Optimal. Leaf size=172 \[ a c e^4 x+\frac {1}{3} e^3 (b c e+a d e+4 a c f) x^3+\frac {1}{5} e^2 (2 a f (2 d e+3 c f)+b e (d e+4 c f)) x^5+\frac {2}{7} e f (a f (3 d e+2 c f)+b e (2 d e+3 c f)) x^7+\frac {1}{9} f^2 (a f (4 d e+c f)+2 b e (3 d e+2 c f)) x^9+\frac {1}{11} f^3 (4 b d e+b c f+a d f) x^{11}+\frac {1}{13} b d f^4 x^{13} \]
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Rubi [A]
time = 0.13, antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {535}
\begin {gather*} \frac {1}{3} e^3 x^3 (4 a c f+a d e+b c e)+\frac {1}{5} e^2 x^5 (2 a f (3 c f+2 d e)+b e (4 c f+d e))+\frac {1}{11} f^3 x^{11} (a d f+b c f+4 b d e)+\frac {1}{9} f^2 x^9 (a f (c f+4 d e)+2 b e (2 c f+3 d e))+\frac {2}{7} e f x^7 (a f (2 c f+3 d e)+b e (3 c f+2 d e))+a c e^4 x+\frac {1}{13} b d f^4 x^{13} \end {gather*}
Antiderivative was successfully verified.
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Rule 535
Rubi steps
\begin {align*} \int \left (a+b x^2\right ) \left (c+d x^2\right ) \left (e+f x^2\right )^4 \, dx &=\int \left (a c e^4+e^3 (b c e+a d e+4 a c f) x^2+e^2 (2 a f (2 d e+3 c f)+b e (d e+4 c f)) x^4+2 e f (a f (3 d e+2 c f)+b e (2 d e+3 c f)) x^6+f^2 (a f (4 d e+c f)+2 b e (3 d e+2 c f)) x^8+f^3 (4 b d e+b c f+a d f) x^{10}+b d f^4 x^{12}\right ) \, dx\\ &=a c e^4 x+\frac {1}{3} e^3 (b c e+a d e+4 a c f) x^3+\frac {1}{5} e^2 (2 a f (2 d e+3 c f)+b e (d e+4 c f)) x^5+\frac {2}{7} e f (a f (3 d e+2 c f)+b e (2 d e+3 c f)) x^7+\frac {1}{9} f^2 (a f (4 d e+c f)+2 b e (3 d e+2 c f)) x^9+\frac {1}{11} f^3 (4 b d e+b c f+a d f) x^{11}+\frac {1}{13} b d f^4 x^{13}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 172, normalized size = 1.00 \begin {gather*} a c e^4 x+\frac {1}{3} e^3 (b c e+a d e+4 a c f) x^3+\frac {1}{5} e^2 (2 a f (2 d e+3 c f)+b e (d e+4 c f)) x^5+\frac {2}{7} e f (a f (3 d e+2 c f)+b e (2 d e+3 c f)) x^7+\frac {1}{9} f^2 (a f (4 d e+c f)+2 b e (3 d e+2 c f)) x^9+\frac {1}{11} f^3 (4 b d e+b c f+a d f) x^{11}+\frac {1}{13} b d f^4 x^{13} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.15, size = 176, normalized size = 1.02
method | result | size |
default | \(\frac {b d \,f^{4} x^{13}}{13}+\frac {\left (\left (a d +b c \right ) f^{4}+4 b d e \,f^{3}\right ) x^{11}}{11}+\frac {\left (a c \,f^{4}+4 \left (a d +b c \right ) e \,f^{3}+6 b d \,e^{2} f^{2}\right ) x^{9}}{9}+\frac {\left (4 a c e \,f^{3}+6 \left (a d +b c \right ) e^{2} f^{2}+4 b d \,e^{3} f \right ) x^{7}}{7}+\frac {\left (6 a c \,e^{2} f^{2}+4 \left (a d +b c \right ) e^{3} f +b d \,e^{4}\right ) x^{5}}{5}+\frac {\left (4 a c \,e^{3} f +\left (a d +b c \right ) e^{4}\right ) x^{3}}{3}+a c \,e^{4} x\) | \(176\) |
