Optimal. Leaf size=70 \[ -\frac {x^{3/2}}{2 b (a+b x)^2}-\frac {3 \sqrt {x}}{4 b^2 (a+b x)}+\frac {3 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 \sqrt {a} b^{5/2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {43, 65, 211}
\begin {gather*} \frac {3 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 \sqrt {a} b^{5/2}}-\frac {3 \sqrt {x}}{4 b^2 (a+b x)}-\frac {x^{3/2}}{2 b (a+b x)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 65
Rule 211
Rubi steps
\begin {align*} \int \frac {x^{3/2}}{(a+b x)^3} \, dx &=-\frac {x^{3/2}}{2 b (a+b x)^2}+\frac {3 \int \frac {\sqrt {x}}{(a+b x)^2} \, dx}{4 b}\\ &=-\frac {x^{3/2}}{2 b (a+b x)^2}-\frac {3 \sqrt {x}}{4 b^2 (a+b x)}+\frac {3 \int \frac {1}{\sqrt {x} (a+b x)} \, dx}{8 b^2}\\ &=-\frac {x^{3/2}}{2 b (a+b x)^2}-\frac {3 \sqrt {x}}{4 b^2 (a+b x)}+\frac {3 \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {x}\right )}{4 b^2}\\ &=-\frac {x^{3/2}}{2 b (a+b x)^2}-\frac {3 \sqrt {x}}{4 b^2 (a+b x)}+\frac {3 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 \sqrt {a} b^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 59, normalized size = 0.84 \begin {gather*} -\frac {\sqrt {x} (3 a+5 b x)}{4 b^2 (a+b x)^2}+\frac {3 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 \sqrt {a} b^{5/2}} \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 = 17.35, size = 590, normalized size = 8.43 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\text {DirectedInfinity}\left [\frac {1}{\sqrt {x}}\right ],a\text {==}0\text {\&\&}b\text {==}0\right \},\left \{\frac {-2}{b^3 \sqrt {x}},a\text {==}0\right \},\left \{\frac {2 x^{\frac {5}{2}}}{5 a^3},b\text {==}0\right \}\right \},\frac {-3 a^2 \text {Log}\left [\sqrt {x}+\sqrt {-\frac {a}{b}}\right ]}{8 a^2 b^3 \sqrt {-\frac {a}{b}}+16 a b^4 x \sqrt {-\frac {a}{b}}+8 b^5 x^2 \sqrt {-\frac {a}{b}}}+\frac {3 a^2 \text {Log}\left [\sqrt {x}-\sqrt {-\frac {a}{b}}\right ]}{8 a^2 b^3 \sqrt {-\frac {a}{b}}+16 a b^4 x \sqrt {-\frac {a}{b}}+8 b^5 x^2 \sqrt {-\frac {a}{b}}}-\frac {6 a b \sqrt {x} \sqrt {-\frac {a}{b}}}{8 a^2 b^3 \sqrt {-\frac {a}{b}}+16 a b^4 x \sqrt {-\frac {a}{b}}+8 b^5 x^2 \sqrt {-\frac {a}{b}}}-\frac {6 a b x \text {Log}\left [\sqrt {x}+\sqrt {-\frac {a}{b}}\right ]}{8 a^2 b^3 \sqrt {-\frac {a}{b}}+16 a b^4 x \sqrt {-\frac {a}{b}}+8 b^5 x^2 \sqrt {-\frac {a}{b}}}+\frac {6 a b x \text {Log}\left [\sqrt {x}-\sqrt {-\frac {a}{b}}\right ]}{8 a^2 b^3 \sqrt {-\frac {a}{b}}+16 a b^4 x \sqrt {-\frac {a}{b}}+8 b^5 x^2 \sqrt {-\frac {a}{b}}}-\frac {10 b^2 x^{\frac {3}{2}} \sqrt {-\frac {a}{b}}}{8 a^2 b^3 \sqrt {-\frac {a}{b}}+16 a b^4 x \sqrt {-\frac {a}{b}}+8 b^5 x^2 \sqrt {-\frac {a}{b}}}-\frac {3 b^2 x^2 \text {Log}\left [\sqrt {x}+\sqrt {-\frac {a}{b}}\right ]}{8 a^2 b^3 \sqrt {-\frac {a}{b}}+16 a b^4 x \sqrt {-\frac {a}{b}}+8 b^5 x^2 \sqrt {-\frac {a}{b}}}+\frac {3 b^2 x^2 \text {Log}\left [\sqrt {x}-\sqrt {-\frac {a}{b}}\right ]}{8 a^2 b^3 \sqrt {-\frac {a}{b}}+16 a b^4 x \sqrt {-\frac {a}{b}}+8 b^5 x^2 \sqrt {-\frac {a}{b}}}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.10, size = 50, normalized size = 0.71
method | result | size |
derivativedivides | \(\frac {-\frac {5 x^{\frac {3}{2}}}{4 b}-\frac {3 a \sqrt {x}}{4 b^{2}}}{\left (b x +a \right )^{2}}+\frac {3 \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 b^{2} \sqrt {a b}}\) | \(50\) |
