3.1124 \(\int \frac{x^6}{\left (a+b x^4\right )^{3/4}} \, dx\)

Optimal. Leaf size=80 \[ \frac{3 a \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{8 b^{7/4}}-\frac{3 a \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{8 b^{7/4}}+\frac{x^3 \sqrt [4]{a+b x^4}}{4 b} \]

[Out]

(x^3*(a + b*x^4)^(1/4))/(4*b) + (3*a*ArcTan[(b^(1/4)*x)/(a + b*x^4)^(1/4)])/(8*b
^(7/4)) - (3*a*ArcTanh[(b^(1/4)*x)/(a + b*x^4)^(1/4)])/(8*b^(7/4))

_______________________________________________________________________________________

Rubi [A]  time = 0.0796956, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{3 a \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{8 b^{7/4}}-\frac{3 a \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{8 b^{7/4}}+\frac{x^3 \sqrt [4]{a+b x^4}}{4 b} \]

Antiderivative was successfully verified.

[In]  Int[x^6/(a + b*x^4)^(3/4),x]

[Out]

(x^3*(a + b*x^4)^(1/4))/(4*b) + (3*a*ArcTan[(b^(1/4)*x)/(a + b*x^4)^(1/4)])/(8*b
^(7/4)) - (3*a*ArcTanh[(b^(1/4)*x)/(a + b*x^4)^(1/4)])/(8*b^(7/4))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 10.6395, size = 73, normalized size = 0.91 \[ \frac{3 a \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a + b x^{4}}} \right )}}{8 b^{\frac{7}{4}}} - \frac{3 a \operatorname{atanh}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a + b x^{4}}} \right )}}{8 b^{\frac{7}{4}}} + \frac{x^{3} \sqrt [4]{a + b x^{4}}}{4 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**6/(b*x**4+a)**(3/4),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [C]  time = 0.0482144, size = 64, normalized size = 0.8 \[ \frac{x^3 \left (-a \left (\frac{b x^4}{a}+1\right )^{3/4} \, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};-\frac{b x^4}{a}\right )+a+b x^4\right )}{4 b \left (a+b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^6/(a + b*x^4)^(3/4),x]

[Out]

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

_______________________________________________________________________________________

Maple [F]  time = 0.05, size = 0, normalized size = 0. \[ \int{{x}^{6} \left ( b{x}^{4}+a \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^6/(b*x^4+a)^(3/4),x)

[Out]

int(x^6/(b*x^4+a)^(3/4),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(x^6/(b*x^4 + a)^(3/4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.26681, size = 254, normalized size = 3.18 \[ \frac{4 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} x^{3} + 12 \, b \left (\frac{a^{4}}{b^{7}}\right )^{\frac{1}{4}} \arctan \left (\frac{b^{2} x \left (\frac{a^{4}}{b^{7}}\right )^{\frac{1}{4}}}{x \sqrt{\frac{b^{4} x^{2} \sqrt{\frac{a^{4}}{b^{7}}} + \sqrt{b x^{4} + a} a^{2}}{x^{2}}} +{\left (b x^{4} + a\right )}^{\frac{1}{4}} a}\right ) - 3 \, b \left (\frac{a^{4}}{b^{7}}\right )^{\frac{1}{4}} \log \left (\frac{3 \,{\left (b^{2} x \left (\frac{a^{4}}{b^{7}}\right )^{\frac{1}{4}} +{\left (b x^{4} + a\right )}^{\frac{1}{4}} a\right )}}{x}\right ) + 3 \, b \left (\frac{a^{4}}{b^{7}}\right )^{\frac{1}{4}} \log \left (-\frac{3 \,{\left (b^{2} x \left (\frac{a^{4}}{b^{7}}\right )^{\frac{1}{4}} -{\left (b x^{4} + a\right )}^{\frac{1}{4}} a\right )}}{x}\right )}{16 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/(b*x^4 + a)^(3/4),x, algorithm="fricas")

[Out]

1/16*(4*(b*x^4 + a)^(1/4)*x^3 + 12*b*(a^4/b^7)^(1/4)*arctan(b^2*x*(a^4/b^7)^(1/4
)/(x*sqrt((b^4*x^2*sqrt(a^4/b^7) + sqrt(b*x^4 + a)*a^2)/x^2) + (b*x^4 + a)^(1/4)
*a)) - 3*b*(a^4/b^7)^(1/4)*log(3*(b^2*x*(a^4/b^7)^(1/4) + (b*x^4 + a)^(1/4)*a)/x
) + 3*b*(a^4/b^7)^(1/4)*log(-3*(b^2*x*(a^4/b^7)^(1/4) - (b*x^4 + a)^(1/4)*a)/x))
/b

_______________________________________________________________________________________

Sympy [A]  time = 4.61782, size = 37, normalized size = 0.46 \[ \frac{x^{7} \Gamma \left (\frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac{3}{4}} \Gamma \left (\frac{11}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**6/(b*x**4+a)**(3/4),x)

[Out]

x**7*gamma(7/4)*hyper((3/4, 7/4), (11/4,), b*x**4*exp_polar(I*pi)/a)/(4*a**(3/4)
*gamma(11/4))

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{6}}{{\left (b x^{4} + a\right )}^{\frac{3}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/(b*x^4 + a)^(3/4),x, algorithm="giac")

[Out]

integrate(x^6/(b*x^4 + a)^(3/4), x)