3.304 \(\int x^2 \left (3+4 x^4\right )^{5/4} \, dx\)

Optimal. Leaf size=93 \[ -\frac{45 \tan ^{-1}\left (\frac{\sqrt{2} x}{\sqrt [4]{4 x^4+3}}\right )}{128 \sqrt{2}}+\frac{45 \tanh ^{-1}\left (\frac{\sqrt{2} x}{\sqrt [4]{4 x^4+3}}\right )}{128 \sqrt{2}}+\frac{1}{8} \left (4 x^4+3\right )^{5/4} x^3+\frac{15}{32} \sqrt [4]{4 x^4+3} x^3 \]

[Out]

(15*x^3*(3 + 4*x^4)^(1/4))/32 + (x^3*(3 + 4*x^4)^(5/4))/8 - (45*ArcTan[(Sqrt[2]*
x)/(3 + 4*x^4)^(1/4)])/(128*Sqrt[2]) + (45*ArcTanh[(Sqrt[2]*x)/(3 + 4*x^4)^(1/4)
])/(128*Sqrt[2])

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Rubi [A]  time = 0.0860697, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{45 \tan ^{-1}\left (\frac{\sqrt{2} x}{\sqrt [4]{4 x^4+3}}\right )}{128 \sqrt{2}}+\frac{45 \tanh ^{-1}\left (\frac{\sqrt{2} x}{\sqrt [4]{4 x^4+3}}\right )}{128 \sqrt{2}}+\frac{1}{8} \left (4 x^4+3\right )^{5/4} x^3+\frac{15}{32} \sqrt [4]{4 x^4+3} x^3 \]

Antiderivative was successfully verified.

[In]  Int[x^2*(3 + 4*x^4)^(5/4),x]

[Out]

(15*x^3*(3 + 4*x^4)^(1/4))/32 + (x^3*(3 + 4*x^4)^(5/4))/8 - (45*ArcTan[(Sqrt[2]*
x)/(3 + 4*x^4)^(1/4)])/(128*Sqrt[2]) + (45*ArcTanh[(Sqrt[2]*x)/(3 + 4*x^4)^(1/4)
])/(128*Sqrt[2])

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Rubi in Sympy [A]  time = 4.22124, size = 85, normalized size = 0.91 \[ \frac{x^{3} \left (4 x^{4} + 3\right )^{\frac{5}{4}}}{8} + \frac{15 x^{3} \sqrt [4]{4 x^{4} + 3}}{32} - \frac{45 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{\sqrt [4]{4 x^{4} + 3}} \right )}}{256} + \frac{45 \sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} x}{\sqrt [4]{4 x^{4} + 3}} \right )}}{256} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*(4*x**4+3)**(5/4),x)

[Out]

x**3*(4*x**4 + 3)**(5/4)/8 + 15*x**3*(4*x**4 + 3)**(1/4)/32 - 45*sqrt(2)*atan(sq
rt(2)*x/(4*x**4 + 3)**(1/4))/256 + 45*sqrt(2)*atanh(sqrt(2)*x/(4*x**4 + 3)**(1/4
))/256

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Mathematica [C]  time = 0.0301574, size = 51, normalized size = 0.55 \[ \frac{1}{32} x^3 \left (5 \sqrt [4]{3} \, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};-\frac{4 x^4}{3}\right )+\sqrt [4]{4 x^4+3} \left (16 x^4+27\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^2*(3 + 4*x^4)^(5/4),x]

[Out]

(x^3*((3 + 4*x^4)^(1/4)*(27 + 16*x^4) + 5*3^(1/4)*Hypergeometric2F1[3/4, 3/4, 7/
4, (-4*x^4)/3]))/32

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Maple [C]  time = 0.034, size = 42, normalized size = 0.5 \[{\frac{{x}^{3} \left ( 16\,{x}^{4}+27 \right ) }{32}\sqrt [4]{4\,{x}^{4}+3}}+{\frac{5\,\sqrt [4]{3}{x}^{3}}{32}{\mbox{$_2$F$_1$}({\frac{3}{4}},{\frac{3}{4}};\,{\frac{7}{4}};\,-{\frac{4\,{x}^{4}}{3}})}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(4*x^4+3)^(5/4),x)

[Out]

1/32*x^3*(16*x^4+27)*(4*x^4+3)^(1/4)+5/32*3^(1/4)*x^3*hypergeom([3/4,3/4],[7/4],
-4/3*x^4)

