3.1010 \(\int x^6 \sqrt [6]{a+b x^2} \, dx\)

Optimal. Leaf size=345 \[ -\frac{81\ 3^{3/4} \sqrt{2-\sqrt{3}} a^4 \sqrt [6]{a+b x^2} \left (1-\sqrt [3]{\frac{a}{a+b x^2}}\right ) \sqrt{\frac{\left (\frac{a}{a+b x^2}\right )^{2/3}+\sqrt [3]{\frac{a}{a+b x^2}}+1}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-\sqrt [3]{\frac{a}{b x^2+a}}+\sqrt{3}+1}{-\sqrt [3]{\frac{a}{b x^2+a}}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{2816 b^4 x \sqrt [3]{\frac{a}{a+b x^2}} \sqrt{-\frac{1-\sqrt [3]{\frac{a}{a+b x^2}}}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}}}+\frac{81 a^3 x \sqrt [6]{a+b x^2}}{2816 b^3}-\frac{9 a^2 x^3 \sqrt [6]{a+b x^2}}{704 b^2}+\frac{3}{22} x^7 \sqrt [6]{a+b x^2}+\frac{3 a x^5 \sqrt [6]{a+b x^2}}{352 b} \]

[Out]

(81*a^3*x*(a + b*x^2)^(1/6))/(2816*b^3) - (9*a^2*x^3*(a + b*x^2)^(1/6))/(704*b^2
) + (3*a*x^5*(a + b*x^2)^(1/6))/(352*b) + (3*x^7*(a + b*x^2)^(1/6))/22 - (81*3^(
3/4)*Sqrt[2 - Sqrt[3]]*a^4*(a + b*x^2)^(1/6)*(1 - (a/(a + b*x^2))^(1/3))*Sqrt[(1
+ (a/(a + b*x^2))^(1/3) + (a/(a + b*x^2))^(2/3))/(1 - Sqrt[3] - (a/(a + b*x^2))
^(1/3))^2]*EllipticF[ArcSin[(1 + Sqrt[3] - (a/(a + b*x^2))^(1/3))/(1 - Sqrt[3] -
(a/(a + b*x^2))^(1/3))], -7 + 4*Sqrt[3]])/(2816*b^4*x*(a/(a + b*x^2))^(1/3)*Sqr
t[-((1 - (a/(a + b*x^2))^(1/3))/(1 - Sqrt[3] - (a/(a + b*x^2))^(1/3))^2)])

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Rubi [A]  time = 0.890334, antiderivative size = 345, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{81\ 3^{3/4} \sqrt{2-\sqrt{3}} a^4 \sqrt [6]{a+b x^2} \left (1-\sqrt [3]{\frac{a}{a+b x^2}}\right ) \sqrt{\frac{\left (\frac{a}{a+b x^2}\right )^{2/3}+\sqrt [3]{\frac{a}{a+b x^2}}+1}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-\sqrt [3]{\frac{a}{b x^2+a}}+\sqrt{3}+1}{-\sqrt [3]{\frac{a}{b x^2+a}}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{2816 b^4 x \sqrt [3]{\frac{a}{a+b x^2}} \sqrt{-\frac{1-\sqrt [3]{\frac{a}{a+b x^2}}}{\left (-\sqrt [3]{\frac{a}{a+b x^2}}-\sqrt{3}+1\right )^2}}}+\frac{81 a^3 x \sqrt [6]{a+b x^2}}{2816 b^3}-\frac{9 a^2 x^3 \sqrt [6]{a+b x^2}}{704 b^2}+\frac{3}{22} x^7 \sqrt [6]{a+b x^2}+\frac{3 a x^5 \sqrt [6]{a+b x^2}}{352 b} \]

Antiderivative was successfully verified.

[In]  Int[x^6*(a + b*x^2)^(1/6),x]

[Out]

(81*a^3*x*(a + b*x^2)^(1/6))/(2816*b^3) - (9*a^2*x^3*(a + b*x^2)^(1/6))/(704*b^2
) + (3*a*x^5*(a + b*x^2)^(1/6))/(352*b) + (3*x^7*(a + b*x^2)^(1/6))/22 - (81*3^(
3/4)*Sqrt[2 - Sqrt[3]]*a^4*(a + b*x^2)^(1/6)*(1 - (a/(a + b*x^2))^(1/3))*Sqrt[(1
+ (a/(a + b*x^2))^(1/3) + (a/(a + b*x^2))^(2/3))/(1 - Sqrt[3] - (a/(a + b*x^2))
^(1/3))^2]*EllipticF[ArcSin[(1 + Sqrt[3] - (a/(a + b*x^2))^(1/3))/(1 - Sqrt[3] -
(a/(a + b*x^2))^(1/3))], -7 + 4*Sqrt[3]])/(2816*b^4*x*(a/(a + b*x^2))^(1/3)*Sqr
t[-((1 - (a/(a + b*x^2))^(1/3))/(1 - Sqrt[3] - (a/(a + b*x^2))^(1/3))^2)])

