Optimal. Leaf size=69 \[ -\frac {2 A (a+b x)^{3/2}}{3 a x^{3/2}}-\frac {2 B \sqrt {a+b x}}{\sqrt {x}}+2 \sqrt {b} B \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {78, 47, 63, 217, 206} \[ -\frac {2 A (a+b x)^{3/2}}{3 a x^{3/2}}-\frac {2 B \sqrt {a+b x}}{\sqrt {x}}+2 \sqrt {b} B \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right ) \]
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 78
Rule 206
Rule 217
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x} (A+B x)}{x^{5/2}} \, dx &=-\frac {2 A (a+b x)^{3/2}}{3 a x^{3/2}}+B \int \frac {\sqrt {a+b x}}{x^{3/2}} \, dx\\ &=-\frac {2 B \sqrt {a+b x}}{\sqrt {x}}-\frac {2 A (a+b x)^{3/2}}{3 a x^{3/2}}+(b B) \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx\\ &=-\frac {2 B \sqrt {a+b x}}{\sqrt {x}}-\frac {2 A (a+b x)^{3/2}}{3 a x^{3/2}}+(2 b B) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {x}\right )\\ &=-\frac {2 B \sqrt {a+b x}}{\sqrt {x}}-\frac {2 A (a+b x)^{3/2}}{3 a x^{3/2}}+(2 b B) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a+b x}}\right )\\ &=-\frac {2 B \sqrt {a+b x}}{\sqrt {x}}-\frac {2 A (a+b x)^{3/2}}{3 a x^{3/2}}+2 \sqrt {b} B \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )\\ \end {align*}
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Mathematica [C] time = 0.10, size = 85, normalized size = 1.23 \[ -\frac {2 (a+b x)^{3/2} (A b-a B)}{3 a b x^{3/2}}-\frac {2 a B \sqrt {a+b x} \, _2F_1\left (-\frac {3}{2},-\frac {3}{2};-\frac {1}{2};-\frac {b x}{a}\right )}{3 b x^{3/2} \sqrt {\frac {b x}{a}+1}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 134, normalized size = 1.94 \[ \left [\frac {3 \, B a \sqrt {b} x^{2} \log \left (2 \, b x + 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) - 2 \, {\left (A a + {\left (3 \, B a + A b\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{3 \, a x^{2}}, -\frac {2 \, {\left (3 \, B a \sqrt {-b} x^{2} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) + {\left (A a + {\left (3 \, B a + A b\right )} x\right )} \sqrt {b x + a} \sqrt {x}\right )}}{3 \, a x^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [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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maple [B] time = 0.02, size = 112, normalized size = 1.62 \[ -\frac {\sqrt {b x +a}\, \left (-3 B a b \,x^{2} \ln \left (\frac {2 b x +a +2 \sqrt {\left (b x +a \right ) x}\, \sqrt {b}}{2 \sqrt {b}}\right )+2 \sqrt {\left (b x +a \right ) x}\, A \,b^{\frac {3}{2}} x +6 \sqrt {\left (b x +a \right ) x}\, B a \sqrt {b}\, x +2 \sqrt {\left (b x +a \right ) x}\, A a \sqrt {b}\right )}{3 \sqrt {\left (b x +a \right ) x}\, a \sqrt {b}\, x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.01, size = 73, normalized size = 1.06 \[ -{\left (\sqrt {b} \log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x + a}}{\sqrt {x}}}{\sqrt {b} + \frac {\sqrt {b x + a}}{\sqrt {x}}}\right ) + \frac {2 \, \sqrt {b x + a}}{\sqrt {x}}\right )} B - \frac {2 \, {\left (b x + a\right )}^{\frac {3}{2}} A}{3 \, a x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\left (A+B\,x\right )\,\sqrt {a+b\,x}}{x^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 24.27, size = 114, normalized size = 1.65 \[ A \left (- \frac {2 \sqrt {b} \sqrt {\frac {a}{b x} + 1}}{3 x} - \frac {2 b^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}}{3 a}\right ) + B \left (- \frac {2 \sqrt {a}}{\sqrt {x} \sqrt {1 + \frac {b x}{a}}} + 2 \sqrt {b} \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )} - \frac {2 b \sqrt {x}}{\sqrt {a} \sqrt {1 + \frac {b x}{a}}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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