Integrand size = 19, antiderivative size = 174 \[ \int \frac {x^2}{\sqrt {b \sqrt {x}+a x}} \, dx=\frac {63 b^4 \sqrt {b \sqrt {x}+a x}}{64 a^5}-\frac {21 b^3 \sqrt {x} \sqrt {b \sqrt {x}+a x}}{32 a^4}+\frac {21 b^2 x \sqrt {b \sqrt {x}+a x}}{40 a^3}-\frac {9 b x^{3/2} \sqrt {b \sqrt {x}+a x}}{20 a^2}+\frac {2 x^2 \sqrt {b \sqrt {x}+a x}}{5 a}-\frac {63 b^5 \text {arctanh}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b \sqrt {x}+a x}}\right )}{64 a^{11/2}} \]
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Time = 0.11 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {2043, 684, 654, 634, 212} \[ \int \frac {x^2}{\sqrt {b \sqrt {x}+a x}} \, dx=-\frac {63 b^5 \text {arctanh}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {a x+b \sqrt {x}}}\right )}{64 a^{11/2}}+\frac {63 b^4 \sqrt {a x+b \sqrt {x}}}{64 a^5}-\frac {21 b^3 \sqrt {x} \sqrt {a x+b \sqrt {x}}}{32 a^4}+\frac {21 b^2 x \sqrt {a x+b \sqrt {x}}}{40 a^3}-\frac {9 b x^{3/2} \sqrt {a x+b \sqrt {x}}}{20 a^2}+\frac {2 x^2 \sqrt {a x+b \sqrt {x}}}{5 a} \]
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Rule 212
Rule 634
Rule 654
Rule 684
Rule 2043
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {x^5}{\sqrt {b x+a x^2}} \, dx,x,\sqrt {x}\right ) \\ & = \frac {2 x^2 \sqrt {b \sqrt {x}+a x}}{5 a}-\frac {(9 b) \text {Subst}\left (\int \frac {x^4}{\sqrt {b x+a x^2}} \, dx,x,\sqrt {x}\right )}{5 a} \\ & = -\frac {9 b x^{3/2} \sqrt {b \sqrt {x}+a x}}{20 a^2}+\frac {2 x^2 \sqrt {b \sqrt {x}+a x}}{5 a}+\frac {\left (63 b^2\right ) \text {Subst}\left (\int \frac {x^3}{\sqrt {b x+a x^2}} \, dx,x,\sqrt {x}\right )}{40 a^2} \\ & = \frac {21 b^2 x \sqrt {b \sqrt {x}+a x}}{40 a^3}-\frac {9 b x^{3/2} \sqrt {b \sqrt {x}+a x}}{20 a^2}+\frac {2 x^2 \sqrt {b \sqrt {x}+a x}}{5 a}-\frac {\left (21 b^3\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {b x+a x^2}} \, dx,x,\sqrt {x}\right )}{16 a^3} \\ & = -\frac {21 b^3 \sqrt {x} \sqrt {b \sqrt {x}+a x}}{32 a^4}+\frac {21 b^2 x \sqrt {b \sqrt {x}+a x}}{40 a^3}-\frac {9 b x^{3/2} \sqrt {b \sqrt {x}+a x}}{20 a^2}+\frac {2 x^2 \sqrt {b \sqrt {x}+a x}}{5 a}+\frac {\left (63 b^4\right ) \text {Subst}\left (\int \frac {x}{\sqrt {b x+a x^2}} \, dx,x,\sqrt {x}\right )}{64 a^4} \\ & = \frac {63 b^4 \sqrt {b \sqrt {x}+a x}}{64 a^5}-\frac {21 b^3 \sqrt {x} \sqrt {b \sqrt {x}+a x}}{32 a^4}+\frac {21 b^2 x \sqrt {b \sqrt {x}+a x}}{40 a^3}-\frac {9 b x^{3/2} \sqrt {b \sqrt {x}+a x}}{20 a^2}+\frac {2 x^2 \sqrt {b \sqrt {x}+a x}}{5 a}-\frac {\left (63 b^5\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b x+a x^2}} \, dx,x,\sqrt {x}\right )}{128 a^5} \\ & = \frac {63 b^4 \sqrt {b \sqrt {x}+a x}}{64 a^5}-\frac {21 b^3 \sqrt {x} \sqrt {b \sqrt {x}+a x}}{32 a^4}+\frac {21 b^2 x \sqrt {b \sqrt {x}+a x}}{40 a^3}-\frac {9 b x^{3/2} \sqrt {b \sqrt {x}+a x}}{20 a^2}+\frac {2 x^2 \sqrt {b \sqrt {x}+a x}}{5 a}-\frac {\left (63 b^5\right ) \text {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {b \sqrt {x}+a x}}\right )}{64 a^5} \\ & = \frac {63 b^4 \sqrt {b \sqrt {x}+a x}}{64 a^5}-\frac {21 b^3 \sqrt {x} \sqrt {b \sqrt {x}+a x}}{32 a^4}+\frac {21 b^2 x \sqrt {b \sqrt {x}+a x}}{40 a^3}-\frac {9 b x^{3/2} \sqrt {b \sqrt {x}+a x}}{20 a^2}+\frac {2 x^2 \sqrt {b \sqrt {x}+a x}}{5 a}-\frac {63 b^5 \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b \sqrt {x}+a x}}\right )}{64 a^{11/2}} \\ \end{align*}
Time = 0.33 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.65 \[ \int \frac {x^2}{\sqrt {b \sqrt {x}+a x}} \, dx=\frac {\sqrt {b \sqrt {x}+a x} \left (315 b^4-210 a b^3 \sqrt {x}+168 a^2 b^2 x-144 a^3 b x^{3/2}+128 a^4 x^2\right )}{320 a^5}-\frac {63 b^5 \text {arctanh}\left (\frac {\sqrt {a} \sqrt {b \sqrt {x}+a x}}{b+a \sqrt {x}}\right )}{64 a^{11/2}} \]
