Integrand size = 21, antiderivative size = 171 \[ \int \frac {x^{5/2}}{\left (b \sqrt {x}+a x\right )^{3/2}} \, dx=-\frac {4 x^{5/2}}{a \sqrt {b \sqrt {x}+a x}}-\frac {315 b^3 \sqrt {b \sqrt {x}+a x}}{32 a^5}+\frac {105 b^2 \sqrt {x} \sqrt {b \sqrt {x}+a x}}{16 a^4}-\frac {21 b x \sqrt {b \sqrt {x}+a x}}{4 a^3}+\frac {9 x^{3/2} \sqrt {b \sqrt {x}+a x}}{2 a^2}+\frac {315 b^4 \text {arctanh}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b \sqrt {x}+a x}}\right )}{32 a^{11/2}} \]
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Time = 0.10 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2043, 682, 684, 654, 634, 212} \[ \int \frac {x^{5/2}}{\left (b \sqrt {x}+a x\right )^{3/2}} \, dx=\frac {315 b^4 \text {arctanh}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {a x+b \sqrt {x}}}\right )}{32 a^{11/2}}-\frac {315 b^3 \sqrt {a x+b \sqrt {x}}}{32 a^5}+\frac {105 b^2 \sqrt {x} \sqrt {a x+b \sqrt {x}}}{16 a^4}-\frac {21 b x \sqrt {a x+b \sqrt {x}}}{4 a^3}+\frac {9 x^{3/2} \sqrt {a x+b \sqrt {x}}}{2 a^2}-\frac {4 x^{5/2}}{a \sqrt {a x+b \sqrt {x}}} \]
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Rule 212
Rule 634
Rule 654
Rule 682
Rule 684
Rule 2043
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {x^6}{\left (b x+a x^2\right )^{3/2}} \, dx,x,\sqrt {x}\right ) \\ & = -\frac {4 x^{5/2}}{a \sqrt {b \sqrt {x}+a x}}+\frac {18 \text {Subst}\left (\int \frac {x^4}{\sqrt {b x+a x^2}} \, dx,x,\sqrt {x}\right )}{a} \\ & = -\frac {4 x^{5/2}}{a \sqrt {b \sqrt {x}+a x}}+\frac {9 x^{3/2} \sqrt {b \sqrt {x}+a x}}{2 a^2}-\frac {(63 b) \text {Subst}\left (\int \frac {x^3}{\sqrt {b x+a x^2}} \, dx,x,\sqrt {x}\right )}{4 a^2} \\ & = -\frac {4 x^{5/2}}{a \sqrt {b \sqrt {x}+a x}}-\frac {21 b x \sqrt {b \sqrt {x}+a x}}{4 a^3}+\frac {9 x^{3/2} \sqrt {b \sqrt {x}+a x}}{2 a^2}+\frac {\left (105 b^2\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {b x+a x^2}} \, dx,x,\sqrt {x}\right )}{8 a^3} \\ & = -\frac {4 x^{5/2}}{a \sqrt {b \sqrt {x}+a x}}+\frac {105 b^2 \sqrt {x} \sqrt {b \sqrt {x}+a x}}{16 a^4}-\frac {21 b x \sqrt {b \sqrt {x}+a x}}{4 a^3}+\frac {9 x^{3/2} \sqrt {b \sqrt {x}+a x}}{2 a^2}-\frac {\left (315 b^3\right ) \text {Subst}\left (\int \frac {x}{\sqrt {b x+a x^2}} \, dx,x,\sqrt {x}\right )}{32 a^4} \\ & = -\frac {4 x^{5/2}}{a \sqrt {b \sqrt {x}+a x}}-\frac {315 b^3 \sqrt {b \sqrt {x}+a x}}{32 a^5}+\frac {105 b^2 \sqrt {x} \sqrt {b \sqrt {x}+a x}}{16 a^4}-\frac {21 b x \sqrt {b \sqrt {x}+a x}}{4 a^3}+\frac {9 x^{3/2} \sqrt {b \sqrt {x}+a x}}{2 a^2}+\frac {\left (315 b^4\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b x+a x^2}} \, dx,x,\sqrt {x}\right )}{64 a^5} \\ & = -\frac {4 x^{5/2}}{a \sqrt {b \sqrt {x}+a x}}-\frac {315 b^3 \sqrt {b \sqrt {x}+a x}}{32 a^5}+\frac {105 b^2 \sqrt {x} \sqrt {b \sqrt {x}+a x}}{16 a^4}-\frac {21 b x \sqrt {b \sqrt {x}+a x}}{4 a^3}+\frac {9 x^{3/2} \sqrt {b \sqrt {x}+a x}}{2 a^2}+\frac {\left (315 b^4\right ) \text {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {b \sqrt {x}+a x}}\right )}{32 a^5} \\ & = -\frac {4 x^{5/2}}{a \sqrt {b \sqrt {x}+a x}}-\frac {315 b^3 \sqrt {b \sqrt {x}+a x}}{32 a^5}+\frac {105 b^2 \sqrt {x} \sqrt {b \sqrt {x}+a x}}{16 a^4}-\frac {21 b x \sqrt {b \sqrt {x}+a x}}{4 a^3}+\frac {9 x^{3/2} \sqrt {b \sqrt {x}+a x}}{2 a^2}+\frac {315 b^4 \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b \sqrt {x}+a x}}\right )}{32 a^{11/2}} \\ \end{align*}
Time = 0.58 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.73 \[ \int \frac {x^{5/2}}{\left (b \sqrt {x}+a x\right )^{3/2}} \, dx=\frac {\sqrt {b \sqrt {x}+a x} \left (-315 b^4-105 a b^3 \sqrt {x}+42 a^2 b^2 x-24 a^3 b x^{3/2}+16 a^4 x^2\right )}{32 a^5 \left (b+a \sqrt {x}\right )}+\frac {315 b^4 \text {arctanh}\left (\frac {\sqrt {a} \sqrt {b \sqrt {x}+a x}}{b+a \sqrt {x}}\right )}{32 a^{11/2}} \]
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Time = 2.14 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.18
method | result | size |
