Optimal. Leaf size=57 \[ \frac {3 \sqrt {x}}{b^2}-\frac {x^{3/2}}{b (a+b x)}-\frac {3 \sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{5/2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 57, 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, 52, 65, 211}
\begin {gather*} -\frac {3 \sqrt {a} \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{5/2}}-\frac {x^{3/2}}{b (a+b x)}+\frac {3 \sqrt {x}}{b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 52
Rule 65
Rule 211
Rubi steps
\begin {align*} \int \frac {x^{3/2}}{(a+b x)^2} \, dx &=-\frac {x^{3/2}}{b (a+b x)}+\frac {3 \int \frac {\sqrt {x}}{a+b x} \, dx}{2 b}\\ &=\frac {3 \sqrt {x}}{b^2}-\frac {x^{3/2}}{b (a+b x)}-\frac {(3 a) \int \frac {1}{\sqrt {x} (a+b x)} \, dx}{2 b^2}\\ &=\frac {3 \sqrt {x}}{b^2}-\frac {x^{3/2}}{b (a+b x)}-\frac {(3 a) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {x}\right )}{b^2}\\ &=\frac {3 \sqrt {x}}{b^2}-\frac {x^{3/2}}{b (a+b x)}-\frac {3 \sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 54, normalized size = 0.95 \begin {gather*} \frac {\sqrt {x} (3 a+2 b x)}{b^2 (a+b x)}-\frac {3 \sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 47, normalized size = 0.82
method | result | size |
derivativedivides | \(\frac {2 \sqrt {x}}{b^{2}}-\frac {2 a \left (-\frac {\sqrt {x}}{2 \left (b x +a \right )}+\frac {3 \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{b^{2}}\) | \(47\) |
default | \(\frac {2 \sqrt {x}}{b^{2}}-\frac {2 a \left (-\frac {\sqrt {x}}{2 \left (b x +a \right )}+\frac {3 \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{b^{2}}\) | \(47\) |
risch | \(\frac {2 \sqrt {x}}{b^{2}}+\frac {a \sqrt {x}}{b^{2} \left (b x +a \right )}-\frac {3 a \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{b^{2} \sqrt {a b}}\) | \(47\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 49, normalized size = 0.86 \begin {gather*} \frac {a \sqrt {x}}{b^{3} x + a b^{2}} - \frac {3 \, a \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} b^{2}} + \frac {2 \, \sqrt {x}}{b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.56, size = 134, normalized size = 2.35 \begin {gather*} \left [\frac {3 \, {\left (b x + a\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x - 2 \, b \sqrt {x} \sqrt {-\frac {a}{b}} - a}{b x + a}\right ) + 2 \, {\left (2 \, b x + 3 \, a\right )} \sqrt {x}}{2 \, {\left (b^{3} x + a b^{2}\right )}}, -\frac {3 \, {\left (b x + a\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b \sqrt {x} \sqrt {\frac {a}{b}}}{a}\right ) - {\left (2 \, b x + 3 \, a\right )} \sqrt {x}}{b^{3} x + a b^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 332 vs.
\(2 (49) = 98\).
time = 4.03, size = 332, normalized size = 5.82 \begin {gather*} \begin {cases} \tilde {\infty } \sqrt {x} & \text {for}\: a = 0 \wedge b = 0 \\\frac {2 x^{\frac {5}{2}}}{5 a^{2}} & \text {for}\: b = 0 \\\frac {2 \sqrt {x}}{b^{2}} & \text {for}\: a = 0 \\- \frac {3 a^{2} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{2 a b^{3} \sqrt {- \frac {a}{b}} + 2 b^{4} x \sqrt {- \frac {a}{b}}} + \frac {3 a^{2} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{2 a b^{3} \sqrt {- \frac {a}{b}} + 2 b^{4} x \sqrt {- \frac {a}{b}}} + \frac {6 a b \sqrt {x} \sqrt {- \frac {a}{b}}}{2 a b^{3} \sqrt {- \frac {a}{b}} + 2 b^{4} x \sqrt {- \frac {a}{b}}} - \frac {3 a b x \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{2 a b^{3} \sqrt {- \frac {a}{b}} + 2 b^{4} x \sqrt {- \frac {a}{b}}} + \frac {3 a b x \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{2 a b^{3} \sqrt {- \frac {a}{b}} + 2 b^{4} x \sqrt {- \frac {a}{b}}} + \frac {4 b^{2} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}}}{2 a b^{3} \sqrt {- \frac {a}{b}} + 2 b^{4} x \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 = 0.53, size = 46, normalized size = 0.81 \begin {gather*} -\frac {3 \, a \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} b^{2}} + \frac {a \sqrt {x}}{{\left (b x + a\right )} b^{2}} + \frac {2 \, \sqrt {x}}{b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.12, size = 46, normalized size = 0.81 \begin {gather*} \frac {2\,\sqrt {x}}{b^2}+\frac {a\,\sqrt {x}}{x\,b^3+a\,b^2}-\frac {3\,\sqrt {a}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )}{b^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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