3.1187 \(\int x^6 \sqrt [4]{a-b x^4} \, dx\)

Optimal. Leaf size=263 \[ -\frac{3 a^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{64 \sqrt{2} b^{7/4}}+\frac{3 a^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{64 \sqrt{2} b^{7/4}}+\frac{3 a^2 \log \left (-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}+1\right )}{128 \sqrt{2} b^{7/4}}-\frac{3 a^2 \log \left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}+1\right )}{128 \sqrt{2} b^{7/4}}+\frac{1}{8} x^7 \sqrt [4]{a-b x^4}-\frac{a x^3 \sqrt [4]{a-b x^4}}{32 b} \]

[Out]

-(a*x^3*(a - b*x^4)^(1/4))/(32*b) + (x^7*(a - b*x^4)^(1/4))/8 - (3*a^2*ArcTan[1
- (Sqrt[2]*b^(1/4)*x)/(a - b*x^4)^(1/4)])/(64*Sqrt[2]*b^(7/4)) + (3*a^2*ArcTan[1
+ (Sqrt[2]*b^(1/4)*x)/(a - b*x^4)^(1/4)])/(64*Sqrt[2]*b^(7/4)) + (3*a^2*Log[1 +
(Sqrt[b]*x^2)/Sqrt[a - b*x^4] - (Sqrt[2]*b^(1/4)*x)/(a - b*x^4)^(1/4)])/(128*Sq
rt[2]*b^(7/4)) - (3*a^2*Log[1 + (Sqrt[b]*x^2)/Sqrt[a - b*x^4] + (Sqrt[2]*b^(1/4)
*x)/(a - b*x^4)^(1/4)])/(128*Sqrt[2]*b^(7/4))

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Rubi [A]  time = 0.365333, antiderivative size = 263, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.562 \[ -\frac{3 a^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{64 \sqrt{2} b^{7/4}}+\frac{3 a^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{64 \sqrt{2} b^{7/4}}+\frac{3 a^2 \log \left (-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}+1\right )}{128 \sqrt{2} b^{7/4}}-\frac{3 a^2 \log \left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}+1\right )}{128 \sqrt{2} b^{7/4}}+\frac{1}{8} x^7 \sqrt [4]{a-b x^4}-\frac{a x^3 \sqrt [4]{a-b x^4}}{32 b} \]

Antiderivative was successfully verified.

[In]  Int[x^6*(a - b*x^4)^(1/4),x]

[Out]

-(a*x^3*(a - b*x^4)^(1/4))/(32*b) + (x^7*(a - b*x^4)^(1/4))/8 - (3*a^2*ArcTan[1
- (Sqrt[2]*b^(1/4)*x)/(a - b*x^4)^(1/4)])/(64*Sqrt[2]*b^(7/4)) + (3*a^2*ArcTan[1
+ (Sqrt[2]*b^(1/4)*x)/(a - b*x^4)^(1/4)])/(64*Sqrt[2]*b^(7/4)) + (3*a^2*Log[1 +
(Sqrt[b]*x^2)/Sqrt[a - b*x^4] - (Sqrt[2]*b^(1/4)*x)/(a - b*x^4)^(1/4)])/(128*Sq
rt[2]*b^(7/4)) - (3*a^2*Log[1 + (Sqrt[b]*x^2)/Sqrt[a - b*x^4] + (Sqrt[2]*b^(1/4)
*x)/(a - b*x^4)^(1/4)])/(128*Sqrt[2]*b^(7/4))

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Rubi in Sympy [A]  time = 41.1077, size = 240, normalized size = 0.91 \[ \frac{3 \sqrt{2} a^{2} \log{\left (- \frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a - b x^{4}}} + \frac{\sqrt{b} x^{2}}{\sqrt{a - b x^{4}}} + 1 \right )}}{256 b^{\frac{7}{4}}} - \frac{3 \sqrt{2} a^{2} \log{\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a - b x^{4}}} + \frac{\sqrt{b} x^{2}}{\sqrt{a - b x^{4}}} + 1 \right )}}{256 b^{\frac{7}{4}}} + \frac{3 \sqrt{2} a^{2} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a - b x^{4}}} - 1 \right )}}{128 b^{\frac{7}{4}}} + \frac{3 \sqrt{2} a^{2} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a - b x^{4}}} + 1 \right )}}{128 b^{\frac{7}{4}}} - \frac{a x^{3} \sqrt [4]{a - b x^{4}}}{32 b} + \frac{x^{7} \sqrt [4]{a - b x^{4}}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**6*(-b*x**4+a)**(1/4),x)

