Optimal. Leaf size=114 \[ -\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{4 x^4 (a+b x)}-\frac {a A \sqrt {a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)}-\frac {b B \sqrt {a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)} \]
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Rubi [A] time = 0.05, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {770, 76} \begin {gather*} -\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{4 x^4 (a+b x)}-\frac {a A \sqrt {a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)}-\frac {b B \sqrt {a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 76
Rule 770
Rubi steps
\begin {align*} \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{x^6} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right ) (A+B x)}{x^6} \, dx}{a b+b^2 x}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {a A b}{x^6}+\frac {b (A b+a B)}{x^5}+\frac {b^2 B}{x^4}\right ) \, dx}{a b+b^2 x}\\ &=-\frac {a A \sqrt {a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)}-\frac {(A b+a B) \sqrt {a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)}-\frac {b B \sqrt {a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 49, normalized size = 0.43 \begin {gather*} -\frac {\sqrt {(a+b x)^2} (3 a (4 A+5 B x)+5 b x (3 A+4 B x))}{60 x^5 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 1.52, size = 448, normalized size = 3.93 \begin {gather*} \frac {4 b^4 \sqrt {a^2+2 a b x+b^2 x^2} \left (-12 a^5 A b-15 a^5 b B x-63 a^4 A b^2 x-80 a^4 b^2 B x^2-132 a^3 A b^3 x^2-170 a^3 b^3 B x^3-138 a^2 A b^4 x^3-180 a^2 b^4 B x^4-72 a A b^5 x^4-95 a b^5 B x^5-15 A b^6 x^5-20 b^6 B x^6\right )+4 \sqrt {b^2} b^4 \left (12 a^6 A+15 a^6 B x+75 a^5 A b x+95 a^5 b B x^2+195 a^4 A b^2 x^2+250 a^4 b^2 B x^3+270 a^3 A b^3 x^3+350 a^3 b^3 B x^4+210 a^2 A b^4 x^4+275 a^2 b^4 B x^5+87 a A b^5 x^5+115 a b^5 B x^6+15 A b^6 x^6+20 b^6 B x^7\right )}{15 \sqrt {b^2} x^5 \sqrt {a^2+2 a b x+b^2 x^2} \left (-16 a^4 b^4-64 a^3 b^5 x-96 a^2 b^6 x^2-64 a b^7 x^3-16 b^8 x^4\right )+15 x^5 \left (16 a^5 b^5+80 a^4 b^6 x+160 a^3 b^7 x^2+160 a^2 b^8 x^3+80 a b^9 x^4+16 b^{10} x^5\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 27, normalized size = 0.24 \begin {gather*} -\frac {20 \, B b x^{2} + 12 \, A a + 15 \, {\left (B a + A b\right )} x}{60 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 77, normalized size = 0.68 \begin {gather*} -\frac {{\left (5 \, B a b^{4} - 3 \, A b^{5}\right )} \mathrm {sgn}\left (b x + a\right )}{60 \, a^{4}} - \frac {20 \, B b x^{2} \mathrm {sgn}\left (b x + a\right ) + 15 \, B a x \mathrm {sgn}\left (b x + a\right ) + 15 \, A b x \mathrm {sgn}\left (b x + a\right ) + 12 \, A a \mathrm {sgn}\left (b x + a\right )}{60 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 44, normalized size = 0.39 \begin {gather*} -\frac {\left (20 B b \,x^{2}+15 A b x +15 B a x +12 A a \right ) \sqrt {\left (b x +a \right )^{2}}}{60 \left (b x +a \right ) x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.65, size = 315, normalized size = 2.76 \begin {gather*} \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} B b^{4}}{2 \, a^{4}} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} A b^{5}}{2 \, a^{5}} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} B b^{3}}{2 \, a^{3} x} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} A b^{4}}{2 \, a^{4} x} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} B b^{2}}{2 \, a^{4} x^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} A b^{3}}{2 \, a^{5} x^{2}} + \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} B b}{12 \, a^{3} x^{3}} - \frac {9 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} A b^{2}}{20 \, a^{4} x^{3}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} B}{4 \, a^{2} x^{4}} + \frac {7 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} A b}{20 \, a^{3} x^{4}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} A}{5 \, a^{2} x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.12, size = 43, normalized size = 0.38 \begin {gather*} -\frac {\sqrt {{\left (a+b\,x\right )}^2}\,\left (12\,A\,a+15\,A\,b\,x+15\,B\,a\,x+20\,B\,b\,x^2\right )}{60\,x^5\,\left (a+b\,x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.43, size = 31, normalized size = 0.27 \begin {gather*} \frac {- 12 A a - 20 B b x^{2} + x \left (- 15 A b - 15 B a\right )}{60 x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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