Optimal. Leaf size=70 \[ \frac {3 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 \sqrt {a} b^{5/2}}-\frac {3 \sqrt {x}}{4 b^2 (a+b x)}-\frac {x^{3/2}}{2 b (a+b x)^2} \]
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Rubi [A] time = 0.02, 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 = {47, 63, 205} \[ -\frac {3 \sqrt {x}}{4 b^2 (a+b x)}+\frac {3 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 \sqrt {a} b^{5/2}}-\frac {x^{3/2}}{2 b (a+b x)^2} \]
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 205
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 \operatorname {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 = 0.03, size = 59, normalized size = 0.84 \[ \frac {3 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 \sqrt {a} b^{5/2}}-\frac {\sqrt {x} (3 a+5 b x)}{4 b^2 (a+b x)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 185, normalized size = 2.64 \[ \left [-\frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt {-a b} \log \left (\frac {b x - a - 2 \, \sqrt {-a b} \sqrt {x}}{b x + a}\right ) + 2 \, {\left (5 \, a b^{2} x + 3 \, a^{2} b\right )} \sqrt {x}}{8 \, {\left (a b^{5} x^{2} + 2 \, a^{2} b^{4} x + a^{3} b^{3}\right )}}, -\frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b}}{b \sqrt {x}}\right ) + {\left (5 \, a b^{2} x + 3 \, a^{2} b\right )} \sqrt {x}}{4 \, {\left (a b^{5} x^{2} + 2 \, a^{2} b^{4} x + a^{3} b^{3}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.91, size = 47, normalized size = 0.67 \[ \frac {3 \, \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} b^{2}} - \frac {5 \, b x^{\frac {3}{2}} + 3 \, a \sqrt {x}}{4 \, {\left (b x + a\right )}^{2} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 50, normalized size = 0.71 \[ \frac {3 \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \sqrt {a b}\, b^{2}}+\frac {-\frac {5 x^{\frac {3}{2}}}{4 b}-\frac {3 a \sqrt {x}}{4 b^{2}}}{\left (b x +a \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.96, size = 61, normalized size = 0.87 \[ -\frac {5 \, b x^{\frac {3}{2}} + 3 \, a \sqrt {x}}{4 \, {\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )}} + \frac {3 \, \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} b^{2}} \]
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 \[ \frac {3\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )}{4\,\sqrt {a}\,b^{5/2}}-\frac {\frac {5\,x^{3/2}}{4\,b}+\frac {3\,a\,\sqrt {x}}{4\,b^2}}{a^2+2\,a\,b\,x+b^2\,x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 29.37, size = 726, normalized size = 10.37 \[ \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 {6 i a^{\frac {3}{2}} b \sqrt {x} \sqrt {\frac {1}{b}}}{8 i a^{\frac {5}{2}} b^{3} \sqrt {\frac {1}{b}} + 16 i a^{\frac {3}{2}} b^{4} x \sqrt {\frac {1}{b}} + 8 i \sqrt {a} b^{5} x^{2} \sqrt {\frac {1}{b}}} - \frac {10 i \sqrt {a} b^{2} x^{\frac {3}{2}} \sqrt {\frac {1}{b}}}{8 i a^{\frac {5}{2}} b^{3} \sqrt {\frac {1}{b}} + 16 i a^{\frac {3}{2}} b^{4} x \sqrt {\frac {1}{b}} + 8 i \sqrt {a} b^{5} x^{2} \sqrt {\frac {1}{b}}} + \frac {3 a^{2} \log {\left (- i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{8 i a^{\frac {5}{2}} b^{3} \sqrt {\frac {1}{b}} + 16 i a^{\frac {3}{2}} b^{4} x \sqrt {\frac {1}{b}} + 8 i \sqrt {a} b^{5} x^{2} \sqrt {\frac {1}{b}}} - \frac {3 a^{2} \log {\left (i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{8 i a^{\frac {5}{2}} b^{3} \sqrt {\frac {1}{b}} + 16 i a^{\frac {3}{2}} b^{4} x \sqrt {\frac {1}{b}} + 8 i \sqrt {a} b^{5} x^{2} \sqrt {\frac {1}{b}}} + \frac {6 a b x \log {\left (- i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{8 i a^{\frac {5}{2}} b^{3} \sqrt {\frac {1}{b}} + 16 i a^{\frac {3}{2}} b^{4} x \sqrt {\frac {1}{b}} + 8 i \sqrt {a} b^{5} x^{2} \sqrt {\frac {1}{b}}} - \frac {6 a b x \log {\left (i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{8 i a^{\frac {5}{2}} b^{3} \sqrt {\frac {1}{b}} + 16 i a^{\frac {3}{2}} b^{4} x \sqrt {\frac {1}{b}} + 8 i \sqrt {a} b^{5} x^{2} \sqrt {\frac {1}{b}}} + \frac {3 b^{2} x^{2} \log {\left (- i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{8 i a^{\frac {5}{2}} b^{3} \sqrt {\frac {1}{b}} + 16 i a^{\frac {3}{2}} b^{4} x \sqrt {\frac {1}{b}} + 8 i \sqrt {a} b^{5} x^{2} \sqrt {\frac {1}{b}}} - \frac {3 b^{2} x^{2} \log {\left (i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{8 i a^{\frac {5}{2}} b^{3} \sqrt {\frac {1}{b}} + 16 i a^{\frac {3}{2}} b^{4} x \sqrt {\frac {1}{b}} + 8 i \sqrt {a} b^{5} x^{2} \sqrt {\frac {1}{b}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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