Optimal. Leaf size=146 \[ -\frac {35 b^3 \sqrt {b \sqrt {x}+a x}}{32 a^4}+\frac {35 b^2 \sqrt {x} \sqrt {b \sqrt {x}+a x}}{48 a^3}-\frac {7 b x \sqrt {b \sqrt {x}+a x}}{12 a^2}+\frac {x^{3/2} \sqrt {b \sqrt {x}+a x}}{2 a}+\frac {35 b^4 \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b \sqrt {x}+a x}}\right )}{32 a^{9/2}} \]
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Rubi [A]
time = 0.09, antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {2043, 684, 654,
634, 212} \begin {gather*} \frac {35 b^4 \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {a x+b \sqrt {x}}}\right )}{32 a^{9/2}}-\frac {35 b^3 \sqrt {a x+b \sqrt {x}}}{32 a^4}+\frac {35 b^2 \sqrt {x} \sqrt {a x+b \sqrt {x}}}{48 a^3}-\frac {7 b x \sqrt {a x+b \sqrt {x}}}{12 a^2}+\frac {x^{3/2} \sqrt {a x+b \sqrt {x}}}{2 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 634
Rule 654
Rule 684
Rule 2043
Rubi steps
\begin {align*} \int \frac {x^{3/2}}{\sqrt {b \sqrt {x}+a x}} \, dx &=2 \text {Subst}\left (\int \frac {x^4}{\sqrt {b x+a x^2}} \, dx,x,\sqrt {x}\right )\\ &=\frac {x^{3/2} \sqrt {b \sqrt {x}+a x}}{2 a}-\frac {(7 b) \text {Subst}\left (\int \frac {x^3}{\sqrt {b x+a x^2}} \, dx,x,\sqrt {x}\right )}{4 a}\\ &=-\frac {7 b x \sqrt {b \sqrt {x}+a x}}{12 a^2}+\frac {x^{3/2} \sqrt {b \sqrt {x}+a x}}{2 a}+\frac {\left (35 b^2\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {b x+a x^2}} \, dx,x,\sqrt {x}\right )}{24 a^2}\\ &=\frac {35 b^2 \sqrt {x} \sqrt {b \sqrt {x}+a x}}{48 a^3}-\frac {7 b x \sqrt {b \sqrt {x}+a x}}{12 a^2}+\frac {x^{3/2} \sqrt {b \sqrt {x}+a x}}{2 a}-\frac {\left (35 b^3\right ) \text {Subst}\left (\int \frac {x}{\sqrt {b x+a x^2}} \, dx,x,\sqrt {x}\right )}{32 a^3}\\ &=-\frac {35 b^3 \sqrt {b \sqrt {x}+a x}}{32 a^4}+\frac {35 b^2 \sqrt {x} \sqrt {b \sqrt {x}+a x}}{48 a^3}-\frac {7 b x \sqrt {b \sqrt {x}+a x}}{12 a^2}+\frac {x^{3/2} \sqrt {b \sqrt {x}+a x}}{2 a}+\frac {\left (35 b^4\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b x+a x^2}} \, dx,x,\sqrt {x}\right )}{64 a^4}\\ &=-\frac {35 b^3 \sqrt {b \sqrt {x}+a x}}{32 a^4}+\frac {35 b^2 \sqrt {x} \sqrt {b \sqrt {x}+a x}}{48 a^3}-\frac {7 b x \sqrt {b \sqrt {x}+a x}}{12 a^2}+\frac {x^{3/2} \sqrt {b \sqrt {x}+a x}}{2 a}+\frac {\left (35 b^4\right ) \text {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {b \sqrt {x}+a x}}\right )}{32 a^4}\\ &=-\frac {35 b^3 \sqrt {b \sqrt {x}+a x}}{32 a^4}+\frac {35 b^2 \sqrt {x} \sqrt {b \sqrt {x}+a x}}{48 a^3}-\frac {7 b x \sqrt {b \sqrt {x}+a x}}{12 a^2}+\frac {x^{3/2} \sqrt {b \sqrt {x}+a x}}{2 a}+\frac {35 b^4 \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b \sqrt {x}+a x}}\right )}{32 a^{9/2}}\\ \end {align*}
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Mathematica [A]
time = 0.20, size = 108, normalized size = 0.74 \begin {gather*} \frac {\sqrt {b \sqrt {x}+a x} \left (-105 b^3+70 a b^2 \sqrt {x}-56 a^2 b x+48 a^3 x^{3/2}\right )}{96 a^4}-\frac {35 b^4 \log \left (a^4 b+2 a^5 \sqrt {x}-2 a^{9/2} \sqrt {b \sqrt {x}+a x}\right )}{64 a^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.39, size = 203, normalized size = 1.39
method | result | size |
derivativedivides | \(\frac {x^{\frac {3}{2}} \sqrt {b \sqrt {x}+a x}}{2 a}-\frac {7 b \left (\frac {x \sqrt {b \sqrt {x}+a x}}{3 a}-\frac {5 b \left (\frac {\sqrt {x}\, \sqrt {b \sqrt {x}+a x}}{2 a}-\frac {3 b \left (\frac {\sqrt {b \sqrt {x}+a x}}{a}-\frac {b \ln \left (\frac {\frac {b}{2}+a \sqrt {x}}{\sqrt {a}}+\sqrt {b \sqrt {x}+a x}\right )}{2 a^{\frac {3}{2}}}\right )}{4 a}\right )}{6 a}\right )}{4 a}\) | \(125\) |
default | \(\frac {\sqrt {b \sqrt {x}+a x}\, \left (96 \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} \sqrt {x}\, a^{\frac {7}{2}}+348 \sqrt {x}\, \sqrt {b \sqrt {x}+a x}\, a^{\frac {5}{2}} b^{2}-208 \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} a^{\frac {5}{2}} b +174 \sqrt {b \sqrt {x}+a x}\, a^{\frac {3}{2}} b^{3}-384 \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, a^{\frac {3}{2}} b^{3}+192 a \ln \left (\frac {2 a \sqrt {x}+2 \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, \sqrt {a}+b}{2 \sqrt {a}}\right ) b^{4}-87 \ln \left (\frac {2 a \sqrt {x}+2 \sqrt {b \sqrt {x}+a x}\, \sqrt {a}+b}{2 \sqrt {a}}\right ) a \,b^{4}\right )}{192 \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, a^{\frac {11}{2}}}\) | \(203\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \int \frac {x^{\frac {3}{2}}}{\sqrt {a x + b \sqrt {x}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.69, size = 97, normalized size = 0.66 \begin {gather*} \frac {1}{96} \, \sqrt {a x + b \sqrt {x}} {\left (2 \, {\left (4 \, \sqrt {x} {\left (\frac {6 \, \sqrt {x}}{a} - \frac {7 \, b}{a^{2}}\right )} + \frac {35 \, b^{2}}{a^{3}}\right )} \sqrt {x} - \frac {105 \, b^{3}}{a^{4}}\right )} - \frac {35 \, b^{4} \log \left ({\left | -2 \, \sqrt {a} {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )} - b \right |}\right )}{64 \, a^{\frac {9}{2}}} \end {gather*}
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 \begin {gather*} \int \frac {x^{3/2}}{\sqrt {a\,x+b\,\sqrt {x}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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