norman | \(\frac {b d \,f^{4} x^{13}}{13}+\left (\frac {1}{11} a d \,f^{4}+\frac {1}{11} b c \,f^{4}+\frac {4}{11} b d e \,f^{3}\right ) x^{11}+\left (\frac {1}{9} a c \,f^{4}+\frac {4}{9} a d e \,f^{3}+\frac {4}{9} b c e \,f^{3}+\frac {2}{3} b d \,e^{2} f^{2}\right ) x^{9}+\left (\frac {4}{7} a c e \,f^{3}+\frac {6}{7} a d \,e^{2} f^{2}+\frac {6}{7} b c \,e^{2} f^{2}+\frac {4}{7} b d \,e^{3} f \right ) x^{7}+\left (\frac {6}{5} a c \,e^{2} f^{2}+\frac {4}{5} a d \,e^{3} f +\frac {4}{5} b c \,e^{3} f +\frac {1}{5} b d \,e^{4}\right ) x^{5}+\left (\frac {4}{3} a c \,e^{3} f +\frac {1}{3} a d \,e^{4}+\frac {1}{3} b c \,e^{4}\right ) x^{3}+a c \,e^{4} x\) | \(190\) |
gosper | \(\frac {1}{13} b d \,f^{4} x^{13}+\frac {1}{11} x^{11} a d \,f^{4}+\frac {1}{11} x^{11} b c \,f^{4}+\frac {4}{11} x^{11} b d e \,f^{3}+\frac {1}{9} x^{9} a c \,f^{4}+\frac {4}{9} x^{9} a d e \,f^{3}+\frac {4}{9} x^{9} b c e \,f^{3}+\frac {2}{3} x^{9} b d \,e^{2} f^{2}+\frac {4}{7} x^{7} a c e \,f^{3}+\frac {6}{7} x^{7} a d \,e^{2} f^{2}+\frac {6}{7} x^{7} b c \,e^{2} f^{2}+\frac {4}{7} x^{7} b d \,e^{3} f +\frac {6}{5} x^{5} a c \,e^{2} f^{2}+\frac {4}{5} x^{5} a d \,e^{3} f +\frac {4}{5} x^{5} b c \,e^{3} f +\frac {1}{5} x^{5} b d \,e^{4}+\frac {4}{3} x^{3} a c \,e^{3} f +\frac {1}{3} x^{3} a d \,e^{4}+\frac {1}{3} x^{3} b c \,e^{4}+a c \,e^{4} x\) | \(219\) |
risch | \(\frac {1}{13} b d \,f^{4} x^{13}+\frac {1}{11} x^{11} a d \,f^{4}+\frac {1}{11} x^{11} b c \,f^{4}+\frac {4}{11} x^{11} b d e \,f^{3}+\frac {1}{9} x^{9} a c \,f^{4}+\frac {4}{9} x^{9} a d e \,f^{3}+\frac {4}{9} x^{9} b c e \,f^{3}+\frac {2}{3} x^{9} b d \,e^{2} f^{2}+\frac {4}{7} x^{7} a c e \,f^{3}+\frac {6}{7} x^{7} a d \,e^{2} f^{2}+\frac {6}{7} x^{7} b c \,e^{2} f^{2}+\frac {4}{7} x^{7} b d \,e^{3} f +\frac {6}{5} x^{5} a c \,e^{2} f^{2}+\frac {4}{5} x^{5} a d \,e^{3} f +\frac {4}{5} x^{5} b c \,e^{3} f +\frac {1}{5} x^{5} b d \,e^{4}+\frac {4}{3} x^{3} a c \,e^{3} f +\frac {1}{3} x^{3} a d \,e^{4}+\frac {1}{3} x^{3} b c \,e^{4}+a c \,e^{4} x\) | \(219\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 175, normalized size = 1.02 \begin {gather*} \frac {1}{13} \, b d f^{4} x^{13} + \frac {1}{11} \, {\left (4 \, b d f^{3} e + {\left (b c + a d\right )} f^{4}\right )} x^{11} + \frac {1}{9} \, {\left (a c f^{4} + 6 \, b d f^{2} e^{2} + 4 \, {\left (b c e + a d e\right )} f^{3}\right )} x^{9} + \frac {2}{7} \, {\left (2 \, a c f^{3} e + 2 \, b d f e^{3} + 3 \, {\left (b c e^{2} + a d e^{2}\right )} f^{2}\right )} x^{7} + \frac {1}{5} \, {\left (6 \, a c f^{2} e^{2} + b d e^{4} + 4 \, {\left (b c e^{3} + a d e^{3}\right )} f\right )} x^{5} + \frac {1}{3} \, {\left (4 \, a c f e^{3} + b c e^{4} + a d e^{4}\right )} x^{3} + a c x e^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.97, size = 179, normalized size = 1.04 \begin {gather*} \frac {1}{13} \, b d f^{4} x^{13} + \frac {1}{11} \, {\left (b c + a d\right )} f^{4} x^{11} + \frac {1}{9} \, a c f^{4} x^{9} + \frac {1}{15} \, {\left (3 \, b