Optimal. Leaf size=204 \[ -\frac {231 b^5 \sqrt {b \sqrt {x}+a x}}{256 a^6}+\frac {77 b^4 \sqrt {x} \sqrt {b \sqrt {x}+a x}}{128 a^5}-\frac {77 b^3 x \sqrt {b \sqrt {x}+a x}}{160 a^4}+\frac {33 b^2 x^{3/2} \sqrt {b \sqrt {x}+a x}}{80 a^3}-\frac {11 b x^2 \sqrt {b \sqrt {x}+a x}}{30 a^2}+\frac {x^{5/2} \sqrt {b \sqrt {x}+a x}}{3 a}+\frac {231 b^6 \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b \sqrt {x}+a x}}\right )}{256 a^{13/2}} \]
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Rubi [A]
time = 0.12, 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 = {2043, 684, 654,
634, 212} \begin {gather*} \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} \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^{5/2}}{\sqrt {b \sqrt {x}+a x}} \, dx &=2 \text {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) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 = 0.24, size = 126, normalized size = 0.62 \begin {gather*} \frac {\sqrt {b \sqrt {x}+a x} \left (-3465 b^5+2310 a b^4 \sqrt {x}-1848 a^2 b^3 x+1584 a^3 b^2 x^{3/2}-1408 a^4 b x^2+1280 a^5 x^{5/2}\right )}{3840 a^6}-\frac {231 b^6 \log \left (b+2 a \sqrt {x}-2 \sqrt {a} \sqrt {b \sqrt {x}+a x}\right )}{512 a^{13/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.39, size = 245, normalized size = 1.20
method | result | size |
derivativedivides | \(\frac {x^{\frac {5}{2}} \sqrt {b \sqrt {x}+a x}}{3 a}-\frac {11 b \left (\frac {x^{2} \sqrt {b \sqrt {x}+a x}}{5 a}-\frac {9 b \left (\frac {x^{\frac {3}{2}} \sqrt {b \sqrt {x}+a x}}{4 a}-\frac {7 b \left (\frac {x \sqrt {b \sqrt {x}+a x}}{3 a}-\frac {5 b \left (\frac {\sqrt {x}\, \sqrt {b \sqrt {x}+a x}}{2 a}-\frac {3 b \left (\frac {\sqrt {b \sqrt {x}+a x}}{a}-\frac {b \ln \left (\frac {\frac {b}{2}+a \sqrt {x}}{\sqrt {a}}+\sqrt {b \sqrt {x}+a x}\right )}{2 a^{\frac {3}{2}}}\right )}{4 a}\right )}{6 a}\right )}{8 a}\right )}{10 a}\right )}{6 a}\) | \(177\) |
default | \(\frac {\sqrt {b \sqrt {x}+a x}\, \left (2560 x^{\frac {3}{2}} \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} a^{\frac {11}{2}}+8544 \sqrt {x}\, \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} a^{\frac {7}{2}} b^{2}-5376 \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} a^{\frac {9}{2}} b x +16860 \sqrt {x}\, \sqrt {b \sqrt {x}+a x}\, a^{\frac {5}{2}} b^{4}-12240 \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} a^{\frac {5}{2}} b^{3}+8430 \sqrt {b \sqrt {x}+a x}\, a^{\frac {3}{2}} b^{5}-15360 \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, a^{\frac {3}{2}} b^{5}+7680 a \ln \left (\frac {2 a \sqrt {x}+2 \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, \sqrt {a}+b}{2 \sqrt {a}}\right ) b^{6}-4215 \ln \left (\frac {2 a \sqrt {x}+2 \sqrt {b \sqrt {x}+a x}\, \sqrt {a}+b}{2 \sqrt {a}}\right ) a \,b^{6}\right )}{7680 \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, a^{\frac {15}{2}}}\) | \(245\) |
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 {5}{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 = 0.55, size = 125, normalized size = 0.61 \begin {gather*} \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}}} \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 = -0.00 \begin {gather*} \int \frac {x^{5/2}}{\sqrt {a\,x+b\,\sqrt {x}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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