3.2967 \(\int x^2 \sqrt{a+b \left (c x^3\right )^{3/2}} \, dx\)

Optimal. Leaf size=642 \[ \frac{4 \sqrt{2} a^{4/3} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^3}\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt{c x^3}+b^{2/3} c x^3}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^3}\right )^2}} F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} \sqrt{c x^3}+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} \sqrt{c x^3}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{7 \sqrt [4]{3} b^{2/3} c \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^3}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^3}\right )^2}} \sqrt{a+b \left (c x^3\right )^{3/2}}}-\frac{2 \sqrt [4]{3} \sqrt{2-\sqrt{3}} a^{4/3} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^3}\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt{c x^3}+b^{2/3} c x^3}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^3}\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} \sqrt{c x^3}+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} \sqrt{c x^3}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{7 b^{2/3} c \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^3}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^3}\right )^2}} \sqrt{a+b \left (c x^3\right )^{3/2}}}+\frac{4 a \sqrt{a+b \left (c x^3\right )^{3/2}}}{7 b^{2/3} c \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^3}\right )}+\frac{4}{21} x^3 \sqrt{a+b \left (c x^3\right )^{3/2}} \]

[Out]

(4*x^3*Sqrt[a + b*(c*x^3)^(3/2)])/21 + (4*a*Sqrt[a + b*(c*x^3)^(3/2)])/(7*b^(2/3
)*c*((1 + Sqrt[3])*a^(1/3) + b^(1/3)*Sqrt[c*x^3])) - (2*3^(1/4)*Sqrt[2 - Sqrt[3]
]*a^(4/3)*(a^(1/3) + b^(1/3)*Sqrt[c*x^3])*Sqrt[(a^(2/3) + b^(2/3)*c*x^3 - a^(1/3
)*b^(1/3)*Sqrt[c*x^3])/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*Sqrt[c*x^3])^2]*Elliptic
E[ArcSin[((1 - Sqrt[3])*a^(1/3) + b^(1/3)*Sqrt[c*x^3])/((1 + Sqrt[3])*a^(1/3) +
b^(1/3)*Sqrt[c*x^3])], -7 - 4*Sqrt[3]])/(7*b^(2/3)*c*Sqrt[(a^(1/3)*(a^(1/3) + b^
(1/3)*Sqrt[c*x^3]))/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*Sqrt[c*x^3])^2]*Sqrt[a + b*
(c*x^3)^(3/2)]) + (4*Sqrt[2]*a^(4/3)*(a^(1/3) + b^(1/3)*Sqrt[c*x^3])*Sqrt[(a^(2/
3) + b^(2/3)*c*x^3 - a^(1/3)*b^(1/3)*Sqrt[c*x^3])/((1 + Sqrt[3])*a^(1/3) + b^(1/
3)*Sqrt[c*x^3])^2]*EllipticF[ArcSin[((1 - Sqrt[3])*a^(1/3) + b^(1/3)*Sqrt[c*x^3]
)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*Sqrt[c*x^3])], -7 - 4*Sqrt[3]])/(7*3^(1/4)*b^
(2/3)*c*Sqrt[(a^(1/3)*(a^(1/3) + b^(1/3)*Sqrt[c*x^3]))/((1 + Sqrt[3])*a^(1/3) +
b^(1/3)*Sqrt[c*x^3])^2]*Sqrt[a + b*(c*x^3)^(3/2)])

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Rubi [A]  time = 0.935807, antiderivative size = 642, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ \frac{4 \sqrt{2} a^{4/3} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^3}\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt{c x^3}+b^{2/3} c x^3}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^3}\right )^2}} F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} \sqrt{c x^3}+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} \sqrt{c x^3}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{7 \sqrt [4]{3} b^{2/3} c \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^3}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^3}\right )^2}} \sqrt{a+b \left (c x^3\right )^{3/2}}}-\frac{2 \sqrt [4]{3} \sqrt{2-\sqrt{3}} a^{4/3} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^3}\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt{c x^3}+b^{2/3} c x^3}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^3}\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} \sqrt{c x^3}+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} \sqrt{c x^3}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{7 b^{2/3} c \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^3}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^3}\right )^2}} \sqrt{a+b \left (c x^3\right )^{3/2}}}+\frac{4 a \sqrt{a+b \left (c x^3\right )^{3/2}}}{7 b^{2/3} c \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^3}\right )}+\frac{4}{21} x^3 \sqrt{a+b \left (c x^3\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[x^2*Sqrt[a + b*(c*x^3)^(3/2)],x]

