Optimal. Leaf size=642 \[ \frac{2 \sqrt{2} 3^{3/4} a^{4/3} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt{c x^2}+b^{2/3} c x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} \sqrt{c x^2}+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} \sqrt{c x^2}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{7 b^{2/3} c \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )^2}} \sqrt{a+b \left (c x^2\right )^{3/2}}}-\frac{3 \sqrt [4]{3} \sqrt{2-\sqrt{3}} a^{4/3} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt{c x^2}+b^{2/3} c x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} \sqrt{c x^2}+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} \sqrt{c x^2}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{7 b^{2/3} c \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )^2}} \sqrt{a+b \left (c x^2\right )^{3/2}}}+\frac{6 a \sqrt{a+b \left (c x^2\right )^{3/2}}}{7 b^{2/3} c \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )}+\frac{2}{7} x^2 \sqrt{a+b \left (c x^2\right )^{3/2}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 1.03985, antiderivative size = 642, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316 \[ \frac{2 \sqrt{2} 3^{3/4} a^{4/3} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt{c x^2}+b^{2/3} c x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} \sqrt{c x^2}+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} \sqrt{c x^2}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{7 b^{2/3} c \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )^2}} \sqrt{a+b \left (c x^2\right )^{3/2}}}-\frac{3 \sqrt [4]{3} \sqrt{2-\sqrt{3}} a^{4/3} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt{c x^2}+b^{2/3} c x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} \sqrt{c x^2}+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} \sqrt{c x^2}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{7 b^{2/3} c \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )^2}} \sqrt{a+b \left (c x^2\right )^{3/2}}}+\frac{6 a \sqrt{a+b \left (c x^2\right )^{3/2}}}{7 b^{2/3} c \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )}+\frac{2}{7} x^2 \sqrt{a+b \left (c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[x*Sqrt[a + b*(c*x^2)^(3/2)],x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 46.1792, size = 561, normalized size = 0.87 \[ - \frac{3 \sqrt [4]{3} a^{\frac{4}{3}} \sqrt{\frac{a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} \sqrt{c x^{2}} + b^{\frac{2}{3}} c x^{2}}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} \sqrt{c x^{2}}\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (\sqrt [3]{a} + \sqrt [3]{b} \sqrt{c x^{2}}\right ) E\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{b} \sqrt{c x^{2}}}{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} \sqrt{c x^{2}}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{7 b^{\frac{2}{3}} c \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a} + \sqrt [3]{b} \sqrt{c x^{2}}\right )}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} \sqrt{c x^{2}}\right )^{2}}} \sqrt{a + b \left (c x^{2}\right )^{\frac{3}{2}}}} + \frac{2 \sqrt{2} \cdot 3^{\frac{3}{4}} a^{\frac{4}{3}} \sqrt{\frac{a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} \sqrt{c x^{2}} + b^{\frac{2}{3}} c x^{2}}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} \sqrt{c x^{2}}\right )^{2}}} \left (\sqrt [3]{a} + \sqrt [3]{b} \sqrt{c x^{2}}\right ) F\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{b} \sqrt{c x^{2}}}{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} \sqrt{c x^{2}}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{7 b^{\frac{2}{3}} c \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a} + \sqrt [3]{b} \sqrt{c x^{2}}\right )}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} \sqrt{c x^{2}}\right )^{2}}} \sqrt{a + b \left (c x^{2}\right )^{\frac{3}{2}}}} + \frac{6 a \sqrt{a + b \left (c x^{2}\right )^{\frac{3}{2}}}}{7 b^{\frac{2}{3}} c \left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} \sqrt{c x^{2}}\right )} + \frac{2 x^{2} \sqrt{a + b \left (c x^{2}\right )^{\frac{3}{2}}}}{7} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(a+b*(c*x**2)**(3/2))**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [C] time = 0.119353, size = 89, normalized size = 0.14 \[ \frac{x^2 \left (3 a \sqrt{\frac{a+b \left (c x^2\right )^{3/2}}{a}} \, _2F_1\left (\frac{1}{2},\frac{2}{3};\frac{5}{3};-\frac{b \left (c x^2\right )^{3/2}}{a}\right )+4 \left (a+b \left (c x^2\right )^{3/2}\right )\right )}{14 \sqrt{a+b \left (c x^2\right )^{3/2}}} \]
Antiderivative was successfully verified.
