Optimal. Leaf size=70 \[ \frac {3 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{5/2} \sqrt {b}}+\frac {3 \sqrt {x}}{4 a^2 (a+b x)}+\frac {\sqrt {x}}{2 a (a+b x)^2} \]
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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 = {51, 63, 205} \begin {gather*} \frac {3 \sqrt {x}}{4 a^2 (a+b x)}+\frac {3 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{5/2} \sqrt {b}}+\frac {\sqrt {x}}{2 a (a+b x)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 205
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {x} (a+b x)^3} \, dx &=\frac {\sqrt {x}}{2 a (a+b x)^2}+\frac {3 \int \frac {1}{\sqrt {x} (a+b x)^2} \, dx}{4 a}\\ &=\frac {\sqrt {x}}{2 a (a+b x)^2}+\frac {3 \sqrt {x}}{4 a^2 (a+b x)}+\frac {3 \int \frac {1}{\sqrt {x} (a+b x)} \, dx}{8 a^2}\\ &=\frac {\sqrt {x}}{2 a (a+b x)^2}+\frac {3 \sqrt {x}}{4 a^2 (a+b x)}+\frac {3 \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {x}\right )}{4 a^2}\\ &=\frac {\sqrt {x}}{2 a (a+b x)^2}+\frac {3 \sqrt {x}}{4 a^2 (a+b x)}+\frac {3 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{5/2} \sqrt {b}}\\ \end {align*}
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Mathematica [C] time = 0.00, size = 25, normalized size = 0.36 \begin {gather*} \frac {2 \sqrt {x} \, _2F_1\left (\frac {1}{2},3;\frac {3}{2};-\frac {b x}{a}\right )}{a^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.08, size = 63, normalized size = 0.90 \begin {gather*} \frac {3 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{5/2} \sqrt {b}}+\frac {5 a \sqrt {x}+3 b x^{3/2}}{4 a^2 (a+b x)^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.98, size = 186, normalized size = 2.66 \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 (3 \, a b^{2} x + 5 \, a^{2} b\right )} \sqrt {x}}{8 \, {\left (a^{3} b^{3} x^{2} + 2 \, a^{4} b^{2} x + a^{5} b\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 (3 \, a b^{2} x + 5 \, a^{2} b\right )} \sqrt {x}}{4 \, {\left (a^{3} b^{3} x^{2} + 2 \, a^{4} b^{2} x + a^{5} b\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.86, size = 47, normalized size = 0.67 \begin {gather*} \frac {3 \, \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} a^{2}} + \frac {3 \, b x^{\frac {3}{2}} + 5 \, a \sqrt {x}}{4 \, {\left (b x + a\right )}^{2} a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 53, normalized size = 0.76 \begin {gather*} \frac {\sqrt {x}}{2 \left (b x +a \right )^{2} a}+\frac {3 \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \sqrt {a b}\, a^{2}}+\frac {3 \sqrt {x}}{4 \left (b x +a \right ) a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.96, size = 60, normalized size = 0.86 \begin {gather*} \frac {3 \, b x^{\frac {3}{2}} + 5 \, a \sqrt {x}}{4 \, {\left (a^{2} b^{2} x^{2} + 2 \, a^{3} b x + a^{4}\right )}} + \frac {3 \, \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.13, size = 57, normalized size = 0.81 \begin {gather*} \frac {\frac {5\,\sqrt {x}}{4\,a}+\frac {3\,b\,x^{3/2}}{4\,a^2}}{a^2+2\,a\,b\,x+b^2\,x^2}+\frac {3\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )}{4\,a^{5/2}\,\sqrt {b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 25.69, size = 712, normalized size = 10.17 \begin {gather*} \begin {cases} \frac {\tilde {\infty }}{x^{\frac {5}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {2}{5 b^{3} x^{\frac {5}{2}}} & \text {for}\: a = 0 \\\frac {2 \sqrt {x}}{a^{3}} & \text {for}\: b = 0 \\\frac {10 i a^{\frac {3}{2}} b \sqrt {x} \sqrt {\frac {1}{b}}}{8 i a^{\frac {9}{2}} b \sqrt {\frac {1}{b}} + 16 i a^{\frac {7}{2}} b^{2} x \sqrt {\frac {1}{b}} + 8 i a^{\frac {5}{2}} b^{3} x^{2} \sqrt {\frac {1}{b}}} + \frac {6 i \sqrt {a} b^{2} x^{\frac {3}{2}} \sqrt {\frac {1}{b}}}{8 i a^{\frac {9}{2}} b \sqrt {\frac {1}{b}} + 16 i a^{\frac {7}{2}} b^{2} x \sqrt {\frac {1}{b}} + 8 i a^{\frac {5}{2}} b^{3} 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 {9}{2}} b \sqrt {\frac {1}{b}} + 16 i a^{\frac {7}{2}} b^{2} x \sqrt {\frac {1}{b}} + 8 i a^{\frac {5}{2}} b^{3} 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 {9}{2}} b \sqrt {\frac {1}{b}} + 16 i a^{\frac {7}{2}} b^{2} x \sqrt {\frac {1}{b}} + 8 i a^{\frac {5}{2}} b^{3} 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 {9}{2}} b \sqrt {\frac {1}{b}} + 16 i a^{\frac {7}{2}} b^{2} x \sqrt {\frac {1}{b}} + 8 i a^{\frac {5}{2}} b^{3} 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 {9}{2}} b \sqrt {\frac {1}{b}} + 16 i a^{\frac {7}{2}} b^{2} x \sqrt {\frac {1}{b}} + 8 i a^{\frac {5}{2}} b^{3} 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 {9}{2}} b \sqrt {\frac {1}{b}} + 16 i a^{\frac {7}{2}} b^{2} x \sqrt {\frac {1}{b}} + 8 i a^{\frac {5}{2}} b^{3} 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 {9}{2}} b \sqrt {\frac {1}{b}} + 16 i a^{\frac {7}{2}} b^{2} x \sqrt {\frac {1}{b}} + 8 i a^{\frac {5}{2}} b^{3} x^{2} \sqrt {\frac {1}{b}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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