[Out]

(4*x^3*Sqrt[a + b*(c*x^3)^(3/2)])/21 + (4*a*Sqrt[a + b*(c*x^3)^(3/2)])/(7*b^(2/3
)*c*((1 + Sqrt[3])*a^(1/3) + b^(1/3)*Sqrt[c*x^3])) - (2*3^(1/4)*Sqrt[2 - Sqrt[3]
]*a^(4/3)*(a^(1/3) + b^(1/3)*Sqrt[c*x^3])*Sqrt[(a^(2/3) + b^(2/3)*c*x^3 - a^(1/3
)*b^(1/3)*Sqrt[c*x^3])/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*Sqrt[c*x^3])^2]*Elliptic
E[ArcSin[((1 - Sqrt[3])*a^(1/3) + b^(1/3)*Sqrt[c*x^3])/((1 + Sqrt[3])*a^(1/3) +
b^(1/3)*Sqrt[c*x^3])], -7 - 4*Sqrt[3]])/(7*b^(2/3)*c*Sqrt[(a^(1/3)*(a^(1/3) + b^
(1/3)*Sqrt[c*x^3]))/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*Sqrt[c*x^3])^2]*Sqrt[a + b*
(c*x^3)^(3/2)]) + (4*Sqrt[2]*a^(4/3)*(a^(1/3) + b^(1/3)*Sqrt[c*x^3])*Sqrt[(a^(2/
3) + b^(2/3)*c*x^3 - a^(1/3)*b^(1/3)*Sqrt[c*x^3])/((1 + Sqrt[3])*a^(1/3) + b^(1/
3)*Sqrt[c*x^3])^2]*EllipticF[ArcSin[((1 - Sqrt[3])*a^(1/3) + b^(1/3)*Sqrt[c*x^3]
)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*Sqrt[c*x^3])], -7 - 4*Sqrt[3]])/(7*3^(1/4)*b^
(2/3)*c*Sqrt[(a^(1/3)*(a^(1/3) + b^(1/3)*Sqrt[c*x^3]))/((1 + Sqrt[3])*a^(1/3) +
b^(1/3)*Sqrt[c*x^3])^2]*Sqrt[a + b*(c*x^3)^(3/2)])

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Rubi in Sympy [A]  time = 53.6134, size = 561, normalized size = 0.87 \[ - \frac{2 \sqrt [4]{3} a^{\frac{4}{3}} \sqrt{\frac{a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} \sqrt{c x^{3}} + b^{\frac{2}{3}} c x^{3}}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} \sqrt{c x^{3}}\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (\sqrt [3]{a} + \sqrt [3]{b} \sqrt{c x^{3}}\right ) E\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{b} \sqrt{c x^{3}}}{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} \sqrt{c x^{3}}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{7 b^{\frac{2}{3}} c \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a} + \sqrt [3]{b} \sqrt{c x^{3}}\right )}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} \sqrt{c x^{3}}\right )^{2}}} \sqrt{a + b \left (c x^{3}\right )^{\frac{3}{2}}}} + \frac{4 \sqrt{2} \cdot 3^{\frac{3}{4}} a^{\frac{4}{3}} \sqrt{\frac{a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} \sqrt{c x^{3}} + b^{\frac{2}{3}} c x^{3}}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} \sqrt{c x^{3}}\right )^{2}}} \left (\sqrt [3]{a} + \sqrt [3]{b} \sqrt{c x^{3}}\right ) F\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{b} \sqrt{c x^{3}}}{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} \sqrt{c x^{3}}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{21 b^{\frac{2}{3}} c \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a} + \sqrt [3]{b} \sqrt{c x^{3}}\right )}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} \sqrt{c x^{3}}\right )^{2}}} \sqrt{a + b \left (c x^{3}\right )^{\frac{3}{2}}}} + \frac{4 a \sqrt{a + b \left (c x^{3}\right )^{\frac{3}{2}}}}{7 b^{\frac{2}{3}} c \left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} \sqrt{c x^{3}}\right )} + \frac{4 x^{3} \sqrt{a + b \left (c x^{3}\right )^{\frac{3}{2}}}}{21} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*(a+b*(c*x**3)**(3/2))**(1/2),x)

[Out]

