Optimal. Leaf size=259 \[ a^3 d x+\frac {1}{2} a^3 e x^2+\frac {1}{7} x^7 \left (3 a^2 c f+3 a b^2 f+6 a b c d+b^3 d\right )+\frac {1}{3} a^2 x^3 (a f+3 b d)+\frac {3}{4} a^2 b e x^4+\frac {3}{11} c x^{11} \left (a c f+b^2 f+b c d\right )+\frac {3}{5} a x^5 \left (a b f+a c d+b^2 d\right )+\frac {3}{10} c e x^{10} \left (a c+b^2\right )+\frac {1}{8} b e x^8 \left (6 a c+b^2\right )+\frac {1}{2} a e x^6 \left (a c+b^2\right )+\frac {1}{9} x^9 \left (6 a b c f+3 a c^2 d+b^3 f+3 b^2 c d\right )+\frac {1}{13} c^2 x^{13} (3 b f+c d)+\frac {1}{4} b c^2 e x^{12}+\frac {1}{14} c^3 e x^{14}+\frac {1}{15} c^3 f x^{15} \]
________________________________________________________________________________________
Rubi [A] time = 0.33, antiderivative size = 259, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 63, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.016, Rules used = {1671} \begin {gather*} \frac {1}{7} x^7 \left (3 a^2 c f+3 a b^2 f+6 a b c d+b^3 d\right )+\frac {1}{3} a^2 x^3 (a f+3 b d)+\frac {3}{4} a^2 b e x^4+a^3 d x+\frac {1}{2} a^3 e x^2+\frac {1}{9} x^9 \left (6 a b c f+3 a c^2 d+3 b^2 c d+b^3 f\right )+\frac {3}{11} c x^{11} \left (a c f+b^2 f+b c d\right )+\frac {3}{5} a x^5 \left (a b f+a c d+b^2 d\right )+\frac {3}{10} c e x^{10} \left (a c+b^2\right )+\frac {1}{8} b e x^8 \left (6 a c+b^2\right )+\frac {1}{2} a e x^6 \left (a c+b^2\right )+\frac {1}{13} c^2 x^{13} (3 b f+c d)+\frac {1}{4} b c^2 e x^{12}+\frac {1}{14} c^3 e x^{14}+\frac {1}{15} c^3 f x^{15} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 1671
Rubi steps
\begin {align*} \int \left (a+b x^2+c x^4\right )^2 \left (a d+a e x+(b d+a f) x^2+b e x^3+(c d+b f) x^4+c e x^5+c f x^6\right ) \, dx &=\int \left (a^3 d+a^3 e x+a^2 (3 b d+a f) x^2+3 a^2 b e x^3+3 a \left (b^2 d+a c d+a b f\right ) x^4+3 a \left (b^2+a c\right ) e x^5+\left (b^3 d+6 a b c d+3 a b^2 f+3 a^2 c f\right ) x^6+b \left (b^2+6 a c\right ) e x^7+\left (3 b^2 c d+3 a c^2 d+b^3 f+6 a b c f\right ) x^8+3 c \left (b^2+a c\right ) e x^9+3 c \left (b c d+b^2 f+a c f\right ) x^{10}+3 b c^2 e x^{11}+c^2 (c d+3 b f) x^{12}+c^3 e x^{13}+c^3 f x^{14}\right ) \, dx\\ &=a^3 d x+\frac {1}{2} a^3 e x^2+\frac {1}{3} a^2 (3 b d+a f) x^3+\frac {3}{4} a^2 b e x^4+\frac {3}{5} a \left (b^2 d+a c d+a b f\right ) x^5+\frac {1}{2} a \left (b^2+a c\right ) e x^6+\frac {1}{7} \left (b^3 d+6 a b c d+3 a b^2 f+3 a^2 c f\right ) x^7+\frac {1}{8} b \left (b^2+6 a c\right ) e x^8+\frac {1}{9} \left (3 b^2 c d+3 