Optimal. Leaf size=226 \[ a c^2 e^3 x+\frac {1}{3} c e^2 (b c e+2 a d e+3 a c f) x^3+\frac {1}{5} e \left (b c e (2 d e+3 c f)+a \left (d^2 e^2+6 c d e f+3 c^2 f^2\right )\right ) x^5+\frac {1}{7} \left (a f \left (3 d^2 e^2+6 c d e f+c^2 f^2\right )+b e \left (d^2 e^2+6 c d e f+3 c^2 f^2\right )\right ) x^7+\frac {1}{9} f \left (a d f (3 d e+2 c f)+b \left (3 d^2 e^2+6 c d e f+c^2 f^2\right )\right ) x^9+\frac {1}{11} d f^2 (3 b d e+2 b c f+a d f) x^{11}+\frac {1}{13} b d^2 f^3 x^{13} \]
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Rubi [A]
time = 0.14, antiderivative size = 226, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {535}
\begin {gather*} \frac {1}{9} f x^9 \left (a d f (2 c f+3 d e)+b \left (c^2 f^2+6 c d e f+3 d^2 e^2\right )\right )+\frac {1}{7} x^7 \left (a f \left (c^2 f^2+6 c d e f+3 d^2 e^2\right )+b e \left (3 c^2 f^2+6 c d e f+d^2 e^2\right )\right )+\frac {1}{5} e x^5 \left (a \left (3 c^2 f^2+6 c d e f+d^2 e^2\right )+b c e (3 c f+2 d e)\right )+\frac {1}{3} c e^2 x^3 (3 a c f+2 a d e+b c e)+\frac {1}{11} d f^2 x^{11} (a d f+2 b c f+3 b d e)+a c^2 e^3 x+\frac {1}{13} b d^2 f^3 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 )^2 \left (e+f x^2\right )^3 \, dx &=\int \left (a c^2 e^3+c e^2 (b c e+2 a d e+3 a c f) x^2+e \left (b c e (2 d e+3 c f)+a \left (d^2 e^2+6 c d e f+3 c^2 f^2\right )\right ) x^4+\left (a f \left (3 d^2 e^2+6 c d e f+c^2 f^2\right )+b e \left (d^2 e^2+6 c d e f+3 c^2 f^2\right )\right ) x^6+f \left (a d f (3 d e+2 c f)+b \left (3 d^2 e^2+6 c d e f+c^2 f^2\right )\right ) x^8+d f^2 (3 b d e+2 b c f+a d f) x^{10}+b d^2 f^3 x^{12}\right ) \, dx\\ &=a c^2 e^3 x+\frac {1}{3} c e^2 (b c e+2 a d e+3 a c f) x^3+\frac {1}{5} e \left (b c e (2 d e+3 c f)+a \left (d^2 e^2+6 c d e f+3 c^2 f^2\right )\right ) x^5+\frac {1}{7} \left (a f \left (3 d^2 e^2+6 c d e f+c^2 f^2\right )+b e \left (d^2 e^2+6 c d e f+3 c^2 f^2\right )\right ) x^7+\frac {1}{9} f \left (a d f (3 d e+2 c f)+b \left (3 d^2 e^2+6 c d e f+c^2 f^2\right )\right ) x^9+\frac {1}{11} d f^2 (3 b d e+2 b c f+a d f) x^{11}+\frac {1}{13} b d^2 f^3 x^{13}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 226, normalized size = 1.00 \begin {gather*} a c^2 e^3 x+\frac {1}{3} c e^2 (b c e+2 a d e+3 a c f) x^3+\frac {1}{5} e \left (b c e (2 d e+3 c f)+a \left (d^2 e^2+6 c d e f+3 c^2 f^2\right )\right ) x^5+\frac {1}{7} \left (a f \left (3 d^2 e^2+6 c d e f+c^2 f^2\right )+b e \left (d^2 e^2+6 c d e f+3 c^2 f^2\right )\right ) x^7+\frac {1}{9} f \left (a d f (3 d e+2 c f)+b \left (3 d^2 e^2+6 c d e f+c^2 f^2\right )\right ) x^9+\frac {1}{11} d f^2 (3 b d e+2 b c f+a d f) x^{11}+\frac {1}{13} b d^2 f^3 x^{13} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.17, size = 237, normalized size = 1.05
