Optimal. Leaf size=196 \[ a^2 d x+\frac {1}{2} a^2 e x^2+\frac {1}{7} x^7 \left (2 a c f+b^2 f+2 b c d\right )+\frac {1}{5} x^5 \left (2 a b f+2 a c d+b^2 d\right )+\frac {1}{8} x^8 \left (2 a c g+b^2 g+2 b c e\right )+\frac {1}{6} x^6 \left (2 a b g+2 a c e+b^2 e\right )+\frac {1}{3} a x^3 (a f+2 b d)+\frac {1}{4} a x^4 (a g+2 b e)+\frac {1}{9} c x^9 (2 b f+c d)+\frac {1}{10} c x^{10} (2 b g+c e)+\frac {1}{11} c^2 f x^{11}+\frac {1}{12} c^2 g x^{12} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.17, antiderivative size = 196, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {1671} \[ a^2 d x+\frac {1}{2} a^2 e x^2+\frac {1}{7} x^7 \left (2 a c f+b^2 f+2 b c d\right )+\frac {1}{5} x^5 \left (2 a b f+2 a c d+b^2 d\right )+\frac {1}{8} x^8 \left (2 a c g+b^2 g+2 b c e\right )+\frac {1}{6} x^6 \left (2 a b g+2 a c e+b^2 e\right )+\frac {1}{3} a x^3 (a f+2 b d)+\frac {1}{4} a x^4 (a g+2 b e)+\frac {1}{9} c x^9 (2 b f+c d)+\frac {1}{10} c x^{10} (2 b g+c e)+\frac {1}{11} c^2 f x^{11}+\frac {1}{12} c^2 g x^{12} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1671
Rubi steps
\begin {align*} \int \left (d+e x+f x^2+g x^3\right ) \left (a+b x^2+c x^4\right )^2 \, dx &=\int \left (a^2 d+a^2 e x+a (2 b d+a f) x^2+a (2 b e+a g) x^3+\left (b^2 d+2 a c d+2 a b f\right ) x^4+\left (b^2 e+2 a c e+2 a b g\right ) x^5+\left (2 b c d+b^2 f+2 a c f\right ) x^6+\left (2 b c e+b^2 g+2 a c g\right ) x^7+c (c d+2 b f) x^8+c (c e+2 b g) x^9+c^2 f x^{10}+c^2 g x^{11}\right ) \, dx\\ &=a^2 d x+\frac {1}{2} a^2 e x^2+\frac {1}{3} a (2 b d+a f) x^3+\frac {1}{4} a (2 b e+a g) x^4+\frac {1}{5} \left (b^2 d+2 a c d+2 a b f\right ) x^5+\frac {1}{6} \left (b^2 e+2 a c e+2 a b g\right ) x^6+\frac {1}{7} \left (2 b c d+b^2 f+2 a c f\right ) x^7+\frac {1}{8} \left (2 b c e+b^2 g+2 a c g\right ) x^8+\frac {1}{9} c (c d+2 b f) x^9+\frac {1}{10} c (c e+2 b g) x^{10}+\frac {1}{11} c^2 f x^{11}+\frac {1}{12} c^2 g x^{12}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.06, size = 196, normalized size = 1.00 \[ a^2 d x+\frac {1}{2} a^2 e x^2+\frac {1}{7} x^7 \left (2 a c f+b^2 f+2 b c d\right )+\frac {1}{5} x^5 \left (2 a b f+2 a c d+b^2 d\right )+\frac {1}{8} x^8 \left (2 a c g+b^2 g+2 b c e\right )+\frac {1}{6} x^6 \left (2 a b g+2 a c e+b^2 e\right )+\frac {1}{3} a x^3 (a f+2 b d)+\frac {1}{4} a x^4 (a g+2 b e)+\frac {1}{9} c x^9 (2 b f+c d)+\frac {1}{10} c x^{10} (2 b g+c e)+\frac {1}{11} c^2 f x^{11}+\frac {1}{12} c^2 g x^{12} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.78, size = 202, normalized size = 1.03 \[ \frac {1}{12} x^{12} g c^{2} + \frac {1}{11} x^{11} f c^{2} + \frac {1}{10} x^{10} e c^{2} + \frac {1}{5} x^{10} g c b + \frac {1}{9} x^{9} d c^{2} + \frac {2}{9} x^{9} f c b + \frac {1}{4} x^{8} e c b + \frac {1}{8} x^{8} g b^{2} + \frac {1}{4} x^{8} g c a + \frac {2}{7} x^{7} d c b + \frac {1}{7} x^{7} f b^{2} + \frac {2}{7} x^{7} f c a + \frac {1}{6} x^{6} e b^{2} + \frac {1}{3} x^{6} e c a + \frac {1}{3} x^{6} g b a + \frac {1}{5} x^{5} d b^{2} + \frac {2}{5} x^{5} d c a + \frac {2}{5} x^{5} f b a + \frac {1}{2} x^{4} e b a + \frac {1}{4} x^{4} g a^{2} + \frac {2}{3} x^{3} d b a + \frac {1}{3} x^{3} f a^{2} + \frac {1}{2} x^{2} e a^{2} + x d a^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.30, size = 208, normalized size = 1.06 \[ \frac {1}{12} \, c^{2} g x^{12} + \frac {1}{11} \, c^{2} f x^{11} + \frac {1}{5} \, b c g x^{10} + \frac {1}{10} \, c^{2} x^{10} e + \frac {1}{9} \, c^{2} d x^{9} + \frac {2}{9} \, b c f x^{9} + \frac {1}{8} \, b^{2} g x^{8} + \frac {1}{4} \, a c g x^{8} + \frac {1}{4} \, b c x^{8} e + \frac {2}{7} \, b c d x^{7} + \frac {1}{7} \, b^{2} f x^{7} + \frac {2}{7} \, a c f x^{7} + \frac {1}{3} \, a b g x^{6} + \frac {1}{6} \, b^{2} x^{6} e + \frac {1}{3} \, a c x^{6} e + \frac {1}{5} \, b^{2} d x^{5} + \frac {2}{5} \, a c d x^{5} + \frac {2}{5} \, a b f x^{5} + \frac {1}{4} \, a^{2} g x^{4} + \frac {1}{2} \, a b x^{4} e + \frac {2}{3} \, a b d x^{3} + \frac {1}{3} \, a^{2} f x^{3} + \frac {1}{2} \, a^{2} x^{2} e + a^{2} d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 183, normalized size = 0.93 \[ \frac {c^{2} g \,x^{12}}{12}+\frac {c^{2} f \,x^{11}}{11}+\frac {\left (2 g b c +e \,c^{2}\right ) x^{10}}{10}+\frac {\left (2 f b c +c^{2} d \right ) x^{9}}{9}+\frac {\left (2 b c e +\left (2 a c +b^{2}\right ) g \right ) x^{8}}{8}+\frac {\left (2 b c d +\left (2 a c +b^{2}\right ) f \right ) x^{7}}{7}+\frac {\left (2 a b g +\left (2 a c +b^{2}\right ) e \right ) x^{6}}{6}+\frac {a^{2} e \,x^{2}}{2}+\frac {\left (2 a b f +\left (2 a c +b^{2}\right ) d \right ) x^{5}}{5}+a^{2} d x +\frac {\left (g \,a^{2}+2 a b e \right ) x^{4}}{4}+\frac {\left (f \,a^{2}+2 a b d \right ) x^{3}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.40, size = 182, normalized size = 0.93 \[ \frac {1}{12} \, c^{2} g x^{12} + \frac {1}{11} \, c^{2} f x^{11} + \frac {1}{10} \, {\left (c^{2} e + 2 \, b c g\right )} x^{10} + \frac {1}{9} \, {\left (c^{2} d + 2 \, b c f\right )} x^{9} + \frac {1}{8} \, {\left (2 \, b c e + {\left (b^{2} + 2 \, a c\right )} g\right )} x^{8} + \frac {1}{7} \, {\left (2 \, b c d + {\left (b^{2} + 2 \, a c\right )} f\right )} x^{7} + \frac {1}{6} \, {\left (2 \, a b g + {\left (b^{2} + 2 \, a c\right )} e\right )} x^{6} + \frac {1}{5} \, {\left (2 \, a b f + {\left (b^{2} + 2 \, a c\right )} d\right )} x^{5} + \frac {1}{2} \, a^{2} e x^{2} + \frac {1}{4} \, {\left (2 \, a b e + a^{2} g\right )} x^{4} + a^{2} d x + \frac {1}{3} \, {\left (2 \, a b d + a^{2} f\right )} x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.72, size = 182, normalized size = 0.93 \[ x^5\,\left (\frac {d\,b^2}{5}+\frac {2\,a\,f\,b}{5}+\frac {2\,a\,c\,d}{5}\right )+x^6\,\left (\frac {e\,b^2}{6}+\frac {a\,g\,b}{3}+\frac {a\,c\,e}{3}\right )+x^7\,\left (\frac {f\,b^2}{7}+\frac {2\,c\,d\,b}{7}+\frac {2\,a\,c\,f}{7}\right )+x^8\,\left (\frac {g\,b^2}{8}+\frac {c\,e\,b}{4}+\frac {a\,c\,g}{4}\right )+x^3\,\left (\frac {f\,a^2}{3}+\frac {2\,b\,d\,a}{3}\right )+x^4\,\left (\frac {g\,a^2}{4}+\frac {b\,e\,a}{2}\right )+x^9\,\left (\frac {d\,c^2}{9}+\frac {2\,b\,f\,c}{9}\right )+x^{10}\,\left (\frac {e\,c^2}{10}+\frac {b\,g\,c}{5}\right )+\frac {a^2\,e\,x^2}{2}+\frac {c^2\,f\,x^{11}}{11}+\frac {c^2\,g\,x^{12}}{12}+a^2\,d\,x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.10, size = 209, normalized size = 1.07 \[ a^{2} d x + \frac {a^{2} e x^{2}}{2} + \frac {c^{2} f x^{11}}{11} + \frac {c^{2} g x^{12}}{12} + x^{10} \left (\frac {b c g}{5} + \frac {c^{2} e}{10}\right ) + x^{9} \left (\frac {2 b c f}{9} + \frac {c^{2} d}{9}\right ) + x^{8} \left (\frac {a c g}{4} + \frac {b^{2} g}{8} + \frac {b c e}{4}\right ) + x^{7} \left (\frac {2 a c f}{7} + \frac {b^{2} f}{7} + \frac {2 b c d}{7}\right ) + x^{6} \left (\frac {a b g}{3} + \frac {a c e}{3} + \frac {b^{2} e}{6}\right ) + x^{5} \left (\frac {2 a b f}{5} + \frac {2 a c d}{5} + \frac {b^{2} d}{5}\right ) + x^{4} \left (\frac {a^{2} g}{4} + \frac {a b e}{2}\right ) + x^{3} \left (\frac {a^{2} f}{3} + \frac {2 a b d}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________