Optimal. Leaf size=118 \[ -\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{3 x^{3/2} (a+b x)}-\frac {2 a A \sqrt {a^2+2 a b x+b^2 x^2}}{5 x^{5/2} (a+b x)}-\frac {2 b B \sqrt {a^2+2 a b x+b^2 x^2}}{\sqrt {x} (a+b x)} \]
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Rubi [A] time = 0.04, antiderivative size = 118, 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} \begin {gather*} -\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{3 x^{3/2} (a+b x)}-\frac {2 a A \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}}{\sqrt {x} (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^{7/2}} \, 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^{7/2}} \, 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^{7/2}}+\frac {b (A b+a B)}{x^{5/2}}+\frac {b^2 B}{x^{3/2}}\right ) \, dx}{a b+b^2 x}\\ &=-\frac {2 a A \sqrt {a^2+2 a b x+b^2 x^2}}{5 x^{5/2} (a+b x)}-\frac {2 (A b+a B) \sqrt {a^2+2 a b x+b^2 x^2}}{3 x^{3/2} (a+b x)}-\frac {2 b B \sqrt {a^2+2 a b x+b^2 x^2}}{\sqrt {x} (a+b x)}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 48, normalized size = 0.41 \begin {gather*} -\frac {2 \sqrt {(a+b x)^2} (a (3 A+5 B x)+5 b x (A+3 B x))}{15 x^{5/2} (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 16.10, size = 49, normalized size = 0.42 \begin {gather*} -\frac {2 \sqrt {(a+b x)^2} \left (3 a A+5 a B x+5 A b x+15 b B x^2\right )}{15 x^{5/2} (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 27, normalized size = 0.23 \begin {gather*} -\frac {2 \, {\left (15 \, B b x^{2} + 3 \, A a + 5 \, {\left (B a + A b\right )} x\right )}}{15 \, x^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 51, normalized size = 0.43 \begin {gather*} -\frac {2 \, {\left (15 \, B b x^{2} \mathrm {sgn}\left (b x + a\right ) + 5 \, B a x \mathrm {sgn}\left (b x + a\right ) + 5 \, A b x \mathrm {sgn}\left (b x + a\right ) + 3 \, A a \mathrm {sgn}\left (b x + a\right )\right )}}{15 \, x^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 44, normalized size = 0.37 \begin {gather*} -\frac {2 \left (15 B b \,x^{2}+5 A b x +5 B a x +3 A a \right ) \sqrt {\left (b x +a \right )^{2}}}{15 \left (b x +a \right ) x^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.57, size = 34, normalized size = 0.29 \begin {gather*} -\frac {2 \, {\left (3 \, b x^{2} + a x\right )} B}{3 \, x^{\frac {5}{2}}} - \frac {2 \, {\left (5 \, b x^{2} + 3 \, a x\right )} A}{15 \, x^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.36, size = 54, normalized size = 0.46 \begin {gather*} -\frac {\sqrt {{\left (a+b\,x\right )}^2}\,\left (2\,B\,x^2+\frac {2\,A\,a}{5\,b}+\frac {x\,\left (10\,A\,b+10\,B\,a\right )}{15\,b}\right )}{x^{7/2}+\frac {a\,x^{5/2}}{b}} \end {gather*}
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 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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