Optimal. Leaf size=123 \[ \frac {1}{4} d x^4 \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac {1}{2} c x^2 \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )-\frac {a^2 c^3}{2 x^2}+a c^2 \log (x) (3 a d+2 b c)+\frac {1}{6} b d^2 x^6 (2 a d+3 b c)+\frac {1}{8} b^2 d^3 x^8 \]
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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} \begin {gather*} \frac {1}{4} d x^4 \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac {1}{2} c x^2 \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )-\frac {a^2 c^3}{2 x^2}+a c^2 \log (x) (3 a d+2 b c)+\frac {1}{6} b d^2 x^6 (2 a d+3 b c)+\frac {1}{8} b^2 d^3 x^8 \end {gather*}
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^3} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^2 (c+d x)^3}{x^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right )+\frac {a^2 c^3}{x^2}+\frac {a c^2 (2 b c+3 a d)}{x}+d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x+b d^2 (3 b c+2 a d) x^2+b^2 d^3 x^3\right ) \, dx,x,x^2\right )\\ &=-\frac {a^2 c^3}{2 x^2}+\frac {1}{2} c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^2+\frac {1}{4} d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^4+\frac {1}{6} b d^2 (3 b c+2 a d) x^6+\frac {1}{8} b^2 d^3 x^8+a c^2 (2 b c+3 a d) \log (x)\\ \end {align*}
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Mathematica [A] time = 0.05, size = 120, normalized size = 0.98 \begin {gather*} \frac {6 a^2 \left (-2 c^3+6 c d^2 x^4+d^3 x^6\right )+4 a b d x^4 \left (18 c^2+9 c d x^2+2 d^2 x^4\right )+3 b^2 x^4 \left (4 c^3+6 c^2 d x^2+4 c d^2 x^4+d^3 x^6\right )}{24 x^2}+a c^2 \log (x) (3 a d+2 b c) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^3}{x^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.66, size = 131, normalized size = 1.07 \begin {gather*} \frac {3 \, b^{2} d^{3} x^{10} + 4 \, {\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{8} + 6 \, {\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{6} - 12 \, a^{2} c^{3} + 12 \, {\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{4} + 24 \, {\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{2} \log \relax (x)}{24 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.32, size = 160, normalized size = 1.30 \begin {gather*} \frac {1}{8} \, b^{2} d^{3} x^{8} + \frac {1}{2} \, b^{2} c d^{2} x^{6} + \frac {1}{3} \, a b d^{3} x^{6} + \frac {3}{4} \, b^{2} c^{2} d x^{4} + \frac {3}{2} \, a b c d^{2} x^{4} + \frac {1}{4} \, a^{2} d^{3} x^{4} + \frac {1}{2} \, b^{2} c^{3} x^{2} + 3 \, a b c^{2} d x^{2} + \frac {3}{2} \, a^{2} c d^{2} x^{2} + \frac {1}{2} \, {\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} \log \left (x^{2}\right ) - \frac {2 \, a b c^{3} x^{2} + 3 \, a^{2} c^{2} d x^{2} + a^{2} c^{3}}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 134, normalized size = 1.09 \begin {gather*} \frac {b^{2} d^{3} x^{8}}{8}+\frac {a b \,d^{3} x^{6}}{3}+\frac {b^{2} c \,d^{2} x^{6}}{2}+\frac {a^{2} d^{3} x^{4}}{4}+\frac {3 a b c \,d^{2} x^{4}}{2}+\frac {3 b^{2} c^{2} d \,x^{4}}{4}+\frac {3 a^{2} c \,d^{2} x^{2}}{2}+3 a b \,c^{2} d \,x^{2}+\frac {b^{2} c^{3} x^{2}}{2}+3 a^{2} c^{2} d \ln \relax (x )+2 a b \,c^{3} \ln \relax (x )-\frac {a^{2} c^{3}}{2 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.06, size = 128, normalized size = 1.04 \begin {gather*} \frac {1}{8} \, b^{2} d^{3} x^{8} + \frac {1}{6} \, {\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{6} + \frac {1}{4} \, {\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{4} - \frac {a^{2} c^{3}}{2 \, x^{2}} + \frac {1}{2} \, {\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{2} + \frac {1}{2} \, {\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} \log \left (x^{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 121, normalized size = 0.98 \begin {gather*} x^2\,\left (\frac {3\,a^2\,c\,d^2}{2}+3\,a\,b\,c^2\,d+\frac {b^2\,c^3}{2}\right )+x^4\,\left (\frac {a^2\,d^3}{4}+\frac {3\,a\,b\,c\,d^2}{2}+\frac {3\,b^2\,c^2\,d}{4}\right )+\ln \relax (x)\,\left (3\,d\,a^2\,c^2+2\,b\,a\,c^3\right )-\frac {a^2\,c^3}{2\,x^2}+\frac {b^2\,d^3\,x^8}{8}+\frac {b\,d^2\,x^6\,\left (2\,a\,d+3\,b\,c\right )}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.33, size = 133, normalized size = 1.08 \begin {gather*} - \frac {a^{2} c^{3}}{2 x^{2}} + a c^{2} \left (3 a d + 2 b c\right ) \log {\relax (x )} + \frac {b^{2} d^{3} x^{8}}{8} + x^{6} \left (\frac {a b d^{3}}{3} + \frac {b^{2} c d^{2}}{2}\right ) + x^{4} \left (\frac {a^{2} d^{3}}{4} + \frac {3 a b c d^{2}}{2} + \frac {3 b^{2} c^{2} d}{4}\right ) + x^{2} \left (\frac {3 a^{2} c d^{2}}{2} + 3 a b c^{2} d + \frac {b^{2} c^{3}}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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