method | result | size |
default | \(\frac {b \,d^{2} f^{3} x^{13}}{13}+\frac {\left (\left (a \,d^{2}+2 b c d \right ) f^{3}+3 b \,d^{2} e \,f^{2}\right ) x^{11}}{11}+\frac {\left (\left (2 a c d +b \,c^{2}\right ) f^{3}+3 \left (a \,d^{2}+2 b c d \right ) e \,f^{2}+3 b \,d^{2} e^{2} f \right ) x^{9}}{9}+\frac {\left (c^{2} a \,f^{3}+3 \left (2 a c d +b \,c^{2}\right ) e \,f^{2}+3 \left (a \,d^{2}+2 b c d \right ) e^{2} f +b \,d^{2} e^{3}\right ) x^{7}}{7}+\frac {\left (3 c^{2} a e \,f^{2}+3 \left (2 a c d +b \,c^{2}\right ) e^{2} f +\left (a \,d^{2}+2 b c d \right ) e^{3}\right ) x^{5}}{5}+\frac {\left (3 c^{2} a \,e^{2} f +\left (2 a c d +b \,c^{2}\right ) e^{3}\right ) x^{3}}{3}+a \,c^{2} e^{3} x\) | \(237\) |
norman | \(\frac {b \,d^{2} f^{3} x^{13}}{13}+\left (\frac {1}{11} a \,d^{2} f^{3}+\frac {2}{11} b c d \,f^{3}+\frac {3}{11} b \,d^{2} e \,f^{2}\right ) x^{11}+\left (\frac {2}{9} a c d \,f^{3}+\frac {1}{3} a \,d^{2} e \,f^{2}+\frac {1}{9} b \,c^{2} f^{3}+\frac {2}{3} b c d e \,f^{2}+\frac {1}{3} b \,d^{2} e^{2} f \right ) x^{9}+\left (\frac {1}{7} c^{2} a \,f^{3}+\frac {6}{7} a c d e \,f^{2}+\frac {3}{7} a \,d^{2} e^{2} f +\frac {3}{7} b \,c^{2} e \,f^{2}+\frac {6}{7} b c d \,e^{2} f +\frac {1}{7} b \,d^{2} e^{3}\right ) x^{7}+\left (\frac {3}{5} c^{2} a e \,f^{2}+\frac {6}{5} a c d \,e^{2} f +\frac {1}{5} a \,d^{2} e^{3}+\frac {3}{5} b \,c^{2} e^{2} f +\frac {2}{5} b c d \,e^{3}\right ) x^{5}+\left (c^{2} a \,e^{2} f +\frac {2}{3} a c d \,e^{3}+\frac {1}{3} b \,c^{2} e^{3}\right ) x^{3}+a \,c^{2} e^{3} x\) | \(249\) |
gosper | \(\frac {1}{13} b \,d^{2} f^{3} x^{13}+\frac {1}{11} x^{11} a \,d^{2} f^{3}+\frac {2}{11} x^{11} b c d \,f^{3}+\frac {3}{11} x^{11} b \,d^{2} e \,f^{2}+\frac {2}{9} x^{9} a c d \,f^{3}+\frac {1}{3} x^{9} a \,d^{2} e \,f^{2}+\frac {1}{9} x^{9} b \,c^{2} f^{3}+\frac {2}{3} x^{9} b c d e \,f^{2}+\frac {1}{3} x^{9} b \,d^{2} e^{2} f +\frac {1}{7} x^{7} c^{2} a \,f^{3}+\frac {6}{7} x^{7} a c d e \,f^{2}+\frac {3}{7} x^{7} a \,d^{2} e^{2} f +\frac {3}{7} x^{7} b \,c^{2} e \,f^{2}+\frac {6}{7} x^{7} b c d \,e^{2} f +\frac {1}{7} x^{7} b \,d^{2} e^{3}+\frac {3}{5} x^{5} c^{2} a e \,f^{2}+\frac {6}{5} x^{5} a c d \,e^{2} f +\frac {1}{5} x^{5} a \,d^{2} e^{3}+\frac {3}{5} x^{5} b \,c^{2} e^{2} f +\frac {2}{5} x^{5} b c d \,e^{3}+x^{3} c^{2} a \,e^{2} f +\frac {2}{3} x^{3} a c d \,e^{3}+\frac {1}{3} x^{3} b \,c^{2} e^{3}+a \,c^{2} e^{3} x\) | \(290\) |
