Optimal. Leaf size=212 \[ \frac{1}{5} a^2 x^5 (a f+3 b c)+\frac{1}{7} a^2 x^7 (a h+3 b e)+\frac{1}{3} a^2 b g x^9+\frac{1}{2} a^3 c x^2+\frac{1}{4} a^3 e x^4+\frac{1}{6} a^3 g x^6+\frac{1}{11} b^2 x^{11} (3 a f+b c)+\frac{1}{13} b^2 x^{13} (3 a h+b e)+\frac{1}{4} a b^2 g x^{12}+\frac{3}{8} a b x^8 (a f+b c)+\frac{d \left (a+b x^3\right )^4}{12 b}+\frac{3}{10} a b x^{10} (a h+b e)+\frac{1}{14} b^3 f x^{14}+\frac{1}{15} b^3 g x^{15}+\frac{1}{16} b^3 h x^{16} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.177018, antiderivative size = 212, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {1582, 1850} \[ \frac{1}{5} a^2 x^5 (a f+3 b c)+\frac{1}{7} a^2 x^7 (a h+3 b e)+\frac{1}{3} a^2 b g x^9+\frac{1}{2} a^3 c x^2+\frac{1}{4} a^3 e x^4+\frac{1}{6} a^3 g x^6+\frac{1}{11} b^2 x^{11} (3 a f+b c)+\frac{1}{13} b^2 x^{13} (3 a h+b e)+\frac{1}{4} a b^2 g x^{12}+\frac{3}{8} a b x^8 (a f+b c)+\frac{d \left (a+b x^3\right )^4}{12 b}+\frac{3}{10} a b x^{10} (a h+b e)+\frac{1}{14} b^3 f x^{14}+\frac{1}{15} b^3 g x^{15}+\frac{1}{16} b^3 h x^{16} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1582
Rule 1850
Rubi steps
\begin{align*} \int x \left (a+b x^3\right )^3 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx &=\frac{d \left (a+b x^3\right )^4}{12 b}+\int \left (a+b x^3\right )^3 \left (-d x^2+x \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )\right ) \, dx\\ &=\frac{d \left (a+b x^3\right )^4}{12 b}+\int \left (a^3 c x+a^3 e x^3+a^2 (3 b c+a f) x^4+a^3 g x^5+a^2 (3 b e+a h) x^6+3 a b (b c+a f) x^7+3 a^2 b g x^8+3 a b (b e+a h) x^9+b^2 (b c+3 a f) x^{10}+3 a b^2 g x^{11}+b^2 (b e+3 a h) x^{12}+b^3 f x^{13}+b^3 g x^{14}+b^3 h x^{15}\right ) \, dx\\ &=\frac{1}{2} a^3 c x^2+\frac{1}{4} a^3 e x^4+\frac{1}{5} a^2 (3 b c+a f) x^5+\frac{1}{6} a^3 g x^6+\frac{1}{7} a^2 (3 b e+a h) x^7+\frac{3}{8} a b (b c+a f) x^8+\frac{1}{3} a^2 b g x^9+\frac{3}{10} a b (b e+a h) x^{10}+\frac{1}{11} b^2 (b c+3 a f) x^{11}+\frac{1}{4} a b^2 g x^{12}+\frac{1}{13} b^2 (b e+3 a h) x^{13}+\frac{1}{14} b^3 f x^{14}+\frac{1}{15} b^3 g x^{15}+\frac{1}{16} b^3 h x^{16}+\frac{d \left (a+b x^3\right )^4}{12 b}\\ \end{align*}
