Optimal. Leaf size=123 \[ \frac {1}{6} d x^6 \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac {1}{4} c x^4 \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+a^2 c^3 \log (x)+\frac {1}{2} a c^2 x^2 (3 a d+2 b c)+\frac {1}{8} b d^2 x^8 (2 a d+3 b c)+\frac {1}{10} b^2 d^3 x^{10} \]
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Rubi [A] time = 0.10, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {446, 88} \[ \frac {1}{6} d x^6 \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac {1}{4} c x^4 \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+a^2 c^3 \log (x)+\frac {1}{2} a c^2 x^2 (3 a d+2 b c)+\frac {1}{8} b d^2 x^8 (2 a d+3 b c)+\frac {1}{10} b^2 d^3 x^{10} \]
Antiderivative was successfully verified.
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Rule 88
Rule 446
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^3}{x} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^2 (c+d x)^3}{x} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (a c^2 (2 b c+3 a d)+\frac {a^2 c^3}{x}+c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x+d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^2+b d^2 (3 b c+2 a d) x^3+b^2 d^3 x^4\right ) \, dx,x,x^2\right )\\ &=\frac {1}{2} a c^2 (2 b c+3 a d) x^2+\frac {1}{4} c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^4+\frac {1}{6} d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^6+\frac {1}{8} b d^2 (3 b c+2 a d) x^8+\frac {1}{10} b^2 d^3 x^{10}+a^2 c^3 \log (x)\\ \end {align*}
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Mathematica [A] time = 0.03, size = 123, normalized size = 1.00 \[ \frac {1}{6} d x^6 \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac {1}{4} c x^4 \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+a^2 c^3 \log (x)+\frac {1}{2} a c^2 x^2 (3 a d+2 b c)+\frac {1}{8} b d^2 x^8 (2 a d+3 b c)+\frac {1}{10} b^2 d^3 x^{10} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 125, normalized size = 1.02 \[ \frac {1}{10} \, b^{2} d^{3} x^{10} + \frac {1}{8} \, {\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{8} + \frac {1}{6} \, {\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{6} + a^{2} c^{3} \log \relax (x) + \frac {1}{4} \, {\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{4} + \frac {1}{2} \, {\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.42, size = 134, normalized size = 1.09 \[ \frac {1}{10} \, b^{2} d^{3} x^{10} + \frac {3}{8} \, b^{2} c d^{2} x^{8} + \frac {1}{4} \, a b d^{3} x^{8} + \frac {1}{2} \, b^{2} c^{2} d x^{6} + a b c d^{2} x^{6} + \frac {1}{6} \, a^{2} d^{3} x^{6} + \frac {1}{4} \, b^{2} c^{3} x^{4} + \frac {3}{2} \, a b c^{2} d x^{4} + \frac {3}{4} \, a^{2} c d^{2} x^{4} + a b c^{3} x^{2} + \frac {3}{2} \, a^{2} c^{2} d x^{2} + \frac {1}{2} \, a^{2} c^{3} \log \left (x^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 132, normalized size = 1.07 \[ \frac {b^{2} d^{3} x^{10}}{10}+\frac {a b \,d^{3} x^{8}}{4}+\frac {3 b^{2} c \,d^{2} x^{8}}{8}+\frac {a^{2} d^{3} x^{6}}{6}+a b c \,d^{2} x^{6}+\frac {b^{2} c^{2} d \,x^{6}}{2}+\frac {3 a^{2} c \,d^{2} x^{4}}{4}+\frac {3 a b \,c^{2} d \,x^{4}}{2}+\frac {b^{2} c^{3} x^{4}}{4}+\frac {3 a^{2} c^{2} d \,x^{2}}{2}+a b \,c^{3} x^{2}+a^{2} c^{3} \ln \relax (x ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.10, size = 128, normalized size = 1.04 \[ \frac {1}{10} \, b^{2} d^{3} x^{10} + \frac {1}{8} \, {\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{8} + \frac {1}{6} \, {\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{6} + \frac {1}{2} \, a^{2} c^{3} \log \left (x^{2}\right ) + \frac {1}{4} \, {\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{4} + \frac {1}{2} \, {\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 116, normalized size = 0.94 \[ x^4\,\left (\frac {3\,a^2\,c\,d^2}{4}+\frac {3\,a\,b\,c^2\,d}{2}+\frac {b^2\,c^3}{4}\right )+x^6\,\left (\frac {a^2\,d^3}{6}+a\,b\,c\,d^2+\frac {b^2\,c^2\,d}{2}\right )+\frac {b^2\,d^3\,x^{10}}{10}+a^2\,c^3\,\ln \relax (x)+\frac {a\,c^2\,x^2\,\left (3\,a\,d+2\,b\,c\right )}{2}+\frac {b\,d^2\,x^8\,\left (2\,a\,d+3\,b\,c\right )}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.23, size = 133, normalized size = 1.08 \[ a^{2} c^{3} \log {\relax (x )} + \frac {b^{2} d^{3} x^{10}}{10} + x^{8} \left (\frac {a b d^{3}}{4} + \frac {3 b^{2} c d^{2}}{8}\right ) + x^{6} \left (\frac {a^{2} d^{3}}{6} + a b c d^{2} + \frac {b^{2} c^{2} d}{2}\right ) + x^{4} \left (\frac {3 a^{2} c d^{2}}{4} + \frac {3 a b c^{2} d}{2} + \frac {b^{2} c^{3}}{4}\right ) + x^{2} \left (\frac {3 a^{2} c^{2} d}{2} + a b c^{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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