Optimal. Leaf size=272 \[ \frac{c (d+e x)^9 \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{3 e^7}-\frac{(d+e x)^8 (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{8 e^7}+\frac{3 (d+e x)^7 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{7 e^7}-\frac{(d+e x)^6 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{2 e^7}+\frac{(d+e x)^5 \left (a e^2-b d e+c d^2\right )^3}{5 e^7}-\frac{3 c^2 (d+e x)^{10} (2 c d-b e)}{10 e^7}+\frac{c^3 (d+e x)^{11}}{11 e^7} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.508922, antiderivative size = 272, 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 = {698} \[ \frac{c (d+e x)^9 \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{3 e^7}-\frac{(d+e x)^8 (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{8 e^7}+\frac{3 (d+e x)^7 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{7 e^7}-\frac{(d+e x)^6 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{2 e^7}+\frac{(d+e x)^5 \left (a e^2-b d e+c d^2\right )^3}{5 e^7}-\frac{3 c^2 (d+e x)^{10} (2 c d-b e)}{10 e^7}+\frac{c^3 (d+e x)^{11}}{11 e^7} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 698
Rubi steps
\begin{align*} \int (d+e x)^4 \left (a+b x+c x^2\right )^3 \, dx &=\int \left (\frac{\left (c d^2-b d e+a e^2\right )^3 (d+e x)^4}{e^6}+\frac{3 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^5}{e^6}+\frac{3 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2-5 b c d e+b^2 e^2+a c e^2\right ) (d+e x)^6}{e^6}+\frac{(2 c d-b e) \left (-10 c^2 d^2-b^2 e^2+2 c e (5 b d-3 a e)\right ) (d+e x)^7}{e^6}+\frac{3 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^8}{e^6}-\frac{3 c^2 (2 c d-b e) (d+e x)^9}{e^6}+\frac{c^3 (d+e x)^{10}}{e^6}\right ) \, dx\\ &=\frac{\left (c d^2-b d e+a e^2\right )^3 (d+e x)^5}{5 e^7}-\frac{(2 c d-b e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^6}{2 e^7}+\frac{3 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^7}{7 e^7}-\frac{(2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) (d+e x)^8}{8 e^7}+\frac{c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^9}{3 e^7}-\frac{3 c^2 (2 c d-b e) (d+e x)^{10}}{10 e^7}+\frac{c^3 (d+e x)^{11}}{11 e^7}\\ \end{align*}
Mathematica [A] time = 0.154939, size = 497, normalized size = 1.83 \[ \frac{1}{7} x^7 \left (3 c e^2 \left (a^2 e^2+8 a b d e+6 b^2 d^2\right )+b^2 e^3 (3 a e+4 b d)+6 c^2 d^2 e (3 a e+2 b d)+c^3 d^4\right )+\frac{1}{2} x^6 \left (b \left (a^2 e^4+12 a c d^2 e^2+c^2 d^4\right )+4 b^2 \left (a d e^3+c d^3 e\right )+4 a c d e \left (a e^2+c d^2\right )+2 b^3 d^2 e^2\right )+\frac{1}{5} x^5 \left (a \left (a^2 e^4+18 a c d^2 e^2+3 c^2 d^4\right )+3 b^2 \left (6 a d^2 e^2+c d^4\right )+12 a b d e \left (a e^2+2 c d^2\right )+4 b^3 d^3 e\right )+\frac{1}{4} d x^4 \left (4 a^2 e \left (a e^2+3 c d^2\right )+12 a b^2 d^2 e+6 a b d \left (3 a e^2+c d^2\right )+b^3 d^3\right )+\frac{1}{2} a^2 d^3 x^2 (4 a e+3 b d)+a^3 d^4 x+\frac{1}{3} c e^2 x^9 \left (c e (a e+4 b d)+b^2 e^2+2 c^2 d^2\right )+\frac{1}{8} e x^8 \left (6 c^2 d e (2 a