Mathematica [A] time = 0.0334001, size = 223, normalized size = 1.05 \[ \frac{1}{5} a^2 x^5 (a f+3 b c)+\frac{1}{6} a^2 x^6 (a g+3 b d)+\frac{1}{7} a^2 x^7 (a h+3 b e)+\frac{1}{2} a^3 c x^2+\frac{1}{3} a^3 d x^3+\frac{1}{4} a^3 e x^4+\frac{1}{11} b^2 x^{11} (3 a f+b c)+\frac{1}{12} b^2 x^{12} (3 a g+b d)+\frac{1}{13} b^2 x^{13} (3 a h+b e)+\frac{3}{8} a b x^8 (a f+b c)+\frac{1}{3} a b x^9 (a g+b d)+\frac{3}{10} a b x^{10} (a h+b e)+\frac{1}{14} b^3 f x^{14}+\frac{1}{15} b^3 g x^{15}+\frac{1}{16} b^3 h x^{16} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0., size = 224, normalized size = 1.1 \begin{align*}{\frac{{b}^{3}h{x}^{16}}{16}}+{\frac{{b}^{3}g{x}^{15}}{15}}+{\frac{{b}^{3}f{x}^{14}}{14}}+{\frac{ \left ( 3\,{b}^{2}ah+{b}^{3}e \right ){x}^{13}}{13}}+{\frac{ \left ( 3\,{b}^{2}ag+{b}^{3}d \right ){x}^{12}}{12}}+{\frac{ \left ( 3\,{b}^{2}af+{b}^{3}c \right ){x}^{11}}{11}}+{\frac{ \left ( 3\,b{a}^{2}h+3\,ae{b}^{2} \right ){x}^{10}}{10}}+{\frac{ \left ( 3\,b{a}^{2}g+3\,a{b}^{2}d \right ){x}^{9}}{9}}+{\frac{ \left ( 3\,b{a}^{2}f+3\,ac{b}^{2} \right ){x}^{8}}{8}}+{\frac{ \left ({a}^{3}h+3\,{a}^{2}be \right ){x}^{7}}{7}}+{\frac{ \left ({a}^{3}g+3\,{a}^{2}bd \right ){x}^{6}}{6}}+{\frac{ \left ({a}^{3}f+3\,b{a}^{2}c \right ){x}^{5}}{5}}+{\frac{{a}^{3}e{x}^{4}}{4}}+{\frac{{a}^{3}d{x}^{3}}{3}}+{\frac{{a}^{3}c{x}^{2}}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.954929, size = 293, normalized size = 1.38 \begin{align*} \frac{1}{16} \, b^{3} h x^{16} + \frac{1}{15} \, b^{3} g x^{15} + \frac{1}{14} \, b^{3} f x^{14} + \frac{1}{13} \,{\left (b^{3} e + 3 \, a b^{2} h\right )} x^{13} + \frac{1}{12} \,{\left (b^{3} d + 3 \, a b^{2} g\right )} x^{12} + \frac{1}{11} \,{\left (b^{3} c + 3 \, a b^{2} f\right )} x^{11} + \frac{3}{10} \,{\left (a b^{2} e + a^{2} b h\right )} x^{10} + \frac{1}{3} \,{\left (a b^{2} d + a^{2} b g\right )} x^{9} + \frac{3}{8} \,{\left (a b^{2} c + a^{2} b f\right )} x^{8} + \frac{1}{4} \, a^{3} e x^{4} + \frac{1}{7} \,{\left (3 \, a^{2} b e + a^{3} h\right )} x^{7} + \frac{1}{3} \, a^{3} d x^{3} + \frac{1}{6} \,{\left (3 \, a^{2} b d + a^{3} g\right )} x^{6} + \frac{1}{2} \, a^{3} c x^{2} + \frac{1}{5} \,{\left (3 \, a^{2} b c + a^{3} f\right )} x^{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0.876503, size = 578, normalized size = 2.73 \begin{align*} \frac{1}{16} x^{16} h b^{3} + \frac{1}{15} x^{15} g b^{3} + \frac{1}{14} x^{14} f b^{3} + \frac{1}{13} x^{13} e b^{3} + \frac{3}{13} x^{13} h b^{2} a + \frac{1}{12} x^{12} d b^{3} + \frac{1}{4} x^{12} g