3.376 \(\int x^6 \sqrt{a+b x^3} \, dx\)

Optimal. Leaf size=275 \[ -\frac{48 a^2 x \sqrt{a+b x^3}}{935 b^2}+\frac{32\ 3^{3/4} \sqrt{2+\sqrt{3}} a^3 \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{935 b^{7/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{2}{17} x^7 \sqrt{a+b x^3}+\frac{6 a x^4 \sqrt{a+b x^3}}{187 b} \]

[Out]

(-48*a^2*x*Sqrt[a + b*x^3])/(935*b^2) + (6*a*x^4*Sqrt[a + b*x^3])/(187*b) + (2*x
^7*Sqrt[a + b*x^3])/17 + (32*3^(3/4)*Sqrt[2 + Sqrt[3]]*a^3*(a^(1/3) + b^(1/3)*x)
*Sqrt[(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/((1 + Sqrt[3])*a^(1/3) + b^(1/
3)*x)^2]*EllipticF[ArcSin[((1 - Sqrt[3])*a^(1/3) + b^(1/3)*x)/((1 + Sqrt[3])*a^(
1/3) + b^(1/3)*x)], -7 - 4*Sqrt[3]])/(935*b^(7/3)*Sqrt[(a^(1/3)*(a^(1/3) + b^(1/
3)*x))/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)^2]*Sqrt[a + b*x^3])

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Rubi [A]  time = 0.298456, antiderivative size = 275, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{48 a^2 x \sqrt{a+b x^3}}{935 b^2}+\frac{32\ 3^{3/4} \sqrt{2+\sqrt{3}} a^3 \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{935 b^{7/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{2}{17} x^7 \sqrt{a+b x^3}+\frac{6 a x^4 \sqrt{a+b x^3}}{187 b} \]

Antiderivative was successfully verified.

[In]  Int[x^6*Sqrt[a + b*x^3],x]

[Out]

(-48*a^2*x*Sqrt[a + b*x^3])/(935*b^2) + (6*a*x^4*Sqrt[a + b*x^3])/(187*b) + (2*x
^7*Sqrt[a + b*x^3])/17 + (32*3^(3/4)*Sqrt[2 + Sqrt[3]]*a^3*(a^(1/3) + b^(1/3)*x)
*Sqrt[(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/((1 + Sqrt[3])*a^(1/3) + b^(1/
3)*x)^2]*EllipticF[ArcSin[((1 - Sqrt[3])*a^(1/3) + b^(1/3)*x)/((1 + Sqrt[3])*a^(
1/3) + b^(1/3)*x)], -7 - 4*Sqrt[3]])/(935*b^(7/3)*Sqrt[(a^(1/3)*(a^(1/3) + b^(1/
3)*x))/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)^2]*Sqrt[a + b*x^3])

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Rubi in Sympy [A]  time = 23.0929, size = 246, normalized size = 0.89 \[ \frac{32 \cdot 3^{\frac{3}{4}} a^{3} \sqrt{\frac{a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2}}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} x\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (\sqrt [3]{a} + \sqrt [3]{b} x\right ) F\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{b} x}{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} x} \right )}\middle | -7 - 4 \sqrt{3}\right )}{935 b^{\frac{7}{3}} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a} + \sqrt [3]{b} x\right )}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} x\right )^{2}}} \sqrt{a + b x^{3}}} - \frac{48 a^{2} x \sqrt{a + b x^{3}}}{935 b^{2}} + \frac{6 a x^{4} \sqrt{a + b x^{3}}}{187 b} + \frac{2 x^{7} \sqrt{a + b x^{3}}}{17} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**6*(b*x**3+a)**(1/2),x)

[Out]

32*3**(3/4)*a**3*sqrt((a**(2/3) - a**(1/3)*b**(1/3)*x + b**(2/3)*x**2)/(a**(1/3)
*(1 + sqrt(3)) + b**(1/3)*x)**2)*sqrt(sqrt(3) + 2)*(a**(1/3) + b**(1/3)*x)*ellip
tic_f(asin((-a**(1/3)*(-1 + sqrt(3)) + b**(1/3)*x)/(a**(1/3)*(1 + sqrt(3)) + b**
(1/3)*x)), -7 - 4*sqrt(3))/(935*b**(7/3)*sqrt(a**(1/3)*(a**(1/3) + b**(1/3)*x)/(
a**(1/3)*(1 + sqrt(3)) + b**(1/3)*x)**2)*sqrt(a + b*x**3)) - 48*a**2*x*sqrt(a +
b*x**3)/(935*b**2) + 6*a*x**4*sqrt(a + b*x**3)/(187*b) + 2*x**7*sqrt(a + b*x**3)
/17

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Mathematica [C]  time = 0.508152, size = 184, normalized size = 0.67 \[ \sqrt{a+b x^3} \left (-\frac{48 a^2 x}{935 b^2}+\frac{6 a x^4}{187 b}+\frac{2 x^7}{17}\right )+\frac{32 i 3^{3/4} a^{10/3} \sqrt{(-1)^{5/6} \left (\frac{\sqrt [3]{-b} x}{\sqrt [3]{a}}-1\right )} \sqrt{\frac{(-b)^{2/3} x^2}{a^{2/3}}+\frac{\sqrt [3]{-b} x}{\sqrt [3]{a}}+1} F\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{i \sqrt [3]{-b} x}{\sqrt [3]{a}}-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )}{935 \sqrt [3]{-b} b^2 \sqrt{a+b x^3}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[x^6*Sqrt[a + b*x^3],x]