e+3 b d)+6 b c e^2 (a e+2 b d)+b^3 e^3+4 c^3 d^3\right )+a d^2 x^3 \left (4 a b d e+a \left (2 a e^2+c d^2\right )+b^2 d^2\right )+\frac{1}{10} c^2 e^3 x^{10} (3 b e+4 c d)+\frac{1}{11} c^3 e^4 x^{11} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.04, size = 631, normalized size = 2.3 \begin{align*}{\frac{{c}^{3}{e}^{4}{x}^{11}}{11}}+{\frac{ \left ( 3\,{e}^{4}b{c}^{2}+4\,d{e}^{3}{c}^{3} \right ){x}^{10}}{10}}+{\frac{ \left ( 6\,{d}^{2}{e}^{2}{c}^{3}+12\,d{e}^{3}b{c}^{2}+{e}^{4} \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) \right ){x}^{9}}{9}}+{\frac{ \left ( 4\,{d}^{3}e{c}^{3}+18\,{d}^{2}{e}^{2}b{c}^{2}+4\,d{e}^{3} \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) +{e}^{4} \left ( 4\,abc+b \left ( 2\,ac+{b}^{2} \right ) \right ) \right ){x}^{8}}{8}}+{\frac{ \left ({c}^{3}{d}^{4}+12\,{d}^{3}eb{c}^{2}+6\,{d}^{2}{e}^{2} \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) +4\,d{e}^{3} \left ( 4\,abc+b \left ( 2\,ac+{b}^{2} \right ) \right ) +{e}^{4} \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,{b}^{2}a+{a}^{2}c \right ) \right ){x}^{7}}{7}}+{\frac{ \left ( 3\,{d}^{4}b{c}^{2}+4\,{d}^{3}e \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) +6\,{d}^{2}{e}^{2} \left ( 4\,abc+b \left ( 2\,ac+{b}^{2} \right ) \right ) +4\,d{e}^{3} \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,{b}^{2}a+{a}^{2}c \right ) +3\,{e}^{4}b{a}^{2} \right ){x}^{6}}{6}}+{\frac{ \left ({d}^{4} \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) +4\,{d}^{3}e \left ( 4\,abc+b \left ( 2\,ac+{b}^{2} \right ) \right ) +6\,{d}^{2}{e}^{2} \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,{b}^{2}a+{a}^{2}c \right ) +12\,d{e}^{3}b{a}^{2}+{e}^{4}{a}^{3} \right ){x}^{5}}{5}}+{\frac{ \left ({d}^{4} \left ( 4\,abc+b \left ( 2\,ac+{b}^{2} \right ) \right ) +4\,{d}^{3}e \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,{b}^{2}a+{a}^{2}c \right ) +18\,{d}^{2}{e}^{2}b{a}^{2}+4\,d{e}^{3}{a}^{3} \right ){x}^{4}}{4}}+{\frac{ \left ({d}^{4} \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,{b}^{2}a+{a}^{2}c \right ) +12\,{d}^{3}eb{a}^{2}+6\,{d}^{2}{e}^{2}{a}^{3} \right ){x}^{3}}{3}}+{\frac{ \left ( 4\,{d}^{3}e{a}^{3}+3\,{d}^{4}b{a}^{2} \right ){x}^{2}}{2}}+{d}^{4}{a}^{3}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.00174, size = 653, normalized size = 2.4 \begin{align*} \frac{1}{11} \, c^{3} e^{4} x^{11} + \frac{1}{10} \,{\left (4 \, c^{3} d e^{3} + 3 \, b c^{2} e^{4}\right )} x^{10} + \frac{1}{3} \,{\left (2 \, c^{3} d^{2} e^{2} + 4 \, b c^{2} d e^{3} +{\left (b^{2} c + a c^{2}\right )} e^{4}\right )} x^{9} + \frac{1}{8} \,{\left (4 \, c^{3} d^{3} e + 18 \, b c^{2} d^{2} e^{2} + 12 \,{\left (b^{2} c + a c^{2}\right )} d e^{3} +{\left (b^{3} + 6 \, a b c\right )} e^{4}\right )} x^{8} + a^{3} d^{4} x + \frac{1}{7} \,{\left (c^{3} d^{4} + 12 \, b c^{2} d^{3} e + 18 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{2} + 4 \,{\left (b^{3} + 6 \, a b c\right )} d e^{3} + 3 \,{\left (a b^{2} + a^{2} c\right )} e^{4}\right )} x^{7} + \frac{1}{2} \,{\left (b c^{2} d^{4} + a^{2} b e^{4} + 4 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e + 2 \,{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{2} + 4 \,{\left (a b^{2} + a^{2} c\right )} d e^{3}\right )} x^{6} + \frac{1}{5} \,{\left (12 \, a^{2} b d e^{3} + a^{3} e^{4} + 3 \,{\left (b^{2} c + a c^{2}\right )} d^{4} + 4 \,{\left (b^{3} + 6 \, a b c\right )} d^{3} e + 18 \,{\left (a b^{2} + a^{2} c\right )} d^{2} e^{2}\right )} x^{5} + \frac{1}{4} \,{\left (18 \, a^{2} b d^{2} e^{2} + 4 \, a^{3} d e^{3} +{\left (b^{3} + 6 \, a b c\right )} d^{4} + 12 \,{\left (a b^{2} + a^{2} c\right )} d^{3} e\right )} x^{4} +{\left (4 \, a^{2} b d^{3} e + 2 \, a^{3} d^{2} e^{2} +{\left (a b^{2} + a^{2} c\right )} d^{4}\right )} x^{3} + \frac{1}{2} \,{\left (3 \, a^{2} b d^{4} + 4 \, a^{3} d^{3} e\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.72891, size = 1377, normalized size = 5.06 \begin{align*} \frac{1}{11} x^{11} e^{4} c^{3} + \frac{2}{5} x^{10} e^{3} d c^{3} + \frac{3}{10} x^{10} e^{4} c^{2} b + \frac{2}{3} x^{9} e^{2} d^{2} c^{3} + \frac{4}{3} x^{9} e^{3} d c^{2} b + \frac{1}{3} x^{9} e^{4} c b^{2} + \frac{1}{3} x^{9} e^{4} c^{2} a + \frac{1}{2} x^{8} e d^{3} c^{3} + \frac{9}{4} x^{8} e^{2} d^{2} c^{2} b + \frac{3}{2} x^{8} e^{3} d c b^{2} + \frac{1}{8} x^{8} e^{4} b^{3} + \frac{3}{2} x^{8} e^{3} d c^{2} a + \frac{3}{4} x^{8} e^{4} c b a + \frac{1}{7} x^{7} d^{4} c^{3} + \frac{12}{7} x^{7} e d^{3} c^{2} b + \frac{18}{7} x^{7} e^{2} d^{2} c b^{2} + \frac{4}{7} x^{7} e^{3} d b^{3} + \frac{18}{7} x^{7} e^{2} d^{2} c^{2} a + \frac{24}{7} x^{7} e^{3} d c b a + \frac{3}{7} x^{7} e^{4} b^{2} a + \frac{3}{7} x^{7} e^{4} c a^{2} + \frac{1}{2} x^{6} d^{4} c^{2} b + 2 x^{6} e d^{3} c b^{2} + x^{6} e^{2} d^{2} b^{3} + 2 x^{6} e d^{3} c^{2} a + 6 x^{6} e^{2} d^{2} c b a + 2 x^{6} e^{3} d b^{2} a + 2 x^{6} e^{3} d c a^{2} + \frac{1}{2} x^{6} e^{4} b a^{2} + \frac{3}{5} x^{5} d^{4} c b^{2} + \frac{4}{5} x^{5} e d^{3} b^{3} + \frac{3}{5} x^{5} d^{4} c^{2} a + \frac{24}{5} x^{5} e d^{3} c b a + \frac{18}{5} x^{5} e^{2} d^{2} b^{2} a + \frac{18}{5} x^{5} e^{2} d^{2} c a^{2} + \frac{12}{5} x^{5} e^{3} d b a^{2} + \frac{1}{5} x^{5} e^{4} a^{3} + \frac{1}{4} x^{4} d^{4} b^{3} + \frac{3}{2} x^{4} d^{4} c b a + 3 x^{4} e d^{3} b^{2} a + 3 x^{4} e d^{3} c a^{2} + \frac{9}{2} x^{4} e^{2} d^{2} b a^{2} + x^{4} e^{3} d a^{3} + x^{3} d^{4} b^{2} a + x^{3} d^{4} c a^{2} + 4 x^{3} e d^{3} b a^{2} + 2 x^{3} e^{2} d^{2} a^{3} + \frac{3}{2} x^{2} d^{4} b a^{2} + 2 x^{2} e d^{3} a^{3} + x d^{4} a^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 0.149958, size = 620, normalized size = 2.28 \begin{align*} a^{3} d^{4} x + \frac{c^{3} e^{4} x^{11}}{11} + x^{10} \left (\frac{3 b c^{2} e^{4}}{10} + \frac{2 c^{3} d e^{3}}{5}\right ) + x^{9} \left (\frac{a c^{2} e^{4}}{3} + \frac{b^{2} c e^{4}}{3} + \frac{4 b c^{2} d e^{3}}{3} + \frac{2 c^{3} d^{2} e^{2}}{3}\right ) + x^{8} \left (\frac{3 a b c e^{4}}{4} + \frac{3 a c^{2} d e^{3}}{2} + \frac{b^{3} e^{4}}{8} + \frac{3 b^{2} c