b^{2} a + \frac{1}{11} x^{11} c b^{3} + \frac{3}{11} x^{11} f b^{2} a + \frac{3}{10} x^{10} e b^{2} a + \frac{3}{10} x^{10} h b a^{2} + \frac{1}{3} x^{9} d b^{2} a + \frac{1}{3} x^{9} g b a^{2} + \frac{3}{8} x^{8} c b^{2} a + \frac{3}{8} x^{8} f b a^{2} + \frac{3}{7} x^{7} e b a^{2} + \frac{1}{7} x^{7} h a^{3} + \frac{1}{2} x^{6} d b a^{2} + \frac{1}{6} x^{6} g a^{3} + \frac{3}{5} x^{5} c b a^{2} + \frac{1}{5} x^{5} f a^{3} + \frac{1}{4} x^{4} e a^{3} + \frac{1}{3} x^{3} d a^{3} + \frac{1}{2} x^{2} c a^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.099559, size = 246, normalized size = 1.16 \begin{align*} \frac{a^{3} c x^{2}}{2} + \frac{a^{3} d x^{3}}{3} + \frac{a^{3} e x^{4}}{4} + \frac{b^{3} f x^{14}}{14} + \frac{b^{3} g x^{15}}{15} + \frac{b^{3} h x^{16}}{16} + x^{13} \left (\frac{3 a b^{2} h}{13} + \frac{b^{3} e}{13}\right ) + x^{12} \left (\frac{a b^{2} g}{4} + \frac{b^{3} d}{12}\right ) + x^{11} \left (\frac{3 a b^{2} f}{11} + \frac{b^{3} c}{11}\right ) + x^{10} \left (\frac{3 a^{2} b h}{10} + \frac{3 a b^{2} e}{10}\right ) + x^{9} \left (\frac{a^{2} b g}{3} + \frac{a b^{2} d}{3}\right ) + x^{8} \left (\frac{3 a^{2} b f}{8} + \frac{3 a b^{2} c}{8}\right ) + x^{7} \left (\frac{a^{3} h}{7} + \frac{3 a^{2} b e}{7}\right ) + x^{6} \left (\frac{a^{3} g}{6} + \frac{a^{2} b d}{2}\right ) + x^{5} \left (\frac{a^{3} f}{5} + \frac{3 a^{2} b c}{5}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.0781, size = 315, normalized size = 1.49 \begin{align*} \frac{1}{16} \, b^{3} h x^{16} + \frac{1}{15} \, b^{3} g x^{15} + \frac{1}{14} \, b^{3} f x^{14} + \frac{3}{13} \, a b^{2} h x^{13} + \frac{1}{13} \, b^{3} x^{13} e + \frac{1}{12} \, b^{3} d x^{12} + \frac{1}{4} \, a b^{2} g x^{12} + \frac{1}{11} \, b^{3} c x^{11} + \frac{3}{11} \, a b^{2} f x^{11} + \frac{3}{10} \, a^{2} b h x^{10} + \frac{3}{10} \, a b^{2} x^{10} e + \frac{1}{3} \, a b^{2} d x^{9} + \frac{1}{3} \, a^{2} b g x^{9} + \frac{3}{8} \, a b^{2} c x^{8} + \frac{3}{8} \, a^{2} b f x^{8} + \frac{1}{7} \, a^{3} h x^{7} + \frac{3}{7} \, a^{2} b x^{7} e + \frac{1}{2} \, a^{2} b d x^{6} + \frac{1}{6} \, a^{3} g x^{6} + \frac{3}{5} \, a^{2} b c x^{5} + \frac{1}{5} \, a^{3} f x^{5} + \frac{1}{4} \, a^{3} x^{4} e + \frac{1}{3} \, a^{3} d x^{3} + \frac{1}{2} \, a^{3} c x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]