Optimal. Leaf size=329 \[ \frac{138567 a^9 \sqrt{a x+b x^{2/3}}}{131072 b^{10} x^{2/3}}-\frac{46189 a^8 \sqrt{a x+b x^{2/3}}}{65536 b^9 x}+\frac{46189 a^7 \sqrt{a x+b x^{2/3}}}{81920 b^8 x^{4/3}}-\frac{138567 a^6 \sqrt{a x+b x^{2/3}}}{286720 b^7 x^{5/3}}+\frac{46189 a^5 \sqrt{a x+b x^{2/3}}}{107520 b^6 x^2}-\frac{4199 a^4 \sqrt{a x+b x^{2/3}}}{10752 b^5 x^{7/3}}+\frac{323 a^3 \sqrt{a x+b x^{2/3}}}{896 b^4 x^{8/3}}-\frac{323 a^2 \sqrt{a x+b x^{2/3}}}{960 b^3 x^3}-\frac{138567 a^{10} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{a x+b x^{2/3}}}\right )}{131072 b^{21/2}}+\frac{19 a \sqrt{a x+b x^{2/3}}}{60 b^2 x^{10/3}}-\frac{3 \sqrt{a x+b x^{2/3}}}{10 b x^{11/3}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.577139, antiderivative size = 329, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {2025, 2029, 206} \[ \frac{138567 a^9 \sqrt{a x+b x^{2/3}}}{131072 b^{10} x^{2/3}}-\frac{46189 a^8 \sqrt{a x+b x^{2/3}}}{65536 b^9 x}+\frac{46189 a^7 \sqrt{a x+b x^{2/3}}}{81920 b^8 x^{4/3}}-\frac{138567 a^6 \sqrt{a x+b x^{2/3}}}{286720 b^7 x^{5/3}}+\frac{46189 a^5 \sqrt{a x+b x^{2/3}}}{107520 b^6 x^2}-\frac{4199 a^4 \sqrt{a x+b x^{2/3}}}{10752 b^5 x^{7/3}}+\frac{323 a^3 \sqrt{a x+b x^{2/3}}}{896 b^4 x^{8/3}}-\frac{323 a^2 \sqrt{a x+b x^{2/3}}}{960 b^3 x^3}-\frac{138567 a^{10} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{a x+b x^{2/3}}}\right )}{131072 b^{21/2}}+\frac{19 a \sqrt{a x+b x^{2/3}}}{60 b^2 x^{10/3}}-\frac{3 \sqrt{a x+b x^{2/3}}}{10 b x^{11/3}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2025
Rule 2029
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{x^4 \sqrt{b x^{2/3}+a x}} \, dx &=-\frac{3 \sqrt{b x^{2/3}+a x}}{10 b x^{11/3}}-\frac{(19 a) \int \frac{1}{x^{11/3} \sqrt{b x^{2/3}+a x}} \, dx}{20 b}\\ &=-\frac{3 \sqrt{b x^{2/3}+a x}}{10 b x^{11/3}}+\frac{19 a \sqrt{b x^{2/3}+a x}}{60 b^2 x^{10/3}}+\frac{\left (323 a^2\right ) \int \frac{1}{x^{10/3} \sqrt{b x^{2/3}+a x}} \, dx}{360 b^2}\\ &=-\frac{3 \sqrt{b x^{2/3}+a x}}{10 b x^{11/3}}+\frac{19 a \sqrt{b x^{2/3}+a x}}{60 b^2 x^{10/3}}-\frac{323 a^2 \sqrt{b x^{2/3}+a x}}{960 b^3 x^3}-\frac{\left (323 a^3\right ) \int \frac{1}{x^3 \sqrt{b x^{2/3}+a x}} \, dx}{384 b^3}\\ &=-\frac{3 \sqrt{b x^{2/3}+a x}}{10 b x^{11/3}}+\frac{19 a \sqrt{b x^{2/3}+a x}}{60 b^2 x^{10/3}}-\frac{323 