Optimal. Leaf size=127 \[ \frac {1}{5} a^2 c^3 x^5+\frac {1}{7} a c^2 (2 b c+3 a d) x^7+\frac {1}{9} c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^9+\frac {1}{11} d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^{11}+\frac {1}{13} b d^2 (3 b c+2 a d) x^{13}+\frac {1}{15} b^2 d^3 x^{15} \]
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Rubi [A]
time = 0.06, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {459}
\begin {gather*} \frac {1}{11} d x^{11} \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac {1}{9} c x^9 \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+\frac {1}{5} a^2 c^3 x^5+\frac {1}{7} a c^2 x^7 (3 a d+2 b c)+\frac {1}{13} b d^2 x^{13} (2 a d+3 b c)+\frac {1}{15} b^2 d^3 x^{15} \end {gather*}
Antiderivative was successfully verified.
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Rule 459
Rubi steps
\begin {align*} \int x^4 \left (a+b x^2\right )^2 \left (c+d x^2\right )^3 \, dx &=\int \left (a^2 c^3 x^4+a c^2 (2 b c+3 a d) x^6+c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^8+d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^{10}+b d^2 (3 b c+2 a d) x^{12}+b^2 d^3 x^{14}\right ) \, dx\\ &=\frac {1}{5} a^2 c^3 x^5+\frac {1}{7} a c^2 (2 b c+3 a d) x^7+\frac {1}{9} c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^9+\frac {1}{11} d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^{11}+\frac {1}{13} b d^2 (3 b c+2 a d) x^{13}+\frac {1}{15} b^2 d^3 x^{15}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 127, normalized size = 1.00 \begin {gather*} \frac {1}{5} a^2 c^3 x^5+\frac {1}{7} a c^2 (2 b c+3 a d) x^7+\frac {1}{9} c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^9+\frac {1}{11} d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^{11}+\frac {1}{13} b d^2 (3 b c+2 a d) x^{13}+\frac {1}{15} b^2 d^3 x^{15} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 128, normalized size = 1.01
method | result | size |
norman | \(\frac {a^{2} c^{3} x^{5}}{5}+\left (\frac {3}{7} a^{2} c^{2} d +\frac {2}{7} a b \,c^{3}\right ) x^{7}+\left (\frac {1}{3} a^{2} c \,d^{2}+\frac {2}{3} a b \,c^{2} d +\frac {1}{9} b^{2} c^{3}\right ) x^{9}+\left (\frac {1}{11} a^{2} d^{3}+\frac {6}{11} a b c \,d^{2}+\frac {3}{11} b^{2} c^{2} d \right ) x^{11}+\left (\frac {2}{13} a b \,d^{3}+\frac {3}{13} b^{2} c \,d^{2}\right ) x^{13}+\frac {b^{2} d^{3} x^{15}}{15}\) | \(126\) |
default | \(\frac {b^{2} d^{3} x^{15}}{15}+\frac {\left (2 a b \,d^{3}+3 b^{2} c \,d^{2}\right ) x^{13}}{13}+\frac {\left (a^{2} d^{3}+6 a b c \,d^{2}+3 b^{2} c^{2} d \right ) x^{11}}{11}+\frac {\left (3 a^{2} c \,d^{2}+6 a b \,c^{2} d +b^{2} c^{3}\right ) x^{9}}{9}+\frac {\left (3 a^{2} c^{2} d +2 a b \,c^{3}\right ) x^{7}}{7}+\frac {a^{2} c^{3} x^{5}}{5}\) | \(128\) |
gosper | \(\frac {1}{5} a^{2} c^{3} x^{5}+\frac {3}{7} x^{7} a^{2} c^{2} d +\frac {2}{7} x^{7} a b \,c^{3}+\frac {1}{3} x^{9} a^{2} c \,d^{2}+\frac {2}{3} x^{9} a b \,c^{2} d +\frac {1}{9} x^{9} b^{2} c^{3}+\frac {1}{11} x^{11} a^{2} d^{3}+\frac {6}{11} x^{11} a b c \,d^{2}+\frac {3}{11} x^{11} b^{2} c^{2} d +\frac {2}{13} x^{13} a b \,d^{3}+\frac {3}{13} x^{13} b^{2} c \,d^{2}+\frac {1}{15} b^{2} d^{3} x^{15}\) | \(136\) |
