Optimal. Leaf size=413 \[ -\frac{154 a^{17/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right ),\frac{1}{2}\right )}{1105 b^{15/4} \sqrt{a x+b \sqrt [3]{x}}}-\frac{308 a^{9/2} \sqrt [3]{x} \left (a x^{2/3}+b\right )}{1105 b^4 \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{a x+b \sqrt [3]{x}}}+\frac{44 a^2 \sqrt{a x+b \sqrt [3]{x}}}{663 b^2 x^{5/3}}+\frac{308 a^{17/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{1105 b^{15/4} \sqrt{a x+b \sqrt [3]{x}}}+\frac{308 a^4 \sqrt{a x+b \sqrt [3]{x}}}{1105 b^4 \sqrt [3]{x}}-\frac{308 a^3 \sqrt{a x+b \sqrt [3]{x}}}{3315 b^3 x}-\frac{12 a \sqrt{a x+b \sqrt [3]{x}}}{221 b x^{7/3}}-\frac{6 \sqrt{a x+b \sqrt [3]{x}}}{17 x^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.543533, antiderivative size = 413, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421, Rules used = {2018, 2020, 2025, 2032, 329, 305, 220, 1196} \[ -\frac{308 a^{9/2} \sqrt [3]{x} \left (a x^{2/3}+b\right )}{1105 b^4 \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{a x+b \sqrt [3]{x}}}+\frac{44 a^2 \sqrt{a x+b \sqrt [3]{x}}}{663 b^2 x^{5/3}}-\frac{154 a^{17/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{1105 b^{15/4} \sqrt{a x+b \sqrt [3]{x}}}+\frac{308 a^{17/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{1105 b^{15/4} \sqrt{a x+b \sqrt [3]{x}}}+\frac{308 a^4 \sqrt{a x+b \sqrt [3]{x}}}{1105 b^4 \sqrt [3]{x}}-\frac{308 a^3 \sqrt{a x+b \sqrt [3]{x}}}{3315 b^3 x}-\frac{12 a \sqrt{a x+b \sqrt [3]{x}}}{221 b x^{7/3}}-\frac{6 \sqrt{a x+b \sqrt [3]{x}}}{17 x^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2018
Rule 2020
Rule 2025
Rule 2032
Rule 329
Rule 305
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int \frac{\sqrt{b \sqrt [3]{x}+a x}}{x^4} \, dx &=3 \operatorname{Subst}\left (\int \frac{\sqrt{b x+a x^3}}{x^{10}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{6 \sqrt{b \sqrt [3]{x}+a x}}{17 x^3}+\frac{1}{17} (6 a) \operatorname{Subst}\left (\int \frac{1}{x^7 \sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{6 \sqrt{b \sqrt [3]{x}+a x}}{17 x^3}-\frac{12 a \sqrt{b \sqrt [3]{x}+a x}}{221 b x^{7/3}}-\frac{\left (66 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{x^5 \sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{221 b}\\ &=-\frac{6 \sqrt{b \sqrt [3]{x}+a x}}{17 x^3}-\frac{12 a \sqrt{b \sqrt [3]{x}+a x}}{221 b x^{7/3}}+\frac{44 a^2 \sqrt{b \sqrt [3]{x}+a x}}{663 b^2 x^{5/3}}+\frac{\left (154 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^3 \sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{663 b^2}\\ &=-\frac{6 \sqrt{b \sqrt [3]{x}+a x}}{17 x^3}-\frac{12 a \sqrt{b \sqrt [3]{x}+a x}}{221 b x^{7/3}}+\frac{44 a^2 \sqrt{b \sqrt [3]{x}+a x}}{663 b^2 x^{5/3}}-\frac{308 a^3 \sqrt{b \sqrt [3]{x}+a x}}{3315 b^3 x}-\frac{\left (154 a^4\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{1105 b^3}\\ &=-\frac{6 \sqrt{b \sqrt [3]{x}+a x}}{17 x^3}-\frac{12 a \sqrt{b \sqrt [3]{x}+a x}}{221 b x^{7/3}}+\frac{44 a^2 \sqrt{b \sqrt [3]{x}+a x}}{663 b^2 x^{5/3}}-\frac{308 a^3 \sqrt{b \sqrt [3]{x}+a x}}{3315 b^3 x}+\frac{308 a^4 \sqrt{b \sqrt [3]{x}+a x}}{1105 b^4 \sqrt [3]{x}}-\frac{\left (154 a^5\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{1105 b^4}\\ &=-\frac{6 \sqrt{b \sqrt [3]{x}+a x}}{17 x^3}-\frac{12 a \sqrt{b \sqrt [3]{x}+a x}}{221 b x^{7/3}}+\frac{44 a^2 \sqrt{b \sqrt [3]{x}+a x}}{663 b^2 x^{5/3}}-\frac{308 a^3 \sqrt{b \sqrt [3]{x}+a x}}{3315 b^3 x}+\frac{308 a^4 \sqrt{b \sqrt [3]{x}+a x}}{1105 b^4 \sqrt [3]{x}}-\frac{\left (154 a^5 \sqrt{b+a x^{2/3}} \sqrt [6]{x}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{x}}{\sqrt{b+a x^2}} \, dx,x,\sqrt [3]{x}\right )}{1105 b^4 \sqrt{b \sqrt [3]{x}+a x}}\\ &=-\frac{6 \sqrt{b \sqrt [3]{x}+a x}}{17 x^3}-\frac{12 a \sqrt{b \sqrt [3]{x}+a x}}{221 b x^{7/3}}+\frac{44 a^2 \sqrt{b \sqrt [3]{x}+a x}}{663 b^2 x^{5/3}}-\frac{308 a^3 \sqrt{b \sqrt [3]{x}+a x}}{3315 b^3 x}+\frac{308 a^4 \sqrt{b \sqrt [3]{x}+a x}}{1105 b^4 \sqrt [3]{x}}-\frac{\left (308 a^5 \sqrt{b+a x^{2/3}} \sqrt [6]{x}\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{b+a x^4}} \, dx,x,\sqrt [6]{x}\right )}{1105 b^4 \sqrt{b \sqrt [3]{x}+a x}}\\ &=-\frac{6 \sqrt{b \sqrt [3]{x}+a x}}{17 x^3}-\frac{12 a \sqrt{b \sqrt [3]{x}+a x}}{221 b x^{7/3}}+\frac{44 a^2 \sqrt{b \sqrt [3]{x}+a x}}{663 b^2 x^{5/3}}-\frac{308 a^3 \sqrt{b \sqrt [3]{x}+a x}}{3315 b^3 x}+\frac{308 a^4 \sqrt{b \sqrt [3]{x}+a x}}{1105 b^4 \sqrt [3]{x}}-\frac{\left (308 a^{9/2} \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 )}{1105 b^{7/2} \sqrt{b \sqrt [3]{x}+a x}}+\frac{\left (308 a^{9/2} \sqrt{b+a x^{2/3}} \sqrt [6]{x}\right ) \operatorname{Subst}\left (\int \frac{1-\frac{\sqrt{a} x^2}{\sqrt{b}}}{\sqrt{b+a x^4}} \, dx,x,\sqrt [6]{x}\right )}{1105 b^{7/2} \sqrt{b \sqrt [3]{x}+a x}}\\ &=-\frac{308 a^{9/2} \left (b+a x^{2/3}\right ) \sqrt [3]{x}}{1105 b^4 \left (\sqrt{b}+\sqrt{a} \sqrt [3]{x}\right ) \sqrt{b \sqrt [3]{x}+a x}}-\frac{6 \sqrt{b \sqrt [3]{x}+a x}}{17 x^3}-\frac{12 a \sqrt{b \sqrt [3]{x}+a x}}{221 b x^{7/3}}+\frac{44 a^2 \sqrt{b \sqrt [3]{x}+a x}}{663 b^2 x^{5/3}}-\frac{308 a^3 \sqrt{b \sqrt [3]{x}+a x}}{3315 b^3 x}+\frac{308 a^4 \sqrt{b \sqrt [3]{x}+a x}}{1105 b^4 \sqrt [3]{x}}+\frac{308 a^{17/4} \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} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{1105 b^{15/4} \sqrt{b \sqrt [3]{x}+a x}}-\frac{154 a^{17/4} \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 )}{1105 b^{15/4} \sqrt{b \sqrt [3]{x}+a x}}\\ \end{align*}
Mathematica [C] time = 0.0470283, size = 59, normalized size = 0.14 \[ -\frac{6 \sqrt{a x+b \sqrt [3]{x}} \, _2F_1\left (-\frac{17}{4},-\frac{1}{2};-\frac{13}{4};-\frac{a x^{2/3}}{b}\right )}{17 x^3 \sqrt{\frac{a x^{2/3}}{b}+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.018, size = 281, normalized size = 0.7 \begin{align*} -{\frac{6}{17\,{x}^{3}}\sqrt{b\sqrt [3]{x}+ax}}-{\frac{12\,a}{221\,b}\sqrt{b\sqrt [3]{x}+ax}{x}^{-{\frac{7}{3}}}}+{\frac{44\,{a}^{2}}{663\,{b}^{2}}\sqrt{b\sqrt [3]{x}+ax}{x}^{-{\frac{5}{3}}}}-{\frac{308\,{a}^{3}}{3315\,{b}^{3}x}\sqrt{b\sqrt [3]{x}+ax}}+{\frac{308\,{a}^{4}}{1105\,{b}^{4}} \left ( b+a{x}^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt{\sqrt [3]{x} \left ( b+a{x}^{{\frac{2}{3}}} \right ) }}}}-{\frac{154\,{a}^{4}}{1105\,{b}^{4}}\sqrt{-ab}\sqrt{{a \left ( \sqrt [3]{x}+{\frac{1}{a}\sqrt{-ab}} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{-2\,{\frac{a}{\sqrt{-ab}} \left ( \sqrt [3]{x}-{\frac{\sqrt{-ab}}{a}} \right ) }}\sqrt{-{a\sqrt [3]{x}{\frac{1}{\sqrt{-ab}}}}} \left ( -2\,{\frac{\sqrt{-ab}}{a}{\it EllipticE} \left ( \sqrt{{\frac{a}{\sqrt{-ab}} \left ( \sqrt [3]{x}+{\frac{\sqrt{-ab}}{a}} \right ) }},1/2\,\sqrt{2} \right ) }+{\frac{1}{a}\sqrt{-ab}{\it EllipticF} \left ( \sqrt{{a \left ( \sqrt [3]{x}+{\frac{1}{a}\sqrt{-ab}} \right ){\frac{1}{\sqrt{-ab}}}}},{\frac{\sqrt{2}}{2}} \right ) } \right ){\frac{1}{\sqrt{b\sqrt [3]{x}+ax}}}} \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{\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{\sqrt{a x + b x^{\frac{1}{3}}}}{x^{4}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a x + b \sqrt [3]{x}}}{x^{4}}\, dx \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{\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]