a c^2 d+b^3 f+6 a b c f\right ) x^9+\frac {3}{10} c \left (b^2+a c\right ) e x^{10}+\frac {3}{11} c \left (b c d+b^2 f+a c f\right ) x^{11}+\frac {1}{4} b c^2 e x^{12}+\frac {1}{13} c^2 (c d+3 b f) x^{13}+\frac {1}{14} c^3 e x^{14}+\frac {1}{15} c^3 f x^{15}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.05, size = 259, normalized size = 1.00 \begin {gather*} a^3 d x+\frac {1}{2} a^3 e x^2+\frac {1}{7} x^7 \left (3 a^2 c f+3 a b^2 f+6 a b c d+b^3 d\right )+\frac {1}{3} a^2 x^3 (a f+3 b d)+\frac {3}{4} a^2 b e x^4+\frac {3}{11} c x^{11} \left (a c f+b^2 f+b c d\right )+\frac {3}{5} a x^5 \left (a b f+a c d+b^2 d\right )+\frac {3}{10} c e x^{10} \left (a c+b^2\right )+\frac {1}{8} b e x^8 \left (6 a c+b^2\right )+\frac {1}{2} a e x^6 \left (a c+b^2\right )+\frac {1}{9} x^9 \left (6 a b c f+3 a c^2 d+b^3 f+3 b^2 c d\right )+\frac {1}{13} c^2 x^{13} (3 b f+c d)+\frac {1}{4} b c^2 e x^{12}+\frac {1}{14} c^3 e x^{14}+\frac {1}{15} c^3 f x^{15} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a+b x^2+c x^4\right )^2 \left (a d+a e x+(b d+a f) x^2+b e x^3+(c d+b f) x^4+c e x^5+c f x^6\right ) \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.02, size = 285, normalized size = 1.10 \begin {gather*} \frac {1}{15} x^{15} f c^{3} + \frac {1}{14} x^{14} e c^{3} + \frac {1}{13} x^{13} d c^{3} + \frac {3}{13} x^{13} f c^{2} b + \frac {1}{4} x^{12} e c^{2} b + \frac {3}{11} x^{11} d c^{2} b + \frac {3}{11} x^{11} f c b^{2} + \frac {3}{11} x^{11} f c^{2} a + \frac {3}{10} x^{10} e c b^{2} + \frac {3}{10} x^{10} e c^{2} a + \frac {1}{3} x^{9} d c b^{2} + \frac {1}{9} x^{9} f b^{3} + \frac {1}{3} x^{9} d c^{2} a + \frac {2}{3} x^{9} f c b a + \frac {1}{8} x^{8} e b^{3} + \frac {3}{4} x^{8} e c b a + \frac {1}{7} x^{7} d b^{3} + \frac {6}{7} x^{7} d c b a + \frac {3}{7} x^{7} f b^{2} a + \frac {3}{7} x^{7} f c a^{2} + \frac {1}{2} x^{6} e b^{2} a + \frac {1}{2} x^{6} e c a^{2} + \frac {3}{5} x^{5} d b^{2} a + \frac {3}{5} x^{5} d c a^{2} + \frac {3}{5} x^{5} f b a^{2} + \frac {3}{4} x^{4} e b a^{2} + x^{3} d b a^{2} + \frac {1}{3} x^{3} f a^{3} + \frac {1}{2} x^{2} e a^{3} + x d a^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.31, size = 295, normalized size = 1.14 \begin {gather*} \frac {1}{15} \, c^{3} f x^{15} + \frac {1}{14} \, c^{3} x^{14} e + \frac {1}{13} \, c^{3} d x^{13} + \frac {3}{13} \, b c^{2} f x^{13} + \frac {1}{4} \, b c^{2} x^{12} e + \frac {3}{11} \, b c^{2} d x^{11} + \frac {3}{11} \, b^{2} c f x^{11} + \frac {3}{11} \, a c^{2} f x^{11} + \frac {3}{10} \, b^{2} c x^{10} e + \frac {3}{10} \, a c^{2} x^{10} e + \frac {1}{3} \, b^{2} c d x^{9} + \frac {1}{3} \, a c^{2} d x^{9} + \frac {1}{9} \, b^{3} f x^{9} + \frac {2}{3} \, a b c f x^{9} + \frac {1}{8} \, b^{3} x^{8} e + \frac {3}{4} \, a b c x^{8} e + \frac {1}{7} \, b^{3} d x^{7} + \frac {6}{7} \, a b c d x^{7} + \frac {3}{7} \, a b^{2} f x^{7} + \frac {3}{7} \, a^{2} c f x^{7} + \frac {1}{2} \, a b^{2} x^{6} e + \frac {1}{2} \, a^{2} c x^{6} e + \frac {3}{5} \, a b^{2} d x^{5} + \frac {3}{5} \, a^{2} c d x^{5} + \frac {3}{5} \, a^{2} b f x^{5} + \frac {3}{4} \, a^{2} b x^{4} e + a^{2} b d x^{3} + \frac {1}{3} \, a^{3} f x^{3} + \frac {1}{2} \, a^{3} x^{2} e + a^{3} d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 354, normalized size = 1.37 \begin {gather*} \frac {c^{3} f \,x^{15}}{15}+\frac {c^{3} e \,x^{14}}{14}+\frac {b \,c^{2} e \,x^{12}}{4}+\frac {\left (2 b \,c^{2} f +\left (b f +c d \right ) c^{2}\right ) x^{13}}{13}+\frac {\left (2 \left (b f +c d \right ) b c +\left (a f +b d \right ) c^{2}+\left (2 a c +b^{2}\right ) c f \right ) x^{11}}{11}+\frac {\left (a \,c^{2} e +2 b^{2} c e +\left (2 a c +b^{2}\right ) c e \right ) x^{10}}{10}+\frac {\left (2 a b c f +a \,c^{2} d +2 \left (a f +b d \right ) b c +\left (2 a c +b^{2}\right ) \left (b f +c d \right )\right ) x^{9}}{9}+\frac {3 a^{2} b e \,x^{4}}{4}+\frac {\left (4 a b c e +\left (2 a c +b^{2}\right ) b e \right ) x^{8}}{8}+\frac {\left (a^{2} c f +2 a b c d +2 \left (b f +c d \right ) a b +\left (2 a c +b^{2}\right ) \left (a f +b d \right )\right ) x^{7}}{7}+\frac {a^{3} e \,x^{2}}{2}+\frac {\left (a^{2} c e +2 a \,b^{2} e +\left (2 a c +b^{2}\right ) a e \right ) x^{6}}{6}+a^{3} d x +\frac {\left (\left (b f +c d \right ) a^{2}+2 \left (a f +b d \right ) a b +\left (2 a c +b^{2}\right ) a d \right ) x^{5}}{5}+\frac {\left (2 a^{2} b d +\left (a f +b d \right ) a^{2}\right ) x^{3}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.70, size = 251, normalized size = 0.97 \begin {gather*} \frac {1}{15} \, c^{3} f x^{15} + \frac {1}{14} \, c^{3} e x^{14} + \frac {1}{4} \, b c^{2} e x^{12} + \frac {1}{13} \, {\left (c^{3} d + 3 \, b c^{2} f\right )} x^{13} + \frac {3}{10} \, {\left (b^{2} c + a c^{2}\right )} e x^{10} + \frac {3}{11} \, {\left (b c^{2} d + {\left (b^{2} c + a c^{2}\right )} f\right )} x^{11} + \frac {1}{8} \, {\left (b^{3} + 6 \, a b c\right )} e x^{8} + \frac {1}{9} \, {\left (3 \, {\left (b^{2} c + a