \left (6 a b c e+2 a c^2 d+3 b^2 c d+2 b^3 e\right )+\frac{1}{9} c^3 x^9 (4 b e+c d)+\frac{1}{10} c^4 e x^{10} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.290969, antiderivative size = 268, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {631} \[ \frac{1}{6} x^6 \left (6 a^2 c^2 e+12 a b^2 c e+12 a b c^2 d+4 b^3 c d+b^4 e\right )+\frac{1}{5} x^5 \left (12 a^2 b c e+6 a^2 c^2 d+12 a b^2 c d+4 a b^3 e+b^4 d\right )+\frac{1}{2} a x^4 \left (2 a^2 c e+3 a b^2 e+6 a b c d+2 b^3 d\right )+\frac{2}{3} a^2 x^3 \left (2 a b e+2 a c d+3 b^2 d\right )+\frac{1}{2} a^3 x^2 (a e+4 b d)+a^4 d x+\frac{1}{4} c^2 x^8 \left (2 a c e+3 b^2 e+2 b c d\right )+\frac{2}{7} c x^7 \left (6 a b c e+2 a c^2 d+3 b^2 c d+2 b^3 e\right )+\frac{1}{9} c^3 x^9 (4 b e+c d)+\frac{1}{10} c^4 e x^{10} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 631
Rubi steps
\begin{align*} \int (d+e x) \left (a+b x+c x^2\right )^4 \, dx &=\int \left (a^4 d+a^3 (4 b d+a e) x+2 a^2 \left (3 b^2 d+2 a c d+2 a b e\right ) x^2+2 a \left (2 b^3 d+6 a b c d+3 a b^2 e+2 a^2 c e\right ) x^3+\left (b^4 d+12 a b^2 c d+6 a^2 c^2 d+4 a b^3 e+12 a^2 b c e\right ) x^4+\left (4 b^3 c d+12 a b c^2 d+b^4 e+12 a b^2 c e+6 a^2 c^2 e\right ) x^5+2 c \left (3 b^2 c d+2 a c^2 d+2 b^3 e+6 a b c e\right ) x^6+2 c^2 \left (2 b c d+3 b^2 e+2 a c e\right ) x^7+c^3 (c d+4 b e) x^8+c^4 e x^9\right ) \, dx\\ &=a^4 d x+\frac{1}{2} a^3 (4 b d+a e) x^2+\frac{2}{3} a^2 \left (3 b^2 d+2 a c d+2 a b e\right ) x^3+\frac{1}{2} a \left (2 b^3 d+6 a b c d+3 a b^2 e+2 a^2 c e\right ) x^4+\frac{1}{5} \left (b^4 d+12 a b^2 c d+6 a^2 c^2 d+4 a b^3 e+12 a^2 b c e\right ) x^5+\frac{1}{6} \left (4 b^3 c d+12 a b c^2 d+b^4 e+12 a b^2 c e+6 a^2 c^2 e\right ) x^6+\frac{2}{7} c \left (3 b^2 c d+2 a c^2 d+2 b^3 e+6 a b c e\right ) x^7+\frac{1}{4} c^2 \left (2 b c d+3 b^2 e+2 a c e\right ) x^8+\frac{1}{9} c^3 (c d+4 b e) x^9+\frac{1}{10} c^4 e x^{10}\\ \end{align*}
Mathematica [A] time = 0.0694483, size = 268, normalized size = 1. \[ \frac{1}{6} x^6 \left (6 a^2 c^2 e+12 a b^2 c e+12 a b c^2 d+4 b^3 c d+b^4 e\right )+\frac{1}{5} x^5 \left (12 a^2 b c e+6 a^2 c^2 d+12 a b^2 c d+4 a b^3 e+b^4 d\right )+\frac{1}{2} a x^4 \left (2 a^2 c e+3 a b^2 e+6 a b c d+2 b^3 d\right )+\frac{2}{3} a^2 x^3 \left (2 a b e+2 a c d+3 b^2 d\right )+\frac{1}{2} a^3 x^2 (a e+4 b d)+a^4 d x+\frac{1}{4} c^2 x^8 \left (2 a c e+3 b^2 e+2 b c d\right )+\frac{2}{7} c x^7 \left (6 a b c e+2 a c^2 d+3 b^2 c d+2 b^3 e\right )+\frac{1}{9} c^3 x^9 (4 b e+c d)+\frac{1}{10} c^4 e x^{10} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.038, size = 343, normalized