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Rubi in Sympy [A]  time = 29.8853, size = 320, normalized size = 0.93 \[ - \frac{81 \cdot 3^{\frac{3}{4}} a^{4} \sqrt{\frac{\left (- \frac{b x^{2}}{a + b x^{2}} + 1\right )^{\frac{2}{3}} + \sqrt [3]{- \frac{b x^{2}}{a + b x^{2}} + 1} + 1}{\left (- \sqrt [3]{- \frac{b x^{2}}{a + b x^{2}} + 1} - \sqrt{3} + 1\right )^{2}}} \sqrt{- \sqrt{3} + 2} \sqrt [6]{a + b x^{2}} \left (- \sqrt [3]{- \frac{b x^{2}}{a + b x^{2}} + 1} + 1\right ) F\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{- \frac{b x^{2}}{a + b x^{2}} + 1} + 1 + \sqrt{3}}{- \sqrt [3]{- \frac{b x^{2}}{a + b x^{2}} + 1} - \sqrt{3} + 1} \right )}\middle | -7 + 4 \sqrt{3}\right )}{2816 b^{4} x \sqrt [3]{\frac{a}{a + b x^{2}}} \sqrt{\frac{\sqrt [3]{- \frac{b x^{2}}{a + b x^{2}} + 1} - 1}{\left (- \sqrt [3]{- \frac{b x^{2}}{a + b x^{2}} + 1} - \sqrt{3} + 1\right )^{2}}}} + \frac{81 a^{3} x \sqrt [6]{a + b x^{2}}}{2816 b^{3}} - \frac{9 a^{2} x^{3} \sqrt [6]{a + b x^{2}}}{704 b^{2}} + \frac{3 a x^{5} \sqrt [6]{a + b x^{2}}}{352 b} + \frac{3 x^{7} \sqrt [6]{a + b x^{2}}}{22} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-81*3**(3/4)*a**4*sqrt(((-b*x**2/(a + b*x**2) + 1)**(2/3) + (-b*x**2/(a + b*x**2
) + 1)**(1/3) + 1)/(-(-b*x**2/(a + b*x**2) + 1)**(1/3) - sqrt(3) + 1)**2)*sqrt(-
sqrt(3) + 2)*(a + b*x**2)**(1/6)*(-(-b*x**2/(a + b*x**2) + 1)**(1/3) + 1)*ellipt
ic_f(asin((-(-b*x**2/(a + b*x**2) + 1)**(1/3) + 1 + sqrt(3))/(-(-b*x**2/(a + b*x
**2) + 1)**(1/3) - sqrt(3) + 1)), -7 + 4*sqrt(3))/(2816*b**4*x*(a/(a + b*x**2))*
*(1/3)*sqrt(((-b*x**2/(a + b*x**2) + 1)**(1/3) - 1)/(-(-b*x**2/(a + b*x**2) + 1)
**(1/3) - sqrt(3) + 1)**2)) + 81*a**3*x*(a + b*x**2)**(1/6)/(2816*b**3) - 9*a**2
*x**3*(a + b*x**2)**(1/6)/(704*b**2) + 3*a*x**5*(a + b*x**2)**(1/6)/(352*b) + 3*
x**7*(a + b*x**2)**(1/6)/22

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Mathematica [C]  time = 0.0790908, size = 101, normalized size = 0.29 \[ \frac{3 \left (-27 a^4 x \left (\frac{b x^2}{a}+1\right )^{5/6} \, _2F_1\left (\frac{1}{2},\frac{5}{6};\frac{3}{2};-\frac{b x^2}{a}\right )+27 a^4 x+15 a^3 b x^3-4 a^2 b^2 x^5+136 a b^3 x^7+128 b^4 x^9\right )}{2816 b^3 \left (a+b x^2\right )^{5/6}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^6*(a + b*x^2)^(1/6),x]

[Out]

(3*(27*a^4*x + 15*a^3*b*x^3 - 4*a^2*b^2*x^5 + 136*a*b^3*x^7 + 128*b^4*x^9 - 27*a
^4*x*(1 + (b*x^2)/a)^(5/6)*Hypergeometric2F1[1/2, 5/6, 3/2, -((b*x^2)/a)]))/(281
6*b^3*(a + b*x^2)^(5/6))

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Maple [F]  time = 0.049, size = 0, normalized size = 0. \[ \int{x}^{6}\sqrt [6]{b{x}^{2}+a}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^6*(b*x^2+a)^(1/6),x)

[Out]

int(x^6*(b*x^2+a)^(1/6),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{2} + a\right )}^{\frac{1}{6}} x^{6}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(1/6)*x^6,x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)^(1/6)*x^6, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b x^{2} + a\right )}^{\frac{1}{6}} x^{6}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(1/6)*x^6,x, algorithm="fricas")

[Out]

integral((b*x^2 + a)^(1/6)*x^6, x)

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Sympy [A]  time = 5.17152, size = 29, normalized size = 0.08 \[ \frac{\sqrt [6]{a} x^{7}{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{6}, \frac{7}{2} \\ \frac{9}{2} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**(1/6)*x**7*hyper((-1/6, 7/2), (9/2,), b*x**2*exp_polar(I*pi)/a)/7

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{2} + a\right )}^{\frac{1}{6}} x^{6}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(1/6)*x^6,x, algorithm="giac")

[Out]

integrate((b*x^2 + a)^(1/6)*x^6, x)