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 = 0.012025, antiderivative size = 44, normalized size of antiderivative = 1., 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} \[ \frac{(a+b x)^5 (A b-6 a B)}{30 a^2 x^5}-\frac{A (a+b x)^5}{6 a x^6} \]
Antiderivative was successfully verified.
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Rule 27
Rule 78
Rule 37
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*}
Mathematica [A] time = 0.0241847, size = 85, normalized size = 1.93 \[ -\frac{15 a^2 b^2 x^2 (3 A+4 B x)+6 a^3 b x (4 A+5 B x)+a^4 (5 A+6 B x)+20 a b^3 x^3 (2 A+3 B x)+15 b^4 x^4 (A+2 B x)}{30 x^6} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.006, size = 88, normalized size = 2. \begin{align*} -{\frac{2\,a{b}^{2} \left ( 2\,Ab+3\,aB \right ) }{3\,{x}^{3}}}-{\frac{{b}^{3} \left ( Ab+4\,aB \right ) }{2\,{x}^{2}}}-{\frac{{b}^{4}B}{x}}-{\frac{{a}^{2}b \left ( 3\,Ab+2\,aB \right ) }{2\,{x}^{4}}}-{\frac{{a}^{3} \left ( 4\,Ab+aB \right ) }{5\,{x}^{5}}}-{\frac{A{a}^{4}}{6\,{x}^{6}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.01383, size = 134, normalized size = 3.05 \begin{align*} -\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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.19156, size = 217, normalized size = 4.93 \begin{align*} -\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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 2.74023, size = 102, normalized size = 2.32 \begin{align*} - \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.14564, size = 134, normalized size = 3.05 \begin{align*} -\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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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