Optimal. Leaf size=73 \[ -\frac {\sqrt {x}}{2 b (a+b x)^2}+\frac {\sqrt {x}}{4 a b (a+b x)}+\frac {\tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{3/2} b^{3/2}} \]
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Rubi [A]
time = 0.01, 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 = {43, 44, 65, 211}
\begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 44
Rule 65
Rule 211
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 {\text {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 [A]
time = 0.09, size = 60, normalized size = 0.82 \begin {gather*} -\frac {\sqrt {x} (a-b x)}{4 a b (a+b x)^2}+\frac {\tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{3/2} b^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 6.82, size = 612, normalized size = 8.38 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\text {DirectedInfinity}\left [\frac {1}{x^{\frac {3}{2}}}\right ],a\text {==}0\text {\&\&}b\text {==}0\right \},\left \{\frac {-2}{3 b^3 x^{\frac {3}{2}}},a\text {==}0\right \},\left \{\frac {2 x^{\frac {3}{2}}}{3 a^3},b\text {==}0\right \}\right \},-\frac {a^2 \text {Log}\left [\sqrt {x}+\sqrt {-\frac {a}{b}}\right ]}{8 a^3 b^2 \sqrt {-\frac {a}{b}}+16 a^2 b^3 x \sqrt {-\frac {a}{b}}+8 a b^4 x^2 \sqrt {-\frac {a}{b}}}+\frac {a^2 \text {Log}\left [\sqrt {x}-\sqrt {-\frac {a}{b}}\right ]}{8 a^3 b^2 \sqrt {-\frac {a}{b}}+16 a^2 b^3 x \sqrt {-\frac {a}{b}}+8 a b^4 x^2 \sqrt {-\frac {a}{b}}}-\frac {2 a b \sqrt {x} \sqrt {-\frac {a}{b}}}{8 a^3 b^2 \sqrt {-\frac {a}{b}}+16 a^2 b^3 x \sqrt {-\frac {a}{b}}+8 a b^4 x^2 \sqrt {-\frac {a}{b}}}-\frac {2 a b x \text {Log}\left [\sqrt {x}+\sqrt {-\frac {a}{b}}\right ]}{8 a^3 b^2 \sqrt {-\frac {a}{b}}+16 a^2 b^3 x \sqrt {-\frac {a}{b}}+8 a b^4 x^2 \sqrt {-\frac {a}{b}}}+\frac {2 a b x \text {Log}\left [\sqrt {x}-\sqrt {-\frac {a}{b}}\right ]}{8 a^3 b^2 \sqrt {-\frac {a}{b}}+16 a^2 b^3 x \sqrt {-\frac {a}{b}}+8 a b^4 x^2 \sqrt {-\frac {a}{b}}}+\frac {2 b^2 x^{\frac {3}{2}} \sqrt {-\frac {a}{b}}}{8 a^3 b^2 \sqrt {-\frac {a}{b}}+16 a^2 b^3 x \sqrt {-\frac {a}{b}}+8 a b^4 x^2 \sqrt {-\frac {a}{b}}}-\frac {b^2 x^2 \text {Log}\left [\sqrt {x}+\sqrt {-\frac {a}{b}}\right ]}{8 a^3 b^2 \sqrt {-\frac {a}{b}}+16 a^2 b^3 x \sqrt {-\frac {a}{b}}+8 a b^4 x^2 \sqrt {-\frac {a}{b}}}+\frac {b^2 x^2 \text {Log}\left [\sqrt {x}-\sqrt {-\frac {a}{b}}\right ]}{8 a^3 b^2 \sqrt {-\frac {a}{b}}+16 a^2 b^3 x \sqrt {-\frac {a}{b}}+8 a b^4 x^2 \sqrt {-\frac {a}{b}}}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.13, size = 52, normalized size = 0.71
method | result | size |
derivativedivides | \(\frac {\frac {x^{\frac {3}{2}}}{4 a}-\frac {\sqrt {x}}{4 b}}{\left (b x +a \right )^{2}}+\frac {\arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 a b \sqrt {a b}}\) | \(52\) |
default | \(\frac {\frac {x^{\frac {3}{2}}}{4 a}-\frac {\sqrt {x}}{4 b}}{\left (b x +a \right )^{2}}+\frac {\arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 a b \sqrt {a b}}\) | \(52\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.34, size = 64, normalized size = 0.88 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.31, size = 186, normalized size = 2.55 \begin {gather*} \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 ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 7.17, size = 627, normalized size = 8.59 \begin {gather*} \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 {a^{2} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{8 a^{3} b^{2} \sqrt {- \frac {a}{b}} + 16 a^{2} b^{3} x \sqrt {- \frac {a}{b}} + 8 a b^{4} x^{2} \sqrt {- \frac {a}{b}}} - \frac {a^{2} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{8 a^{3} b^{2} \sqrt {- \frac {a}{b}} + 16 a^{2} b^{3} x \sqrt {- \frac {a}{b}} + 8 a b^{4} x^{2} \sqrt {- \frac {a}{b}}} - \frac {2 a b \sqrt {x} \sqrt {- \frac {a}{b}}}{8 a^{3} b^{2} \sqrt {- \frac {a}{b}} + 16 a^{2} b^{3} x \sqrt {- \frac {a}{b}} + 8 a b^{4} x^{2} \sqrt {- \frac {a}{b}}} + \frac {2 a b x \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{8 a^{3} b^{2} \sqrt {- \frac {a}{b}} + 16 a^{2} b^{3} x \sqrt {- \frac {a}{b}} + 8 a b^{4} x^{2} \sqrt {- \frac {a}{b}}} - \frac {2 a b x \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{8 a^{3} b^{2} \sqrt {- \frac {a}{b}} + 16 a^{2} b^{3} x \sqrt {- \frac {a}{b}} + 8 a b^{4} x^{2} \sqrt {- \frac {a}{b}}} + \frac {2 b^{2} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}}}{8 a^{3} b^{2} \sqrt {- \frac {a}{b}} + 16 a^{2} b^{3} x \sqrt {- \frac {a}{b}} + 8 a b^{4} x^{2} \sqrt {- \frac {a}{b}}} + \frac {b^{2} x^{2} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{8 a^{3} b^{2} \sqrt {- \frac {a}{b}} + 16 a^{2} b^{3} x \sqrt {- \frac {a}{b}} + 8 a b^{4} x^{2} \sqrt {- \frac {a}{b}}} - \frac {b^{2} x^{2} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{8 a^{3} b^{2} \sqrt {- \frac {a}{b}} + 16 a^{2} b^{3} x \sqrt {- \frac {a}{b}} + 8 a b^{4} x^{2} \sqrt {- \frac {a}{b}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 68, normalized size = 0.93 \begin {gather*} 2 \left (\frac {\sqrt {x} x b-\sqrt {x} a}{8 b a \left (x b+a\right )^{2}}+\frac {\arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 b a\cdot 2 \sqrt {a b}}\right ) \end {gather*}
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 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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