Optimal. Leaf size=117 \[ -\frac {a^5 A}{18 x^{18}}-\frac {a^4 (a B+5 A b)}{16 x^{16}}-\frac {5 a^3 b (a B+2 A b)}{14 x^{14}}-\frac {5 a^2 b^2 (a B+A b)}{6 x^{12}}-\frac {b^4 (5 a B+A b)}{8 x^8}-\frac {a b^3 (2 a B+A b)}{2 x^{10}}-\frac {b^5 B}{6 x^6} \]
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Rubi [A] time = 0.08, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {446, 76} \[ -\frac {5 a^2 b^2 (a B+A b)}{6 x^{12}}-\frac {a^4 (a B+5 A b)}{16 x^{16}}-\frac {5 a^3 b (a B+2 A b)}{14 x^{14}}-\frac {a^5 A}{18 x^{18}}-\frac {a b^3 (2 a B+A b)}{2 x^{10}}-\frac {b^4 (5 a B+A b)}{8 x^8}-\frac {b^5 B}{6 x^6} \]
Antiderivative was successfully verified.
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Rule 76
Rule 446
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^{19}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^5 (A+B x)}{x^{10}} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {a^5 A}{x^{10}}+\frac {a^4 (5 A b+a B)}{x^9}+\frac {5 a^3 b (2 A b+a B)}{x^8}+\frac {10 a^2 b^2 (A b+a B)}{x^7}+\frac {5 a b^3 (A b+2 a B)}{x^6}+\frac {b^4 (A b+5 a B)}{x^5}+\frac {b^5 B}{x^4}\right ) \, dx,x,x^2\right )\\ &=-\frac {a^5 A}{18 x^{18}}-\frac {a^4 (5 A b+a B)}{16 x^{16}}-\frac {5 a^3 b (2 A b+a B)}{14 x^{14}}-\frac {5 a^2 b^2 (A b+a B)}{6 x^{12}}-\frac {a b^3 (A b+2 a B)}{2 x^{10}}-\frac {b^4 (A b+5 a B)}{8 x^8}-\frac {b^5 B}{6 x^6}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 121, normalized size = 1.03 \[ -\frac {7 a^5 \left (8 A+9 B x^2\right )+45 a^4 b x^2 \left (7 A+8 B x^2\right )+120 a^3 b^2 x^4 \left (6 A+7 B x^2\right )+168 a^2 b^3 x^6 \left (5 A+6 B x^2\right )+126 a b^4 x^8 \left (4 A+5 B x^2\right )+42 b^5 x^{10} \left (3 A+4 B x^2\right )}{1008 x^{18}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 121, normalized size = 1.03 \[ -\frac {168 \, B b^{5} x^{12} + 126 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + 504 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 840 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + 56 \, A a^{5} + 360 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 63 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{1008 \, x^{18}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.34, size = 127, normalized size = 1.09 \[ -\frac {168 \, B b^{5} x^{12} + 630 \, B a b^{4} x^{10} + 126 \, A b^{5} x^{10} + 1008 \, B a^{2} b^{3} x^{8} + 504 \, A a b^{4} x^{8} + 840 \, B a^{3} b^{2} x^{6} + 840 \, A a^{2} b^{3} x^{6} + 360 \, B a^{4} b x^{4} + 720 \, A a^{3} b^{2} x^{4} + 63 \, B a^{5} x^{2} + 315 \, A a^{4} b x^{2} + 56 \, A a^{5}}{1008 \, x^{18}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 104, normalized size = 0.89 \[ -\frac {B \,b^{5}}{6 x^{6}}-\frac {\left (A b +5 B a \right ) b^{4}}{8 x^{8}}-\frac {\left (A b +2 B a \right ) a \,b^{3}}{2 x^{10}}-\frac {5 \left (A b +B a \right ) a^{2} b^{2}}{6 x^{12}}-\frac {5 \left (2 A b +B a \right ) a^{3} b}{14 x^{14}}-\frac {A \,a^{5}}{18 x^{18}}-\frac {\left (5 A b +B a \right ) a^{4}}{16 x^{16}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.12, size = 121, normalized size = 1.03 \[ -\frac {168 \, B b^{5} x^{12} + 126 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + 504 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 840 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + 56 \, A a^{5} + 360 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 63 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{1008 \, x^{18}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 121, normalized size = 1.03 \[ -\frac {\frac {A\,a^5}{18}+x^8\,\left (B\,a^2\,b^3+\frac {A\,a\,b^4}{2}\right )+x^4\,\left (\frac {5\,B\,a^4\,b}{14}+\frac {5\,A\,a^3\,b^2}{7}\right )+x^2\,\left (\frac {B\,a^5}{16}+\frac {5\,A\,b\,a^4}{16}\right )+x^{10}\,\left (\frac {A\,b^5}{8}+\frac {5\,B\,a\,b^4}{8}\right )+x^6\,\left (\frac {5\,B\,a^3\,b^2}{6}+\frac {5\,A\,a^2\,b^3}{6}\right )+\frac {B\,b^5\,x^{12}}{6}}{x^{18}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 65.26, size = 134, normalized size = 1.15 \[ \frac {- 56 A a^{5} - 168 B b^{5} x^{12} + x^{10} \left (- 126 A b^{5} - 630 B a b^{4}\right ) + x^{8} \left (- 504 A a b^{4} - 1008 B a^{2} b^{3}\right ) + x^{6} \left (- 840 A a^{2} b^{3} - 840 B a^{3} b^{2}\right ) + x^{4} \left (- 720 A a^{3} b^{2} - 360 B a^{4} b\right ) + x^{2} \left (- 315 A a^{4} b - 63 B a^{5}\right )}{1008 x^{18}} \]
Verification of antiderivative is not currently implemented for this CAS.
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