3.1000 \(\int x^6 \sqrt [4]{a+b x^4} \, dx\)

Optimal. Leaf size=103 \[ \frac{3 a^2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{7/4}}-\frac{3 a^2 \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{7/4}}+\frac{1}{8} x^7 \sqrt [4]{a+b x^4}+\frac{a x^3 \sqrt [4]{a+b x^4}}{32 b} \]

[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[(b^
(1/4)*x)/(a + b*x^4)^(1/4)])/(64*b^(7/4)) - (3*a^2*ArcTanh[(b^(1/4)*x)/(a + b*x^
4)^(1/4)])/(64*b^(7/4))

_______________________________________________________________________________________

Rubi [A]  time = 0.112435, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ \frac{3 a^2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{7/4}}-\frac{3 a^2 \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{7/4}}+\frac{1}{8} x^7 \sqrt [4]{a+b x^4}+\frac{a x^3 \sqrt [4]{a+b x^4}}{32 b} \]

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[(b^
(1/4)*x)/(a + b*x^4)^(1/4)])/(64*b^(7/4)) - (3*a^2*ArcTanh[(b^(1/4)*x)/(a + b*x^
4)^(1/4)])/(64*b^(7/4))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 13.9107, size = 94, normalized size = 0.91 \[ \frac{3 a^{2} \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a + b x^{4}}} \right )}}{64 b^{\frac{7}{4}}} - \frac{3 a^{2} \operatorname{atanh}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a + b x^{4}}} \right )}}{64 b^{\frac{7}{4}}} + \frac{a x^{3} \sqrt [4]{a + b x^{4}}}{32 b} + \frac{x^{7} \sqrt [4]{a + b x^{4}}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*a**2*atan(b**(1/4)*x/(a + b*x**4)**(1/4))/(64*b**(7/4)) - 3*a**2*atanh(b**(1/4
)*x/(a + b*x**4)**(1/4))/(64*b**(7/4)) + a*x**3*(a + b*x**4)**(1/4)/(32*b) + x**
7*(a + b*x**4)**(1/4)/8

_______________________________________________________________________________________

Mathematica [C]  time = 0.0525284, size = 78, normalized size = 0.76 \[ \frac{x^3 \left (-a^2 \left (\frac{b x^4}{a}+1\right )^{3/4} \, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};-\frac{b x^4}{a}\right )+a^2+5 a b x^4+4 b^2 x^8\right )}{32 b \left (a+b x^4\right )^{3/4}} \]

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))

_______________________________________________________________________________________

Maple [F]  time = 0.046, 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)

_______________________________________________________________________________________

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

_______________________________________________________________________________________

Fricas [A]  time = 0.28558, size = 274, normalized size = 2.66 \[ \frac{12 \, \left (\frac{a^{8}}{b^{7}}\right )^{\frac{1}{4}} b \arctan \left (\frac{\left (\frac{a^{8}}{b^{7}}\right )^{\frac{1}{4}} b^{2} x}{{\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{2} + x \sqrt{\frac{\sqrt{\frac{a^{8}}{b^{7}}} b^{4} x^{2} + \sqrt{b x^{4} + a} a^{4}}{x^{2}}}}\right ) - 3 \, \left (\frac{a^{8}}{b^{7}}\right )^{\frac{1}{4}} b \log \left (\frac{3 \,{\left (\left (\frac{a^{8}}{b^{7}}\right )^{\frac{1}{4}} b^{2} x +{\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{2}\right )}}{x}\right ) + 3 \, \left (\frac{a^{8}}{b^{7}}\right )^{\frac{1}{4}} b \log \left (-\frac{3 \,{\left (\left (\frac{a^{8}}{b^{7}}\right )^{\frac{1}{4}} b^{2} x -{\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{2}\right )}}{x}\right ) + 4 \,{\left (4 \, b x^{7} + a x^{3}\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{128 \, b} \]

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

_______________________________________________________________________________________

Sympy [A]  time = 5.88931, size = 39, normalized size = 0.38 \[ \frac{\sqrt [4]{a} x^{7} \Gamma \left (\frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{11}{4}\right )} \]

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(I*pi)/a)/(
4*gamma(11/4))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.237515, size = 348, normalized size = 3.38 \[ \frac{1}{256} \,{\left (\frac{8 \, x^{8}{\left (\frac{{\left (b x^{4} + a\right )}^{\frac{1}{4}}{\left (b + \frac{a}{x^{4}}\right )}}{x} + \frac{3 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} b}{x}\right )}}{a^{2} b} - \frac{6 \, \sqrt{2} \left (-b\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (-b\right )^{\frac{1}{4}} + \frac{2 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{x}\right )}}{2 \, \left (-b\right )^{\frac{1}{4}}}\right )}{b^{2}} - \frac{6 \, \sqrt{2} \left (-b\right )^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (-b\right )^{\frac{1}{4}} - \frac{2 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{x}\right )}}{2 \, \left (-b\right )^{\frac{1}{4}}}\right )}{b^{2}} - \frac{3 \, \sqrt{2} \left (-b\right )^{\frac{1}{4}}{\rm ln}\left (\sqrt{-b} + \frac{\sqrt{2}{\left (b x^{4} + a\right )}^{\frac{1}{4}} \left (-b\right )^{\frac{1}{4}}}{x} + \frac{\sqrt{b x^{4} + a}}{x^{2}}\right )}{b^{2}} + \frac{3 \, \sqrt{2} \left (-b\right )^{\frac{1}{4}}{\rm ln}\left (\sqrt{-b} - \frac{\sqrt{2}{\left (b x^{4} + a\right )}^{\frac{1}{4}} \left (-b\right )^{\frac{1}{4}}}{x} + \frac{\sqrt{b x^{4} + a}}{x^{2}}\right )}{b^{2}}\right )} a^{2} \]

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