[Out]

3*sqrt(2)*a**2*log(-sqrt(2)*b**(1/4)*x/(a - b*x**4)**(1/4) + sqrt(b)*x**2/sqrt(a
- b*x**4) + 1)/(256*b**(7/4)) - 3*sqrt(2)*a**2*log(sqrt(2)*b**(1/4)*x/(a - b*x*
*4)**(1/4) + sqrt(b)*x**2/sqrt(a - b*x**4) + 1)/(256*b**(7/4)) + 3*sqrt(2)*a**2*
atan(sqrt(2)*b**(1/4)*x/(a - b*x**4)**(1/4) - 1)/(128*b**(7/4)) + 3*sqrt(2)*a**2
*atan(sqrt(2)*b**(1/4)*x/(a - b*x**4)**(1/4) + 1)/(128*b**(7/4)) - a*x**3*(a - b
*x**4)**(1/4)/(32*b) + x**7*(a - b*x**4)**(1/4)/8

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Mathematica [C]  time = 0.0746264, size = 80, normalized size = 0.3 \[ \frac{x^3 \left (a^2 \left (1-\frac{b x^4}{a}\right )^{3/4} \, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};\frac{b x^4}{a}\right )-a^2+5 a b x^4-4 b^2 x^8\right )}{32 b \left (a-b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^6*(a - b*x^4)^(1/4),x]

[Out]

(x^3*(-a^2 + 5*a*b*x^4 - 4*b^2*x^8 + a^2*(1 - (b*x^4)/a)^(3/4)*Hypergeometric2F1
[3/4, 3/4, 7/4, (b*x^4)/a]))/(32*b*(a - b*x^4)^(3/4))

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Maple [F]  time = 0.036, size = 0, normalized size = 0. \[ \int{x}^{6}\sqrt [4]{-b{x}^{4}+a}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^6*(-b*x^4+a)^(1/4),x)

[Out]

int(x^6*(-b*x^4+a)^(1/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x^4 + a)^(1/4)*x^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.259153, size = 292, normalized size = 1.11 \[ \frac{12 \, \left (-\frac{a^{8}}{b^{7}}\right )^{\frac{1}{4}} b \arctan \left (\frac{\left (-\frac{a^{8}}{b^{7}}\right )^{\frac{1}{4}} b^{2} x}{{\left (-b x^{4} + a\right )}^{\frac{1}{4}} a^{2} + x \sqrt{\frac{\sqrt{-\frac{a^{8}}{b^{7}}} b^{4} x^{2} + \sqrt{-b x^{4} + a} a^{4}}{x^{2}}}}\right ) - 3 \, \left (-\frac{a^{8}}{b^{7}}\right )^{\frac{1}{4}} b \log \left (\frac{3 \,{\left (\left (-\frac{a^{8}}{b^{7}}\right )^{\frac{1}{4}} b^{2} x +{\left (-b x^{4} + a\right )}^{\frac{1}{4}} a^{2}\right )}}{x}\right ) + 3 \, \left (-\frac{a^{8}}{b^{7}}\right )^{\frac{1}{4}} b \log \left (-\frac{3 \,{\left (\left (-\frac{a^{8}}{b^{7}}\right )^{\frac{1}{4}} b^{2} x -{\left (-b x^{4} + a\right )}^{\frac{1}{4}} a^{2}\right )}}{x}\right ) + 4 \,{\left (4 \, b x^{7} - a x^{3}\right )}{\left (-b x^{4} + a\right )}^{\frac{1}{4}}}{128 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x^4 + a)^(1/4)*x^6,x, algorithm="fricas")

[Out]