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Maxima [A]  time = 1.63117, size = 177, normalized size = 1.9 \[ \frac{45}{256} \, \sqrt{2} \arctan \left (\frac{\sqrt{2}{\left (4 \, x^{4} + 3\right )}^{\frac{1}{4}}}{2 \, x}\right ) - \frac{45}{512} \, \sqrt{2} \log \left (-\frac{2 \,{\left (\sqrt{2} - \frac{{\left (4 \, x^{4} + 3\right )}^{\frac{1}{4}}}{x}\right )}}{2 \, \sqrt{2} + \frac{2 \,{\left (4 \, x^{4} + 3\right )}^{\frac{1}{4}}}{x}}\right ) + \frac{9 \,{\left (\frac{20 \,{\left (4 \, x^{4} + 3\right )}^{\frac{1}{4}}}{x} - \frac{9 \,{\left (4 \, x^{4} + 3\right )}^{\frac{5}{4}}}{x^{5}}\right )}}{32 \,{\left (\frac{8 \,{\left (4 \, x^{4} + 3\right )}}{x^{4}} - \frac{{\left (4 \, x^{4} + 3\right )}^{2}}{x^{8}} - 16\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((4*x^4 + 3)^(5/4)*x^2,x, algorithm="maxima")

[Out]

45/256*sqrt(2)*arctan(1/2*sqrt(2)*(4*x^4 + 3)^(1/4)/x) - 45/512*sqrt(2)*log(-2*(
sqrt(2) - (4*x^4 + 3)^(1/4)/x)/((2*sqrt(2)) + 2*(4*x^4 + 3)^(1/4)/x)) + 9/32*(20
*(4*x^4 + 3)^(1/4)/x - 9*(4*x^4 + 3)^(5/4)/x^5)/(8*(4*x^4 + 3)/x^4 - (4*x^4 + 3)
^2/x^8 - 16)

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Fricas [A]  time = 0.211196, size = 147, normalized size = 1.58 \[ \frac{1}{512} \, \sqrt{2}{\left (8 \, \sqrt{2}{\left (16 \, x^{7} + 27 \, x^{3}\right )}{\left (4 \, x^{4} + 3\right )}^{\frac{1}{4}} + 90 \, \arctan \left (\frac{\sqrt{2}{\left (4 \, x^{4} + 3\right )}^{\frac{1}{4}}}{2 \, x}\right ) + 45 \, \log \left (-\frac{2 \, \sqrt{2} x^{2} + 4 \,{\left (4 \, x^{4} + 3\right )}^{\frac{1}{4}} x + \sqrt{2} \sqrt{4 \, x^{4} + 3}}{2 \, x^{2} - \sqrt{4 \, x^{4} + 3}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((4*x^4 + 3)^(5/4)*x^2,x, algorithm="fricas")

[Out]

1/512*sqrt(2)*(8*sqrt(2)*(16*x^7 + 27*x^3)*(4*x^4 + 3)^(1/4) + 90*arctan(1/2*sqr
t(2)*(4*x^4 + 3)^(1/4)/x) + 45*log(-(2*sqrt(2)*x^2 + 4*(4*x^4 + 3)^(1/4)*x + sqr
t(2)*sqrt(4*x^4 + 3))/(2*x^2 - sqrt(4*x^4 + 3))))

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Sympy [A]  time = 4.21478, size = 41, normalized size = 0.44 \[ \frac{3 \sqrt [4]{3} x^{3} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{4}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{4 x^{4} e^{i \pi }}{3}} \right )}}{4 \Gamma \left (\frac{7}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2*(4*x**4+3)**(5/4),x)

[Out]

3*3**(1/4)*x**3*gamma(3/4)*hyper((-5/4, 3/4), (7/4,), 4*x**4*exp_polar(I*pi)/3)/
(4*gamma(7/4))

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GIAC/XCAS [A]  time = 0.215104, size = 149, normalized size = 1.6 \[ \frac{1}{32} \, x^{8}{\left (\frac{9 \,{\left (4 \, x^{4} + 3\right )}^{\frac{1}{4}}{\left (\frac{3}{x^{4}} + 4\right )}}{x} - \frac{20 \,{\left (4 \, x^{4} + 3\right )}^{\frac{1}{4}}}{x}\right )} + \frac{45}{256} \, \sqrt{2} \arctan \left (\frac{\sqrt{2}{\left (4 \, x^{4} + 3\right )}^{\frac{1}{4}}}{2 \, x}\right ) - \frac{45}{512} \, \sqrt{2}{\rm ln}\left (-\frac{\sqrt{2} - \frac{{\left (4 \, x^{4} + 3\right )}^{\frac{1}{4}}}{x}}{\sqrt{2} + \frac{{\left (4 \, x^{4} + 3\right )}^{\frac{1}{4}}}{x}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((4*x^4 + 3)^(5/4)*x^2,x, algorithm="giac")

[Out]

1/32*x^8*(9*(4*x^4 + 3)^(1/4)*(3/x^4 + 4)/x - 20*(4*x^4 + 3)^(1/4)/x) + 45/256*s
qrt(2)*arctan(1/2*sqrt(2)*(4*x^4 + 3)^(1/4)/x) - 45/512*sqrt(2)*ln(-(sqrt(2) - (
4*x^4 + 3)^(1/4)/x)/(sqrt(2) + (4*x^4 + 3)^(1/4)/x))