Optimal. Leaf size=42 \[ \frac {\left (a+b x^2\right )^6 (A b-a B)}{12 b^2}+\frac {B \left (a+b x^2\right )^7}{14 b^2} \]
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Rubi [A] time = 0.07, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {444, 43} \begin {gather*} \frac {\left (a+b x^2\right )^6 (A b-a B)}{12 b^2}+\frac {B \left (a+b x^2\right )^7}{14 b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 444
Rubi steps
\begin {align*} \int x \left (a+b x^2\right )^5 \left (A+B x^2\right ) \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int (a+b x)^5 (A+B x) \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {(A b-a B) (a+b x)^5}{b}+\frac {B (a+b x)^6}{b}\right ) \, dx,x,x^2\right )\\ &=\frac {(A b-a B) \left (a+b x^2\right )^6}{12 b^2}+\frac {B \left (a+b x^2\right )^7}{14 b^2}\\ \end {align*}
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Mathematica [B] time = 0.04, size = 107, normalized size = 2.55 \begin {gather*} \frac {1}{84} x^2 \left (42 a^5 A+21 a^4 x^2 (a B+5 A b)+70 a^3 b x^4 (a B+2 A b)+105 a^2 b^2 x^6 (a B+A b)+7 b^4 x^{10} (5 a B+A b)+42 a b^3 x^8 (2 a B+A b)+6 b^5 B x^{12}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x \left (a+b x^2\right )^5 \left (A+B x^2\right ) \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.37, size = 124, normalized size = 2.95 \begin {gather*} \frac {1}{14} x^{14} b^{5} B + \frac {5}{12} x^{12} b^{4} a B + \frac {1}{12} x^{12} b^{5} A + x^{10} b^{3} a^{2} B + \frac {1}{2} x^{10} b^{4} a A + \frac {5}{4} x^{8} b^{2} a^{3} B + \frac {5}{4} x^{8} b^{3} a^{2} A + \frac {5}{6} x^{6} b a^{4} B + \frac {5}{3} x^{6} b^{2} a^{3} A + \frac {1}{4} x^{4} a^{5} B + \frac {5}{4} x^{4} b a^{4} A + \frac {1}{2} x^{2} a^{5} A \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.34, size = 124, normalized size = 2.95 \begin {gather*} \frac {1}{14} \, B b^{5} x^{14} + \frac {5}{12} \, B a b^{4} x^{12} + \frac {1}{12} \, A b^{5} x^{12} + B a^{2} b^{3} x^{10} + \frac {1}{2} \, A a b^{4} x^{10} + \frac {5}{4} \, B a^{3} b^{2} x^{8} + \frac {5}{4} \, A a^{2} b^{3} x^{8} + \frac {5}{6} \, B a^{4} b x^{6} + \frac {5}{3} \, A a^{3} b^{2} x^{6} + \frac {1}{4} \, B a^{5} x^{4} + \frac {5}{4} \, A a^{4} b x^{4} + \frac {1}{2} \, A a^{5} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.00, size = 124, normalized size = 2.95 \begin {gather*} \frac {B \,b^{5} x^{14}}{14}+\frac {\left (b^{5} A +5 a \,b^{4} B \right ) x^{12}}{12}+\frac {\left (5 a \,b^{4} A +10 a^{2} b^{3} B \right ) x^{10}}{10}+\frac {A \,a^{5} x^{2}}{2}+\frac {\left (10 a^{2} b^{3} A +10 a^{3} b^{2} B \right ) x^{8}}{8}+\frac {\left (10 a^{3} b^{2} A +5 a^{4} b B \right ) x^{6}}{6}+\frac {\left (5 a^{4} b A +a^{5} B \right ) x^{4}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.01, size = 119, normalized size = 2.83 \begin {gather*} \frac {1}{14} \, B b^{5} x^{14} + \frac {1}{12} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{12} + \frac {1}{2} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{10} + \frac {5}{4} \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{8} + \frac {1}{2} \, A a^{5} x^{2} + \frac {5}{6} \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} + \frac {1}{4} \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 107, normalized size = 2.55 \begin {gather*} x^4\,\left (\frac {B\,a^5}{4}+\frac {5\,A\,b\,a^4}{4}\right )+x^{12}\,\left (\frac {A\,b^5}{12}+\frac {5\,B\,a\,b^4}{12}\right )+\frac {A\,a^5\,x^2}{2}+\frac {B\,b^5\,x^{14}}{14}+\frac {5\,a^2\,b^2\,x^8\,\left (A\,b+B\,a\right )}{4}+\frac {5\,a^3\,b\,x^6\,\left (2\,A\,b+B\,a\right )}{6}+\frac {a\,b^3\,x^{10}\,\left (A\,b+2\,B\,a\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.09, size = 133, normalized size = 3.17 \begin {gather*} \frac {A a^{5} x^{2}}{2} + \frac {B b^{5} x^{14}}{14} + x^{12} \left (\frac {A b^{5}}{12} + \frac {5 B a b^{4}}{12}\right ) + x^{10} \left (\frac {A a b^{4}}{2} + B a^{2} b^{3}\right ) + x^{8} \left (\frac {5 A a^{2} b^{3}}{4} + \frac {5 B a^{3} b^{2}}{4}\right ) + x^{6} \left (\frac {5 A a^{3} b^{2}}{3} + \frac {5 B a^{4} b}{6}\right ) + x^{4} \left (\frac {5 A a^{4} b}{4} + \frac {B a^{5}}{4}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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