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Time = 2.22 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.87
method | result | size |
derivativedivides | \(\frac {2 x^{2} \sqrt {b \sqrt {x}+a x}}{5 a}-\frac {9 b \left (\frac {x^{\frac {3}{2}} \sqrt {b \sqrt {x}+a x}}{4 a}-\frac {7 b \left (\frac {x \sqrt {b \sqrt {x}+a x}}{3 a}-\frac {5 b \left (\frac {\sqrt {x}\, \sqrt {b \sqrt {x}+a x}}{2 a}-\frac {3 b \left (\frac {\sqrt {b \sqrt {x}+a x}}{a}-\frac {b \ln \left (\frac {\frac {b}{2}+a \sqrt {x}}{\sqrt {a}}+\sqrt {b \sqrt {x}+a x}\right )}{2 a^{\frac {3}{2}}}\right )}{4 a}\right )}{6 a}\right )}{8 a}\right )}{5 a}\) | \(151\) |
default | \(-\frac {\sqrt {b \sqrt {x}+a x}\, \left (544 \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} \sqrt {x}\, a^{\frac {7}{2}} b -256 x \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} a^{\frac {9}{2}}-880 \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} a^{\frac {5}{2}} b^{2}+1300 \sqrt {b \sqrt {x}+a x}\, \sqrt {x}\, a^{\frac {5}{2}} b^{3}+650 \sqrt {b \sqrt {x}+a x}\, a^{\frac {3}{2}} b^{4}-1280 a^{\frac {3}{2}} \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, b^{4}+640 a \ln \left (\frac {2 a \sqrt {x}+2 \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, \sqrt {a}+b}{2 \sqrt {a}}\right ) b^{5}-325 \ln \left (\frac {2 \sqrt {b \sqrt {x}+a x}\, \sqrt {a}+2 a \sqrt {x}+b}{2 \sqrt {a}}\right ) a \,b^{5}\right )}{640 \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, a^{\frac {13}{2}}}\) | \(223\) |
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Timed out. \[ \int \frac {x^2}{\sqrt {b \sqrt {x}+a x}} \, dx=\text {Timed out} \]
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Time = 0.42 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.05 \[ \int \frac {x^2}{\sqrt {b \sqrt {x}+a x}} \, dx=2 \left (\begin {cases} \sqrt {a x + b \sqrt {x}} \left (\frac {x^{2}}{5 a} - \frac {9 b x^{\frac {3}{2}}}{40 a^{2}} + \frac {21 b^{2} x}{80 a^{3}} - \frac {21 b^{3} \sqrt {x}}{64 a^{4}} + \frac {63 b^{4}}{128 a^{5}}\right ) - \frac {63 b^{5} \left (\begin {cases} \frac {\log {\left (2 \sqrt {a} \sqrt {a x + b \sqrt {x}} + 2 a \sqrt {x} + b \right )}}{\sqrt {a}} & \text {for}\: \frac {b^{2}}{a} \neq 0 \\\frac {\left (\sqrt {x} + \frac {b}{2 a}\right ) \log {\left (\sqrt {x} + \frac {b}{2 a} \right )}}{\sqrt {a \left (\sqrt {x} + \frac {b}{2 a}\right )^{2}}} & \text {otherwise} \end {cases}\right )}{256 a^{5}} & \text {for}\: a \neq 0 \\\frac {2 \left (b \sqrt {x}\right )^{\frac {11}{2}}}{11 b^{6}} & \text {for}\: b \neq 0 \\\tilde {\infty } x^{3} & \text {otherwise} \end {cases}\right ) \]
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\[ \int \frac {x^2}{\sqrt {b \sqrt {x}+a x}} \, dx=\int { \frac {x^{2}}{\sqrt {a x + b \sqrt {x}}} \,d x } \]
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Time = 0.32 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.63 \[ \int \frac {x^2}{\sqrt {b \sqrt {x}+a x}} \, dx=\frac {1}{320} \, \sqrt {a x + b \sqrt {x}} {\left (2 \, {\left (4 \, {\left (2 \, \sqrt {x} {\left (\frac {8 \, \sqrt {x}}{a} - \frac {9 \, b}{a^{2}}\right )} + \frac {21 \, b^{2}}{a^{3}}\right )} \sqrt {x} - \frac {105 \, b^{3}}{a^{4}}\right )} \sqrt {x} + \frac {315 \, b^{4}}{a^{5}}\right )} + \frac {63 \, b^{5} \log \left ({\left | 2 \, \sqrt {a} {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )} + b \right |}\right )}{128 \, a^{\frac {11}{2}}} \]
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Timed out. \[ \int \frac {x^2}{\sqrt {b \sqrt {x}+a x}} \, dx=\int \frac {x^2}{\sqrt {a\,x+b\,\sqrt {x}}} \,d x \]
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