d x^{5} + 5 \, {\left (b c + a d\right )} x^{3} + 15 \, a c x\right )} e^{4} + \frac {4}{105} \, {\left (15 \, b d f x^{7} + 21 \, {\left (b c + a d\right )} f x^{5} + 35 \, a c f x^{3}\right )} e^{3} + \frac {2}{105} \, {\left (35 \, b d f^{2} x^{9} + 45 \, {\left (b c + a d\right )} f^{2} x^{7} + 63 \, a c f^{2} x^{5}\right )} e^{2} + \frac {4}{693} \, {\left (63 \, b d f^{3} x^{11} + 77 \, {\left (b c + a d\right )} f^{3} x^{9} + 99 \, a c f^{3} x^{7}\right )} e \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.02, size = 236, normalized size = 1.37 \begin {gather*} a c e^{4} x + \frac {b d f^{4} x^{13}}{13} + x^{11} \left (\frac {a d f^{4}}{11} + \frac {b c f^{4}}{11} + \frac {4 b d e f^{3}}{11}\right ) + x^{9} \left (\frac {a c f^{4}}{9} + \frac {4 a d e f^{3}}{9} + \frac {4 b c e f^{3}}{9} + \frac {2 b d e^{2} f^{2}}{3}\right ) + x^{7} \cdot \left (\frac {4 a c e f^{3}}{7} + \frac {6 a d e^{2} f^{2}}{7} + \frac {6 b c e^{2} f^{2}}{7} + \frac {4 b d e^{3} f}{7}\right ) + x^{5} \cdot \left (\frac {6 a c e^{2} f^{2}}{5} + \frac {4 a d e^{3} f}{5} + \frac {4 b c e^{3} f}{5} + \frac {b d e^{4}}{5}\right ) + x^{3} \cdot \left (\frac {4 a c e^{3} f}{3} + \frac {a d e^{4}}{3} + \frac {b c e^{4}}{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.43, size = 210, normalized size = 1.22 \begin {gather*} \frac {1}{13} \, b d f^{4} x^{13} + \frac {1}{11} \, b c f^{4} x^{11} + \frac {1}{11} \, a d f^{4} x^{11} + \frac {4}{11} \, b d f^{3} x^{11} e + \frac {1}{9} \, a c f^{4} x^{9} + \frac {4}{9} \, b c f^{3} x^{9} e + \frac {4}{9} \, a d f^{3} x^{9} e + \frac {2}{3} \, b d f^{2} x^{9} e^{2} + \frac {4}{7} \, a c f^{3} x^{7} e + \frac {6}{7} \, b c f^{2} x^{7} e^{2} + \frac {6}{7} \, a d f^{2} x^{7} e^{2} + \frac {4}{7} \, b d f x^{7} e^{3} + \frac {6}{5} \, a c f^{2} x^{5} e^{2} + \frac {4}{5} \, b c f x^{5} e^{3} + \frac {4}{5} \, a d f x^{5} e^{3} + \frac {1}{5} \, b d x^{5} e^{4} + \frac {4}{3} \, a c f x^{3} e^{3} + \frac {1}{3} \, b c x^{3} e^{4} + \frac {1}{3} \, a d x^{3} e^{4} + a c x e^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.10, size = 182, normalized size = 1.06 \begin {gather*} x^3\,\left (\frac {a\,d\,e^4}{3}+\frac {b\,c\,e^4}{3}+\frac {4\,a\,c\,e^3\,f}{3}\right )+x^{11}\,\left (\frac {a\,d\,f^4}{11}+\frac {b\,c\,f^4}{11}+\frac {4\,b\,d\,e\,f^3}{11}\right )+x^5\,\left (\frac {b\,d\,e^4}{5}+\frac {4\,a\,d\,e^3\,f}{5}+\frac {4\,b\,c\,e^3\,f}{5}+\frac {6\,a\,c\,e^2\,f^2}{5}\right )+x^9\,\left (\frac {a\,c\,f^4}{9}+\frac {4\,a\,d\,e\,f^3}{9}+\frac {4\,b\,c\,e\,f^3}{9}+\frac {2\,b\,d\,e^2\,f^2}{3}\right )+\frac {2\,e\,f\,x^7\,\left (2\,a\,c\,f^2+2\,b\,d\,e^2+3\,a\,d\,e\,f+3\,b\,c\,e\,f\right )}{7}+a\,c\,e^4\,x+\frac {b\,d\,f^4\,x^{13}}{13} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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