Optimal. Leaf size=628 \[ \frac{71712 \sqrt{2} 3^{3/4} a^{10/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right ),4 \sqrt{3}-7\right )}{1729 b x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}-\frac{215136 a^3 x}{1729 \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}-\frac{15768 a^2 x \left (a-b x^2\right )^{2/3}}{1729}-\frac{107568 \sqrt [4]{3} \sqrt{2+\sqrt{3}} a^{10/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{1729 b x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}-\frac{324}{247} a x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )-\frac{3}{19} x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^2 \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.439609, antiderivative size = 628, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {416, 528, 388, 235, 304, 219, 1879} \[ -\frac{215136 a^3 x}{1729 \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}-\frac{15768 a^2 x \left (a-b x^2\right )^{2/3}}{1729}+\frac{71712 \sqrt{2} 3^{3/4} a^{10/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{1729 b x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}-\frac{107568 \sqrt [4]{3} \sqrt{2+\sqrt{3}} a^{10/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{1729 b x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}-\frac{324}{247} a x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )-\frac{3}{19} x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^2 \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 416
Rule 528
Rule 388
Rule 235
Rule 304
Rule 219
Rule 1879
Rubi steps
\begin{align*} \int \frac{\left (3 a+b x^2\right )^3}{\sqrt [3]{a-b x^2}} \, dx &=-\frac{3}{19} x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^2-\frac{3 \int \frac{\left (3 a+b x^2\right ) \left (-60 a^2 b-36 a b^2 x^2\right )}{\sqrt [3]{a-b x^2}} \, dx}{19 b}\\ &=-\frac{324}{247} a x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )-\frac{3}{19} x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^2+\frac{9 \int \frac{888 a^3 b^2+584 a^2 b^3 x^2}{\sqrt [3]{a-b x^2}} \, dx}{247 b^2}\\ &=-\frac{15768 a^2 x \left (a-b x^2\right )^{2/3}}{1729}-\frac{324}{247} a x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )-\frac{3}{19} x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^2+\frac{\left (71712 a^3\right ) \int \frac{1}{\sqrt [3]{a-b x^2}} \, dx}{1729}\\ &=-\frac{15768 a^2 x \left (a-b x^2\right )^{2/3}}{1729}-\frac{324}{247} a x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )-\frac{3}{19} x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^2-\frac{\left (107568 a^3 \sqrt{-b x^2}\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{-a+x^3}} \, dx,x,\sqrt [3]{a-b x^2}\right )}{1729 b x}\\ &=-\frac{15768 a^2 x \left (a-b x^2\right )^{2/3}}{1729}-\frac{324}{247} a x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )-\frac{3}{19} x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^2+\frac{\left (107568 a^3 \sqrt{-b x^2}\right ) \operatorname{Subst}\left (\int \frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-x}{\sqrt{-a+x^3}} \, dx,x,\sqrt [3]{a-b x^2}\right )}{1729 b x}-\frac{\left (107568 \sqrt{2 \left (2+\sqrt{3}\right )} a^{10/3} \sqrt{-b x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-a+x^3}} \, dx,x,\sqrt [3]{a-b x^2}\right )}{1729 b x}\\ &=-\frac{15768 a^2 x \left (a-b x^2\right )^{2/3}}{1729}-\frac{324}{247} a x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )-\frac{3}{19} x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^2-\frac{215136 a^3 x}{1729 \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}-\frac{107568 \sqrt [4]{3} \sqrt{2+\sqrt{3}} a^{10/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{1729 b x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}+\frac{71712 \sqrt{2} 3^{3/4} a^{10/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{1729 b x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}\\ \end{align*}
Mathematica [C] time = 5.04694, size = 88, normalized size = 0.14 \[ \frac{3 \left (23904 a^3 x \sqrt [3]{1-\frac{b x^2}{a}} \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{3}{2};\frac{b x^2}{a}\right )+7041 a^2 b x^3-8343 a^3 x+1211 a b^2 x^5+91 b^3 x^7\right )}{1729 \sqrt [3]{a-b x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.026, size = 0, normalized size = 0. \begin{align*} \int{ \left ( b{x}^{2}+3\,a \right ) ^{3}{\frac{1}{\sqrt [3]{-b{x}^{2}+a}}}}\, dx \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{{\left (b x^{2} + 3 \, a\right )}^{3}}{{\left (-b x^{2} + a\right )}^{\frac{1}{3}}}\,{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 (b^{3} x^{6} + 9 \, a b^{2} x^{4} + 27 \, a^{2} b x^{2} + 27 \, a^{3}\right )}{\left (-b x^{2} + a\right )}^{\frac{2}{3}}}{b x^{2} - a}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 3.45422, size = 129, normalized size = 0.21 \begin{align*} 27 a^{\frac{8}{3}} x{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{\frac{b x^{2} e^{2 i \pi }}{a}} \right )} + 9 a^{\frac{5}{3}} b x^{3}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{\frac{b x^{2} e^{2 i \pi }}{a}} \right )} + \frac{9 a^{\frac{2}{3}} b^{2} x^{5}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{5}{2} \\ \frac{7}{2} \end{matrix}\middle |{\frac{b x^{2} e^{2 i \pi }}{a}} \right )}}{5} + \frac{b^{3} x^{7}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{7}{2} \\ \frac{9}{2} \end{matrix}\middle |{\frac{b x^{2} e^{2 i \pi }}{a}} \right )}}{7 \sqrt [3]{a}} \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{{\left (b x^{2} + 3 \, a\right )}^{3}}{{\left (-b x^{2} + a\right )}^{\frac{1}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]