3.1031 \(\int x^4 \left (a+b x^4\right )^{3/4} \, dx\)

Optimal. Leaf size=101 \[ -\frac{3 a^2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{5/4}}-\frac{3 a^2 \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{5/4}}+\frac{3 a x \left (a+b x^4\right )^{3/4}}{32 b}+\frac{1}{8} x^5 \left (a+b x^4\right )^{3/4} \]

[Out]

(3*a*x*(a + b*x^4)^(3/4))/(32*b) + (x^5*(a + b*x^4)^(3/4))/8 - (3*a^2*ArcTan[(b^
(1/4)*x)/(a + b*x^4)^(1/4)])/(64*b^(5/4)) - (3*a^2*ArcTanh[(b^(1/4)*x)/(a + b*x^
4)^(1/4)])/(64*b^(5/4))

_______________________________________________________________________________________

Rubi [A]  time = 0.0882331, antiderivative size = 101, 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 \[ -\frac{3 a^2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{5/4}}-\frac{3 a^2 \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{5/4}}+\frac{3 a x \left (a+b x^4\right )^{3/4}}{32 b}+\frac{1}{8} x^5 \left (a+b x^4\right )^{3/4} \]

Antiderivative was successfully verified.

[In]  Int[x^4*(a + b*x^4)^(3/4),x]

[Out]

(3*a*x*(a + b*x^4)^(3/4))/(32*b) + (x^5*(a + b*x^4)^(3/4))/8 - (3*a^2*ArcTan[(b^
(1/4)*x)/(a + b*x^4)^(1/4)])/(64*b^(5/4)) - (3*a^2*ArcTanh[(b^(1/4)*x)/(a + b*x^
4)^(1/4)])/(64*b^(5/4))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 11.0421, size = 94, normalized size = 0.93 \[ - \frac{3 a^{2} \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a + b x^{4}}} \right )}}{64 b^{\frac{5}{4}}} - \frac{3 a^{2} \operatorname{atanh}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a + b x^{4}}} \right )}}{64 b^{\frac{5}{4}}} + \frac{3 a x \left (a + b x^{4}\right )^{\frac{3}{4}}}{32 b} + \frac{x^{5} \left (a + b x^{4}\right )^{\frac{3}{4}}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*a**2*atan(b**(1/4)*x/(a + b*x**4)**(1/4))/(64*b**(5/4)) - 3*a**2*atanh(b**(1/
4)*x/(a + b*x**4)**(1/4))/(64*b**(5/4)) + 3*a*x*(a + b*x**4)**(3/4)/(32*b) + x**
5*(a + b*x**4)**(3/4)/8

_______________________________________________________________________________________

Mathematica [A]  time = 0.105817, size = 109, normalized size = 1.08 \[ \left (a+b x^4\right )^{3/4} \left (\frac{3 a x}{32 b}+\frac{x^5}{8}\right )-\frac{3 a^2 \left (-\log \left (1-\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )+\log \left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}+1\right )+2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )\right )}{128 b^{5/4}} \]

Antiderivative was successfully verified.

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

[Out]

(a + b*x^4)^(3/4)*((3*a*x)/(32*b) + x^5/8) - (3*a^2*(2*ArcTan[(b^(1/4)*x)/(a + b
*x^4)^(1/4)] - Log[1 - (b^(1/4)*x)/(a + b*x^4)^(1/4)] + Log[1 + (b^(1/4)*x)/(a +
b*x^4)^(1/4)]))/(128*b^(5/4))

_______________________________________________________________________________________

Maple [F]  time = 0.037, size = 0, normalized size = 0. \[ \int{x}^{4} \left ( b{x}^{4}+a \right ) ^{{\frac{3}{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*(b*x^4+a)^(3/4),x)

[Out]

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.279809, size = 277, normalized size = 2.74 \[ -\frac{12 \, \left (\frac{a^{8}}{b^{5}}\right )^{\frac{1}{4}} b \arctan \left (\frac{\left (\frac{a^{8}}{b^{5}}\right )^{\frac{3}{4}} b^{4} x}{{\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{6} + x \sqrt{\frac{\sqrt{\frac{a^{8}}{b^{5}}} a^{8} b^{3} x^{2} + \sqrt{b x^{4} + a} a^{12}}{x^{2}}}}\right ) + 3 \, \left (\frac{a^{8}}{b^{5}}\right )^{\frac{1}{4}} b \log \left (\frac{27 \,{\left ({\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{6} + \left (\frac{a^{8}}{b^{5}}\right )^{\frac{3}{4}} b^{4} x\right )}}{x}\right ) - 3 \, \left (\frac{a^{8}}{b^{5}}\right )^{\frac{1}{4}} b \log \left (\frac{27 \,{\left ({\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{6} - \left (\frac{a^{8}}{b^{5}}\right )^{\frac{3}{4}} b^{4} x\right )}}{x}\right ) - 4 \,{\left (4 \, b x^{5} + 3 \, a x\right )}{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{128 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/128*(12*(a^8/b^5)^(1/4)*b*arctan((a^8/b^5)^(3/4)*b^4*x/((b*x^4 + a)^(1/4)*a^6
+ x*sqrt((sqrt(a^8/b^5)*a^8*b^3*x^2 + sqrt(b*x^4 + a)*a^12)/x^2))) + 3*(a^8/b^5
)^(1/4)*b*log(27*((b*x^4 + a)^(1/4)*a^6 + (a^8/b^5)^(3/4)*b^4*x)/x) - 3*(a^8/b^5
)^(1/4)*b*log(27*((b*x^4 + a)^(1/4)*a^6 - (a^8/b^5)^(3/4)*b^4*x)/x) - 4*(4*b*x^5
+ 3*a*x)*(b*x^4 + a)^(3/4))/b

_______________________________________________________________________________________

Sympy [A]  time = 7.18052, size = 39, normalized size = 0.39 \[ \frac{a^{\frac{3}{4}} x^{5} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{9}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**(3/4)*x**5*gamma(5/4)*hyper((-3/4, 5/4), (9/4,), b*x**4*exp_polar(I*pi)/a)/(4
*gamma(9/4))

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^4 + a)^(3/4)*x^4, x)