-2*3**(1/4)*a**(4/3)*sqrt((a**(2/3) - a**(1/3)*b**(1/3)*sqrt(c*x**3) + b**(2/3)*
c*x**3)/(a**(1/3)*(1 + sqrt(3)) + b**(1/3)*sqrt(c*x**3))**2)*sqrt(-sqrt(3) + 2)*
(a**(1/3) + b**(1/3)*sqrt(c*x**3))*elliptic_e(asin((-a**(1/3)*(-1 + sqrt(3)) + b
**(1/3)*sqrt(c*x**3))/(a**(1/3)*(1 + sqrt(3)) + b**(1/3)*sqrt(c*x**3))), -7 - 4*
sqrt(3))/(7*b**(2/3)*c*sqrt(a**(1/3)*(a**(1/3) + b**(1/3)*sqrt(c*x**3))/(a**(1/3
)*(1 + sqrt(3)) + b**(1/3)*sqrt(c*x**3))**2)*sqrt(a + b*(c*x**3)**(3/2))) + 4*sq
rt(2)*3**(3/4)*a**(4/3)*sqrt((a**(2/3) - a**(1/3)*b**(1/3)*sqrt(c*x**3) + b**(2/
3)*c*x**3)/(a**(1/3)*(1 + sqrt(3)) + b**(1/3)*sqrt(c*x**3))**2)*(a**(1/3) + b**(
1/3)*sqrt(c*x**3))*elliptic_f(asin((-a**(1/3)*(-1 + sqrt(3)) + b**(1/3)*sqrt(c*x
**3))/(a**(1/3)*(1 + sqrt(3)) + b**(1/3)*sqrt(c*x**3))), -7 - 4*sqrt(3))/(21*b**
(2/3)*c*sqrt(a**(1/3)*(a**(1/3) + b**(1/3)*sqrt(c*x**3))/(a**(1/3)*(1 + sqrt(3))
+ b**(1/3)*sqrt(c*x**3))**2)*sqrt(a + b*(c*x**3)**(3/2))) + 4*a*sqrt(a + b*(c*x
**3)**(3/2))/(7*b**(2/3)*c*(a**(1/3)*(1 + sqrt(3)) + b**(1/3)*sqrt(c*x**3))) + 4
*x**3*sqrt(a + b*(c*x**3)**(3/2))/21

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Mathematica [C]  time = 0.142015, size = 89, normalized size = 0.14 \[ \frac{x^3 \left (3 a \sqrt{\frac{a+b \left (c x^3\right )^{3/2}}{a}} \, _2F_1\left (\frac{1}{2},\frac{2}{3};\frac{5}{3};-\frac{b \left (c x^3\right )^{3/2}}{a}\right )+4 \left (a+b \left (c x^3\right )^{3/2}\right )\right )}{21 \sqrt{a+b \left (c x^3\right )^{3/2}}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2*Sqrt[a + b*(c*x^3)^(3/2)],x]

[Out]

(x^3*(4*(a + b*(c*x^3)^(3/2)) + 3*a*Sqrt[(a + b*(c*x^3)^(3/2))/a]*Hypergeometric
2F1[1/2, 2/3, 5/3, -((b*(c*x^3)^(3/2))/a)]))/(21*Sqrt[a + b*(c*x^3)^(3/2)])