size = 1.3 \begin{align*}{\frac{{c}^{4}e{x}^{10}}{10}}+{\frac{ \left ( 4\,eb{c}^{3}+d{c}^{4} \right ){x}^{9}}{9}}+{\frac{ \left ( 4\,db{c}^{3}+e \left ( 2\, \left ( 2\,ac+{b}^{2} \right ){c}^{2}+4\,{b}^{2}{c}^{2} \right ) \right ){x}^{8}}{8}}+{\frac{ \left ( d \left ( 2\, \left ( 2\,ac+{b}^{2} \right ){c}^{2}+4\,{b}^{2}{c}^{2} \right ) +e \left ( 4\,ba{c}^{2}+4\, \left ( 2\,ac+{b}^{2} \right ) bc \right ) \right ){x}^{7}}{7}}+{\frac{ \left ( d \left ( 4\,ba{c}^{2}+4\, \left ( 2\,ac+{b}^{2} \right ) bc \right ) +e \left ( 2\,{a}^{2}{c}^{2}+8\,ac{b}^{2}+ \left ( 2\,ac+{b}^{2} \right ) ^{2} \right ) \right ){x}^{6}}{6}}+{\frac{ \left ( d \left ( 2\,{a}^{2}{c}^{2}+8\,ac{b}^{2}+ \left ( 2\,ac+{b}^{2} \right ) ^{2} \right ) +e \left ( 4\,{a}^{2}bc+4\,ab \left ( 2\,ac+{b}^{2} \right ) \right ) \right ){x}^{5}}{5}}+{\frac{ \left ( d \left ( 4\,{a}^{2}bc+4\,ab \left ( 2\,ac+{b}^{2} \right ) \right ) +e \left ( 2\,{a}^{2} \left ( 2\,ac+{b}^{2} \right ) +4\,{b}^{2}{a}^{2} \right ) \right ){x}^{4}}{4}}+{\frac{ \left ( d \left ( 2\,{a}^{2} \left ( 2\,ac+{b}^{2} \right ) +4\,{b}^{2}{a}^{2} \right ) +4\,e{a}^{3}b \right ){x}^{3}}{3}}+{\frac{ \left ( e{a}^{4}+4\,d{a}^{3}b \right ){x}^{2}}{2}}+{a}^{4}dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.01288, size = 373, normalized size = 1.39 \begin{align*} \frac{1}{10} \, c^{4} e x^{10} + \frac{1}{9} \,{\left (c^{4} d + 4 \, b c^{3} e\right )} x^{9} + \frac{1}{4} \,{\left (2 \, b c^{3} d +{\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e\right )} x^{8} + \frac{2}{7} \,{\left ({\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d + 2 \,{\left (b^{3} c + 3 \, a b c^{2}\right )} e\right )} x^{7} + \frac{1}{6} \,{\left (4 \,{\left (b^{3} c + 3 \, a b c^{2}\right )} d +{\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} e\right )} x^{6} + a^{4} d x + \frac{1}{5} \,{\left ({\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d + 4 \,{\left (a b^{3} + 3 \, a^{2} b c\right )} e\right )} x^{5} + \frac{1}{2} \,{\left (2 \,{\left (a b^{3} + 3 \, a^{2} b c\right )} d +{\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e\right )} x^{4} + \frac{2}{3} \,{\left (2 \, a^{3} b e +{\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d\right )} x^{3} + \frac{1}{2} \,{\left (4 \, a^{3} b d + a^{4} e\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.75107, size = 713, normalized size = 2.66 \begin{align*} \frac{1}{10} x^{10} e c^{4} + \frac{1}{9} x^{9} d c^{4} + \frac{4}{9} x^{9} e c^{3} b + \frac{1}{2} x^{8} d c^{3} b + \frac{3}{4} x^{8} e c^{2} b^{2} + \frac{1}{2} x^{8} e c^{3} a + \frac{6}{7} x^{7} d