Optimal. Leaf size=195 \[ \frac{512 a^3 \sqrt{a x+b \sqrt{x}}}{77 b^5 x^{3/2}}-\frac{1280 a^2 \sqrt{a x+b \sqrt{x}}}{231 b^4 x^2}+\frac{4096 a^5 \sqrt{a x+b \sqrt{x}}}{231 b^7 \sqrt{x}}-\frac{2048 a^4 \sqrt{a x+b \sqrt{x}}}{231 b^6 x}+\frac{160 a \sqrt{a x+b \sqrt{x}}}{33 b^3 x^{5/2}}-\frac{48 \sqrt{a x+b \sqrt{x}}}{11 b^2 x^3}+\frac{4}{b x^{5/2} \sqrt{a x+b \sqrt{x}}} \]
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Rubi [A] time = 0.299999, antiderivative size = 195, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {2015, 2016, 2014} \[ \frac{512 a^3 \sqrt{a x+b \sqrt{x}}}{77 b^5 x^{3/2}}-\frac{1280 a^2 \sqrt{a x+b \sqrt{x}}}{231 b^4 x^2}+\frac{4096 a^5 \sqrt{a x+b \sqrt{x}}}{231 b^7 \sqrt{x}}-\frac{2048 a^4 \sqrt{a x+b \sqrt{x}}}{231 b^6 x}+\frac{160 a \sqrt{a x+b \sqrt{x}}}{33 b^3 x^{5/2}}-\frac{48 \sqrt{a x+b \sqrt{x}}}{11 b^2 x^3}+\frac{4}{b x^{5/2} \sqrt{a x+b \sqrt{x}}} \]
Antiderivative was successfully verified.
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Rule 2015
Rule 2016
Rule 2014
Rubi steps
\begin{align*} \int \frac{1}{x^3 \left (b \sqrt{x}+a x\right )^{3/2}} \, dx &=\frac{4}{b x^{5/2} \sqrt{b \sqrt{x}+a x}}+\frac{12 \int \frac{1}{x^{7/2} \sqrt{b \sqrt{x}+a x}} \, dx}{b}\\ &=\frac{4}{b x^{5/2} \sqrt{b \sqrt{x}+a x}}-\frac{48 \sqrt{b \sqrt{x}+a x}}{11 b^2 x^3}-\frac{(120 a) \int \frac{1}{x^3 \sqrt{b \sqrt{x}+a x}} \, dx}{11 b^2}\\ &=\frac{4}{b x^{5/2} \sqrt{b \sqrt{x}+a x}}-\frac{48 \sqrt{b \sqrt{x}+a x}}{11 b^2 x^3}+\frac{160 a \sqrt{b \sqrt{x}+a x}}{33 b^3 x^{5/2}}+\frac{\left (320 a^2\right ) \int \frac{1}{x^{5/2} \sqrt{b \sqrt{x}+a x}} \, dx}{33 b^3}\\ &=\frac{4}{b x^{5/2} \sqrt{b \sqrt{x}+a x}}-\frac{48 \sqrt{b \sqrt{x}+a x}}{11 b^2 x^3}+\frac{160 a \sqrt{b \sqrt{x}+a x}}{33 b^3 x^{5/2}}-\frac{1280 a^2 \sqrt{b \sqrt{x}+a x}}{231 b^4 x^2}-\frac{\left (640 a^3\right ) \int \frac{1}{x^2 \sqrt{b \sqrt{x}+a x}} \, dx}{77 b^4}\\ &=\frac{4}{b x^{5/2} \sqrt{b \sqrt{x}+a x}}-\frac{48 \sqrt{b \sqrt{x}+a x}}{11 b^2 x^3}+\frac{160 a \sqrt{b \sqrt{x}+a x}}{33 b^3 x^{5/2}}-\frac{1280 a^2 \sqrt{b \sqrt{x}+a x}}{231 b^4 x^2}+\frac{512 a^3 \sqrt{b \sqrt{x}+a x}}{77 b^5 x^{3/2}}+\frac{\left (512 a^4\right ) \int \frac{1}{x^{3/2} \sqrt{b \sqrt{x}+a x}} \, dx}{77 b^5}\\ &=\frac{4}{b x^{5/2} \sqrt{b \sqrt{x}+a x}}-\frac{48 \sqrt{b \sqrt{x}+a x}}{11 b^2 x^3}+\frac{160 a \sqrt{b \sqrt{x}+a x}}{33 b^3 x^{5/2}}-\frac{1280 a^2 \sqrt{b \sqrt{x}+a x}}{231 b^4 x^2}+\frac{512 a^3 \sqrt{b \sqrt{x}+a x}}{77 b^5 x^{3/2}}-\frac{2048 a^4 \sqrt{b \sqrt{x}+a x}}{231 b^6 x}-\frac{\left (1024 a^5\right ) \int \frac{1}{x \sqrt{b \sqrt{x}+a x}} \, dx}{231 b^6}\\ &=\frac{4}{b x^{5/2} \sqrt{b \sqrt{x}+a x}}-\frac{48 \sqrt{b \sqrt{x}+a x}}{11 b^2 x^3}+\frac{160 a \sqrt{b \sqrt{x}+a x}}{33 b^3 x^{5/2}}-\frac{1280 a^2 \sqrt{b \sqrt{x}+a x}}{231 b^4 x^2}+\frac{512 a^3 \sqrt{b \sqrt{x}+a x}}{77 b^5 x^{3/2}}-\frac{2048 a^4 \sqrt{b \sqrt{x}+a x}}{231 b^6 x}+\frac{4096 a^5 \sqrt{b \sqrt{x}+a x}}{231 b^7 \sqrt{x}}\\ \end{align*}