a^2 \sqrt{b x^{2/3}+a x}}{960 b^3 x^3}+\frac{323 a^3 \sqrt{b x^{2/3}+a x}}{896 b^4 x^{8/3}}+\frac{\left (4199 a^4\right ) \int \frac{1}{x^{8/3} \sqrt{b x^{2/3}+a x}} \, dx}{5376 b^4}\\ &=-\frac{3 \sqrt{b x^{2/3}+a x}}{10 b x^{11/3}}+\frac{19 a \sqrt{b x^{2/3}+a x}}{60 b^2 x^{10/3}}-\frac{323 a^2 \sqrt{b x^{2/3}+a x}}{960 b^3 x^3}+\frac{323 a^3 \sqrt{b x^{2/3}+a x}}{896 b^4 x^{8/3}}-\frac{4199 a^4 \sqrt{b x^{2/3}+a x}}{10752 b^5 x^{7/3}}-\frac{\left (46189 a^5\right ) \int \frac{1}{x^{7/3} \sqrt{b x^{2/3}+a x}} \, dx}{64512 b^5}\\ &=-\frac{3 \sqrt{b x^{2/3}+a x}}{10 b x^{11/3}}+\frac{19 a \sqrt{b x^{2/3}+a x}}{60 b^2 x^{10/3}}-\frac{323 a^2 \sqrt{b x^{2/3}+a x}}{960 b^3 x^3}+\frac{323 a^3 \sqrt{b x^{2/3}+a x}}{896 b^4 x^{8/3}}-\frac{4199 a^4 \sqrt{b x^{2/3}+a x}}{10752 b^5 x^{7/3}}+\frac{46189 a^5 \sqrt{b x^{2/3}+a x}}{107520 b^6 x^2}+\frac{\left (46189 a^6\right ) \int \frac{1}{x^2 \sqrt{b x^{2/3}+a x}} \, dx}{71680 b^6}\\ &=-\frac{3 \sqrt{b x^{2/3}+a x}}{10 b x^{11/3}}+\frac{19 a \sqrt{b x^{2/3}+a x}}{60 b^2 x^{10/3}}-\frac{323 a^2 \sqrt{b x^{2/3}+a x}}{960 b^3 x^3}+\frac{323 a^3 \sqrt{b x^{2/3}+a x}}{896 b^4 x^{8/3}}-\frac{4199 a^4 \sqrt{b x^{2/3}+a x}}{10752 b^5 x^{7/3}}+\frac{46189 a^5 \sqrt{b x^{2/3}+a x}}{107520 b^6 x^2}-\frac{138567 a^6 \sqrt{b x^{2/3}+a x}}{286720 b^7 x^{5/3}}-\frac{\left (46189 a^7\right ) \int \frac{1}{x^{5/3} \sqrt{b x^{2/3}+a x}} \, dx}{81920 b^7}\\ &=-\frac{3 \sqrt{b x^{2/3}+a x}}{10 b x^{11/3}}+\frac{19 a \sqrt{b x^{2/3}+a x}}{60 b^2 x^{10/3}}-\frac{323 a^2 \sqrt{b x^{2/3}+a x}}{960 b^3 x^3}+\frac{323 a^3 \sqrt{b x^{2/3}+a x}}{896 b^4 x^{8/3}}-\frac{4199 a^4 \sqrt{b x^{2/3}+a x}}{10752 b^5 x^{7/3}}+\frac{46189 a^5 \sqrt{b x^{2/3}+a x}}{107520 b^6 x^2}-\frac{138567 a^6 \sqrt{b x^{2/3}+a x}}{286720 b^7 x^{5/3}}+\frac{46189 a^7 \sqrt{b x^{2/3}+a x}}{81920 b^8 x^{4/3}}+\frac{\left (46189 a^8\right ) \int \frac{1}{x^{4/3} \sqrt{b x^{2/3}+a x}} \, dx}{98304 b^8}\\ &=-\frac{3 \sqrt{b x^{2/3}+a x}}{10 b x^{11/3}}+\frac{19 a \sqrt{b x^{2/3}+a x}}{60 b^2 x^{10/3}}-\frac{323 a^2 \sqrt{b x^{2/3}+a x}}{960 b^3 x^3}+\frac{323 a^3 \sqrt{b x^{2/3}+a x}}{896 b^4 x^{8/3}}-\frac{4199 a^4 \sqrt{b x^{2/3}+a x}}{10752 b^5 x^{7/3}}+\frac{46189 a^5 \sqrt{b x^{2/3}+a x}}{107520 b^6 x^2}-\frac{138567 a^6 \sqrt{b x^{2/3}+a x}}{286720 b^7 