[Out]

Sqrt[a + b*x^3]*((-48*a^2*x)/(935*b^2) + (6*a*x^4)/(187*b) + (2*x^7)/17) + (((32
*I)/935)*3^(3/4)*a^(10/3)*Sqrt[(-1)^(5/6)*(-1 + ((-b)^(1/3)*x)/a^(1/3))]*Sqrt[1
+ ((-b)^(1/3)*x)/a^(1/3) + ((-b)^(2/3)*x^2)/a^(2/3)]*EllipticF[ArcSin[Sqrt[-(-1)
^(5/6) - (I*(-b)^(1/3)*x)/a^(1/3)]/3^(1/4)], (-1)^(1/3)])/((-b)^(1/3)*b^2*Sqrt[a
+ b*x^3])

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Maple [A]  time = 0.024, size = 337, normalized size = 1.2 \[{\frac{2\,{x}^{7}}{17}\sqrt{b{x}^{3}+a}}+{\frac{6\,a{x}^{4}}{187\,b}\sqrt{b{x}^{3}+a}}-{\frac{48\,x{a}^{2}}{935\,{b}^{2}}\sqrt{b{x}^{3}+a}}-{\frac{{\frac{32\,i}{935}}{a}^{3}\sqrt{3}}{{b}^{3}}\sqrt [3]{-a{b}^{2}}\sqrt{{i\sqrt{3}b \left ( x+{\frac{1}{2\,b}\sqrt [3]{-a{b}^{2}}}-{\frac{{\frac{i}{2}}\sqrt{3}}{b}\sqrt [3]{-a{b}^{2}}} \right ){\frac{1}{\sqrt [3]{-a{b}^{2}}}}}}\sqrt{{1 \left ( x-{\frac{1}{b}\sqrt [3]{-a{b}^{2}}} \right ) \left ( -{\frac{3}{2\,b}\sqrt [3]{-a{b}^{2}}}+{\frac{{\frac{i}{2}}\sqrt{3}}{b}\sqrt [3]{-a{b}^{2}}} \right ) ^{-1}}}\sqrt{{-i\sqrt{3}b \left ( x+{\frac{1}{2\,b}\sqrt [3]{-a{b}^{2}}}+{\frac{{\frac{i}{2}}\sqrt{3}}{b}\sqrt [3]{-a{b}^{2}}} \right ){\frac{1}{\sqrt [3]{-a{b}^{2}}}}}}{\it EllipticF} \left ({\frac{\sqrt{3}}{3}\sqrt{{i\sqrt{3}b \left ( x+{\frac{1}{2\,b}\sqrt [3]{-a{b}^{2}}}-{\frac{{\frac{i}{2}}\sqrt{3}}{b}\sqrt [3]{-a{b}^{2}}} \right ){\frac{1}{\sqrt [3]{-a{b}^{2}}}}}}},\sqrt{{\frac{i\sqrt{3}}{b}\sqrt [3]{-a{b}^{2}} \left ( -{\frac{3}{2\,b}\sqrt [3]{-a{b}^{2}}}+{\frac{{\frac{i}{2}}\sqrt{3}}{b}\sqrt [3]{-a{b}^{2}}} \right ) ^{-1}}} \right ){\frac{1}{\sqrt{b{x}^{3}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^6*(b*x^3+a)^(1/2),x)

[Out]

2/17*x^7*(b*x^3+a)^(1/2)+6/187*a*x^4*(b*x^3+a)^(1/2)/b-48/935*a^2*x*(b*x^3+a)^(1
/2)/b^2-32/935*I*a^3/b^3*3^(1/2)*(-a*b^2)^(1/3)*(I*(x+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)*((x-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*(x+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)/(b*
x^3+a)^(1/2)*EllipticF(1/3*3^(1/2)*(I*(x+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. \[ \int \sqrt{b x^{3} + a} x^{6}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x^3 + a)*x^6,x, algorithm="maxima")

[Out]

integrate(sqrt(b*x^3 + a)*x^6, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\sqrt{b x^{3} + a} x^{6}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x^3 + a)*x^6,x, algorithm="fricas")

[Out]

integral(sqrt(b*x^3 + a)*x^6, x)

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Sympy [A]  time = 2.86875, size = 39, normalized size = 0.14 \[ \frac{\sqrt{a} x^{7} \Gamma \left (\frac{7}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{7}{3} \\ \frac{10}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac{10}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**6*(b*x**3+a)**(1/2),x)

[Out]

sqrt(a)*x**7*gamma(7/3)*hyper((-1/2, 7/3), (10/3,), b*x**3*exp_polar(I*pi)/a)/(3
*gamma(10/3))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{b x^{3} + a} x^{6}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x^3 + a)*x^6,x, algorithm="giac")

[Out]

integrate(sqrt(b*x^3 + a)*x^6, x)