Optimal. Leaf size=60 \[ A b^3 \log (x)+\frac {3}{2} A b^2 c x^2+\frac {3}{4} A b c^2 x^4+\frac {1}{6} A c^3 x^6+\frac {B \left (b+c x^2\right )^4}{8 c} \]
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Rubi [A] time = 0.05, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1584, 446, 80, 43} \[ \frac {3}{2} A b^2 c x^2+A b^3 \log (x)+\frac {3}{4} A b c^2 x^4+\frac {1}{6} A c^3 x^6+\frac {B \left (b+c x^2\right )^4}{8 c} \]
Antiderivative was successfully verified.
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Rule 43
Rule 80
Rule 446
Rule 1584
Rubi steps
\begin {align*} \int \frac {\left (A+B x^2\right ) \left (b x^2+c x^4\right )^3}{x^7} \, dx &=\int \frac {\left (A+B x^2\right ) \left (b+c x^2\right )^3}{x} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(A+B x) (b+c x)^3}{x} \, dx,x,x^2\right )\\ &=\frac {B \left (b+c x^2\right )^4}{8 c}+\frac {1}{2} A \operatorname {Subst}\left (\int \frac {(b+c x)^3}{x} \, dx,x,x^2\right )\\ &=\frac {B \left (b+c x^2\right )^4}{8 c}+\frac {1}{2} A \operatorname {Subst}\left (\int \left (3 b^2 c+\frac {b^3}{x}+3 b c^2 x+c^3 x^2\right ) \, dx,x,x^2\right )\\ &=\frac {3}{2} A b^2 c x^2+\frac {3}{4} A b c^2 x^4+\frac {1}{6} A c^3 x^6+\frac {B \left (b+c x^2\right )^4}{8 c}+A b^3 \log (x)\\ \end {align*}
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Mathematica [A] time = 0.02, size = 71, normalized size = 1.18 \[ A b^3 \log (x)+\frac {1}{2} b^2 x^2 (3 A c+b B)+\frac {1}{6} c^2 x^6 (A c+3 b B)+\frac {3}{4} b c x^4 (A c+b B)+\frac {1}{8} B c^3 x^8 \]
Antiderivative was successfully verified.
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fricas [A] time = 0.91, size = 71, normalized size = 1.18 \[ \frac {1}{8} \, B c^{3} x^{8} + \frac {1}{6} \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + \frac {3}{4} \, {\left (B b^{2} c + A b c^{2}\right )} x^{4} + A b^{3} \log \relax (x) + \frac {1}{2} \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 78, normalized size = 1.30 \[ \frac {1}{8} \, B c^{3} x^{8} + \frac {1}{2} \, B b c^{2} x^{6} + \frac {1}{6} \, A c^{3} x^{6} + \frac {3}{4} \, B b^{2} c x^{4} + \frac {3}{4} \, A b c^{2} x^{4} + \frac {1}{2} \, B b^{3} x^{2} + \frac {3}{2} \, A b^{2} c x^{2} + \frac {1}{2} \, A b^{3} \log \left (x^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 76, normalized size = 1.27 \[ \frac {B \,c^{3} x^{8}}{8}+\frac {A \,c^{3} x^{6}}{6}+\frac {B b \,c^{2} x^{6}}{2}+\frac {3 A b \,c^{2} x^{4}}{4}+\frac {3 B \,b^{2} c \,x^{4}}{4}+\frac {3 A \,b^{2} c \,x^{2}}{2}+\frac {B \,b^{3} x^{2}}{2}+A \,b^{3} \ln \relax (x ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.31, size = 74, normalized size = 1.23 \[ \frac {1}{8} \, B c^{3} x^{8} + \frac {1}{6} \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + \frac {3}{4} \, {\left (B b^{2} c + A b c^{2}\right )} x^{4} + \frac {1}{2} \, A b^{3} \log \left (x^{2}\right ) + \frac {1}{2} \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 67, normalized size = 1.12 \[ x^2\,\left (\frac {B\,b^3}{2}+\frac {3\,A\,c\,b^2}{2}\right )+x^6\,\left (\frac {A\,c^3}{6}+\frac {B\,b\,c^2}{2}\right )+\frac {B\,c^3\,x^8}{8}+A\,b^3\,\ln \relax (x)+\frac {3\,b\,c\,x^4\,\left (A\,c+B\,b\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.18, size = 80, normalized size = 1.33 \[ A b^{3} \log {\relax (x )} + \frac {B c^{3} x^{8}}{8} + x^{6} \left (\frac {A c^{3}}{6} + \frac {B b c^{2}}{2}\right ) + x^{4} \left (\frac {3 A b c^{2}}{4} + \frac {3 B b^{2} c}{4}\right ) + x^{2} \left (\frac {3 A b^{2} c}{2} + \frac {B b^{3}}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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