Optimal. Leaf size=113 \[ \frac {15 b^2 \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {a x+b \sqrt {x}}}\right )}{2 a^{7/2}}-\frac {15 b \sqrt {a x+b \sqrt {x}}}{2 a^3}+\frac {5 \sqrt {x} \sqrt {a x+b \sqrt {x}}}{a^2}-\frac {4 x^{3/2}}{a \sqrt {a x+b \sqrt {x}}} \]
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Rubi [A] time = 0.10, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2018, 668, 670, 640, 620, 206} \[ \frac {15 b^2 \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {a x+b \sqrt {x}}}\right )}{2 a^{7/2}}-\frac {15 b \sqrt {a x+b \sqrt {x}}}{2 a^3}+\frac {5 \sqrt {x} \sqrt {a x+b \sqrt {x}}}{a^2}-\frac {4 x^{3/2}}{a \sqrt {a x+b \sqrt {x}}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 620
Rule 640
Rule 668
Rule 670
Rule 2018
Rubi steps
\begin {align*} \int \frac {x^{3/2}}{\left (b \sqrt {x}+a x\right )^{3/2}} \, dx &=2 \operatorname {Subst}\left (\int \frac {x^4}{\left (b x+a x^2\right )^{3/2}} \, dx,x,\sqrt {x}\right )\\ &=-\frac {4 x^{3/2}}{a \sqrt {b \sqrt {x}+a x}}+\frac {10 \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {b x+a x^2}} \, dx,x,\sqrt {x}\right )}{a}\\ &=-\frac {4 x^{3/2}}{a \sqrt {b \sqrt {x}+a x}}+\frac {5 \sqrt {x} \sqrt {b \sqrt {x}+a x}}{a^2}-\frac {(15 b) \operatorname {Subst}\left (\int \frac {x}{\sqrt {b x+a x^2}} \, dx,x,\sqrt {x}\right )}{2 a^2}\\ &=-\frac {4 x^{3/2}}{a \sqrt {b \sqrt {x}+a x}}-\frac {15 b \sqrt {b \sqrt {x}+a x}}{2 a^3}+\frac {5 \sqrt {x} \sqrt {b \sqrt {x}+a x}}{a^2}+\frac {\left (15 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b x+a x^2}} \, dx,x,\sqrt {x}\right )}{4 a^3}\\ &=-\frac {4 x^{3/2}}{a \sqrt {b \sqrt {x}+a x}}-\frac {15 b \sqrt {b \sqrt {x}+a x}}{2 a^3}+\frac {5 \sqrt {x} \sqrt {b \sqrt {x}+a x}}{a^2}+\frac {\left (15 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {b \sqrt {x}+a x}}\right )}{2 a^3}\\ &=-\frac {4 x^{3/2}}{a \sqrt {b \sqrt {x}+a x}}-\frac {15 b \sqrt {b \sqrt {x}+a x}}{2 a^3}+\frac {5 \sqrt {x} \sqrt {b \sqrt {x}+a x}}{a^2}+\frac {15 b^2 \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b \sqrt {x}+a x}}\right )}{2 a^{7/2}}\\ \end {align*}
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Mathematica [C] time = 0.07, size = 62, normalized size = 0.55 \[ \frac {4 x^2 \sqrt {\frac {a \sqrt {x}}{b}+1} \, _2F_1\left (\frac {3}{2},\frac {7}{2};\frac {9}{2};-\frac {a \sqrt {x}}{b}\right )}{7 b \sqrt {a x+b \sqrt {x}}} \]
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 [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 440, normalized size = 3.89 \[ \frac {\sqrt {a x +b \sqrt {x}}\, \left (16 a^{3} b^{2} x \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )-a^{3} b^{2} x \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {a x +b \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )+32 a^{2} b^{3} \sqrt {x}\, \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )-2 a^{2} b^{3} \sqrt {x}\, \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {a x +b \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )+4 \sqrt {a x +b \sqrt {x}}\, a^{\frac {9}{2}} x^{\frac {3}{2}}+16 a \,b^{4} \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )-a \,b^{4} \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {a x +b \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )-32 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, a^{\frac {7}{2}} b x +10 \sqrt {a x +b \sqrt {x}}\, a^{\frac {7}{2}} b x -64 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, a^{\frac {5}{2}} b^{2} \sqrt {x}+8 \sqrt {a x +b \sqrt {x}}\, a^{\frac {5}{2}} b^{2} \sqrt {x}-32 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, a^{\frac {3}{2}} b^{3}+2 \sqrt {a x +b \sqrt {x}}\, a^{\frac {3}{2}} b^{3}+16 \left (\left (a \sqrt {x}+b \right ) \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {5}{2}} b \right )}{4 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, \left (a \sqrt {x}+b \right )^{2} a^{\frac {9}{2}}} \]
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 \[ \int \frac {x^{\frac {3}{2}}}{{\left (a x + b \sqrt {x}\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {x^{3/2}}{{\left (a\,x+b\,\sqrt {x}\right )}^{3/2}} \,d x \]
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 \[ \int \frac {x^{\frac {3}{2}}}{\left (a x + b \sqrt {x}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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