[In] Integrate[x*Sqrt[a + b*(c*x^2)^(3/2)],x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.015, size = 495, normalized size = 0.8 \[{\frac{1}{2\,c} \left ({\frac{4\,c{x}^{2}}{7}\sqrt{a+b \left ( c{x}^{2} \right ) ^{{\frac{3}{2}}}}}-{\frac{{\frac{4\,i}{7}}a\sqrt{3}}{b}\sqrt [3]{-a{b}^{2}}\sqrt{{i\sqrt{3}b \left ( \sqrt{c{x}^{2}}+{\frac{1}{2\,b}\sqrt [3]{-a{b}^{2}}}-{\frac{{\frac{i}{2}}\sqrt{3}}{b}\sqrt [3]{-a{b}^{2}}} \right ){\frac{1}{\sqrt [3]{-a{b}^{2}}}}}}\sqrt{{1 \left ( \sqrt{c{x}^{2}}-{\frac{1}{b}\sqrt [3]{-a{b}^{2}}} \right ) \left ( -{\frac{3}{2\,b}\sqrt [3]{-a{b}^{2}}}+{\frac{{\frac{i}{2}}\sqrt{3}}{b}\sqrt [3]{-a{b}^{2}}} \right ) ^{-1}}}\sqrt{{-i\sqrt{3}b \left ( \sqrt{c{x}^{2}}+{\frac{1}{2\,b}\sqrt [3]{-a{b}^{2}}}+{\frac{{\frac{i}{2}}\sqrt{3}}{b}\sqrt [3]{-a{b}^{2}}} \right ){\frac{1}{\sqrt [3]{-a{b}^{2}}}}}} \left ( \left ( -{\frac{3}{2\,b}\sqrt [3]{-a{b}^{2}}}+{\frac{{\frac{i}{2}}\sqrt{3}}{b}\sqrt [3]{-a{b}^{2}}} \right ){\it EllipticE} \left ({\frac{\sqrt{3}}{3}\sqrt{{i\sqrt{3}b \left ( \sqrt{c{x}^{2}}+{\frac{1}{2\,b}\sqrt [3]{-a{b}^{2}}}-{\frac{{\frac{i}{2}}\sqrt{3}}{b}\sqrt [3]{-a{b}^{2}}} \right ){\frac{1}{\sqrt [3]{-a{b}^{2}}}}}}},\sqrt{{\frac{i\sqrt{3}}{b}\sqrt [3]{-a{b}^{2}} \left ( -{\frac{3}{2\,b}\sqrt [3]{-a{b}^{2}}}+{\frac{{\frac{i}{2}}\sqrt{3}}{b}\sqrt [3]{-a{b}^{2}}} \right ) ^{-1}}} \right ) +{\frac{1}{b}\sqrt [3]{-a{b}^{2}}{\it EllipticF} \left ({\frac{\sqrt{3}}{3}\sqrt{{i\sqrt{3}b \left ( \sqrt{c{x}^{2}}+{\frac{1}{2\,b}\sqrt [3]{-a{b}^{2}}}-{\frac{{\frac{i}{2}}\sqrt{3}}{b}\sqrt [3]{-a{b}^{2}}} \right ){\frac{1}{\sqrt [3]{-a{b}^{2}}}}}}},\sqrt{{\frac{i\sqrt{3}}{b}\sqrt [3]{-a{b}^{2}} \left ( -{\frac{3}{2\,b}\sqrt [3]{-a{b}^{2}}}+{\frac{{\frac{i}{2}}\sqrt{3}}{b}\sqrt [3]{-a{b}^{2}}} \right ) ^{-1}}} \right ) } \right ){\frac{1}{\sqrt{a+b \left ( c{x}^{2} \right ) ^{{\frac{3}{2}}}}}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(a+b*(c*x^2)^(3/2))^(1/2),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{\left (c x^{2}\right )^{\frac{3}{2}} b + a} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((c*x^2)^(3/2)*b + a)*x,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\sqrt{\sqrt{c x^{2}} b c x^{2} + a} x, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((c*x^2)^(3/2)*b + a)*x,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int x \sqrt{a + b \left (c x^{2}\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(a+b*(c*x**2)**(3/2))**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{\left (c x^{2}\right )^{\frac{3}{2}} b + a} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((c*x^2)^(3/2)*b + a)*x,x, algorithm="giac")
[Out]