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Maple [A]  time = 0.02, size = 495, normalized size = 0.8 \[{\frac{1}{3\,c} \left ({\frac{4\,c{x}^{3}}{7}\sqrt{a+b \left ( c{x}^{3} \right ) ^{{\frac{3}{2}}}}}-{\frac{{\frac{4\,i}{7}}a\sqrt{3}}{b}\sqrt [3]{-a{b}^{2}}\sqrt{{i\sqrt{3}b \left ( \sqrt{c{x}^{3}}+{\frac{1}{2\,b}\sqrt [3]{-a{b}^{2}}}-{\frac{{\frac{i}{2}}\sqrt{3}}{b}\sqrt [3]{-a{b}^{2}}} \right ){\frac{1}{\sqrt [3]{-a{b}^{2}}}}}}\sqrt{{1 \left ( \sqrt{c{x}^{3}}-{\frac{1}{b}\sqrt [3]{-a{b}^{2}}} \right ) \left ( -{\frac{3}{2\,b}\sqrt [3]{-a{b}^{2}}}+{\frac{{\frac{i}{2}}\sqrt{3}}{b}\sqrt [3]{-a{b}^{2}}} \right ) ^{-1}}}\sqrt{{-i\sqrt{3}b \left ( \sqrt{c{x}^{3}}+{\frac{1}{2\,b}\sqrt [3]{-a{b}^{2}}}+{\frac{{\frac{i}{2}}\sqrt{3}}{b}\sqrt [3]{-a{b}^{2}}} \right ){\frac{1}{\sqrt [3]{-a{b}^{2}}}}}} \left ( \left ( -{\frac{3}{2\,b}\sqrt [3]{-a{b}^{2}}}+{\frac{{\frac{i}{2}}\sqrt{3}}{b}\sqrt [3]{-a{b}^{2}}} \right ){\it EllipticE} \left ({\frac{\sqrt{3}}{3}\sqrt{{i\sqrt{3}b \left ( \sqrt{c{x}^{3}}+{\frac{1}{2\,b}\sqrt [3]{-a{b}^{2}}}-{\frac{{\frac{i}{2}}\sqrt{3}}{b}\sqrt [3]{-a{b}^{2}}} \right ){\frac{1}{\sqrt [3]{-a{b}^{2}}}}}}},\sqrt{{\frac{i\sqrt{3}}{b}\sqrt [3]{-a{b}^{2}} \left ( -{\frac{3}{2\,b}\sqrt [3]{-a{b}^{2}}}+{\frac{{\frac{i}{2}}\sqrt{3}}{b}\sqrt [3]{-a{b}^{2}}} \right ) ^{-1}}} \right ) +{\frac{1}{b}\sqrt [3]{-a{b}^{2}}{\it EllipticF} \left ({\frac{\sqrt{3}}{3}\sqrt{{i\sqrt{3}b \left ( \sqrt{c{x}^{3}}+{\frac{1}{2\,b}\sqrt [3]{-a{b}^{2}}}-{\frac{{\frac{i}{2}}\sqrt{3}}{b}\sqrt [3]{-a{b}^{2}}} \right ){\frac{1}{\sqrt [3]{-a{b}^{2}}}}}}},\sqrt{{\frac{i\sqrt{3}}{b}\sqrt [3]{-a{b}^{2}} \left ( -{\frac{3}{2\,b}\sqrt [3]{-a{b}^{2}}}+{\frac{{\frac{i}{2}}\sqrt{3}}{b}\sqrt [3]{-a{b}^{2}}} \right ) ^{-1}}} \right ) } \right ){\frac{1}{\sqrt{a+b \left ( c{x}^{3} \right ) ^{{\frac{3}{2}}}}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(a+b*(c*x^3)^(3/2))^(1/2),x)

[Out]

1/3/c*(4/7*c*x^3*(a+b*(c*x^3)^(3/2))^(1/2)-4/7*I*a*3^(1/2)/b*(-a*b^2)^(1/3)*(I*(
(c*x^3)^(1/2)+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a
*b^2)^(1/3))^(1/2)*(((c*x^3)^(1/2)-1/b*(-a*b^2)^(1/3))/(-3/2/b*(-a*b^2)^(1/3)+1/
2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2)*(-I*((c*x^3)^(1/2)+1/2/b*(-a*b^2)^(1/3)+1/2
*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2)/(a+b*(c*x^3)^(3/2))
^(1/2)*((-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*EllipticE(1/3*3^(
1/2)*(I*((c*x^3)^(1/2)+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1
/2)*b/(-a*b^2)^(1/3))^(1/2),(I*3^(1/2)/b*(-a*b^2)^(1/3)/(-3/2/b*(-a*b^2)^(1/3)+1
/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2))+1/b*(-a*b^2)^(1/3)*EllipticF(1/3*3^(1/2)*
(I*((c*x^3)^(1/2)+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b
/(-a*b^2)^(1/3))^(1/2),(I*3^(1/2)/b*(-a*b^2)^(1/3)/(-3/2/b*(-a*b^2)^(1/3)+1/2*I*
3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2))))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{\left (c x^{3}\right )^{\frac{3}{2}} b + a} x^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((c*x^3)^(3/2)*b + a)*x^2,x, algorithm="maxima")

[Out]

integrate(sqrt((c*x^3)^(3/2)*b + a)*x^2, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\sqrt{\sqrt{c x^{3}} b c x^{3} + a} x^{2}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((c*x^3)^(3/2)*b + a)*x^2,x, algorithm="fricas")

[Out]

integral(sqrt(sqrt(c*x^3)*b*c*x^3 + a)*x^2, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x^{2} \sqrt{a + b \left (c x^{3}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2*(a+b*(c*x**3)**(3/2))**(1/2),x)

[Out]

Integral(x**2*sqrt(a + b*(c*x**3)**(3/2)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{\left (c x^{3}\right )^{\frac{3}{2}} b + a} x^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((c*x^3)^(3/2)*b + a)*x^2,x, algorithm="giac")

[Out]

integrate(sqrt((c*x^3)^(3/2)*b + a)*x^2, x)