Optimal. Leaf size=76 \[ -\frac {b \left (a+b x^2\right )^6 (A b-4 a B)}{336 a^3 x^{12}}+\frac {\left (a+b x^2\right )^6 (A b-4 a B)}{56 a^2 x^{14}}-\frac {A \left (a+b x^2\right )^6}{16 a x^{16}} \]
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Rubi [A] time = 0.05, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {446, 78, 45, 37} \[ -\frac {b \left (a+b x^2\right )^6 (A b-4 a B)}{336 a^3 x^{12}}+\frac {\left (a+b x^2\right )^6 (A b-4 a B)}{56 a^2 x^{14}}-\frac {A \left (a+b x^2\right )^6}{16 a x^{16}} \]
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rule 78
Rule 446
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^{17}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^5 (A+B x)}{x^9} \, dx,x,x^2\right )\\ &=-\frac {A \left (a+b x^2\right )^6}{16 a x^{16}}+\frac {(-2 A b+8 a B) \operatorname {Subst}\left (\int \frac {(a+b x)^5}{x^8} \, dx,x,x^2\right )}{16 a}\\ &=-\frac {A \left (a+b x^2\right )^6}{16 a x^{16}}+\frac {(A b-4 a B) \left (a+b x^2\right )^6}{56 a^2 x^{14}}+\frac {(b (A b-4 a B)) \operatorname {Subst}\left (\int \frac {(a+b x)^5}{x^7} \, dx,x,x^2\right )}{56 a^2}\\ &=-\frac {A \left (a+b x^2\right )^6}{16 a x^{16}}+\frac {(A b-4 a B) \left (a+b x^2\right )^6}{56 a^2 x^{14}}-\frac {b (A b-4 a B) \left (a+b x^2\right )^6}{336 a^3 x^{12}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 121, normalized size = 1.59 \[ -\frac {3 a^5 \left (7 A+8 B x^2\right )+20 a^4 b x^2 \left (6 A+7 B x^2\right )+56 a^3 b^2 x^4 \left (5 A+6 B x^2\right )+84 a^2 b^3 x^6 \left (4 A+5 B x^2\right )+70 a b^4 x^8 \left (3 A+4 B x^2\right )+28 b^5 x^{10} \left (2 A+3 B x^2\right )}{336 x^{16}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 121, normalized size = 1.59 \[ -\frac {84 \, B b^{5} x^{12} + 56 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + 210 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 336 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + 21 \, A a^{5} + 140 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 24 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{336 \, x^{16}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.32, size = 127, normalized size = 1.67 \[ -\frac {84 \, B b^{5} x^{12} + 280 \, B a b^{4} x^{10} + 56 \, A b^{5} x^{10} + 420 \, B a^{2} b^{3} x^{8} + 210 \, A a b^{4} x^{8} + 336 \, B a^{3} b^{2} x^{6} + 336 \, A a^{2} b^{3} x^{6} + 140 \, B a^{4} b x^{4} + 280 \, A a^{3} b^{2} x^{4} + 24 \, B a^{5} x^{2} + 120 \, A a^{4} b x^{2} + 21 \, A a^{5}}{336 \, x^{16}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 104, normalized size = 1.37 \[ -\frac {B \,b^{5}}{4 x^{4}}-\frac {\left (A b +5 B a \right ) b^{4}}{6 x^{6}}-\frac {5 \left (A b +2 B a \right ) a \,b^{3}}{8 x^{8}}-\frac {\left (A b +B a \right ) a^{2} b^{2}}{x^{10}}-\frac {5 \left (2 A b +B a \right ) a^{3} b}{12 x^{12}}-\frac {A \,a^{5}}{16 x^{16}}-\frac {\left (5 A b +B a \right ) a^{4}}{14 x^{14}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.09, size = 121, normalized size = 1.59 \[ -\frac {84 \, B b^{5} x^{12} + 56 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + 210 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 336 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + 21 \, A a^{5} + 140 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 24 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{336 \, x^{16}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 120, normalized size = 1.58 \[ -\frac {\frac {A\,a^5}{16}+x^8\,\left (\frac {5\,B\,a^2\,b^3}{4}+\frac {5\,A\,a\,b^4}{8}\right )+x^4\,\left (\frac {5\,B\,a^4\,b}{12}+\frac {5\,A\,a^3\,b^2}{6}\right )+x^2\,\left (\frac {B\,a^5}{14}+\frac {5\,A\,b\,a^4}{14}\right )+x^{10}\,\left (\frac {A\,b^5}{6}+\frac {5\,B\,a\,b^4}{6}\right )+x^6\,\left (B\,a^3\,b^2+A\,a^2\,b^3\right )+\frac {B\,b^5\,x^{12}}{4}}{x^{16}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 32.11, size = 134, normalized size = 1.76 \[ \frac {- 21 A a^{5} - 84 B b^{5} x^{12} + x^{10} \left (- 56 A b^{5} - 280 B a b^{4}\right ) + x^{8} \left (- 210 A a b^{4} - 420 B a^{2} b^{3}\right ) + x^{6} \left (- 336 A a^{2} b^{3} - 336 B a^{3} b^{2}\right ) + x^{4} \left (- 280 A a^{3} b^{2} - 140 B a^{4} b\right ) + x^{2} \left (- 120 A a^{4} b - 24 B a^{5}\right )}{336 x^{16}} \]
Verification of antiderivative is not currently implemented for this CAS.
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