Optimal. Leaf size=130 \[ a c^3 e x+\frac {1}{3} c^2 (b c e+3 a d e+a c f) x^3+\frac {1}{5} c (3 a d (d e+c f)+b c (3 d e+c f)) x^5+\frac {1}{7} d (3 b c (d e+c f)+a d (d e+3 c f)) x^7+\frac {1}{9} d^2 (b d e+3 b c f+a d f) x^9+\frac {1}{11} b d^3 f x^{11} \]
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Rubi [A]
time = 0.09, antiderivative size = 130, 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} c^2 x^3 (a c f+3 a d e+b c e)+\frac {1}{9} d^2 x^9 (a d f+3 b c f+b d e)+\frac {1}{7} d x^7 (a d (3 c f+d e)+3 b c (c f+d e))+\frac {1}{5} c x^5 (3 a d (c f+d e)+b c (c f+3 d e))+a c^3 e x+\frac {1}{11} b d^3 f x^{11} \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 )^3 \left (e+f x^2\right ) \, dx &=\int \left (a c^3 e+c^2 (b c e+3 a d e+a c f) x^2+c (3 a d (d e+c f)+b c (3 d e+c f)) x^4+d (3 b c (d e+c f)+a d (d e+3 c f)) x^6+d^2 (b d e+3 b c f+a d f) x^8+b d^3 f x^{10}\right ) \, dx\\ &=a c^3 e x+\frac {1}{3} c^2 (b c e+3 a d e+a c f) x^3+\frac {1}{5} c (3 a d (d e+c f)+b c (3 d e+c f)) x^5+\frac {1}{7} d (3 b c (d e+c f)+a d (d e+3 c f)) x^7+\frac {1}{9} d^2 (b d e+3 b c f+a d f) x^9+\frac {1}{11} b d^3 f x^{11}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 130, normalized size = 1.00 \begin {gather*} a c^3 e x+\frac {1}{3} c^2 (b c e+3 a d e+a c f) x^3+\frac {1}{5} c (3 a d (d e+c f)+b c (3 d e+c f)) x^5+\frac {1}{7} d (3 b c (d e+c f)+a d (d e+3 c f)) x^7+\frac {1}{9} d^2 (b d e+3 b c f+a d f) x^9+\frac {1}{11} b d^3 f x^{11} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.15, size = 149, normalized size = 1.15
method | result | size |
norman | \(\frac {b \,d^{3} f \,x^{11}}{11}+\left (\frac {1}{9} a \,d^{3} f +\frac {1}{3} b c \,d^{2} f +\frac {1}{9} b \,d^{3} e \right ) x^{9}+\left (\frac {3}{7} a c \,d^{2} f +\frac {1}{7} a \,d^{3} e +\frac {3}{7} b \,c^{2} d f +\frac {3}{7} b c \,d^{2} e \right ) x^{7}+\left (\frac {3}{5} a \,c^{2} d f +\frac {3}{5} a c \,d^{2} e +\frac {1}{5} b \,c^{3} f +\frac {3}{5} b \,c^{2} d e \right ) x^{5}+\left (\frac {1}{3} c^{3} a f +a \,c^{2} d e +\frac {1}{3} b \,c^{3} e \right ) x^{3}+a \,c^{3} e x\) | \(144\) |
default | \(\frac {b \,d^{3} f \,x^{11}}{11}+\frac {\left (\left (a \,d^{3}+3 b c \,d^{2}\right ) f +b \,d^{3} e \right ) x^{9}}{9}+\frac {\left (\left (3 a c \,d^{2}+3 b \,c^{2} d \right ) f +\left (a \,d^{3}+3 b c \,d^{2}\right ) e \right ) x^{7}}{7}+\frac {\left (\left (3 a \,c^{2} d +b \,c^{3}\right ) f +\left (3 a c \,d^{2}+3 b \,c^{2} d \right ) e \right ) x^{5}}{5}+\frac {\left (c^{3} a f +\left (3 a \,c^{2} d +b \,c^{3}\right ) e \right ) x^{3}}{3}+a \,c^{3} e x\) | \(149\) |
gosper | \(\frac {1}{11} b \,d^{3} f \,x^{11}+\frac {1}{9} x^{9} a \,d^{3} f +\frac {1}{3} x^{9} b c \,d^{2} f +\frac {1}{9} x^{9} b \,d^{3} e +\frac {3}{7} x^{7} a c \,d^{2} f +\frac {1}{7} x^{7} a \,d^{3} e +\frac {3}{7} x^{7} b \,c^{2} d f +\frac {3}{7} x^{7} b c \,d^{2} e +\frac {3}{5} x^{5} a \,c^{2} d f +\frac {3}{5} x^{5} a c \,d^{2} e +\frac {1}{5} x^{5} b \,c^{3} f +\frac {3}{5} x^{5} b \,c^{2} d e +\frac {1}{3} x^{3} c^{3} a f +x^{3} a \,c^{2} d e +\frac {1}{3} x^{3} b \,c^{3} e +a \,c^{3} e x\) | \(166\) |
