Optimal. Leaf size=44 \[ \frac {(a+b x)^5 (A b-6 a B)}{30 a^2 x^5}-\frac {A (a+b x)^5}{6 a x^6} \]
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Rubi [A] time = 0.01, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {27, 78, 37} \begin {gather*} \frac {(a+b x)^5 (A b-6 a B)}{30 a^2 x^5}-\frac {A (a+b x)^5}{6 a x^6} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 37
Rule 78
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{x^7} \, dx &=\int \frac {(a+b x)^4 (A+B x)}{x^7} \, dx\\ &=-\frac {A (a+b x)^5}{6 a x^6}+\frac {(-A b+6 a B) \int \frac {(a+b x)^4}{x^6} \, dx}{6 a}\\ &=-\frac {A (a+b x)^5}{6 a x^6}+\frac {(A b-6 a B) (a+b x)^5}{30 a^2 x^5}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 85, normalized size = 1.93 \begin {gather*} -\frac {a^4 (5 A+6 B x)+6 a^3 b x (4 A+5 B x)+15 a^2 b^2 x^2 (3 A+4 B x)+20 a b^3 x^3 (2 A+3 B x)+15 b^4 x^4 (A+2 B x)}{30 x^6} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{x^7} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.40, size = 99, normalized size = 2.25 \begin {gather*} -\frac {30 \, B b^{4} x^{5} + 5 \, A a^{4} + 15 \, {\left (4 \, B a b^{3} + A b^{4}\right )} x^{4} + 20 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{3} + 15 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{2} + 6 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} x}{30 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 99, normalized size = 2.25 \begin {gather*} -\frac {30 \, B b^{4} x^{5} + 60 \, B a b^{3} x^{4} + 15 \, A b^{4} x^{4} + 60 \, B a^{2} b^{2} x^{3} + 40 \, A a b^{3} x^{3} + 30 \, B a^{3} b x^{2} + 45 \, A a^{2} b^{2} x^{2} + 6 \, B a^{4} x + 24 \, A a^{3} b x + 5 \, A a^{4}}{30 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 88, normalized size = 2.00 \begin {gather*} -\frac {B \,b^{4}}{x}-\frac {\left (A b +4 B a \right ) b^{3}}{2 x^{2}}-\frac {2 \left (2 A b +3 B a \right ) a \,b^{2}}{3 x^{3}}-\frac {A \,a^{4}}{6 x^{6}}-\frac {\left (3 A b +2 B a \right ) a^{2} b}{2 x^{4}}-\frac {\left (4 A b +B a \right ) a^{3}}{5 x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.58, size = 99, normalized size = 2.25 \begin {gather*} -\frac {30 \, B b^{4} x^{5} + 5 \, A a^{4} + 15 \, {\left (4 \, B a b^{3} + A b^{4}\right )} x^{4} + 20 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{3} + 15 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{2} + 6 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} x}{30 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 95, normalized size = 2.16 \begin {gather*} -\frac {x\,\left (\frac {B\,a^4}{5}+\frac {4\,A\,b\,a^3}{5}\right )+\frac {A\,a^4}{6}+x^2\,\left (B\,a^3\,b+\frac {3\,A\,a^2\,b^2}{2}\right )+x^3\,\left (2\,B\,a^2\,b^2+\frac {4\,A\,a\,b^3}{3}\right )+x^4\,\left (\frac {A\,b^4}{2}+2\,B\,a\,b^3\right )+B\,b^4\,x^5}{x^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 2.35, size = 107, normalized size = 2.43 \begin {gather*} \frac {- 5 A a^{4} - 30 B b^{4} x^{5} + x^{4} \left (- 15 A b^{4} - 60 B a b^{3}\right ) + x^{3} \left (- 40 A a b^{3} - 60 B a^{2} b^{2}\right ) + x^{2} \left (- 45 A a^{2} b^{2} - 30 B a^{3} b\right ) + x \left (- 24 A a^{3} b - 6 B a^{4}\right )}{30 x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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