Optimal. Leaf size=120 \[ -\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{5 x^{5/2} (a+b x)}-\frac {2 a A \sqrt {a^2+2 a b x+b^2 x^2}}{7 x^{7/2} (a+b x)}-\frac {2 b B \sqrt {a^2+2 a b x+b^2 x^2}}{3 x^{3/2} (a+b x)} \]
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Rubi [A] time = 0.04, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {770, 76} \[ -\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{5 x^{5/2} (a+b x)}-\frac {2 a A \sqrt {a^2+2 a b x+b^2 x^2}}{7 x^{7/2} (a+b x)}-\frac {2 b B \sqrt {a^2+2 a b x+b^2 x^2}}{3 x^{3/2} (a+b x)} \]
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^{9/2}} \, 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^{9/2}} \, 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^{9/2}}+\frac {b (A b+a B)}{x^{7/2}}+\frac {b^2 B}{x^{5/2}}\right ) \, dx}{a b+b^2 x}\\ &=-\frac {2 a A \sqrt {a^2+2 a b x+b^2 x^2}}{7 x^{7/2} (a+b x)}-\frac {2 (A b+a B) \sqrt {a^2+2 a b x+b^2 x^2}}{5 x^{5/2} (a+b x)}-\frac {2 b B \sqrt {a^2+2 a b x+b^2 x^2}}{3 x^{3/2} (a+b x)}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 51, normalized size = 0.42 \[ -\frac {2 \sqrt {(a+b x)^2} (3 a (5 A+7 B x)+7 b x (3 A+5 B x))}{105 x^{7/2} (a+b x)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.95, size = 27, normalized size = 0.22 \[ -\frac {2 \, {\left (35 \, B b x^{2} + 15 \, A a + 21 \, {\left (B a + A b\right )} x\right )}}{105 \, x^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 51, normalized size = 0.42 \[ -\frac {2 \, {\left (35 \, B b x^{2} \mathrm {sgn}\left (b x + a\right ) + 21 \, B a x \mathrm {sgn}\left (b x + a\right ) + 21 \, A b x \mathrm {sgn}\left (b x + a\right ) + 15 \, A a \mathrm {sgn}\left (b x + a\right )\right )}}{105 \, x^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 44, normalized size = 0.37 \[ -\frac {2 \left (35 B b \,x^{2}+21 A b x +21 B a x +15 A a \right ) \sqrt {\left (b x +a \right )^{2}}}{105 \left (b x +a \right ) x^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.66, size = 35, normalized size = 0.29 \[ -\frac {2 \, {\left (5 \, b x^{2} + 3 \, a x\right )} B}{15 \, x^{\frac {7}{2}}} - \frac {2 \, {\left (7 \, b x^{2} + 5 \, a x\right )} A}{35 \, x^{\frac {9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.35, size = 54, normalized size = 0.45 \[ -\frac {\sqrt {{\left (a+b\,x\right )}^2}\,\left (\frac {2\,B\,x^2}{3}+\frac {2\,A\,a}{7\,b}+\frac {x\,\left (42\,A\,b+42\,B\,a\right )}{105\,b}\right )}{x^{9/2}+\frac {a\,x^{7/2}}{b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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