risch | \(\frac {1}{13} b \,d^{2} f^{3} x^{13}+\frac {1}{11} x^{11} a \,d^{2} f^{3}+\frac {2}{11} x^{11} b c d \,f^{3}+\frac {3}{11} x^{11} b \,d^{2} e \,f^{2}+\frac {2}{9} x^{9} a c d \,f^{3}+\frac {1}{3} x^{9} a \,d^{2} e \,f^{2}+\frac {1}{9} x^{9} b \,c^{2} f^{3}+\frac {2}{3} x^{9} b c d e \,f^{2}+\frac {1}{3} x^{9} b \,d^{2} e^{2} f +\frac {1}{7} x^{7} c^{2} a \,f^{3}+\frac {6}{7} x^{7} a c d e \,f^{2}+\frac {3}{7} x^{7} a \,d^{2} e^{2} f +\frac {3}{7} x^{7} b \,c^{2} e \,f^{2}+\frac {6}{7} x^{7} b c d \,e^{2} f +\frac {1}{7} x^{7} b \,d^{2} e^{3}+\frac {3}{5} x^{5} c^{2} a e \,f^{2}+\frac {6}{5} x^{5} a c d \,e^{2} f +\frac {1}{5} x^{5} a \,d^{2} e^{3}+\frac {3}{5} x^{5} b \,c^{2} e^{2} f +\frac {2}{5} x^{5} b c d \,e^{3}+x^{3} c^{2} a \,e^{2} f +\frac {2}{3} x^{3} a c d \,e^{3}+\frac {1}{3} x^{3} b \,c^{2} e^{3}+a \,c^{2} e^{3} x\) | \(290\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 240, normalized size = 1.06 \begin {gather*} \frac {1}{13} \, b d^{2} f^{3} x^{13} + \frac {1}{11} \, {\left (3 \, b d^{2} f^{2} e + {\left (2 \, b c d + a d^{2}\right )} f^{3}\right )} x^{11} + \frac {1}{9} \, {\left (3 \, b d^{2} f e^{2} + {\left (b c^{2} + 2 \, a c d\right )} f^{3} + 3 \, {\left (2 \, b c d e + a d^{2} e\right )} f^{2}\right )} x^{9} + \frac {1}{7} \, {\left (a c^{2} f^{3} + b d^{2} e^{3} + 3 \, {\left (b c^{2} e + 2 \, a c d e\right )} f^{2} + 3 \, {\left (2 \, b c d e^{2} + a d^{2} e^{2}\right )} f\right )} x^{7} + \frac {1}{5} \, {\left (3 \, a c^{2} f^{2} e + 2 \, b c d e^{3} + a d^{2} e^{3} + 3 \, {\left (b c^{2} e^{2} + 2 \, a c d e^{2}\right )} f\right )} x^{5} + a c^{2} x e^{3} + \frac {1}{3} \, {\left (3 \, a c^{2} f e^{2} + b c^{2} e^{3} + 2 \, a c d e^{3}\right )} x^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.88, size = 242, normalized size = 1.07 \begin {gather*} \frac {1}{13} \, b d^{2} f^{3} x^{13} + \frac {1}{11} \, {\left (2 \, b c d + a d^{2}\right )} f^{3} x^{11} + \frac {1}{7} \, a c^{2} f^{3} x^{7} + \frac {1}{9} \, {\left (b c^{2} + 2 \, a c d\right )} f^{3} x^{9} + \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^{3} + \frac {1}{105} \, {\left (35 \, b d^{2} f x^{9} + 45 \, {\left (2 \, b c d + a d^{2}\right )} f x^{7} + 105 \, a c^{2} f x^{3} + 63 \, {\left (b c^{2} + 2 \, a c d\right )} f x^{5}\right )} e^{2} + \frac {1}{1155} \, {\left (315 \, b d^{2} f^{2} x^{11} + 385 \, {\left (2 \, b c d + a d^{2}\right )} f^{2} x^{9} + 693 \, a c^{2} f^{2} x^{5} + 495 \, {\left (b c^{2} + 2 \, a c d\right )} f^{2} x^{7}\right )} e \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 304, normalized size = 1.35 \begin {gather*} a c^{2} e^{3} x + \frac {b d^{2} f^{3} x^{13}}{13} + x^{11} \left (\frac {a d^{2} f^{3}}{11} + \frac {2 b c d f^{3}}{11} + \frac {3 b d^{2} e f^{2}}{11}\right ) + x^{9} \cdot \left (\frac {2 a c d f^{3}}{9} + \frac {a d^{2} e f^{2}}{3} + \frac {b c^{2} f^{3}}{9} + \frac {2 b c d e f^{2}}{3} + \frac {b d^{2} e^{2} f}{3}\right ) + x^{7} \left (\frac {a c^{2} f^{3}}{7} + \frac {6 a c d e f^{2}}{7} + \frac {3 a d^{2} e^{2} f}{7} + \frac {3 b c^{2} e f^{2}}{7} + \frac {6 b c d e^{2} f}{7} + \frac {b d^{2} e^{3}}{7}\right ) + x^{5} \cdot \left (\frac {3 a c^{2} e f^{2}}{5} + \frac {6 a c d e^{2} f}{5} + \frac {a d^{2} e^{3}}{5} + \frac {3 b c^{2} e^{2} f}{5} + \frac {2 b c d e^{3}}{5}\right ) + x^{3} \left (a c^{2} e^{2} f + \frac {2 a c d e^{3}}{3} + \frac {b c^{2} e^{3}}{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.98, size = 283, normalized size = 1.25 \begin {gather*} \frac {1}{13} \, b d^{2} f^{3} x^{13} + \frac {2}{11} \, b c d f^{3} x^{11} + \frac {1}{11} \, a d^{2} f^{3} x^{11} + \frac {3}{11} \, b d^{2} f^{2} x^{11} e + \frac {1}{9} \, b c^{2} f^{3} x^{9} + \frac {2}{9} \, a c d f^{3} x^{9} + \frac {2}{3} \, b c d f^{2} x^{9} e + \frac {1}{3} \, a d^{2} f^{2} x^{9} e + \frac {1}{3} \, b d^{2} f x^{9} e^{2} + \frac {1}{7} \, a c^{2} f^{3} x^{7} + \frac {3}{7} \, b c^{2} f^{2} x^{7} e + \frac {6}{7} \, a c d f^{2} x^{7} e + \frac {6}{7} \, b c d f x^{7} e^{2} + \frac {3}{7} \, a d^{2} f x^{7} e^{2} + \frac {1}{7} \, b d^{2} x^{7} e^{3} + \frac {3}{5} \, a c^{2} f^{2} x^{5} e + \frac {3}{5} \, b c^{2} f x^{5} e^{2} + \frac {6}{5} \, a c d f x^{5} e^{2} + \frac {2}{5} \, b c d x^{5} e^{3} + \frac {1}{5} \, a d^{2} x^{5} e^{3} + a c^{2} f x^{3} e^{2} + \frac {1}{3} \, b c^{2} x^{3} e^{3} + \frac {2}{3} \, a c d x^{3} e^{3} + a c^{2} x e^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.86, size = 233, normalized size = 1.03 \begin {gather*} x^5\,\left (\frac {3\,b\,c^2\,e^2\,f}{5}+\frac {3\,a\,c^2\,e\,f^2}{5}+\frac {2\,b\,c\,d\,e^3}{5}+\frac {6\,a\,c\,d\,e^2\,f}{5}+\frac {a\,d^2\,e^3}{5}\right )+x^9\,\left (\frac {b\,c^2\,f^3}{9}+\frac {2\,b\,c\,d\,e\,f^2}{3}+\frac {2\,a\,c\,d\,f^3}{9}+\frac {b\,d^2\,e^2\,f}{3}+\frac {a\,d^2\,e\,f^2}{3}\right )+x^7\,\left (\frac {3\,b\,c^2\,e\,f^2}{7}+\frac {a\,c^2\,f^3}{7}+\frac {6\,b\,c\,d\,e^2\,f}{7}+\frac {6\,a\,c\,d\,e\,f^2}{7}+\frac {b\,d^2\,e^3}{7}+\frac {3\,a\,d^2\,e^2\,f}{7}\right )+\frac {b\,d^2\,f^3\,x^{13}}{13}+\frac {c\,e^2\,x^3\,\left (3\,a\,c\,f+2\,a\,d\,e+b\,c\,e\right )}{3}+\frac {d\,f^2\,x^{11}\,\left (a\,d\,f+2\,b\,c\,f+3\,b\,d\,e\right )}{11}+a\,c^2\,e^3\,x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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