Optimal. Leaf size=57 \[ -\frac {3 \sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{5/2}}-\frac {x^{3/2}}{b (a+b x)}+\frac {3 \sqrt {x}}{b^2} \]
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Rubi [A] time = 0.02, 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 = {47, 50, 63, 205} \[ -\frac {3 \sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{5/2}}-\frac {x^{3/2}}{b (a+b x)}+\frac {3 \sqrt {x}}{b^2} \]
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 63
Rule 205
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) \operatorname {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 [C] time = 0.00, size = 27, normalized size = 0.47 \[ \frac {2 x^{5/2} \, _2F_1\left (2,\frac {5}{2};\frac {7}{2};-\frac {b x}{a}\right )}{5 a^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 134, normalized size = 2.35 \[ \left [\frac {3 \, {\left (b x + a\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x - 2 \, b \sqrt {x} \sqrt {-\frac {a}{b}} - a}{b x + a}\right ) + 2 \, {\left (2 \, b x + 3 \, a\right )} \sqrt {x}}{2 \, {\left (b^{3} x + a b^{2}\right )}}, -\frac {3 \, {\left (b x + a\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b \sqrt {x} \sqrt {\frac {a}{b}}}{a}\right ) - {\left (2 \, b x + 3 \, a\right )} \sqrt {x}}{b^{3} x + a b^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.96, size = 46, normalized size = 0.81 \[ -\frac {3 \, a \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} b^{2}} + \frac {a \sqrt {x}}{{\left (b x + a\right )} b^{2}} + \frac {2 \, \sqrt {x}}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 47, normalized size = 0.82 \[ -\frac {3 a \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b}\, b^{2}}+\frac {a \sqrt {x}}{\left (b x +a \right ) b^{2}}+\frac {2 \sqrt {x}}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.91, size = 49, normalized size = 0.86 \[ \frac {a \sqrt {x}}{b^{3} x + a b^{2}} - \frac {3 \, a \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} b^{2}} + \frac {2 \, \sqrt {x}}{b^{2}} \]
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 \[ \frac {2\,\sqrt {x}}{b^2}+\frac {a\,\sqrt {x}}{x\,b^3+a\,b^2}-\frac {3\,\sqrt {a}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )}{b^{5/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 9.18, size = 411, normalized size = 7.21 \[ \begin {cases} \tilde {\infty } \sqrt {x} & \text {for}\: a = 0 \wedge b = 0 \\\frac {2 x^{\frac {5}{2}}}{5 a^{2}} & \text {for}\: b = 0 \\\frac {2 \sqrt {x}}{b^{2}} & \text {for}\: a = 0 \\\frac {6 i a^{\frac {3}{2}} b \sqrt {x} \sqrt {\frac {1}{b}}}{2 i a^{\frac {3}{2}} b^{3} \sqrt {\frac {1}{b}} + 2 i \sqrt {a} b^{4} x \sqrt {\frac {1}{b}}} + \frac {4 i \sqrt {a} b^{2} x^{\frac {3}{2}} \sqrt {\frac {1}{b}}}{2 i a^{\frac {3}{2}} b^{3} \sqrt {\frac {1}{b}} + 2 i \sqrt {a} b^{4} x \sqrt {\frac {1}{b}}} - \frac {3 a^{2} \log {\left (- i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{2 i a^{\frac {3}{2}} b^{3} \sqrt {\frac {1}{b}} + 2 i \sqrt {a} b^{4} x \sqrt {\frac {1}{b}}} + \frac {3 a^{2} \log {\left (i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{2 i a^{\frac {3}{2}} b^{3} \sqrt {\frac {1}{b}} + 2 i \sqrt {a} b^{4} x \sqrt {\frac {1}{b}}} - \frac {3 a b x \log {\left (- i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{2 i a^{\frac {3}{2}} b^{3} \sqrt {\frac {1}{b}} + 2 i \sqrt {a} b^{4} x \sqrt {\frac {1}{b}}} + \frac {3 a b x \log {\left (i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{2 i a^{\frac {3}{2}} b^{3} \sqrt {\frac {1}{b}} + 2 i \sqrt {a} b^{4} x \sqrt {\frac {1}{b}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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