Optimal. Leaf size=122 \[ a^2 c^3 x+\frac {1}{3} a c^2 (2 b c+3 a d) x^3+\frac {1}{5} c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^5+\frac {1}{7} d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^7+\frac {1}{9} b d^2 (3 b c+2 a d) x^9+\frac {1}{11} b^2 d^3 x^{11} \]
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Rubi [A]
time = 0.05, antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {380}
\begin {gather*} \frac {1}{7} d x^7 \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac {1}{5} c x^5 \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+a^2 c^3 x+\frac {1}{3} a c^2 x^3 (3 a d+2 b c)+\frac {1}{9} b d^2 x^9 (2 a d+3 b c)+\frac {1}{11} b^2 d^3 x^{11} \end {gather*}
Antiderivative was successfully verified.
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Rule 380
Rubi steps
\begin {align*} \int \left (a+b x^2\right )^2 \left (c+d x^2\right )^3 \, dx &=\int \left (a^2 c^3+a c^2 (2 b c+3 a d) x^2+c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^4+d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^6+b d^2 (3 b c+2 a d) x^8+b^2 d^3 x^{10}\right ) \, dx\\ &=a^2 c^3 x+\frac {1}{3} a c^2 (2 b c+3 a d) x^3+\frac {1}{5} c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^5+\frac {1}{7} d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^7+\frac {1}{9} b d^2 (3 b c+2 a d) x^9+\frac {1}{11} b^2 d^3 x^{11}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 122, normalized size = 1.00 \begin {gather*} a^2 c^3 x+\frac {1}{3} a c^2 (2 b c+3 a d) x^3+\frac {1}{5} c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^5+\frac {1}{7} d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^7+\frac {1}{9} b d^2 (3 b c+2 a d) x^9+\frac {1}{11} b^2 d^3 x^{11} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 125, normalized size = 1.02
method | result | size |
norman | \(\frac {b^{2} d^{3} x^{11}}{11}+\left (\frac {2}{9} a b \,d^{3}+\frac {1}{3} b^{2} c \,d^{2}\right ) x^{9}+\left (\frac {1}{7} a^{2} d^{3}+\frac {6}{7} a b c \,d^{2}+\frac {3}{7} b^{2} c^{2} d \right ) x^{7}+\left (\frac {3}{5} a^{2} c \,d^{2}+\frac {6}{5} a b \,c^{2} d +\frac {1}{5} b^{2} c^{3}\right ) x^{5}+\left (a^{2} c^{2} d +\frac {2}{3} a b \,c^{3}\right ) x^{3}+a^{2} c^{3} x\) | \(122\) |
default | \(\frac {b^{2} d^{3} x^{11}}{11}+\frac {\left (2 a b \,d^{3}+3 b^{2} c \,d^{2}\right ) x^{9}}{9}+\frac {\left (a^{2} d^{3}+6 a b c \,d^{2}+3 b^{2} c^{2} d \right ) x^{7}}{7}+\frac {\left (3 a^{2} c \,d^{2}+6 a b \,c^{2} d +b^{2} c^{3}\right ) x^{5}}{5}+\frac {\left (3 a^{2} c^{2} d +2 a b \,c^{3}\right ) x^{3}}{3}+a^{2} c^{3} x\) | \(125\) |
gosper | \(\frac {1}{11} b^{2} d^{3} x^{11}+\frac {2}{9} x^{9} a b \,d^{3}+\frac {1}{3} x^{9} b^{2} c \,d^{2}+\frac {1}{7} x^{7} a^{2} d^{3}+\frac {6}{7} x^{7} a b c \,d^{2}+\frac {3}{7} x^{7} b^{2} c^{2} d +\frac {3}{5} x^{5} a^{2} c \,d^{2}+\frac {6}{5} x^{5} a b \,c^{2} d +\frac {1}{5} x^{5} b^{2} c^{3}+x^{3} a^{2} c^{2} d +\frac {2}{3} x^{3} a b \,c^{3}+a^{2} c^{3} x\) | \(132\) |
risch | \(\frac {1}{11} b^{2} d^{3} x^{11}+\frac {2}{9} x^{9} a b \,d^{3}+\frac {1}{3} x^{9} b^{2} c \,d^{2}+\frac {1}{7} x^{7} a^{2} d^{3}+\frac {6}{7} x^{7} a b c \,d^{2}+\frac {3}{7} x^{7} b^{2} c^{2} d +\frac {3}{5} x^{5} a^{2} c \,d^{2}+\frac {6}{5} x^{5} a b \,c^{2} d +\frac {1}{5} x^{5} b^{2} c^{3}+x^{3} a^{2} c^{2} d +\frac {2}{3} x^{3} a b \,c^{3}+a^{2} c^{3} x\) | \(132\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 124, normalized size = 1.02 \begin {gather*} \frac {1}{11} \, b^{2} d^{3} x^{11} + \frac {1}{9} \, {\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{9} + \frac {1}{7} \, {\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{7} + a^{2} c^{3} x + \frac {1}{5} \, {\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{5} + \frac {1}{3} \, {\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.73, size = 124, normalized size = 1.02 \begin {gather*} \frac {1}{11} \, b^{2} d^{3} x^{11} + \frac {1}{9} \, {\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{9} + \frac {1}{7} \, {\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{7} + a^{2} c^{3} x + \frac {1}{5} \, {\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{5} + \frac {1}{3} \, {\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.02, size = 136, normalized size = 1.11 \begin {gather*} a^{2} c^{3} x + \frac {b^{2} d^{3} x^{11}}{11} + x^{9} \cdot \left (\frac {2 a b d^{3}}{9} + \frac {b^{2} c d^{2}}{3}\right ) + x^{7} \left (\frac {a^{2} d^{3}}{7} + \frac {6 a b c d^{2}}{7} + \frac {3 b^{2} c^{2} d}{7}\right ) + x^{5} \cdot \left (\frac {3 a^{2} c d^{2}}{5} + \frac {6 a b c^{2} d}{5} + \frac {b^{2} c^{3}}{5}\right ) + x^{3} \left (a^{2} c^{2} d + \frac {2 a b c^{3}}{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.96, size = 131, normalized size = 1.07 \begin {gather*} \frac {1}{11} \, b^{2} d^{3} x^{11} + \frac {1}{3} \, b^{2} c d^{2} x^{9} + \frac {2}{9} \, a b d^{3} x^{9} + \frac {3}{7} \, b^{2} c^{2} d x^{7} + \frac {6}{7} \, a b c d^{2} x^{7} + \frac {1}{7} \, a^{2} d^{3} x^{7} + \frac {1}{5} \, b^{2} c^{3} x^{5} + \frac {6}{5} \, a b c^{2} d x^{5} + \frac {3}{5} \, a^{2} c d^{2} x^{5} + \frac {2}{3} \, a b c^{3} x^{3} + a^{2} c^{2} d x^{3} + a^{2} c^{3} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.95, size = 116, normalized size = 0.95 \begin {gather*} x^5\,\left (\frac {3\,a^2\,c\,d^2}{5}+\frac {6\,a\,b\,c^2\,d}{5}+\frac {b^2\,c^3}{5}\right )+x^7\,\left (\frac {a^2\,d^3}{7}+\frac {6\,a\,b\,c\,d^2}{7}+\frac {3\,b^2\,c^2\,d}{7}\right )+a^2\,c^3\,x+\frac {b^2\,d^3\,x^{11}}{11}+\frac {a\,c^2\,x^3\,\left (3\,a\,d+2\,b\,c\right )}{3}+\frac {b\,d^2\,x^9\,\left (2\,a\,d+3\,b\,c\right )}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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