1/128*(12*(-a^8/b^7)^(1/4)*b*arctan((-a^8/b^7)^(1/4)*b^2*x/((-b*x^4 + a)^(1/4)*a
^2 + x*sqrt((sqrt(-a^8/b^7)*b^4*x^2 + sqrt(-b*x^4 + a)*a^4)/x^2))) - 3*(-a^8/b^7
)^(1/4)*b*log(3*((-a^8/b^7)^(1/4)*b^2*x + (-b*x^4 + a)^(1/4)*a^2)/x) + 3*(-a^8/b
^7)^(1/4)*b*log(-3*((-a^8/b^7)^(1/4)*b^2*x - (-b*x^4 + a)^(1/4)*a^2)/x) + 4*(4*b
*x^7 - a*x^3)*(-b*x^4 + a)^(1/4))/b

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Sympy [A]  time = 6.18198, size = 41, normalized size = 0.16 \[ \frac{\sqrt [4]{a} x^{7} \Gamma \left (\frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{\frac{b x^{4} e^{2 i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{11}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**6*(-b*x**4+a)**(1/4),x)

[Out]

a**(1/4)*x**7*gamma(7/4)*hyper((-1/4, 7/4), (11/4,), b*x**4*exp_polar(2*I*pi)/a)
/(4*gamma(11/4))

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GIAC/XCAS [A]  time = 0.264567, size = 312, normalized size = 1.19 \[ \frac{1}{256} \,{\left (\frac{8 \, x^{8}{\left (\frac{{\left (-b x^{4} + a\right )}^{\frac{1}{4}}{\left (b - \frac{a}{x^{4}}\right )}}{x} + \frac{3 \,{\left (-b x^{4} + a\right )}^{\frac{1}{4}} b}{x}\right )}}{a^{2} b} - \frac{6 \, \sqrt{2} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} b^{\frac{1}{4}} + \frac{2 \,{\left (-b x^{4} + a\right )}^{\frac{1}{4}}}{x}\right )}}{2 \, b^{\frac{1}{4}}}\right )}{b^{\frac{7}{4}}} - \frac{6 \, \sqrt{2} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} b^{\frac{1}{4}} - \frac{2 \,{\left (-b x^{4} + a\right )}^{\frac{1}{4}}}{x}\right )}}{2 \, b^{\frac{1}{4}}}\right )}{b^{\frac{7}{4}}} - \frac{3 \, \sqrt{2}{\rm ln}\left (\sqrt{b} + \frac{\sqrt{2}{\left (-b x^{4} + a\right )}^{\frac{1}{4}} b^{\frac{1}{4}}}{x} + \frac{\sqrt{-b x^{4} + a}}{x^{2}}\right )}{b^{\frac{7}{4}}} + \frac{3 \, \sqrt{2}{\rm ln}\left (\sqrt{b} - \frac{\sqrt{2}{\left (-b x^{4} + a\right )}^{\frac{1}{4}} b^{\frac{1}{4}}}{x} + \frac{\sqrt{-b x^{4} + a}}{x^{2}}\right )}{b^{\frac{7}{4}}}\right )} a^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x^4 + a)^(1/4)*x^6,x, algorithm="giac")

[Out]

1/256*(8*x^8*((-b*x^4 + a)^(1/4)*(b - a/x^4)/x + 3*(-b*x^4 + a)^(1/4)*b/x)/(a^2*
b) - 6*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*b^(1/4) + 2*(-b*x^4 + a)^(1/4)/x)/b^(
1/4))/b^(7/4) - 6*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2)*b^(1/4) - 2*(-b*x^4 + a)^
(1/4)/x)/b^(1/4))/b^(7/4) - 3*sqrt(2)*ln(sqrt(b) + sqrt(2)*(-b*x^4 + a)^(1/4)*b^
(1/4)/x + sqrt(-b*x^4 + a)/x^2)/b^(7/4) + 3*sqrt(2)*ln(sqrt(b) - sqrt(2)*(-b*x^4
+ a)^(1/4)*b^(1/4)/x + sqrt(-b*x^4 + a)/x^2)/b^(7/4))*a^2