Mathematica [A] time = 0.0627457, size = 96, normalized size = 0.49 \[ \frac{4 \left (-128 a^4 b^2 x^2+64 a^3 b^3 x^{3/2}-40 a^2 b^4 x+512 a^5 b x^{5/2}+1024 a^6 x^3+28 a b^5 \sqrt{x}-21 b^6\right )}{231 b^7 x^{5/2} \sqrt{a x+b \sqrt{x}}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.012, size = 614, normalized size = 3.2 \begin{align*}{\frac{1}{231\,{b}^{8}}\sqrt{b\sqrt{x}+ax} \left ( 1155\,\ln \left ( 1/2\,{\frac{2\,\sqrt{\sqrt{x} \left ( b+a\sqrt{x} \right ) }\sqrt{a}+2\,a\sqrt{x}+b}{\sqrt{a}}} \right ){x}^{15/2}{a}^{8}b-2310\,\ln \left ( 1/2\,{\frac{2\,a\sqrt{x}+2\,\sqrt{b\sqrt{x}+ax}\sqrt{a}+b}{\sqrt{a}}} \right ){x}^{7}{a}^{7}{b}^{2}-924\,{a}^{15/2}{x}^{13/2} \left ( \sqrt{x} \left ( b+a\sqrt{x} \right ) \right ) ^{3/2}+2310\,\ln \left ( 1/2\,{\frac{2\,\sqrt{\sqrt{x} \left ( b+a\sqrt{x} \right ) }\sqrt{a}+2\,a\sqrt{x}+b}{\sqrt{a}}} \right ){x}^{7}{a}^{7}{b}^{2}+256\, \left ( b\sqrt{x}+ax \right ) ^{3/2}{a}^{7/2}{x}^{9/2}{b}^{4}-1155\,\ln \left ( 1/2\,{\frac{2\,a\sqrt{x}+2\,\sqrt{b\sqrt{x}+ax}\sqrt{a}+b}{\sqrt{a}}} \right ){x}^{13/2}{a}^{6}{b}^{3}+1155\,\ln \left ( 1/2\,{\frac{2\,\sqrt{\sqrt{x} \left ( b+a\sqrt{x} \right ) }\sqrt{a}+2\,a\sqrt{x}+b}{\sqrt{a}}} \right ){x}^{13/2}{a}^{6}{b}^{3}-160\, \left ( b\sqrt{x}+ax \right ) ^{3/2}{a}^{5/2}{x}^{4}{b}^{5}+112\, \left ( b\sqrt{x}+ax \right ) ^{3/2}{a}^{3/2}{x}^{7/2}{b}^{6}-84\, \left ( b\sqrt{x}+ax \right ) ^{3/2}\sqrt{a}{x}^{3}{b}^{7}-512\, \left ( b\sqrt{x}+ax \right ) ^{3/2}{a}^{9/2}{x}^{5}{b}^{3}+2048\, \left ( b\sqrt{x}+ax \right ) ^{3/2}{a}^{11/2}{x}^{11/2}{b}^{2}-4620\,{a}^{15/2}{x}^{7}\sqrt{\sqrt{x} \left ( b+a\sqrt{x} \right ) }b+8716\, \left ( b\sqrt{x}+ax \right ) ^{3/2}{a}^{13/2}{x}^{6}b-4620\,\sqrt{b\sqrt{x}+ax}{a}^{15/2}{x}^{7}b-2310\,\sqrt{b\sqrt{x}+ax}{a}^{13/2}{x}^{13/2}{b}^{2}-2310\,{a}^{13/2}{x}^{13/2}\sqrt{\sqrt{x} \left ( b+a\sqrt{x} \right ) }{b}^{2}+5544\, \left ( b\sqrt{x}+ax \right ) ^{3/2}{a}^{15/2}{x}^{13/2}-2310\,\sqrt{b\sqrt{x}+ax}{a}^{17/2}{x}^{15/2}-2310\,{a}^{17/2}{x}^{15/2}\sqrt{\sqrt{x} \left ( b+a\sqrt{x} \right ) }-1155\,\ln \left ( 1/2\,{\frac{2\,a\sqrt{x}+2\,\sqrt{b\sqrt{x}+ax}\sqrt{a}+b}{\sqrt{a}}} \right ){x}^{15/2}{a}^{8}b \right ){\frac{1}{\sqrt{\sqrt{x} \left ( b+a\sqrt{x} \right ) }}}{\frac{1}{\sqrt{a}}}{x}^{-{\frac{13}{2}}} \left ( b+a\sqrt{x} \right ) ^{-2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a x + b \sqrt{x}\right )}^{\frac{3}{2}} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.89979, size = 246, normalized size = 1.26 \begin{align*} -\frac{4 \,{\left (512 \, a^{6} b x^{3} - 192 \, a^{4} b^{3} x^{2} - 68 \, a^{2} b^{5} x - 21 \, b^{7} -{\left (1024 \, a^{7} x^{3} - 640 \, a^{5} b^{2} x^{2} - 104 \, a^{3} b^{4} x - 49 \, a b^{6}\right )} \sqrt{x}\right )} \sqrt{a x + b \sqrt{x}}}{231 \,{\left (a^{2} b^{7} x^{4} - b^{9} x^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{3} \left (a x + b \sqrt{x}\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a x + b \sqrt{x}\right )}^{\frac{3}{2}} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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