Optimal. Leaf size=94 \[ a c^2 e x+\frac {1}{3} c (b c e+2 a d e+a c f) x^3+\frac {1}{5} (b c (2 d e+c f)+a d (d e+2 c f)) x^5+\frac {1}{7} d (b d e+2 b c f+a d f) x^7+\frac {1}{9} b d^2 f x^9 \]
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Rubi [A]
time = 0.05, antiderivative size = 94, 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}{7} d x^7 (a d f+2 b c f+b d e)+\frac {1}{5} x^5 (a d (2 c f+d e)+b c (c f+2 d e))+\frac {1}{3} c x^3 (a c f+2 a d e+b c e)+a c^2 e x+\frac {1}{9} b d^2 f x^9 \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 )^2 \left (e+f x^2\right ) \, dx &=\int \left (a c^2 e+c (b c e+2 a d e+a c f) x^2+(b c (2 d e+c f)+a d (d e+2 c f)) x^4+d (b d e+2 b c f+a d f) x^6+b d^2 f x^8\right ) \, dx\\ &=a c^2 e x+\frac {1}{3} c (b c e+2 a d e+a c f) x^3+\frac {1}{5} (b c (2 d e+c f)+a d (d e+2 c f)) x^5+\frac {1}{7} d (b d e+2 b c f+a d f) x^7+\frac {1}{9} b d^2 f x^9\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 96, normalized size = 1.02 \begin {gather*} a c^2 e x+\frac {1}{3} c (b c e+2 a d e+a c f) x^3+\frac {1}{5} \left (2 b c d e+a d^2 e+b c^2 f+2 a c d f\right ) x^5+\frac {1}{7} d (b d e+2 b c f+a d f) x^7+\frac {1}{9} b d^2 f x^9 \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.15, size = 101, normalized size = 1.07
method | result | size |
norman | \(\frac {b \,d^{2} f \,x^{9}}{9}+\left (\frac {1}{7} a \,d^{2} f +\frac {2}{7} b c d f +\frac {1}{7} b \,d^{2} e \right ) x^{7}+\left (\frac {2}{5} a c d f +\frac {1}{5} a \,d^{2} e +\frac {1}{5} b \,c^{2} f +\frac {2}{5} b c d e \right ) x^{5}+\left (\frac {1}{3} c^{2} a f +\frac {2}{3} a c d e +\frac {1}{3} b \,c^{2} e \right ) x^{3}+a \,c^{2} e x\) | \(100\) |
default | \(\frac {b \,d^{2} f \,x^{9}}{9}+\frac {\left (\left (a \,d^{2}+2 b c d \right ) f +b \,d^{2} e \right ) x^{7}}{7}+\frac {\left (\left (2 a c d +b \,c^{2}\right ) f +\left (a \,d^{2}+2 b c d \right ) e \right ) x^{5}}{5}+\frac {\left (c^{2} a f +\left (2 a c d +b \,c^{2}\right ) e \right ) x^{3}}{3}+a \,c^{2} e x\) | \(101\) |
gosper | \(\frac {1}{9} b \,d^{2} f \,x^{9}+\frac {1}{7} x^{7} a \,d^{2} f +\frac {2}{7} x^{7} b c d f +\frac {1}{7} x^{7} b \,d^{2} e +\frac {2}{5} x^{5} a c d f +\frac {1}{5} x^{5} a \,d^{2} e +\frac {1}{5} x^{5} b \,c^{2} f +\frac {2}{5} x^{5} b c d e +\frac {1}{3} x^{3} c^{2} a f +\frac {2}{3} x^{3} a c d e +\frac {1}{3} x^{3} b \,c^{2} e +a \,c^{2} e x\) | \(115\) |
risch | \(\frac {1}{9} b \,d^{2} f \,x^{9}+\frac {1}{7} x^{7} a \,d^{2} f +\frac {2}{7} x^{7} b c d f +\frac {1}{7} x^{7} b \,d^{2} e +\frac {2}{5} x^{5} a c d f +\frac {1}{5} x^{5} a \,d^{2} e +\frac {1}{5} x^{5} b \,c^{2} f +\frac {2}{5} x^{5} b c d e +\frac {1}{3} x^{3} c^{2} a f +\frac {2}{3} x^{3} a c d e +\frac {1}{3} x^{3} b \,c^{2} e +a \,c^{2} e x\) | \(115\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 104, normalized size = 1.11 \begin {gather*} \frac {1}{9} \, b d^{2} f x^{9} + \frac {1}{7} \, {\left (b d^{2} e + {\left (2 \, b c d + a d^{2}\right )} f\right )} x^{7} + \frac {1}{5} \, {\left (2 \, b c d e + a d^{2} e + {\left (b c^{2} + 2 \, a c d\right )} f\right )} x^{5} + a c^{2} x e + \frac {1}{3} \, {\left (a c^{2} f + b c^{2} e + 2 \, a c d e\right )} x^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.42, size = 108, normalized size = 1.15 \begin {gather*} \frac {1}{9} \, b d^{2} f x^{9} + \frac {1}{7} \, {\left (2 \, b c d + a d^{2}\right )} f x^{7} + \frac {1}{3} \, a c^{2} f x^{3} + \frac {1}{5} \, {\left (b c^{2} + 2 \, a c d\right )} f x^{5} + \frac {1}{105} \, {\left (15 \, b d^{2} x^{7} + 21 \, {\left (2 \, b c d + a d^{2}\right )} x^{5} + 105 \, a c^{2} x + 35 \, {\left (b c^{2} + 2 \, a c 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 = 0.01, size = 121, normalized size = 1.29 \begin {gather*} a c^{2} e x + \frac {b d^{2} f x^{9}}{9} + x^{7} \left (\frac {a d^{2} f}{7} + \frac {2 b c d f}{7} + \frac {b d^{2} e}{7}\right ) + x^{5} \cdot \left (\frac {2 a c d f}{5} + \frac {a d^{2} e}{5} + \frac {b c^{2} f}{5} + \frac {2 b c d e}{5}\right ) + x^{3} \left (\frac {a c^{2} f}{3} + \frac {2 a c d e}{3} + \frac {b c^{2} e}{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.24, size = 120, normalized size = 1.28 \begin {gather*} \frac {1}{9} \, b d^{2} f x^{9} + \frac {2}{7} \, b c d f x^{7} + \frac {1}{7} \, a d^{2} f x^{7} + \frac {1}{7} \, b d^{2} x^{7} e + \frac {1}{5} \, b c^{2} f x^{5} + \frac {2}{5} \, a c d f x^{5} + \frac {2}{5} \, b c d x^{5} e + \frac {1}{5} \, a d^{2} x^{5} e + \frac {1}{3} \, a c^{2} f x^{3} + \frac {1}{3} \, b c^{2} x^{3} e + \frac {2}{3} \, a c d x^{3} e + a c^{2} x e \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 99, normalized size = 1.05 \begin {gather*} x^5\,\left (\frac {a\,d^2\,e}{5}+\frac {b\,c^2\,f}{5}+\frac {2\,a\,c\,d\,f}{5}+\frac {2\,b\,c\,d\,e}{5}\right )+x^3\,\left (\frac {a\,c^2\,f}{3}+\frac {b\,c^2\,e}{3}+\frac {2\,a\,c\,d\,e}{3}\right )+x^7\,\left (\frac {a\,d^2\,f}{7}+\frac {b\,d^2\,e}{7}+\frac {2\,b\,c\,d\,f}{7}\right )+a\,c^2\,e\,x+\frac {b\,d^2\,f\,x^9}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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