x^{5/3}}+\frac{46189 a^7 \sqrt{b x^{2/3}+a x}}{81920 b^8 x^{4/3}}-\frac{46189 a^8 \sqrt{b x^{2/3}+a x}}{65536 b^9 x}-\frac{\left (46189 a^9\right ) \int \frac{1}{x \sqrt{b x^{2/3}+a x}} \, dx}{131072 b^9}\\ &=-\frac{3 \sqrt{b x^{2/3}+a x}}{10 b x^{11/3}}+\frac{19 a \sqrt{b x^{2/3}+a x}}{60 b^2 x^{10/3}}-\frac{323 a^2 \sqrt{b x^{2/3}+a x}}{960 b^3 x^3}+\frac{323 a^3 \sqrt{b x^{2/3}+a x}}{896 b^4 x^{8/3}}-\frac{4199 a^4 \sqrt{b x^{2/3}+a x}}{10752 b^5 x^{7/3}}+\frac{46189 a^5 \sqrt{b x^{2/3}+a x}}{107520 b^6 x^2}-\frac{138567 a^6 \sqrt{b x^{2/3}+a x}}{286720 b^7 x^{5/3}}+\frac{46189 a^7 \sqrt{b x^{2/3}+a x}}{81920 b^8 x^{4/3}}-\frac{46189 a^8 \sqrt{b x^{2/3}+a x}}{65536 b^9 x}+\frac{138567 a^9 \sqrt{b x^{2/3}+a x}}{131072 b^{10} x^{2/3}}+\frac{\left (46189 a^{10}\right ) \int \frac{1}{x^{2/3} \sqrt{b x^{2/3}+a x}} \, dx}{262144 b^{10}}\\ &=-\frac{3 \sqrt{b x^{2/3}+a x}}{10 b x^{11/3}}+\frac{19 a \sqrt{b x^{2/3}+a x}}{60 b^2 x^{10/3}}-\frac{323 a^2 \sqrt{b x^{2/3}+a x}}{960 b^3 x^3}+\frac{323 a^3 \sqrt{b x^{2/3}+a x}}{896 b^4 x^{8/3}}-\frac{4199 a^4 \sqrt{b x^{2/3}+a x}}{10752 b^5 x^{7/3}}+\frac{46189 a^5 \sqrt{b x^{2/3}+a x}}{107520 b^6 x^2}-\frac{138567 a^6 \sqrt{b x^{2/3}+a x}}{286720 b^7 x^{5/3}}+\frac{46189 a^7 \sqrt{b x^{2/3}+a x}}{81920 b^8 x^{4/3}}-\frac{46189 a^8 \sqrt{b x^{2/3}+a x}}{65536 b^9 x}+\frac{138567 a^9 \sqrt{b x^{2/3}+a x}}{131072 b^{10} x^{2/3}}-\frac{\left (138567 a^{10}\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{\sqrt [3]{x}}{\sqrt{b x^{2/3}+a x}}\right )}{131072 b^{10}}\\ &=-\frac{3 \sqrt{b x^{2/3}+a x}}{10 b x^{11/3}}+\frac{19 a \sqrt{b x^{2/3}+a x}}{60 b^2 x^{10/3}}-\frac{323 a^2 \sqrt{b x^{2/3}+a x}}{960 b^3 x^3}+\frac{323 a^3 \sqrt{b x^{2/3}+a x}}{896 b^4 x^{8/3}}-\frac{4199 a^4 \sqrt{b x^{2/3}+a x}}{10752 b^5 x^{7/3}}+\frac{46189 a^5 \sqrt{b x^{2/3}+a x}}{107520 b^6 x^2}-\frac{138567 a^6 \sqrt{b x^{2/3}+a x}}{286720 b^7 x^{5/3}}+\frac{46189 a^7 \sqrt{b x^{2/3}+a x}}{81920 b^8 x^{4/3}}-\frac{46189 a^8 \sqrt{b x^{2/3}+a x}}{65536 b^9 x}+\frac{138567 a^9 \sqrt{b x^{2/3}+a x}}{131072 b^{10} x^{2/3}}-\frac{138567 a^{10} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{b x^{2/3}+a x}}\right )}{131072 b^{21/2}}\\ \end{align*}
