Optimal. Leaf size=146 \[ \frac {35 b^4 \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {a x+b \sqrt {x}}}\right )}{32 a^{9/2}}-\frac {35 b^3 \sqrt {a x+b \sqrt {x}}}{32 a^4}+\frac {35 b^2 \sqrt {x} \sqrt {a x+b \sqrt {x}}}{48 a^3}-\frac {7 b x \sqrt {a x+b \sqrt {x}}}{12 a^2}+\frac {x^{3/2} \sqrt {a x+b \sqrt {x}}}{2 a} \]
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Rubi [A] time = 0.12, 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 = {2018, 670, 640, 620, 206} \[ -\frac {35 b^3 \sqrt {a x+b \sqrt {x}}}{32 a^4}+\frac {35 b^2 \sqrt {x} \sqrt {a x+b \sqrt {x}}}{48 a^3}+\frac {35 b^4 \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {a x+b \sqrt {x}}}\right )}{32 a^{9/2}}-\frac {7 b x \sqrt {a x+b \sqrt {x}}}{12 a^2}+\frac {x^{3/2} \sqrt {a x+b \sqrt {x}}}{2 a} \]
Antiderivative was successfully verified.
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Rule 206
Rule 620
Rule 640
Rule 670
Rule 2018
Rubi steps
\begin {align*} \int \frac {x^{3/2}}{\sqrt {b \sqrt {x}+a x}} \, dx &=2 \operatorname {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) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 = 0.21, size = 142, normalized size = 0.97 \[ -\frac {35 b^5 \left (\frac {a \sqrt {x}}{b}+1\right ) \left (-\frac {32 a^4 x^2}{35 b^4}+\frac {16 a^3 x^{3/2}}{15 b^3}-\frac {4 a^2 x}{3 b^2}+\frac {2 a \sqrt {x}}{b}-\frac {2 \sqrt {a} \sqrt [4]{x} \sinh ^{-1}\left (\frac {\sqrt {a} \sqrt [4]{x}}{\sqrt {b}}\right )}{\sqrt {b} \sqrt {\frac {a \sqrt {x}}{b}+1}}\right )}{64 a^5 \sqrt {\sqrt {x} \left (a \sqrt {x}+b\right )}} \]
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 = 0.41, size = 97, normalized size = 0.66 \[ \frac {1}{96} \, \sqrt {a x + b \sqrt {x}} {\left (2 \, {\left (4 \, \sqrt {x} {\left (\frac {6 \, \sqrt {x}}{a} - \frac {7 \, b}{a^{2}}\right )} + \frac {35 \, b^{2}}{a^{3}}\right )} \sqrt {x} - \frac {105 \, b^{3}}{a^{4}}\right )} - \frac {35 \, b^{4} \log \left ({\left | -2 \, \sqrt {a} {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )} - b \right |}\right )}{64 \, a^{\frac {9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 203, normalized size = 1.39 \[ \frac {\sqrt {a x +b \sqrt {x}}\, \left (192 a \,b^{4} \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )-87 a \,b^{4} \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {a x +b \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )+348 \sqrt {a x +b \sqrt {x}}\, a^{\frac {5}{2}} b^{2} \sqrt {x}-384 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, a^{\frac {3}{2}} b^{3}+174 \sqrt {a x +b \sqrt {x}}\, a^{\frac {3}{2}} b^{3}+96 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {7}{2}} \sqrt {x}-208 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {5}{2}} b \right )}{192 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, a^{\frac {11}{2}}} \]
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 \[ \int \frac {x^{\frac {3}{2}}}{\sqrt {a x + b \sqrt {x}}}\,{d x} \]
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 \[ \int \frac {x^{3/2}}{\sqrt {a\,x+b\,\sqrt {x}}} \,d x \]
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 \[ \int \frac {x^{\frac {3}{2}}}{\sqrt {a x + b \sqrt {x}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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