c^{2} b^{2} + \frac{4}{7} x^{7} e c b^{3} + \frac{4}{7} x^{7} d c^{3} a + \frac{12}{7} x^{7} e c^{2} b a + \frac{2}{3} x^{6} d c b^{3} + \frac{1}{6} x^{6} e b^{4} + 2 x^{6} d c^{2} b a + 2 x^{6} e c b^{2} a + x^{6} e c^{2} a^{2} + \frac{1}{5} x^{5} d b^{4} + \frac{12}{5} x^{5} d c b^{2} a + \frac{4}{5} x^{5} e b^{3} a + \frac{6}{5} x^{5} d c^{2} a^{2} + \frac{12}{5} x^{5} e c b a^{2} + x^{4} d b^{3} a + 3 x^{4} d c b a^{2} + \frac{3}{2} x^{4} e b^{2} a^{2} + x^{4} e c a^{3} + 2 x^{3} d b^{2} a^{2} + \frac{4}{3} x^{3} d c a^{3} + \frac{4}{3} x^{3} e b a^{3} + 2 x^{2} d b a^{3} + \frac{1}{2} x^{2} e a^{4} + x d a^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.113585, size = 313, normalized size = 1.17 \begin{align*} a^{4} d x + \frac{c^{4} e x^{10}}{10} + x^{9} \left (\frac{4 b c^{3} e}{9} + \frac{c^{4} d}{9}\right ) + x^{8} \left (\frac{a c^{3} e}{2} + \frac{3 b^{2} c^{2} e}{4} + \frac{b c^{3} d}{2}\right ) + x^{7} \left (\frac{12 a b c^{2} e}{7} + \frac{4 a c^{3} d}{7} + \frac{4 b^{3} c e}{7} + \frac{6 b^{2} c^{2} d}{7}\right ) + x^{6} \left (a^{2} c^{2} e + 2 a b^{2} c e + 2 a b c^{2} d + \frac{b^{4} e}{6} + \frac{2 b^{3} c d}{3}\right ) + x^{5} \left (\frac{12 a^{2} b c e}{5} + \frac{6 a^{2} c^{2} d}{5} + \frac{4 a b^{3} e}{5} + \frac{12 a b^{2} c d}{5} + \frac{b^{4} d}{5}\right ) + x^{4} \left (a^{3} c e + \frac{3 a^{2} b^{2} e}{2} + 3 a^{2} b c d + a b^{3} d\right ) + x^{3} \left (\frac{4 a^{3} b e}{3} + \frac{4 a^{3} c d}{3} + 2 a^{2} b^{2} d\right ) + x^{2} \left (\frac{a^{4} e}{2} + 2 a^{3} b d\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.09179, size = 435, normalized size = 1.62 \begin{align*} \frac{1}{10} \, c^{4} x^{10} e + \frac{1}{9} \, c^{4} d x^{9} + \frac{4}{9} \, b c^{3} x^{9} e + \frac{1}{2} \, b c^{3} d x^{8} + \frac{3}{4} \, b^{2} c^{2} x^{8} e + \frac{1}{2} \, a c^{3} x^{8} e + \frac{6}{7} \, b^{2} c^{2} d x^{7} + \frac{4}{7} \, a c^{3} d x^{7} + \frac{4}{7} \, b^{3} c x^{7} e + \frac{12}{7} \, a b c^{2} x^{7} e + \frac{2}{3} \, b^{3} c d x^{6} + 2 \, a b c^{2} d x^{6} + \frac{1}{6} \, b^{4} x^{6} e + 2 \, a b^{2} c x^{6} e + a^{2} c^{2} x^{6} e + \frac{1}{5} \, b^{4} d x^{5} + \frac{12}{5} \, a b^{2} c d x^{5} + \frac{6}{5} \, a^{2} c^{2} d x^{5} + \frac{4}{5} \, a b^{3} x^{5} e + \frac{12}{5} \, a^{2} b c x^{5} e + a b^{3} d x^{4} + 3 \, a^{2} b c d x^{4} + \frac{3}{2} \, a^{2} b^{2} x^{4} e + a^{3} c x^{4} e + 2 \, a^{2} b^{2} d x^{3} + \frac{4}{3} \, a^{3} c d x^{3} + \frac{4}{3} \, a^{3} b x^{3} e + 2 \, a^{3} b d x^{2} + \frac{1}{2} \, a^{4} x^{2} e + a^{4} d x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]