3.1032 \(\int \left (a+b x^4\right )^{3/4} \, dx\)

Optimal. Leaf size=75 \[ \frac{1}{4} x \left (a+b x^4\right )^{3/4}+\frac{3 a \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{8 \sqrt [4]{b}}+\frac{3 a \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{8 \sqrt [4]{b}} \]

[Out]

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

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Rubi [A]  time = 0.0456584, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.454 \[ \frac{1}{4} x \left (a+b x^4\right )^{3/4}+\frac{3 a \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{8 \sqrt [4]{b}}+\frac{3 a \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{8 \sqrt [4]{b}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^4)^(3/4),x]

[Out]

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

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Rubi in Sympy [A]  time = 5.31439, size = 70, normalized size = 0.93 \[ \frac{3 a \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a + b x^{4}}} \right )}}{8 \sqrt [4]{b}} + \frac{3 a \operatorname{atanh}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a + b x^{4}}} \right )}}{8 \sqrt [4]{b}} + \frac{x \left (a + b x^{4}\right )^{\frac{3}{4}}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0608566, size = 94, normalized size = 1.25 \[ \frac{1}{4} x \left (a+b x^4\right )^{3/4}+\frac{3 a \left (-\log \left (1-\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )+\log \left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}+1\right )+2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )\right )}{16 \sqrt [4]{b}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^4)^(3/4),x]

[Out]

(x*(a + b*x^4)^(3/4))/4 + (3*a*(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)]))/(16*
b^(1/4))

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Maple [F]  time = 0.044, size = 0, normalized size = 0. \[ \int \left ( b{x}^{4}+a \right ) ^{{\frac{3}{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^4+a)^(3/4),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.272914, size = 242, normalized size = 3.23 \[ \frac{1}{4} \,{\left (b x^{4} + a\right )}^{\frac{3}{4}} x + \frac{3}{4} \, \left (\frac{a^{4}}{b}\right )^{\frac{1}{4}} \arctan \left (\frac{\left (\frac{a^{4}}{b}\right )^{\frac{3}{4}} b x}{{\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{3} + x \sqrt{\frac{\sqrt{\frac{a^{4}}{b}} a^{4} b x^{2} + \sqrt{b x^{4} + a} a^{6}}{x^{2}}}}\right ) + \frac{3}{16} \, \left (\frac{a^{4}}{b}\right )^{\frac{1}{4}} \log \left (\frac{27 \,{\left ({\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{3} + \left (\frac{a^{4}}{b}\right )^{\frac{3}{4}} b x\right )}}{x}\right ) - \frac{3}{16} \, \left (\frac{a^{4}}{b}\right )^{\frac{1}{4}} \log \left (\frac{27 \,{\left ({\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{3} - \left (\frac{a^{4}}{b}\right )^{\frac{3}{4}} b x\right )}}{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(b*x^4 + a)^(3/4)*x + 3/4*(a^4/b)^(1/4)*arctan((a^4/b)^(3/4)*b*x/((b*x^4 + a
)^(1/4)*a^3 + x*sqrt((sqrt(a^4/b)*a^4*b*x^2 + sqrt(b*x^4 + a)*a^6)/x^2))) + 3/16
*(a^4/b)^(1/4)*log(27*((b*x^4 + a)^(1/4)*a^3 + (a^4/b)^(3/4)*b*x)/x) - 3/16*(a^4
/b)^(1/4)*log(27*((b*x^4 + a)^(1/4)*a^3 - (a^4/b)^(3/4)*b*x)/x)

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Sympy [A]  time = 4.58564, size = 37, normalized size = 0.49 \[ \frac{a^{\frac{3}{4}} x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, \frac{1}{4} \\ \frac{5}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{5}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**4+a)**(3/4),x)

[Out]

a**(3/4)*x*gamma(1/4)*hyper((-3/4, 1/4), (5/4,), b*x**4*exp_polar(I*pi)/a)/(4*ga
mma(5/4))

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GIAC/XCAS [A]  time = 0.235747, size = 302, normalized size = 4.03 \[ \frac{1}{32} \,{\left (\frac{6 \, \sqrt{2} \left (-b\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (-b\right )^{\frac{1}{4}} + \frac{2 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{x}\right )}}{2 \, \left (-b\right )^{\frac{1}{4}}}\right )}{b} + \frac{6 \, \sqrt{2} \left (-b\right )^{\frac{3}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (-b\right )^{\frac{1}{4}} - \frac{2 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{x}\right )}}{2 \, \left (-b\right )^{\frac{1}{4}}}\right )}{b} - \frac{3 \, \sqrt{2} \left (-b\right )^{\frac{3}{4}}{\rm ln}\left (\sqrt{-b} + \frac{\sqrt{2}{\left (b x^{4} + a\right )}^{\frac{1}{4}} \left (-b\right )^{\frac{1}{4}}}{x} + \frac{\sqrt{b x^{4} + a}}{x^{2}}\right )}{b} + \frac{3 \, \sqrt{2} \left (-b\right )^{\frac{3}{4}}{\rm ln}\left (\sqrt{-b} - \frac{\sqrt{2}{\left (b x^{4} + a\right )}^{\frac{1}{4}} \left (-b\right )^{\frac{1}{4}}}{x} + \frac{\sqrt{b x^{4} + a}}{x^{2}}\right )}{b} + \frac{8 \,{\left (b x^{4} + a\right )}^{\frac{3}{4}} x}{a}\right )} a \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/32*(6*sqrt(2)*(-b)^(3/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(-b)^(1/4) + 2*(b*x^4 + a
)^(1/4)/x)/(-b)^(1/4))/b + 6*sqrt(2)*(-b)^(3/4)*arctan(-1/2*sqrt(2)*(sqrt(2)*(-b
)^(1/4) - 2*(b*x^4 + a)^(1/4)/x)/(-b)^(1/4))/b - 3*sqrt(2)*(-b)^(3/4)*ln(sqrt(-b
) + sqrt(2)*(b*x^4 + a)^(1/4)*(-b)^(1/4)/x + sqrt(b*x^4 + a)/x^2)/b + 3*sqrt(2)*
(-b)^(3/4)*ln(sqrt(-b) - sqrt(2)*(b*x^4 + a)^(1/4)*(-b)^(1/4)/x + sqrt(b*x^4 + a
)/x^2)/b + 8*(b*x^4 + a)^(3/4)*x/a)*a