Optimal. Leaf size=73 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{3/2} b^{3/2}}+\frac {\sqrt {x}}{4 a b (a+b x)}-\frac {\sqrt {x}}{2 b (a+b x)^2} \]
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Rubi [A] time = 0.02, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {47, 51, 63, 205} \[ \frac {\tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{3/2} b^{3/2}}+\frac {\sqrt {x}}{4 a b (a+b x)}-\frac {\sqrt {x}}{2 b (a+b x)^2} \]
Antiderivative was successfully verified.
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Rule 47
Rule 51
Rule 63
Rule 205
Rubi steps
\begin {align*} \int \frac {\sqrt {x}}{(a+b x)^3} \, dx &=-\frac {\sqrt {x}}{2 b (a+b x)^2}+\frac {\int \frac {1}{\sqrt {x} (a+b x)^2} \, dx}{4 b}\\ &=-\frac {\sqrt {x}}{2 b (a+b x)^2}+\frac {\sqrt {x}}{4 a b (a+b x)}+\frac {\int \frac {1}{\sqrt {x} (a+b x)} \, dx}{8 a b}\\ &=-\frac {\sqrt {x}}{2 b (a+b x)^2}+\frac {\sqrt {x}}{4 a b (a+b x)}+\frac {\operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {x}\right )}{4 a b}\\ &=-\frac {\sqrt {x}}{2 b (a+b x)^2}+\frac {\sqrt {x}}{4 a b (a+b x)}+\frac {\tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{3/2} b^{3/2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 27, normalized size = 0.37 \[ \frac {2 x^{3/2} \, _2F_1\left (\frac {3}{2},3;\frac {5}{2};-\frac {b x}{a}\right )}{3 a^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 186, normalized size = 2.55 \[ \left [-\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt {-a b} \log \left (\frac {b x - a - 2 \, \sqrt {-a b} \sqrt {x}}{b x + a}\right ) - 2 \, {\left (a b^{2} x - a^{2} b\right )} \sqrt {x}}{8 \, {\left (a^{2} b^{4} x^{2} + 2 \, a^{3} b^{3} x + a^{4} b^{2}\right )}}, -\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b}}{b \sqrt {x}}\right ) - {\left (a b^{2} x - a^{2} b\right )} \sqrt {x}}{4 \, {\left (a^{2} b^{4} x^{2} + 2 \, a^{3} b^{3} x + a^{4} b^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.07, size = 52, normalized size = 0.71 \[ \frac {\arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} a b} + \frac {b x^{\frac {3}{2}} - a \sqrt {x}}{4 \, {\left (b x + a\right )}^{2} a b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 52, normalized size = 0.71 \[ \frac {\arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \sqrt {a b}\, a b}+\frac {\frac {x^{\frac {3}{2}}}{4 a}-\frac {\sqrt {x}}{4 b}}{\left (b x +a \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.01, size = 64, normalized size = 0.88 \[ \frac {b x^{\frac {3}{2}} - a \sqrt {x}}{4 \, {\left (a b^{3} x^{2} + 2 \, a^{2} b^{2} x + a^{3} b\right )}} + \frac {\arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} a b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.13, size = 56, normalized size = 0.77 \[ \frac {\frac {x^{3/2}}{4\,a}-\frac {\sqrt {x}}{4\,b}}{a^2+2\,a\,b\,x+b^2\,x^2}+\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )}{4\,a^{3/2}\,b^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 15.27, size = 721, normalized size = 9.88 \[ \begin {cases} \frac {\tilde {\infty }}{x^{\frac {3}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {2 x^{\frac {3}{2}}}{3 a^{3}} & \text {for}\: b = 0 \\- \frac {2}{3 b^{3} x^{\frac {3}{2}}} & \text {for}\: a = 0 \\- \frac {2 i a^{\frac {3}{2}} b \sqrt {x} \sqrt {\frac {1}{b}}}{8 i a^{\frac {7}{2}} b^{2} \sqrt {\frac {1}{b}} + 16 i a^{\frac {5}{2}} b^{3} x \sqrt {\frac {1}{b}} + 8 i a^{\frac {3}{2}} b^{4} x^{2} \sqrt {\frac {1}{b}}} + \frac {2 i \sqrt {a} b^{2} x^{\frac {3}{2}} \sqrt {\frac {1}{b}}}{8 i a^{\frac {7}{2}} b^{2} \sqrt {\frac {1}{b}} + 16 i a^{\frac {5}{2}} b^{3} x \sqrt {\frac {1}{b}} + 8 i a^{\frac {3}{2}} b^{4} x^{2} \sqrt {\frac {1}{b}}} + \frac {a^{2} \log {\left (- i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{8 i a^{\frac {7}{2}} b^{2} \sqrt {\frac {1}{b}} + 16 i a^{\frac {5}{2}} b^{3} x \sqrt {\frac {1}{b}} + 8 i a^{\frac {3}{2}} b^{4} x^{2} \sqrt {\frac {1}{b}}} - \frac {a^{2} \log {\left (i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{8 i a^{\frac {7}{2}} b^{2} \sqrt {\frac {1}{b}} + 16 i a^{\frac {5}{2}} b^{3} x \sqrt {\frac {1}{b}} + 8 i a^{\frac {3}{2}} b^{4} x^{2} \sqrt {\frac {1}{b}}} + \frac {2 a b x \log {\left (- i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{8 i a^{\frac {7}{2}} b^{2} \sqrt {\frac {1}{b}} + 16 i a^{\frac {5}{2}} b^{3} x \sqrt {\frac {1}{b}} + 8 i a^{\frac {3}{2}} b^{4} x^{2} \sqrt {\frac {1}{b}}} - \frac {2 a b x \log {\left (i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{8 i a^{\frac {7}{2}} b^{2} \sqrt {\frac {1}{b}} + 16 i a^{\frac {5}{2}} b^{3} x \sqrt {\frac {1}{b}} + 8 i a^{\frac {3}{2}} b^{4} x^{2} \sqrt {\frac {1}{b}}} + \frac {b^{2} x^{2} \log {\left (- i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{8 i a^{\frac {7}{2}} b^{2} \sqrt {\frac {1}{b}} + 16 i a^{\frac {5}{2}} b^{3} x \sqrt {\frac {1}{b}} + 8 i a^{\frac {3}{2}} b^{4} x^{2} \sqrt {\frac {1}{b}}} - \frac {b^{2} x^{2} \log {\left (i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{8 i a^{\frac {7}{2}} b^{2} \sqrt {\frac {1}{b}} + 16 i a^{\frac {5}{2}} b^{3} x \sqrt {\frac {1}{b}} + 8 i a^{\frac {3}{2}} b^{4} x^{2} \sqrt {\frac {1}{b}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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