Optimal. Leaf size=204 \[ \frac {231 b^6 \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {a x+b \sqrt {x}}}\right )}{256 a^{13/2}}-\frac {231 b^5 \sqrt {a x+b \sqrt {x}}}{256 a^6}+\frac {77 b^4 \sqrt {x} \sqrt {a x+b \sqrt {x}}}{128 a^5}-\frac {77 b^3 x \sqrt {a x+b \sqrt {x}}}{160 a^4}+\frac {33 b^2 x^{3/2} \sqrt {a x+b \sqrt {x}}}{80 a^3}-\frac {11 b x^2 \sqrt {a x+b \sqrt {x}}}{30 a^2}+\frac {x^{5/2} \sqrt {a x+b \sqrt {x}}}{3 a} \]
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Rubi [A] time = 0.17, antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 9, 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} \[ \frac {33 b^2 x^{3/2} \sqrt {a x+b \sqrt {x}}}{80 a^3}-\frac {231 b^5 \sqrt {a x+b \sqrt {x}}}{256 a^6}+\frac {77 b^4 \sqrt {x} \sqrt {a x+b \sqrt {x}}}{128 a^5}-\frac {77 b^3 x \sqrt {a x+b \sqrt {x}}}{160 a^4}+\frac {231 b^6 \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {a x+b \sqrt {x}}}\right )}{256 a^{13/2}}-\frac {11 b x^2 \sqrt {a x+b \sqrt {x}}}{30 a^2}+\frac {x^{5/2} \sqrt {a x+b \sqrt {x}}}{3 a} \]
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^{5/2}}{\sqrt {b \sqrt {x}+a x}} \, dx &=2 \operatorname {Subst}\left (\int \frac {x^6}{\sqrt {b x+a x^2}} \, dx,x,\sqrt {x}\right )\\ &=\frac {x^{5/2} \sqrt {b \sqrt {x}+a x}}{3 a}-\frac {(11 b) \operatorname {Subst}\left (\int \frac {x^5}{\sqrt {b x+a x^2}} \, dx,x,\sqrt {x}\right )}{6 a}\\ &=-\frac {11 b x^2 \sqrt {b \sqrt {x}+a x}}{30 a^2}+\frac {x^{5/2} \sqrt {b \sqrt {x}+a x}}{3 a}+\frac {\left (33 b^2\right ) \operatorname {Subst}\left (\int \frac {x^4}{\sqrt {b x+a x^2}} \, dx,x,\sqrt {x}\right )}{20 a^2}\\ &=\frac {33 b^2 x^{3/2} \sqrt {b \sqrt {x}+a x}}{80 a^3}-\frac {11 b x^2 \sqrt {b \sqrt {x}+a x}}{30 a^2}+\frac {x^{5/2} \sqrt {b \sqrt {x}+a x}}{3 a}-\frac {\left (231 b^3\right ) \operatorname {Subst}\left (\int \frac {x^3}{\sqrt {b x+a x^2}} \, dx,x,\sqrt {x}\right )}{160 a^3}\\ &=-\frac {77 b^3 x \sqrt {b \sqrt {x}+a x}}{160 a^4}+\frac {33 b^2 x^{3/2} \sqrt {b \sqrt {x}+a x}}{80 a^3}-\frac {11 b x^2 \sqrt {b \sqrt {x}+a x}}{30 a^2}+\frac {x^{5/2} \sqrt {b \sqrt {x}+a x}}{3 a}+\frac {\left (77 b^4\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {b x+a x^2}} \, dx,x,\sqrt {x}\right )}{64 a^4}\\ &=\frac {77 b^4 \sqrt {x} \sqrt {b \sqrt {x}+a x}}{128 a^5}-\frac {77 b^3 x \sqrt {b \sqrt {x}+a x}}{160 a^4}+\frac {33 b^2 x^{3/2} \sqrt {b \sqrt {x}+a x}}{80 a^3}-\frac {11 b x^2 \sqrt {b \sqrt {x}+a x}}{30 a^2}+\frac {x^{5/2} \sqrt {b \sqrt {x}+a x}}{3 a}-\frac {\left (231 b^5\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {b x+a x^2}} \, dx,x,\sqrt {x}\right )}{256 a^5}\\ &=-\frac {231 b^5 \sqrt {b \sqrt {x}+a x}}{256 a^6}+\frac {77 b^4 \sqrt {x} \sqrt {b \sqrt {x}+a x}}{128 a^5}-\frac {77 b^3 x \sqrt {b \sqrt {x}+a x}}{160 a^4}+\frac {33 b^2 x^{3/2} \sqrt {b \sqrt {x}+a x}}{80 a^3}-\frac {11 b x^2 \sqrt {b \sqrt {x}+a x}}{30 a^2}+\frac {x^{5/2} \sqrt {b \sqrt {x}+a x}}{3 a}+\frac {\left (231 b^6\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b x+a x^2}} \, dx,x,\sqrt {x}\right )}{512 a^6}\\ &=-\frac {231 b^5 \sqrt {b \sqrt {x}+a x}}{256 a^6}+\frac {77 b^4 \sqrt {x} \sqrt {b \sqrt {x}+a x}}{128 a^5}-\frac {77 b^3 x \sqrt {b \sqrt {x}+a x}}{160 