Optimal. Leaf size=75 \[ \frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (A b-a B)}{2 a^2 x^2}-\frac {A \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{3 a^2 x^3} \]
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Rubi [A] time = 0.04, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {769, 646, 37} \[ \frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (A b-a B)}{2 a^2 x^2}-\frac {A \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{3 a^2 x^3} \]
Antiderivative was successfully verified.
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Rule 37
Rule 646
Rule 769
Rubi steps
\begin {align*} \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{x^4} \, dx &=-\frac {A \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{3 a^2 x^3}-\frac {\left (2 A b^2-2 a b B\right ) \int \frac {\sqrt {a^2+2 a b x+b^2 x^2}}{x^3} \, dx}{2 a b}\\ &=-\frac {A \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{3 a^2 x^3}-\frac {\left (\left (2 A b^2-2 a b B\right ) \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \frac {a b+b^2 x}{x^3} \, dx}{2 a b \left (a b+b^2 x\right )}\\ &=\frac {(A b-a B) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{2 a^2 x^2}-\frac {A \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{3 a^2 x^3}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 46, normalized size = 0.61 \[ -\frac {\sqrt {(a+b x)^2} (a (2 A+3 B x)+3 b x (A+2 B x))}{6 x^3 (a+b x)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 27, normalized size = 0.36 \[ -\frac {6 \, B b x^{2} + 2 \, A a + 3 \, {\left (B a + A b\right )} x}{6 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 77, normalized size = 1.03 \[ -\frac {{\left (3 \, B a b^{2} - A b^{3}\right )} \mathrm {sgn}\left (b x + a\right )}{6 \, a^{2}} - \frac {6 \, B b x^{2} \mathrm {sgn}\left (b x + a\right ) + 3 \, B a x \mathrm {sgn}\left (b x + a\right ) + 3 \, A b x \mathrm {sgn}\left (b x + a\right ) + 2 \, A a \mathrm {sgn}\left (b x + a\right )}{6 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 44, normalized size = 0.59 \[ -\frac {\left (6 B b \,x^{2}+3 A b x +3 B a x +2 A a \right ) \sqrt {\left (b x +a \right )^{2}}}{6 \left (b x +a \right ) x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.49, size = 195, normalized size = 2.60 \[ \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} B b^{2}}{2 \, a^{2}} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} A b^{3}}{2 \, a^{3}} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} B b}{2 \, a x} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} A b^{2}}{2 \, a^{2} x} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} B}{2 \, a^{2} x^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} A b}{2 \, a^{3} x^{2}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} A}{3 \, a^{2} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.11, size = 43, normalized size = 0.57 \[ -\frac {\sqrt {{\left (a+b\,x\right )}^2}\,\left (2\,A\,a+3\,A\,b\,x+3\,B\,a\,x+6\,B\,b\,x^2\right )}{6\,x^3\,\left (a+b\,x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.30, size = 31, normalized size = 0.41 \[ \frac {- 2 A a - 6 B b x^{2} + x \left (- 3 A b - 3 B a\right )}{6 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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