Optimal. Leaf size=251 \[ -\frac{663 a^{19/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right ),\frac{1}{2}\right )}{1463 b^{21/4} \sqrt{a x+b \sqrt [3]{x}}}-\frac{1326 a^4 \sqrt{a x+b \sqrt [3]{x}}}{1463 b^5 x^{2/3}}+\frac{3978 a^3 \sqrt{a x+b \sqrt [3]{x}}}{7315 b^4 x^{4/3}}-\frac{442 a^2 \sqrt{a x+b \sqrt [3]{x}}}{1045 b^3 x^2}+\frac{34 a \sqrt{a x+b \sqrt [3]{x}}}{95 b^2 x^{8/3}}-\frac{6 \sqrt{a x+b \sqrt [3]{x}}}{19 b x^{10/3}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.345415, antiderivative size = 251, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {2018, 2025, 2011, 329, 220} \[ -\frac{1326 a^4 \sqrt{a x+b \sqrt [3]{x}}}{1463 b^5 x^{2/3}}+\frac{3978 a^3 \sqrt{a x+b \sqrt [3]{x}}}{7315 b^4 x^{4/3}}-\frac{442 a^2 \sqrt{a x+b \sqrt [3]{x}}}{1045 b^3 x^2}-\frac{663 a^{19/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{1463 b^{21/4} \sqrt{a x+b \sqrt [3]{x}}}+\frac{34 a \sqrt{a x+b \sqrt [3]{x}}}{95 b^2 x^{8/3}}-\frac{6 \sqrt{a x+b \sqrt [3]{x}}}{19 b x^{10/3}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2018
Rule 2025
Rule 2011
Rule 329
Rule 220
Rubi steps
\begin{align*} \int \frac{1}{x^4 \sqrt{b \sqrt [3]{x}+a x}} \, dx &=3 \operatorname{Subst}\left (\int \frac{1}{x^{10} \sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{6 \sqrt{b \sqrt [3]{x}+a x}}{19 b x^{10/3}}-\frac{(51 a) \operatorname{Subst}\left (\int \frac{1}{x^8 \sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{19 b}\\ &=-\frac{6 \sqrt{b \sqrt [3]{x}+a x}}{19 b x^{10/3}}+\frac{34 a \sqrt{b \sqrt [3]{x}+a x}}{95 b^2 x^{8/3}}+\frac{\left (221 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{x^6 \sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{95 b^2}\\ &=-\frac{6 \sqrt{b \sqrt [3]{x}+a x}}{19 b x^{10/3}}+\frac{34 a \sqrt{b \sqrt [3]{x}+a x}}{95 b^2 x^{8/3}}-\frac{442 a^2 \sqrt{b \sqrt [3]{x}+a x}}{1045 b^3 x^2}-\frac{\left (1989 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^4 \sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{1045 b^3}\\ &=-\frac{6 \sqrt{b \sqrt [3]{x}+a x}}{19 b x^{10/3}}+\frac{34 a \sqrt{b \sqrt [3]{x}+a x}}{95 b^2 x^{8/3}}-\frac{442 a^2 \sqrt{b \sqrt [3]{x}+a x}}{1045 b^3 x^2}+\frac{3978 a^3 \sqrt{b \sqrt [3]{x}+a x}}{7315 b^4 x^{4/3}}+\frac{\left (1989 a^4\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{1463 b^4}\\ &=-\frac{6 \sqrt{b \sqrt [3]{x}+a x}}{19 b x^{10/3}}+\frac{34 a \sqrt{b \sqrt [3]{x}+a x}}{95 b^2 x^{8/3}}-\frac{442 a^2 \sqrt{b \sqrt [3]{x}+a x}}{1045 b^3 x^2}+\frac{3978 a^3 \sqrt{b \sqrt [3]{x}+a x}}{7315 b^4 x^{4/3}}-\frac{1326 a^4 \sqrt{b \sqrt [3]{x}+a x}}{1463 b^5 x^{2/3}}-\frac{\left (663 a^5\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{1463 b^5}\\ &=-\frac{6 \sqrt{b \sqrt [3]{x}+a x}}{19 b x^{10/3}}+\frac{34 a \sqrt{b \sqrt [3]{x}+a x}}{95 b^2 x^{8/3}}-\frac{442 a^2 \sqrt{b \sqrt [3]{x}+a x}}{1045 b^3 x^2}+\frac{3978 a^3 \sqrt{b \sqrt [3]{x}+a x}}{7315 b^4 x^{4/3}}-\frac{1326 a^4 \sqrt{b \sqrt [3]{x}+a x}}{1463 b^5 x^{2/3}}-\frac{\left (663 a^5 \sqrt{b+a x^{2/3}} \sqrt [6]{x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \sqrt{b+a x^2}} \, dx,x,\sqrt [3]{x}\right )}{1463 b^5 \sqrt{b \sqrt [3]{x}+a x}}\\ &=-\frac{6 \sqrt{b \sqrt [3]{x}+a x}}{19 b x^{10/3}}+\frac{34 a \sqrt{b \sqrt [3]{x}+a x}}{95 b^2 x^{8/3}}-\frac{442 a^2 \sqrt{b \sqrt [3]{x}+a x}}{1045 b^3 x^2}+\frac{3978 a^3 \sqrt{b \sqrt [3]{x}+a x}}{7315 b^4 x^{4/3}}-\frac{1326 a^4 \sqrt{b \sqrt [3]{x}+a x}}{1463 b^5 x^{2/3}}-\frac{\left (1326 a^5 \sqrt{b+a x^{2/3}} \sqrt [6]{x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b+a x^4}} \, dx,x,\sqrt [6]{x}\right )}{1463 b^5 \sqrt{b \sqrt [3]{x}+a x}}\\ &=-\frac{6 \sqrt{b \sqrt [3]{x}+a x}}{19 b x^{10/3}}+\frac{34 a \sqrt{b \sqrt [3]{x}+a x}}{95 b^2 x^{8/3}}-\frac{442 a^2 \sqrt{b \sqrt [3]{x}+a x}}{1045 b^3 x^2}+\frac{3978 a^3 \sqrt{b \sqrt [3]{x}+a x}}{7315 b^4 x^{4/3}}-\frac{1326 a^4 \sqrt{b \sqrt [3]{x}+a x}}{1463 b^5 x^{2/3}}-\frac{663 a^{19/4} \left (\sqrt{b}+\sqrt{a} \sqrt [3]{x}\right ) \sqrt{\frac{b+a x^{2/3}}{\left (\sqrt{b}+\sqrt{a} \sqrt [3]{x}\right )^2}} \sqrt [6]{x} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{1463 b^{21/4} \sqrt{b \sqrt [3]{x}+a x}}\\ \end{align*}
Mathematica [C] time = 0.056562, size = 59, normalized size = 0.24 \[ -\frac{6 \sqrt{\frac{a x^{2/3}}{b}+1} \, _2F_1\left (-\frac{19}{4},\frac{1}{2};-\frac{15}{4};-\frac{a x^{2/3}}{b}\right )}{19 x^3 \sqrt{a x+b \sqrt [3]{x}}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.026, size = 179, normalized size = 0.7 \begin{align*} -{\frac{1}{7315\,{b}^{5}} \left ( 3315\,{a}^{4}\sqrt{-ab}\sqrt{{\frac{a\sqrt [3]{x}+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-2\,{\frac{a\sqrt [3]{x}-\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{a\sqrt [3]{x}}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{a\sqrt [3]{x}+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ){x}^{16/3}+2652\,{x}^{5}{a}^{4}b+6630\,{x}^{{\frac{17}{3}}}{a}^{5}+476\,{x}^{11/3}{a}^{2}{b}^{3}-884\,{x}^{13/3}{a}^{3}{b}^{2}-308\,{x}^{3}a{b}^{4}+2310\,{x}^{7/3}{b}^{5} \right ){\frac{1}{\sqrt{\sqrt [3]{x} \left ( b+a{x}^{{\frac{2}{3}}} \right ) }}}{x}^{-{\frac{16}{3}}}} \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{1}{3}}} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (a^{2} x^{2} - a b x^{\frac{4}{3}} + b^{2} x^{\frac{2}{3}}\right )} \sqrt{a x + b x^{\frac{1}{3}}}}{a^{3} x^{7} + b^{3} x^{5}}, x\right ) \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a x + b x^{\frac{1}{3}}} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]