Optimal. Leaf size=117 \[ \frac {c (d+e x)^6 \left (a e^2+3 c d^2\right )}{3 e^5}-\frac {4 c d (d+e x)^5 \left (a e^2+c d^2\right )}{5 e^5}+\frac {(d+e x)^4 \left (a e^2+c d^2\right )^2}{4 e^5}+\frac {c^2 (d+e x)^8}{8 e^5}-\frac {4 c^2 d (d+e x)^7}{7 e^5} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.10, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {697} \[ \frac {c (d+e x)^6 \left (a e^2+3 c d^2\right )}{3 e^5}-\frac {4 c d (d+e x)^5 \left (a e^2+c d^2\right )}{5 e^5}+\frac {(d+e x)^4 \left (a e^2+c d^2\right )^2}{4 e^5}+\frac {c^2 (d+e x)^8}{8 e^5}-\frac {4 c^2 d (d+e x)^7}{7 e^5} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 697
Rubi steps
\begin {align*} \int (d+e x)^3 \left (a+c x^2\right )^2 \, dx &=\int \left (\frac {\left (c d^2+a e^2\right )^2 (d+e x)^3}{e^4}-\frac {4 c d \left (c d^2+a e^2\right ) (d+e x)^4}{e^4}+\frac {2 c \left (3 c d^2+a e^2\right ) (d+e x)^5}{e^4}-\frac {4 c^2 d (d+e x)^6}{e^4}+\frac {c^2 (d+e x)^7}{e^4}\right ) \, dx\\ &=\frac {\left (c d^2+a e^2\right )^2 (d+e x)^4}{4 e^5}-\frac {4 c d \left (c d^2+a e^2\right ) (d+e x)^5}{5 e^5}+\frac {c \left (3 c d^2+a e^2\right ) (d+e x)^6}{3 e^5}-\frac {4 c^2 d (d+e x)^7}{7 e^5}+\frac {c^2 (d+e x)^8}{8 e^5}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 117, normalized size = 1.00 \[ \frac {1}{4} a^2 x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )+\frac {1}{30} a c x^3 \left (20 d^3+45 d^2 e x+36 d e^2 x^2+10 e^3 x^3\right )+\frac {1}{280} c^2 x^5 \left (56 d^3+140 d^2 e x+120 d e^2 x^2+35 e^3 x^3\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.54, size = 131, normalized size = 1.12 \[ \frac {1}{8} x^{8} e^{3} c^{2} + \frac {3}{7} x^{7} e^{2} d c^{2} + \frac {1}{2} x^{6} e d^{2} c^{2} + \frac {1}{3} x^{6} e^{3} c a + \frac {1}{5} x^{5} d^{3} c^{2} + \frac {6}{5} x^{5} e^{2} d c a + \frac {3}{2} x^{4} e d^{2} c a + \frac {1}{4} x^{4} e^{3} a^{2} + \frac {2}{3} x^{3} d^{3} c a + x^{3} e^{2} d a^{2} + \frac {3}{2} x^{2} e d^{2} a^{2} + x d^{3} a^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.16, size = 128, normalized size = 1.09 \[ \frac {1}{8} \, c^{2} x^{8} e^{3} + \frac {3}{7} \, c^{2} d x^{7} e^{2} + \frac {1}{2} \, c^{2} d^{2} x^{6} e + \frac {1}{5} \, c^{2} d^{3} x^{5} + \frac {1}{3} \, a c x^{6} e^{3} + \frac {6}{5} \, a c d x^{5} e^{2} + \frac {3}{2} \, a c d^{2} x^{4} e + \frac {2}{3} \, a c d^{3} x^{3} + \frac {1}{4} \, a^{2} x^{4} e^{3} + a^{2} d x^{3} e^{2} + \frac {3}{2} \, a^{2} d^{2} x^{2} e + a^{2} d^{3} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.04, size = 131, normalized size = 1.12 \[ \frac {c^{2} e^{3} x^{8}}{8}+\frac {3 c^{2} d \,e^{2} x^{7}}{7}+\frac {3 a^{2} d^{2} e \,x^{2}}{2}+a^{2} d^{3} x +\frac {\left (2 e^{3} a c +3 d^{2} e \,c^{2}\right ) x^{6}}{6}+\frac {\left (6 d \,e^{2} a c +c^{2} d^{3}\right ) x^{5}}{5}+\frac {\left (a^{2} e^{3}+6 d^{2} e a c \right ) x^{4}}{4}+\frac {\left (3 d \,e^{2} a^{2}+2 d^{3} a c \right ) x^{3}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.35, size = 130, normalized size = 1.11 \[ \frac {1}{8} \, c^{2} e^{3} x^{8} + \frac {3}{7} \, c^{2} d e^{2} x^{7} + \frac {3}{2} \, a^{2} d^{2} e x^{2} + \frac {1}{6} \, {\left (3 \, c^{2} d^{2} e + 2 \, a c e^{3}\right )} x^{6} + a^{2} d^{3} x + \frac {1}{5} \, {\left (c^{2} d^{3} + 6 \, a c d e^{2}\right )} x^{5} + \frac {1}{4} \, {\left (6 \, a c d^{2} e + a^{2} e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (2 \, a c d^{3} + 3 \, a^{2} d e^{2}\right )} x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.05, size = 127, normalized size = 1.09 \[ x^3\,\left (a^2\,d\,e^2+\frac {2\,c\,a\,d^3}{3}\right )+x^4\,\left (\frac {a^2\,e^3}{4}+\frac {3\,c\,a\,d^2\,e}{2}\right )+x^5\,\left (\frac {c^2\,d^3}{5}+\frac {6\,a\,c\,d\,e^2}{5}\right )+x^6\,\left (\frac {c^2\,d^2\,e}{2}+\frac {a\,c\,e^3}{3}\right )+a^2\,d^3\,x+\frac {c^2\,e^3\,x^8}{8}+\frac {3\,a^2\,d^2\,e\,x^2}{2}+\frac {3\,c^2\,d\,e^2\,x^7}{7} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.09, size = 141, normalized size = 1.21 \[ a^{2} d^{3} x + \frac {3 a^{2} d^{2} e x^{2}}{2} + \frac {3 c^{2} d e^{2} x^{7}}{7} + \frac {c^{2} e^{3} x^{8}}{8} + x^{6} \left (\frac {a c e^{3}}{3} + \frac {c^{2} d^{2} e}{2}\right ) + x^{5} \left (\frac {6 a c d e^{2}}{5} + \frac {c^{2} d^{3}}{5}\right ) + x^{4} \left (\frac {a^{2} e^{3}}{4} + \frac {3 a c d^{2} e}{2}\right ) + x^{3} \left (a^{2} d e^{2} + \frac {2 a c d^{3}}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________