c^{2}\right )} d + {\left (b^{3} + 6 \, a b c\right )} f\right )} x^{9} + \frac {3}{4} \, a^{2} b e x^{4} + \frac {1}{2} \, {\left (a b^{2} + a^{2} c\right )} e x^{6} + \frac {1}{7} \, {\left ({\left (b^{3} + 6 \, a b c\right )} d + 3 \, {\left (a b^{2} + a^{2} c\right )} f\right )} x^{7} + \frac {1}{2} \, a^{3} e x^{2} + \frac {3}{5} \, {\left (a^{2} b f + {\left (a b^{2} + a^{2} c\right )} d\right )} x^{5} + a^{3} d x + \frac {1}{3} \, {\left (3 \, a^{2} b d + a^{3} f\right )} x^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.95, size = 246, normalized size = 0.95 \begin {gather*} x^3\,\left (\frac {f\,a^3}{3}+b\,d\,a^2\right )+x^{13}\,\left (\frac {d\,c^3}{13}+\frac {3\,b\,f\,c^2}{13}\right )+x^5\,\left (\frac {3\,f\,a^2\,b}{5}+\frac {3\,c\,d\,a^2}{5}+\frac {3\,d\,a\,b^2}{5}\right )+x^{11}\,\left (\frac {3\,f\,b^2\,c}{11}+\frac {3\,d\,b\,c^2}{11}+\frac {3\,a\,f\,c^2}{11}\right )+x^7\,\left (\frac {3\,c\,f\,a^2}{7}+\frac {3\,f\,a\,b^2}{7}+\frac {6\,c\,d\,a\,b}{7}+\frac {d\,b^3}{7}\right )+x^9\,\left (\frac {f\,b^3}{9}+\frac {d\,b^2\,c}{3}+\frac {2\,a\,f\,b\,c}{3}+\frac {a\,d\,c^2}{3}\right )+\frac {a^3\,e\,x^2}{2}+\frac {c^3\,e\,x^{14}}{14}+\frac {c^3\,f\,x^{15}}{15}+a^3\,d\,x+\frac {a\,e\,x^6\,\left (b^2+a\,c\right )}{2}+\frac {b\,e\,x^8\,\left (b^2+6\,a\,c\right )}{8}+\frac {3\,c\,e\,x^{10}\,\left (b^2+a\,c\right )}{10}+\frac {3\,a^2\,b\,e\,x^4}{4}+\frac {b\,c^2\,e\,x^{12}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.12, size = 309, normalized size = 1.19 \begin {gather*} a^{3} d x + \frac {a^{3} e x^{2}}{2} + \frac {3 a^{2} b e x^{4}}{4} + \frac {b c^{2} e x^{12}}{4} + \frac {c^{3} e x^{14}}{14} + \frac {c^{3} f x^{15}}{15} + x^{13} \left (\frac {3 b c^{2} f}{13} + \frac {c^{3} d}{13}\right ) + x^{11} \left (\frac {3 a c^{2} f}{11} + \frac {3 b^{2} c f}{11} + \frac {3 b c^{2} d}{11}\right ) + x^{10} \left (\frac {3 a c^{2} e}{10} + \frac {3 b^{2} c e}{10}\right ) + x^{9} \left (\frac {2 a b c f}{3} + \frac {a c^{2} d}{3} + \frac {b^{3} f}{9} + \frac {b^{2} c d}{3}\right ) + x^{8} \left (\frac {3 a b c e}{4} + \frac {b^{3} e}{8}\right ) + x^{7} \left (\frac {3 a^{2} c f}{7} + \frac {3 a b^{2} f}{7} + \frac {6 a b c d}{7} + \frac {b^{3} d}{7}\right ) + x^{6} \left (\frac {a^{2} c e}{2} + \frac {a b^{2} e}{2}\right ) + x^{5} \left (\frac {3 a^{2} b f}{5} + \frac {3 a^{2} c d}{5} + \frac {3 a b^{2} d}{5}\right ) + x^{3} \left (\frac {a^{3} f}{3} + a^{2} b d\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________