a^4}+\frac {33 b^2 x^{3/2} \sqrt {b \sqrt {x}+a x}}{80 a^3}-\frac {11 b x^2 \sqrt {b \sqrt {x}+a x}}{30 a^2}+\frac {x^{5/2} \sqrt {b \sqrt {x}+a x}}{3 a}+\frac {\left (231 b^6\right ) \operatorname {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {b \sqrt {x}+a x}}\right )}{256 a^6}\\ &=-\frac {231 b^5 \sqrt {b \sqrt {x}+a x}}{256 a^6}+\frac {77 b^4 \sqrt {x} \sqrt {b \sqrt {x}+a x}}{128 a^5}-\frac {77 b^3 x \sqrt {b \sqrt {x}+a x}}{160 a^4}+\frac {33 b^2 x^{3/2} \sqrt {b \sqrt {x}+a x}}{80 a^3}-\frac {11 b x^2 \sqrt {b \sqrt {x}+a x}}{30 a^2}+\frac {x^{5/2} \sqrt {b \sqrt {x}+a x}}{3 a}+\frac {231 b^6 \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b \sqrt {x}+a x}}\right )}{256 a^{13/2}}\\ \end {align*}
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Mathematica [A] time = 0.26, size = 164, normalized size = 0.80 \[ \frac {\left (a \sqrt {x}+b\right ) \left (\sqrt {a} \sqrt {x} \sqrt {\frac {a \sqrt {x}}{b}+1} \left (1280 a^5 x^{5/2}-1408 a^4 b x^2+1584 a^3 b^2 x^{3/2}-1848 a^2 b^3 x+2310 a b^4 \sqrt {x}-3465 b^5\right )+3465 b^{11/2} \sqrt [4]{x} \sinh ^{-1}\left (\frac {\sqrt {a} \sqrt [4]{x}}{\sqrt {b}}\right )\right )}{3840 a^{13/2} \sqrt {\frac {a \sqrt {x}}{b}+1} \sqrt {a x+b \sqrt {x}}} \]
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 = 0.28, size = 125, normalized size = 0.61 \[ \frac {1}{3840} \, \sqrt {a x + b \sqrt {x}} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, \sqrt {x} {\left (\frac {10 \, \sqrt {x}}{a} - \frac {11 \, b}{a^{2}}\right )} + \frac {99 \, b^{2}}{a^{3}}\right )} \sqrt {x} - \frac {231 \, b^{3}}{a^{4}}\right )} \sqrt {x} + \frac {1155 \, b^{4}}{a^{5}}\right )} \sqrt {x} - \frac {3465 \, b^{5}}{a^{6}}\right )} - \frac {231 \, b^{6} \log \left ({\left | -2 \, \sqrt {a} {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )} - b \right |}\right )}{512 \, a^{\frac {13}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 245, normalized size = 1.20 \[ \frac {\sqrt {a x +b \sqrt {x}}\, \left (7680 a \,b^{6} \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )-4215 a \,b^{6} \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {a x +b \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )+2560 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {11}{2}} x^{\frac {3}{2}}+16860 \sqrt {a x +b \sqrt {x}}\, a^{\frac {5}{2}} b^{4} \sqrt {x}-5376 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {9}{2}} b x -15360 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, a^{\frac {3}{2}} b^{5}+8430 \sqrt {a x +b \sqrt {x}}\, a^{\frac {3}{2}} b^{5}+8544 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {7}{2}} b^{2} \sqrt {x}-12240 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {5}{2}} b^{3}\right )}{7680 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, a^{\frac {15}{2}}} \]
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 \[ \int \frac {x^{\frac {5}{2}}}{\sqrt {a x + b \sqrt {x}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^{5/2}}{\sqrt {a\,x+b\,\sqrt {x}}} \,d x \]
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 \[ \int \frac {x^{\frac {5}{2}}}{\sqrt {a x + b \sqrt {x}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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