default | \(\frac {-\frac {5 x^{\frac {3}{2}}}{4 b}-\frac {3 a \sqrt {x}}{4 b^{2}}}{\left (b x +a \right )^{2}}+\frac {3 \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 b^{2} \sqrt {a b}}\) | \(50\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.35, size = 61, normalized size = 0.87 \begin {gather*} -\frac {5 \, b x^{\frac {3}{2}} + 3 \, a \sqrt {x}}{4 \, {\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )}} + \frac {3 \, \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.31, size = 185, normalized size = 2.64 \begin {gather*} \left [-\frac {3 \, {\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 (5 \, a b^{2} x + 3 \, a^{2} b\right )} \sqrt {x}}{8 \, {\left (a b^{5} x^{2} + 2 \, a^{2} b^{4} x + a^{3} b^{3}\right )}}, -\frac {3 \, {\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 (5 \, a b^{2} x + 3 \, a^{2} b\right )} \sqrt {x}}{4 \, {\left (a b^{5} x^{2} + 2 \, a^{2} b^{4} x + a^{3} b^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 16.54, size = 605, normalized size = 8.64 \begin {gather*} \begin {cases} \frac {\tilde {\infty }}{\sqrt {x}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {2 x^{\frac {5}{2}}}{5 a^{3}} & \text {for}\: b = 0 \\- \frac {2}{b^{3} \sqrt {x}} & \text {for}\: a = 0 \\\frac {3 a^{2} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{8 a^{2} b^{3} \sqrt {- \frac {a}{b}} + 16 a b^{4} x \sqrt {- \frac {a}{b}} + 8 b^{5} x^{2} \sqrt {- \frac {a}{b}}} - \frac {3 a^{2} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{8 a^{2} b^{3} \sqrt {- \frac {a}{b}} + 16 a b^{4} x \sqrt {- \frac {a}{b}} + 8 b^{5} x^{2} \sqrt {- \frac {a}{b}}} - \frac {6 a b \sqrt {x} \sqrt {- \frac {a}{b}}}{8 a^{2} b^{3} \sqrt {- \frac {a}{b}} + 16 a b^{4} x \sqrt {- \frac {a}{b}} + 8 b^{5} x^{2} \sqrt {- \frac {a}{b}}} + \frac {6 a b x \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{8 a^{2} b^{3} \sqrt {- \frac {a}{b}} + 16 a b^{4} x \sqrt {- \frac {a}{b}} + 8 b^{5} x^{2} \sqrt {- \frac {a}{b}}} - \frac {6 a b x \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{8 a^{2} b^{3} \sqrt {- \frac {a}{b}} + 16 a b^{4} x \sqrt {- \frac {a}{b}} + 8 b^{5} x^{2} \sqrt {- \frac {a}{b}}} - \frac {10 b^{2} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}}}{8 a^{2} b^{3} \sqrt {- \frac {a}{b}} + 16 a b^{4} x \sqrt {- \frac {a}{b}} + 8 b^{5} x^{2} \sqrt {- \frac {a}{b}}} + \frac {3 b^{2} x^{2} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{8 a^{2} b^{3} \sqrt {- \frac {a}{b}} + 16 a b^{4} x \sqrt {- \frac {a}{b}} + 8 b^{5} x^{2} \sqrt {- \frac {a}{b}}} - \frac {3 b^{2} x^{2} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{8 a^{2} b^{3} \sqrt {- \frac {a}{b}} + 16 a b^{4} x \sqrt {- \frac {a}{b}} + 8 b^{5} 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 = 72, normalized size = 1.03 \begin {gather*} 2 \left (\frac {-5 \sqrt {x} x b-3 \sqrt {x} a}{8 b^{2} \left (x b+a\right )^{2}}+\frac {3 \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 b^{2}\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 = 58, normalized size = 0.83 \begin {gather*} \frac {3\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )}{4\,\sqrt {a}\,b^{5/2}}-\frac {\frac {5\,x^{3/2}}{4\,b}+\frac {3\,a\,\sqrt {x}}{4\,b^2}}{a^2+2\,a\,b\,x+b^2\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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