Optimal. Leaf size=161 \[ a^2 d x^3 \left (a e^2+c d^2\right )+\frac{1}{2} a^2 c e^3 x^6+a^3 d^3 x+\frac{1}{4} a^3 e^3 x^4+\frac{1}{7} c^2 d x^7 \left (9 a e^2+c d^2\right )+\frac{3}{8} a c^2 e^3 x^8+\frac{3}{5} a c d x^5 \left (3 a e^2+c d^2\right )+\frac{3 d^2 e \left (a+c x^2\right )^4}{8 c}+\frac{1}{3} c^3 d e^2 x^9+\frac{1}{10} c^3 e^3 x^{10} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.140378, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {696, 1810} \[ a^2 d x^3 \left (a e^2+c d^2\right )+\frac{1}{2} a^2 c e^3 x^6+a^3 d^3 x+\frac{1}{4} a^3 e^3 x^4+\frac{1}{7} c^2 d x^7 \left (9 a e^2+c d^2\right )+\frac{3}{8} a c^2 e^3 x^8+\frac{3}{5} a c d x^5 \left (3 a e^2+c d^2\right )+\frac{3 d^2 e \left (a+c x^2\right )^4}{8 c}+\frac{1}{3} c^3 d e^2 x^9+\frac{1}{10} c^3 e^3 x^{10} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 696
Rule 1810
Rubi steps
\begin{align*} \int (d+e x)^3 \left (a+c x^2\right )^3 \, dx &=\frac{3 d^2 e \left (a+c x^2\right )^4}{8 c}+\int \left (a+c x^2\right )^3 \left (-3 d^2 e x+(d+e x)^3\right ) \, dx\\ &=\frac{3 d^2 e \left (a+c x^2\right )^4}{8 c}+\int \left (a^3 d^3+3 a^2 d \left (c d^2+a e^2\right ) x^2+a^3 e^3 x^3+3 a c d \left (c d^2+3 a e^2\right ) x^4+3 a^2 c e^3 x^5+c^2 d \left (c d^2+9 a e^2\right ) x^6+3 a c^2 e^3 x^7+3 c^3 d e^2 x^8+c^3 e^3 x^9\right ) \, dx\\ &=a^3 d^3 x+a^2 d \left (c d^2+a e^2\right ) x^3+\frac{1}{4} a^3 e^3 x^4+\frac{3}{5} a c d \left (c d^2+3 a e^2\right ) x^5+\frac{1}{2} a^2 c e^3 x^6+\frac{1}{7} c^2 d \left (c d^2+9 a e^2\right ) x^7+\frac{3}{8} a c^2 e^3 x^8+\frac{1}{3} c^3 d e^2 x^9+\frac{1}{10} c^3 e^3 x^{10}+\frac{3 d^2 e \left (a+c x^2\right )^4}{8 c}\\ \end{align*}
Mathematica [A] time = 0.0475745, size = 155, normalized size = 0.96 \[ \frac{1}{840} x \left (42 a^2 c x^2 \left (45 d^2 e x+20 d^3+36 d e^2 x^2+10 e^3 x^3\right )+210 a^3 \left (6 d^2 e x+4 d^3+4 d e^2 x^2+e^3 x^3\right )+9 a c^2 x^4 \left (140 d^2 e x+56 d^3+120 d e^2 x^2+35 e^3 x^3\right )+c^3 x^6 \left (315 d^2 e x+120 d^3+280 d e^2 x^2+84 e^3 x^3\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.039, size = 189, normalized size = 1.2 \begin{align*}{\frac{{c}^{3}{e}^{3}{x}^{10}}{10}}+{\frac{{c}^{3}d{e}^{2}{x}^{9}}{3}}+{\frac{ \left ( 3\,{e}^{3}a{c}^{2}+3\,{d}^{2}e{c}^{3} \right ){x}^{8}}{8}}+{\frac{ \left ( 9\,d{e}^{2}a{c}^{2}+{c}^{3}{d}^{3} \right ){x}^{7}}{7}}+{\frac{ \left ( 3\,{a}^{2}c{e}^{3}+9\,{d}^{2}ea{c}^{2} \right ){x}^{6}}{6}}+{\frac{ \left ( 9\,d{e}^{2}{a}^{2}c+3\,{d}^{3}a{c}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ({e}^{3}{a}^{3}+9\,{d}^{2}e{a}^{2}c \right ){x}^{4}}{4}}+{\frac{ \left ( 3\,d{e}^{2}{a}^{3}+3\,{d}^{3}{a}^{2}c \right ){x}^{3}}{3}}+{\frac{3\,{d}^{2}e{a}^{3}{x}^{2}}{2}}+{a}^{3}{d}^{3}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.16501, size = 244, normalized size = 1.52 \begin{align*} \frac{1}{10} \, c^{3} e^{3} x^{10} + \frac{1}{3} \, c^{3} d e^{2} x^{9} + \frac{3}{8} \,{\left (c^{3} d^{2} e + a c^{2} e^{3}\right )} x^{8} + \frac{3}{2} \, a^{3} d^{2} e x^{2} + \frac{1}{7} \,{\left (c^{3} d^{3} + 9 \, a c^{2} d e^{2}\right )} x^{7} + a^{3} d^{3} x + \frac{1}{2} \,{\left (3 \, a c^{2} d^{2} e + a^{2} c e^{3}\right )} x^{6} + \frac{3}{5} \,{\left (a c^{2} d^{3} + 3 \, a^{2} c d e^{2}\right )} x^{5} + \frac{1}{4} \,{\left (9 \, a^{2} c d^{2} e + a^{3} e^{3}\right )} x^{4} +{\left (a^{2} c d^{3} + a^{3} d e^{2}\right )} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.70597, size = 414, normalized size = 2.57 \begin{align*} \frac{1}{10} x^{10} e^{3} c^{3} + \frac{1}{3} x^{9} e^{2} d c^{3} + \frac{3}{8} x^{8} e d^{2} c^{3} + \frac{3}{8} x^{8} e^{3} c^{2} a + \frac{1}{7} x^{7} d^{3} c^{3} + \frac{9}{7} x^{7} e^{2} d c^{2} a + \frac{3}{2} x^{6} e d^{2} c^{2} a + \frac{1}{2} x^{6} e^{3} c a^{2} + \frac{3}{5} x^{5} d^{3} c^{2} a + \frac{9}{5} x^{5} e^{2} d c a^{2} + \frac{9}{4} x^{4} e d^{2} c a^{2} + \frac{1}{4} x^{4} e^{3} a^{3} + x^{3} d^{3} c a^{2} + x^{3} e^{2} d a^{3} + \frac{3}{2} x^{2} e d^{2} a^{3} + x d^{3} a^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.122793, size = 202, normalized size = 1.25 \begin{align*} a^{3} d^{3} x + \frac{3 a^{3} d^{2} e x^{2}}{2} + \frac{c^{3} d e^{2} x^{9}}{3} + \frac{c^{3} e^{3} x^{10}}{10} + x^{8} \left (\frac{3 a c^{2} e^{3}}{8} + \frac{3 c^{3} d^{2} e}{8}\right ) + x^{7} \left (\frac{9 a c^{2} d e^{2}}{7} + \frac{c^{3} d^{3}}{7}\right ) + x^{6} \left (\frac{a^{2} c e^{3}}{2} + \frac{3 a c^{2} d^{2} e}{2}\right ) + x^{5} \left (\frac{9 a^{2} c d e^{2}}{5} + \frac{3 a c^{2} d^{3}}{5}\right ) + x^{4} \left (\frac{a^{3} e^{3}}{4} + \frac{9 a^{2} c d^{2} e}{4}\right ) + x^{3} \left (a^{3} d e^{2} + a^{2} c d^{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.31344, size = 248, normalized size = 1.54 \begin{align*} \frac{1}{10} \, c^{3} x^{10} e^{3} + \frac{1}{3} \, c^{3} d x^{9} e^{2} + \frac{3}{8} \, c^{3} d^{2} x^{8} e + \frac{1}{7} \, c^{3} d^{3} x^{7} + \frac{3}{8} \, a c^{2} x^{8} e^{3} + \frac{9}{7} \, a c^{2} d x^{7} e^{2} + \frac{3}{2} \, a c^{2} d^{2} x^{6} e + \frac{3}{5} \, a c^{2} d^{3} x^{5} + \frac{1}{2} \, a^{2} c x^{6} e^{3} + \frac{9}{5} \, a^{2} c d x^{5} e^{2} + \frac{9}{4} \, a^{2} c d^{2} x^{4} e + a^{2} c d^{3} x^{3} + \frac{1}{4} \, a^{3} x^{4} e^{3} + a^{3} d x^{3} e^{2} + \frac{3}{2} \, a^{3} d^{2} x^{2} e + a^{3} d^{3} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]