Optimal. Leaf size=114 \[ -\frac {a^5 A}{6 x^6}-\frac {a^4 (5 A b+a B)}{4 x^4}-\frac {5 a^3 b (2 A b+a B)}{2 x^2}+\frac {5}{2} a b^3 (A b+2 a B) x^2+\frac {1}{4} b^4 (A b+5 a B) x^4+\frac {1}{6} b^5 B x^6+10 a^2 b^2 (A b+a B) \log (x) \]
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Rubi [A]
time = 0.07, antiderivative size = 114, 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 = {457, 77}
\begin {gather*} -\frac {a^5 A}{6 x^6}-\frac {a^4 (a B+5 A b)}{4 x^4}-\frac {5 a^3 b (a B+2 A b)}{2 x^2}+10 a^2 b^2 \log (x) (a B+A b)+\frac {1}{4} b^4 x^4 (5 a B+A b)+\frac {5}{2} a b^3 x^2 (2 a B+A b)+\frac {1}{6} b^5 B x^6 \end {gather*}
Antiderivative was successfully verified.
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Rule 77
Rule 457
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^7} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {(a+b x)^5 (A+B x)}{x^4} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (5 a b^3 (A b+2 a B)+\frac {a^5 A}{x^4}+\frac {a^4 (5 A b+a B)}{x^3}+\frac {5 a^3 b (2 A b+a B)}{x^2}+\frac {10 a^2 b^2 (A b+a B)}{x}+b^4 (A b+5 a B) x+b^5 B x^2\right ) \, dx,x,x^2\right )\\ &=-\frac {a^5 A}{6 x^6}-\frac {a^4 (5 A b+a B)}{4 x^4}-\frac {5 a^3 b (2 A b+a B)}{2 x^2}+\frac {5}{2} a b^3 (A b+2 a B) x^2+\frac {1}{4} b^4 (A b+5 a B) x^4+\frac {1}{6} b^5 B x^6+10 a^2 b^2 (A b+a B) \log (x)\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 116, normalized size = 1.02 \begin {gather*} \frac {1}{12} \left (-\frac {60 a^3 A b^2}{x^2}+60 a^2 b^3 B x^2+15 a b^4 x^2 \left (2 A+B x^2\right )-\frac {15 a^4 b \left (A+2 B x^2\right )}{x^4}+b^5 x^4 \left (3 A+2 B x^2\right )-\frac {a^5 \left (2 A+3 B x^2\right )}{x^6}+120 a^2 b^2 (A b+a B) \log (x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 111, normalized size = 0.97
method | result | size |
default | \(\frac {b^{5} B \,x^{6}}{6}+\frac {A \,b^{5} x^{4}}{4}+\frac {5 B a \,b^{4} x^{4}}{4}+\frac {5 A a \,b^{4} x^{2}}{2}+5 B \,a^{2} b^{3} x^{2}-\frac {a^{4} \left (5 A b +B a \right )}{4 x^{4}}-\frac {a^{5} A}{6 x^{6}}-\frac {5 a^{3} b \left (2 A b +B a \right )}{2 x^{2}}+10 a^{2} b^{2} \left (A b +B a \right ) \ln \left (x \right )\) | \(111\) |
norman | \(\frac {\left (\frac {1}{4} b^{5} A +\frac {5}{4} a \,b^{4} B \right ) x^{10}+\left (\frac {5}{2} a \,b^{4} A +5 a^{2} b^{3} B \right ) x^{8}+\left (-5 a^{3} b^{2} A -\frac {5}{2} a^{4} b B \right ) x^{4}+\left (-\frac {5}{4} a^{4} b A -\frac {1}{4} a^{5} B \right ) x^{2}-\frac {a^{5} A}{6}+\frac {b^{5} B \,x^{12}}{6}}{x^{6}}+\left (10 a^{2} b^{3} A +10 a^{3} b^{2} B \right ) \ln \left (x \right )\) | \(122\) |
risch | \(\frac {b^{5} B \,x^{6}}{6}+\frac {A \,b^{5} x^{4}}{4}+\frac {5 B a \,b^{4} x^{4}}{4}+\frac {5 A a \,b^{4} x^{2}}{2}+5 B \,a^{2} b^{3} x^{2}+\frac {\left (-5 a^{3} b^{2} A -\frac {5}{2} a^{4} b B \right ) x^{4}+\left (-\frac {5}{4} a^{4} b A -\frac {1}{4} a^{5} B \right ) x^{2}-\frac {a^{5} A}{6}}{x^{6}}+10 A \ln \left (x \right ) a^{2} b^{3}+10 B \ln \left (x \right ) a^{3} b^{2}\) | \(124\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 123, normalized size = 1.08 \begin {gather*} \frac {1}{6} \, B b^{5} x^{6} + \frac {1}{4} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{4} + \frac {5}{2} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{2} + 5 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} \log \left (x^{2}\right ) - \frac {2 \, A a^{5} + 30 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 3 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{12 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.96, size = 123, normalized size = 1.08 \begin {gather*} \frac {2 \, B b^{5} x^{12} + 3 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + 30 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 120 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} \log \left (x\right ) - 2 \, A a^{5} - 30 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} - 3 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{12 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.72, size = 128, normalized size = 1.12 \begin {gather*} \frac {B b^{5} x^{6}}{6} + 10 a^{2} b^{2} \left (A b + B a\right ) \log {\left (x \right )} + x^{4} \left (\frac {A b^{5}}{4} + \frac {5 B a b^{4}}{4}\right ) + x^{2} \cdot \left (\frac {5 A a b^{4}}{2} + 5 B a^{2} b^{3}\right ) + \frac {- 2 A a^{5} + x^{4} \left (- 60 A a^{3} b^{2} - 30 B a^{4} b\right ) + x^{2} \left (- 15 A a^{4} b - 3 B a^{5}\right )}{12 x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.65, size = 151, normalized size = 1.32 \begin {gather*} \frac {1}{6} \, B b^{5} x^{6} + \frac {5}{4} \, B a b^{4} x^{4} + \frac {1}{4} \, A b^{5} x^{4} + 5 \, B a^{2} b^{3} x^{2} + \frac {5}{2} \, A a b^{4} x^{2} + 5 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} \log \left (x^{2}\right ) - \frac {110 \, B a^{3} b^{2} x^{6} + 110 \, A a^{2} b^{3} x^{6} + 30 \, B a^{4} b x^{4} + 60 \, A a^{3} b^{2} x^{4} + 3 \, B a^{5} x^{2} + 15 \, A a^{4} b x^{2} + 2 \, A a^{5}}{12 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 118, normalized size = 1.04 \begin {gather*} x^4\,\left (\frac {A\,b^5}{4}+\frac {5\,B\,a\,b^4}{4}\right )-\frac {\frac {A\,a^5}{6}+x^4\,\left (\frac {5\,B\,a^4\,b}{2}+5\,A\,a^3\,b^2\right )+x^2\,\left (\frac {B\,a^5}{4}+\frac {5\,A\,b\,a^4}{4}\right )}{x^6}+\ln \left (x\right )\,\left (10\,B\,a^3\,b^2+10\,A\,a^2\,b^3\right )+\frac {B\,b^5\,x^6}{6}+\frac {5\,a\,b^3\,x^2\,\left (A\,b+2\,B\,a\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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