risch | \(\frac {1}{11} b \,d^{3} f \,x^{11}+\frac {1}{9} x^{9} a \,d^{3} f +\frac {1}{3} x^{9} b c \,d^{2} f +\frac {1}{9} x^{9} b \,d^{3} e +\frac {3}{7} x^{7} a c \,d^{2} f +\frac {1}{7} x^{7} a \,d^{3} e +\frac {3}{7} x^{7} b \,c^{2} d f +\frac {3}{7} x^{7} b c \,d^{2} e +\frac {3}{5} x^{5} a \,c^{2} d f +\frac {3}{5} x^{5} a c \,d^{2} e +\frac {1}{5} x^{5} b \,c^{3} f +\frac {3}{5} x^{5} b \,c^{2} d e +\frac {1}{3} x^{3} c^{3} a f +x^{3} a \,c^{2} d e +\frac {1}{3} x^{3} b \,c^{3} e +a \,c^{3} e x\) | \(166\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 152, normalized size = 1.17 \begin {gather*} \frac {1}{11} \, b d^{3} f x^{11} + \frac {1}{9} \, {\left (b d^{3} e + {\left (3 \, b c d^{2} + a d^{3}\right )} f\right )} x^{9} + \frac {1}{7} \, {\left (3 \, b c d^{2} e + a d^{3} e + 3 \, {\left (b c^{2} d + a c d^{2}\right )} f\right )} x^{7} + \frac {1}{5} \, {\left (3 \, b c^{2} d e + 3 \, a c d^{2} e + {\left (b c^{3} + 3 \, a c^{2} d\right )} f\right )} x^{5} + a c^{3} x e + \frac {1}{3} \, {\left (a c^{3} f + b c^{3} e + 3 \, a c^{2} d e\right )} x^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.13, size = 153, normalized size = 1.18 \begin {gather*} \frac {1}{11} \, b d^{3} f x^{11} + \frac {1}{9} \, {\left (3 \, b c d^{2} + a d^{3}\right )} f x^{9} + \frac {3}{7} \, {\left (b c^{2} d + a c d^{2}\right )} f x^{7} + \frac {1}{3} \, a c^{3} f x^{3} + \frac {1}{5} \, {\left (b c^{3} + 3 \, a c^{2} d\right )} f x^{5} + \frac {1}{315} \, {\left (35 \, b d^{3} x^{9} + 45 \, {\left (3 \, b c d^{2} + a d^{3}\right )} x^{7} + 189 \, {\left (b c^{2} d + a c d^{2}\right )} x^{5} + 315 \, a c^{3} x + 105 \, {\left (b c^{3} + 3 \, a c^{2} d\right )} x^{3}\right )} e \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.02, size = 173, normalized size = 1.33 \begin {gather*} a c^{3} e x + \frac {b d^{3} f x^{11}}{11} + x^{9} \left (\frac {a d^{3} f}{9} + \frac {b c d^{2} f}{3} + \frac {b d^{3} e}{9}\right ) + x^{7} \cdot \left (\frac {3 a c d^{2} f}{7} + \frac {a d^{3} e}{7} + \frac {3 b c^{2} d f}{7} + \frac {3 b c d^{2} e}{7}\right ) + x^{5} \cdot \left (\frac {3 a c^{2} d f}{5} + \frac {3 a c d^{2} e}{5} + \frac {b c^{3} f}{5} + \frac {3 b c^{2} d e}{5}\right ) + x^{3} \left (\frac {a c^{3} f}{3} + a c^{2} d e + \frac {b c^{3} e}{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.77, size = 173, normalized size = 1.33 \begin {gather*} \frac {1}{11} \, b d^{3} f x^{11} + \frac {1}{3} \, b c d^{2} f x^{9} + \frac {1}{9} \, a d^{3} f x^{9} + \frac {1}{9} \, b d^{3} x^{9} e + \frac {3}{7} \, b c^{2} d f x^{7} + \frac {3}{7} \, a c d^{2} f x^{7} + \frac {3}{7} \, b c d^{2} x^{7} e + \frac {1}{7} \, a d^{3} x^{7} e + \frac {1}{5} \, b c^{3} f x^{5} + \frac {3}{5} \, a c^{2} d f x^{5} + \frac {3}{5} \, b c^{2} d x^{5} e + \frac {3}{5} \, a c d^{2} x^{5} e + \frac {1}{3} \, a c^{3} f x^{3} + \frac {1}{3} \, b c^{3} x^{3} e + a c^{2} d x^{3} e + a c^{3} x e \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.81, size = 143, normalized size = 1.10 \begin {gather*} x^5\,\left (\frac {b\,c^3\,f}{5}+\frac {3\,a\,c\,d^2\,e}{5}+\frac {3\,a\,c^2\,d\,f}{5}+\frac {3\,b\,c^2\,d\,e}{5}\right )+x^7\,\left (\frac {a\,d^3\,e}{7}+\frac {3\,a\,c\,d^2\,f}{7}+\frac {3\,b\,c\,d^2\,e}{7}+\frac {3\,b\,c^2\,d\,f}{7}\right )+x^3\,\left (\frac {a\,c^3\,f}{3}+\frac {b\,c^3\,e}{3}+a\,c^2\,d\,e\right )+x^9\,\left (\frac {a\,d^3\,f}{9}+\frac {b\,d^3\,e}{9}+\frac {b\,c\,d^2\,f}{3}\right )+a\,c^3\,e\,x+\frac {b\,d^3\,f\,x^{11}}{11} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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