risch | \(\frac {1}{5} a^{2} c^{3} x^{5}+\frac {3}{7} x^{7} a^{2} c^{2} d +\frac {2}{7} x^{7} a b \,c^{3}+\frac {1}{3} x^{9} a^{2} c \,d^{2}+\frac {2}{3} x^{9} a b \,c^{2} d +\frac {1}{9} x^{9} b^{2} c^{3}+\frac {1}{11} x^{11} a^{2} d^{3}+\frac {6}{11} x^{11} a b c \,d^{2}+\frac {3}{11} x^{11} b^{2} c^{2} d +\frac {2}{13} x^{13} a b \,d^{3}+\frac {3}{13} x^{13} b^{2} c \,d^{2}+\frac {1}{15} b^{2} d^{3} x^{15}\) | \(136\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.32, size = 127, normalized size = 1.00 \begin {gather*} \frac {1}{15} \, b^{2} d^{3} x^{15} + \frac {1}{13} \, {\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{13} + \frac {1}{11} \, {\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{11} + \frac {1}{5} \, a^{2} c^{3} x^{5} + \frac {1}{9} \, {\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{9} + \frac {1}{7} \, {\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.77, size = 127, normalized size = 1.00 \begin {gather*} \frac {1}{15} \, b^{2} d^{3} x^{15} + \frac {1}{13} \, {\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{13} + \frac {1}{11} \, {\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{11} + \frac {1}{5} \, a^{2} c^{3} x^{5} + \frac {1}{9} \, {\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{9} + \frac {1}{7} \, {\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.02, size = 143, normalized size = 1.13 \begin {gather*} \frac {a^{2} c^{3} x^{5}}{5} + \frac {b^{2} d^{3} x^{15}}{15} + x^{13} \cdot \left (\frac {2 a b d^{3}}{13} + \frac {3 b^{2} c d^{2}}{13}\right ) + x^{11} \left (\frac {a^{2} d^{3}}{11} + \frac {6 a b c d^{2}}{11} + \frac {3 b^{2} c^{2} d}{11}\right ) + x^{9} \left (\frac {a^{2} c d^{2}}{3} + \frac {2 a b c^{2} d}{3} + \frac {b^{2} c^{3}}{9}\right ) + x^{7} \cdot \left (\frac {3 a^{2} c^{2} d}{7} + \frac {2 a b c^{3}}{7}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.01, size = 135, normalized size = 1.06 \begin {gather*} \frac {1}{15} \, b^{2} d^{3} x^{15} + \frac {3}{13} \, b^{2} c d^{2} x^{13} + \frac {2}{13} \, a b d^{3} x^{13} + \frac {3}{11} \, b^{2} c^{2} d x^{11} + \frac {6}{11} \, a b c d^{2} x^{11} + \frac {1}{11} \, a^{2} d^{3} x^{11} + \frac {1}{9} \, b^{2} c^{3} x^{9} + \frac {2}{3} \, a b c^{2} d x^{9} + \frac {1}{3} \, a^{2} c d^{2} x^{9} + \frac {2}{7} \, a b c^{3} x^{7} + \frac {3}{7} \, a^{2} c^{2} d x^{7} + \frac {1}{5} \, a^{2} c^{3} x^{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 119, normalized size = 0.94 \begin {gather*} x^9\,\left (\frac {a^2\,c\,d^2}{3}+\frac {2\,a\,b\,c^2\,d}{3}+\frac {b^2\,c^3}{9}\right )+x^{11}\,\left (\frac {a^2\,d^3}{11}+\frac {6\,a\,b\,c\,d^2}{11}+\frac {3\,b^2\,c^2\,d}{11}\right )+\frac {a^2\,c^3\,x^5}{5}+\frac {b^2\,d^3\,x^{15}}{15}+\frac {a\,c^2\,x^7\,\left (3\,a\,d+2\,b\,c\right )}{7}+\frac {b\,d^2\,x^{13}\,\left (2\,a\,d+3\,b\,c\right )}{13} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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