derivativedivides | \(\frac {x^{\frac {5}{2}}}{2 a \sqrt {b \sqrt {x}+a x}}-\frac {9 b \left (\frac {x^{2}}{3 a \sqrt {b \sqrt {x}+a x}}-\frac {7 b \left (\frac {x^{\frac {3}{2}}}{2 a \sqrt {b \sqrt {x}+a x}}-\frac {5 b \left (\frac {x}{a \sqrt {b \sqrt {x}+a x}}-\frac {3 b \left (-\frac {\sqrt {x}}{a \sqrt {b \sqrt {x}+a x}}-\frac {b \left (-\frac {1}{a \sqrt {b \sqrt {x}+a x}}+\frac {b +2 a \sqrt {x}}{b a \sqrt {b \sqrt {x}+a x}}\right )}{2 a}+\frac {\ln \left (\frac {\frac {b}{2}+a \sqrt {x}}{\sqrt {a}}+\sqrt {b \sqrt {x}+a x}\right )}{a^{\frac {3}{2}}}\right )}{2 a}\right )}{4 a}\right )}{6 a}\right )}{4 a}\) | \(201\) |
default | \(\frac {\sqrt {b \sqrt {x}+a x}\, \left (32 x^{\frac {3}{2}} \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} a^{\frac {11}{2}}+276 x^{\frac {3}{2}} \sqrt {b \sqrt {x}+a x}\, a^{\frac {9}{2}} b^{2}-48 a^{\frac {9}{2}} \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} b x +690 x \sqrt {b \sqrt {x}+a x}\, a^{\frac {7}{2}} b^{3}-768 x \,a^{\frac {7}{2}} \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, b^{3}+384 x \,a^{3} \ln \left (\frac {2 a \sqrt {x}+2 \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, \sqrt {a}+b}{2 \sqrt {a}}\right ) b^{4}-192 a^{\frac {7}{2}} \sqrt {x}\, \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} b^{2}-69 x \ln \left (\frac {2 \sqrt {b \sqrt {x}+a x}\, \sqrt {a}+2 a \sqrt {x}+b}{2 \sqrt {a}}\right ) a^{3} b^{4}+552 a^{\frac {5}{2}} \sqrt {x}\, \sqrt {b \sqrt {x}+a x}\, b^{4}-1536 \sqrt {x}\, a^{\frac {5}{2}} \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, b^{4}+768 \sqrt {x}\, a^{2} \ln \left (\frac {2 a \sqrt {x}+2 \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, \sqrt {a}+b}{2 \sqrt {a}}\right ) b^{5}-112 a^{\frac {5}{2}} \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} b^{3}+256 a^{\frac {5}{2}} \left (\sqrt {x}\, \left (a \sqrt {x}+b \right )\right )^{\frac {3}{2}} b^{3}-138 \sqrt {x}\, \ln \left (\frac {2 \sqrt {b \sqrt {x}+a x}\, \sqrt {a}+2 a \sqrt {x}+b}{2 \sqrt {a}}\right ) a^{2} b^{5}+138 a^{\frac {3}{2}} \sqrt {b \sqrt {x}+a x}\, b^{5}-768 a^{\frac {3}{2}} \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, b^{5}+384 a \ln \left (\frac {2 a \sqrt {x}+2 \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, \sqrt {a}+b}{2 \sqrt {a}}\right ) b^{6}-69 \ln \left (\frac {2 \sqrt {b \sqrt {x}+a x}\, \sqrt {a}+2 a \sqrt {x}+b}{2 \sqrt {a}}\right ) a \,b^{6}\right )}{64 a^{\frac {13}{2}} \sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, \left (a \sqrt {x}+b \right )^{2}}\) | \(527\) |
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Timed out. \[ \int \frac {x^{5/2}}{\left (b \sqrt {x}+a x\right )^{3/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {x^{5/2}}{\left (b \sqrt {x}+a x\right )^{3/2}} \, dx=\int \frac {x^{\frac {5}{2}}}{\left (a x + b \sqrt {x}\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {x^{5/2}}{\left (b \sqrt {x}+a x\right )^{3/2}} \, dx=\int { \frac {x^{\frac {5}{2}}}{{\left (a x + b \sqrt {x}\right )}^{\frac {3}{2}}} \,d x } \]
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Time = 0.34 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.78 \[ \int \frac {x^{5/2}}{\left (b \sqrt {x}+a x\right )^{3/2}} \, dx=\frac {1}{32} \, \sqrt {a x + b \sqrt {x}} {\left (2 \, {\left (4 \, \sqrt {x} {\left (\frac {2 \, \sqrt {x}}{a^{2}} - \frac {5 \, b}{a^{3}}\right )} + \frac {41 \, b^{2}}{a^{4}}\right )} \sqrt {x} - \frac {187 \, b^{3}}{a^{5}}\right )} - \frac {315 \, b^{4} \log \left ({\left | -2 \, \sqrt {a} {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )} - b \right |}\right )}{64 \, a^{\frac {11}{2}}} - \frac {4 \, b^{5}}{{\left (\sqrt {a} {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )} + b\right )} a^{\frac {11}{2}}} \]
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Timed out. \[ \int \frac {x^{5/2}}{\left (b \sqrt {x}+a x\right )^{3/2}} \, dx=\int \frac {x^{5/2}}{{\left (a\,x+b\,\sqrt {x}\right )}^{3/2}} \,d x \]
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