Mathematica [C] time = 0.0547898, size = 48, normalized size = 0.15 \[ -\frac{6 a^{10} \sqrt{a x+b x^{2/3}} \, _2F_1\left (\frac{1}{2},11;\frac{3}{2};\frac{\sqrt [3]{x} a}{b}+1\right )}{b^{11} \sqrt [3]{x}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.007, size = 248, normalized size = 0.8 \begin{align*} -{\frac{1}{13762560\,{x}^{6}}\sqrt{b+a\sqrt [3]{x}} \left ( -4358144\,{b}^{19/2}{x}^{10/3}\sqrt{b+a\sqrt [3]{x}}a+4630528\,{b}^{17/2}{x}^{11/3}\sqrt{b+a\sqrt [3]{x}}{a}^{2}+5374720\,{b}^{13/2}{x}^{13/3}\sqrt{b+a\sqrt [3]{x}}{a}^{4}-5912192\,{b}^{11/2}{x}^{14/3}\sqrt{b+a\sqrt [3]{x}}{a}^{5}-7759752\,{b}^{7/2}{x}^{16/3}\sqrt{b+a\sqrt [3]{x}}{a}^{7}+9699690\,{b}^{5/2}{x}^{{\frac{17}{3}}}\sqrt{b+a\sqrt [3]{x}}{a}^{8}+14549535\,{\it Artanh} \left ({\frac{\sqrt{b+a\sqrt [3]{x}}}{\sqrt{b}}} \right ){x}^{{\frac{19}{3}}}{a}^{10}b+4128768\,\sqrt{b+a\sqrt [3]{x}}{b}^{21/2}{x}^{3}-4961280\,{b}^{15/2}{x}^{4}\sqrt{b+a\sqrt [3]{x}}{a}^{3}+6651216\,{b}^{9/2}{x}^{5}\sqrt{b+a\sqrt [3]{x}}{a}^{6}-14549535\,{b}^{3/2}{x}^{6}\sqrt{b+a\sqrt [3]{x}}{a}^{9} \right ){\frac{1}{\sqrt{b{x}^{{\frac{2}{3}}}+ax}}}{b}^{-{\frac{23}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a x + b x^{\frac{2}{3}}} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.35374, size = 285, normalized size = 0.87 \begin{align*} \frac{\frac{14549535 \, a^{11} \arctan \left (\frac{\sqrt{a x^{\frac{1}{3}} + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b^{10}} + \frac{14549535 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{19}{2}} a^{11} - 140645505 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{17}{2}} a^{11} b + 609140532 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{15}{2}} a^{11} b^{2} - 1554721740 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{13}{2}} a^{11} b^{3} + 2585198330 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{11}{2}} a^{11} b^{4} - 2918514950 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{9}{2}} a^{11} b^{5} + 2255541300 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{7}{2}} a^{11} b^{6} - 1168982220 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{5}{2}} a^{11} b^{7} + 382331775 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{3}{2}} a^{11} b^{8} - 68025825 \, \sqrt{a x^{\frac{1}{3}} + b} a^{11} b^{9}}{a^{10} b^{10} x^{\frac{10}{3}}}}{13762560 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]