Optimal. Leaf size=49 \[ -\frac {\left (b+c x^2\right )^4 (5 b B-A c)}{40 b^2 x^8}-\frac {A \left (b+c x^2\right )^4}{10 b x^{10}} \]
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Rubi [A] time = 0.04, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1584, 446, 78, 37} \[ -\frac {\left (b+c x^2\right )^4 (5 b B-A c)}{40 b^2 x^8}-\frac {A \left (b+c x^2\right )^4}{10 b x^{10}} \]
Antiderivative was successfully verified.
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Rule 37
Rule 78
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^{17}} \, dx &=\int \frac {\left (A+B x^2\right ) \left (b+c x^2\right )^3}{x^{11}} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(A+B x) (b+c x)^3}{x^6} \, dx,x,x^2\right )\\ &=-\frac {A \left (b+c x^2\right )^4}{10 b x^{10}}+\frac {(5 b B-A c) \operatorname {Subst}\left (\int \frac {(b+c x)^3}{x^5} \, dx,x,x^2\right )}{10 b}\\ &=-\frac {A \left (b+c x^2\right )^4}{10 b x^{10}}-\frac {(5 b B-A c) \left (b+c x^2\right )^4}{40 b^2 x^8}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 78, normalized size = 1.59 \[ -\frac {A \left (4 b^3+15 b^2 c x^2+20 b c^2 x^4+10 c^3 x^6\right )+5 B x^2 \left (b^3+4 b^2 c x^2+6 b c^2 x^4+4 c^3 x^6\right )}{40 x^{10}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 75, normalized size = 1.53 \[ -\frac {20 \, B c^{3} x^{8} + 10 \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + 20 \, {\left (B b^{2} c + A b c^{2}\right )} x^{4} + 4 \, A b^{3} + 5 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{2}}{40 \, x^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 79, normalized size = 1.61 \[ -\frac {20 \, B c^{3} x^{8} + 30 \, B b c^{2} x^{6} + 10 \, A c^{3} x^{6} + 20 \, B b^{2} c x^{4} + 20 \, A b c^{2} x^{4} + 5 \, B b^{3} x^{2} + 15 \, A b^{2} c x^{2} + 4 \, A b^{3}}{40 \, x^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 66, normalized size = 1.35 \[ -\frac {B \,c^{3}}{2 x^{2}}-\frac {\left (A c +3 b B \right ) c^{2}}{4 x^{4}}-\frac {\left (A c +b B \right ) b c}{2 x^{6}}-\frac {A \,b^{3}}{10 x^{10}}-\frac {\left (3 A c +b B \right ) b^{2}}{8 x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.33, size = 75, normalized size = 1.53 \[ -\frac {20 \, B c^{3} x^{8} + 10 \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + 20 \, {\left (B b^{2} c + A b c^{2}\right )} x^{4} + 4 \, A b^{3} + 5 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{2}}{40 \, x^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 76, normalized size = 1.55 \[ -\frac {x^4\,\left (\frac {B\,b^2\,c}{2}+\frac {A\,b\,c^2}{2}\right )+\frac {A\,b^3}{10}+x^2\,\left (\frac {B\,b^3}{8}+\frac {3\,A\,c\,b^2}{8}\right )+x^6\,\left (\frac {A\,c^3}{4}+\frac {3\,B\,b\,c^2}{4}\right )+\frac {B\,c^3\,x^8}{2}}{x^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.43, size = 83, normalized size = 1.69 \[ \frac {- 4 A b^{3} - 20 B c^{3} x^{8} + x^{6} \left (- 10 A c^{3} - 30 B b c^{2}\right ) + x^{4} \left (- 20 A b c^{2} - 20 B b^{2} c\right ) + x^{2} \left (- 15 A b^{2} c - 5 B b^{3}\right )}{40 x^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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