d e^{3}}{2} + \frac{9 b c^{2} d^{2} e^{2}}{4} + \frac{c^{3} d^{3} e}{2}\right ) + x^{7} \left (\frac{3 a^{2} c e^{4}}{7} + \frac{3 a b^{2} e^{4}}{7} + \frac{24 a b c d e^{3}}{7} + \frac{18 a c^{2} d^{2} e^{2}}{7} + \frac{4 b^{3} d e^{3}}{7} + \frac{18 b^{2} c d^{2} e^{2}}{7} + \frac{12 b c^{2} d^{3} e}{7} + \frac{c^{3} d^{4}}{7}\right ) + x^{6} \left (\frac{a^{2} b e^{4}}{2} + 2 a^{2} c d e^{3} + 2 a b^{2} d e^{3} + 6 a b c d^{2} e^{2} + 2 a c^{2} d^{3} e + b^{3} d^{2} e^{2} + 2 b^{2} c d^{3} e + \frac{b c^{2} d^{4}}{2}\right ) + x^{5} \left (\frac{a^{3} e^{4}}{5} + \frac{12 a^{2} b d e^{3}}{5} + \frac{18 a^{2} c d^{2} e^{2}}{5} + \frac{18 a b^{2} d^{2} e^{2}}{5} + \frac{24 a b c d^{3} e}{5} + \frac{3 a c^{2} d^{4}}{5} + \frac{4 b^{3} d^{3} e}{5} + \frac{3 b^{2} c d^{4}}{5}\right ) + x^{4} \left (a^{3} d e^{3} + \frac{9 a^{2} b d^{2} e^{2}}{2} + 3 a^{2} c d^{3} e + 3 a b^{2} d^{3} e + \frac{3 a b c d^{4}}{2} + \frac{b^{3} d^{4}}{4}\right ) + x^{3} \left (2 a^{3} d^{2} e^{2} + 4 a^{2} b d^{3} e + a^{2} c d^{4} + a b^{2} d^{4}\right ) + x^{2} \left (2 a^{3} d^{3} e + \frac{3 a^{2} b d^{4}}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.12191, size = 815, normalized size = 3. \begin{align*} \frac{1}{11} \, c^{3} x^{11} e^{4} + \frac{2}{5} \, c^{3} d x^{10} e^{3} + \frac{2}{3} \, c^{3} d^{2} x^{9} e^{2} + \frac{1}{2} \, c^{3} d^{3} x^{8} e + \frac{1}{7} \, c^{3} d^{4} x^{7} + \frac{3}{10} \, b c^{2} x^{10} e^{4} + \frac{4}{3} \, b c^{2} d x^{9} e^{3} + \frac{9}{4} \, b c^{2} d^{2} x^{8} e^{2} + \frac{12}{7} \, b c^{2} d^{3} x^{7} e + \frac{1}{2} \, b c^{2} d^{4} x^{6} + \frac{1}{3} \, b^{2} c x^{9} e^{4} + \frac{1}{3} \, a c^{2} x^{9} e^{4} + \frac{3}{2} \, b^{2} c d x^{8} e^{3} + \frac{3}{2} \, a c^{2} d x^{8} e^{3} + \frac{18}{7} \, b^{2} c d^{2} x^{7} e^{2} + \frac{18}{7} \, a c^{2} d^{2} x^{7} e^{2} + 2 \, b^{2} c d^{3} x^{6} e + 2 \, a c^{2} d^{3} x^{6} e + \frac{3}{5} \, b^{2} c d^{4} x^{5} + \frac{3}{5} \, a c^{2} d^{4} x^{5} + \frac{1}{8} \, b^{3} x^{8} e^{4} + \frac{3}{4} \, a b c x^{8} e^{4} + \frac{4}{7} \, b^{3} d x^{7} e^{3} + \frac{24}{7} \, a b c d x^{7} e^{3} + b^{3} d^{2} x^{6} e^{2} + 6 \, a b c d^{2} x^{6} e^{2} + \frac{4}{5} \, b^{3} d^{3} x^{5} e + \frac{24}{5} \, a b c d^{3} x^{5} e + \frac{1}{4} \, b^{3} d^{4} x^{4} + \frac{3}{2} \, a b c d^{4} x^{4} + \frac{3}{7} \, a b^{2} x^{7} e^{4} + \frac{3}{7} \, a^{2} c x^{7} e^{4} + 2 \, a b^{2} d x^{6} e^{3} + 2 \, a^{2} c d x^{6} e^{3} + \frac{18}{5} \, a b^{2} d^{2} x^{5} e^{2} + \frac{18}{5} \, a^{2} c d^{2} x^{5} e^{2} + 3 \, a b^{2} d^{3} x^{4} e + 3 \, a^{2} c d^{3} x^{4} e + a b^{2} d^{4} x^{3} + a^{2} c d^{4} x^{3} + \frac{1}{2} \, a^{2} b x^{6} e^{4} + \frac{12}{5} \, a^{2} b d x^{5} e^{3} + \frac{9}{2} \, a^{2} b d^{2} x^{4} e^{2} + 4 \, a^{2} b d^{3} x^{3} e + \frac{3}{2} \, a^{2} b d^{4} x^{2} + \frac{1}{5} \, a^{3} x^{5} e^{4} + a^{3} d x^{4} e^{3} + 2 \, a^{3} d^{2} x^{3} e^{2} + 2 \, a^{3} d^{3} x^{2} e + a^{3} d^{4} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]