Optimal. Leaf size=77 \[ -\frac {6 b \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {a x+b \sqrt {x}}}\right )}{a^{5/2}}+\frac {6 \sqrt {a x+b \sqrt {x}}}{a^2}-\frac {4 x}{a \sqrt {a x+b \sqrt {x}}} \]
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Rubi [A] time = 0.07, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {2018, 668, 640, 620, 206} \[ \frac {6 \sqrt {a x+b \sqrt {x}}}{a^2}-\frac {6 b \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {a x+b \sqrt {x}}}\right )}{a^{5/2}}-\frac {4 x}{a \sqrt {a x+b \sqrt {x}}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 620
Rule 640
Rule 668
Rule 2018
Rubi steps
\begin {align*} \int \frac {x}{\left (b \sqrt {x}+a x\right )^{3/2}} \, dx &=2 \operatorname {Subst}\left (\int \frac {x^3}{\left (b x+a x^2\right )^{3/2}} \, dx,x,\sqrt {x}\right )\\ &=-\frac {4 x}{a \sqrt {b \sqrt {x}+a x}}+\frac {6 \operatorname {Subst}\left (\int \frac {x}{\sqrt {b x+a x^2}} \, dx,x,\sqrt {x}\right )}{a}\\ &=-\frac {4 x}{a \sqrt {b \sqrt {x}+a x}}+\frac {6 \sqrt {b \sqrt {x}+a x}}{a^2}-\frac {(3 b) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b x+a x^2}} \, dx,x,\sqrt {x}\right )}{a^2}\\ &=-\frac {4 x}{a \sqrt {b \sqrt {x}+a x}}+\frac {6 \sqrt {b \sqrt {x}+a x}}{a^2}-\frac {(6 b) \operatorname {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {b \sqrt {x}+a x}}\right )}{a^2}\\ &=-\frac {4 x}{a \sqrt {b \sqrt {x}+a x}}+\frac {6 \sqrt {b \sqrt {x}+a x}}{a^2}-\frac {6 b \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b \sqrt {x}+a x}}\right )}{a^{5/2}}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 64, normalized size = 0.83 \[ \frac {4 x^{3/2} \sqrt {\frac {a \sqrt {x}}{b}+1} \, _2F_1\left (\frac {3}{2},\frac {5}{2};\frac {7}{2};-\frac {a \sqrt {x}}{b}\right )}{5 b \sqrt {a x+b \sqrt {x}}} \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.32, size = 94, normalized size = 1.22 \[ \frac {3 \, b \log \left ({\left | -2 \, \sqrt {a} {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )} - b \right |}\right )}{a^{\frac {5}{2}}} + \frac {4 \, b^{2}}{{\left (a {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )} + \sqrt {a} b\right )} a^{2}} + \frac {2 \, \sqrt {a x + b \sqrt {x}}}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 236, normalized size = 3.06 \[ \frac {\sqrt {a x +b \sqrt {x}}\, \left (-3 a^{2} b x \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )-6 a \,b^{2} \sqrt {x}\, \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )-3 b^{3} \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )+6 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, a^{\frac {5}{2}} x +12 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, a^{\frac {3}{2}} b \sqrt {x}+6 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, \sqrt {a}\, b^{2}-4 \left (\left (a \sqrt {x}+b \right ) \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {3}{2}}\right )}{\sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, \left (a \sqrt {x}+b \right )^{2} a^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{{\left (a x + b \sqrt {x}\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x}{{\left (a\,x+b\,\sqrt {x}\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{\left (a x + b \sqrt {x}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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