Optimal. Leaf size=53 \[ \frac {2 (a+b x)^{3/2} (2 A b-5 a B)}{15 a^2 x^{3/2}}-\frac {2 A (a+b x)^{3/2}}{5 a x^{5/2}} \]
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Rubi [A] time = 0.02, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {78, 37} \[ \frac {2 (a+b x)^{3/2} (2 A b-5 a B)}{15 a^2 x^{3/2}}-\frac {2 A (a+b x)^{3/2}}{5 a x^{5/2}} \]
Antiderivative was successfully verified.
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Rule 37
Rule 78
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x} (A+B x)}{x^{7/2}} \, dx &=-\frac {2 A (a+b x)^{3/2}}{5 a x^{5/2}}+\frac {\left (2 \left (-A b+\frac {5 a B}{2}\right )\right ) \int \frac {\sqrt {a+b x}}{x^{5/2}} \, dx}{5 a}\\ &=-\frac {2 A (a+b x)^{3/2}}{5 a x^{5/2}}+\frac {2 (2 A b-5 a B) (a+b x)^{3/2}}{15 a^2 x^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 36, normalized size = 0.68 \[ -\frac {2 (a+b x)^{3/2} (3 a A+5 a B x-2 A b x)}{15 a^2 x^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 51, normalized size = 0.96 \[ -\frac {2 \, {\left (3 \, A a^{2} + {\left (5 \, B a b - 2 \, A b^{2}\right )} x^{2} + {\left (5 \, B a^{2} + A a b\right )} x\right )} \sqrt {b x + a}}{15 \, a^{2} x^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.38, size = 73, normalized size = 1.38 \[ -\frac {2 \, {\left (b x + a\right )}^{\frac {3}{2}} b {\left (\frac {{\left (5 \, B a b^{4} - 2 \, A b^{5}\right )} {\left (b x + a\right )}}{a^{2}} - \frac {5 \, {\left (B a^{2} b^{4} - A a b^{5}\right )}}{a^{2}}\right )}}{15 \, {\left ({\left (b x + a\right )} b - a b\right )}^{\frac {5}{2}} {\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 31, normalized size = 0.58 \[ -\frac {2 \left (b x +a \right )^{\frac {3}{2}} \left (-2 A x b +5 B a x +3 A a \right )}{15 a^{2} x^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.84, size = 100, normalized size = 1.89 \[ -\frac {2 \, \sqrt {b x^{2} + a x} B b}{3 \, a x} + \frac {4 \, \sqrt {b x^{2} + a x} A b^{2}}{15 \, a^{2} x} - \frac {2 \, \sqrt {b x^{2} + a x} B}{3 \, x^{2}} - \frac {2 \, \sqrt {b x^{2} + a x} A b}{15 \, a x^{2}} - \frac {2 \, \sqrt {b x^{2} + a x} A}{5 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.67, size = 54, normalized size = 1.02 \[ -\frac {\sqrt {a+b\,x}\,\left (\frac {2\,A}{5}-\frac {x^2\,\left (4\,A\,b^2-10\,B\,a\,b\right )}{15\,a^2}+\frac {x\,\left (10\,B\,a^2+2\,A\,b\,a\right )}{15\,a^2}\right )}{x^{5/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 61.24, size = 110, normalized size = 2.08 \[ A \left (- \frac {2 \sqrt {b} \sqrt {\frac {a}{b x} + 1}}{5 x^{2}} - \frac {2 b^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}}{15 a x} + \frac {4 b^{\frac {5}{2}} \sqrt {\frac {a}{b x} + 1}}{15 a^{2}}\right ) + B \left (- \frac {2 \sqrt {b} \sqrt {\frac {a}{b x} + 1}}{3 x} - \frac {2 b^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}}{3 a}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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