3.1062 \(\int x^2 \left (a+b x^4\right )^{5/4} \, dx\)

Optimal. Leaf size=100 \[ -\frac{5 a^2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{3/4}}+\frac{5 a^2 \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{3/4}}+\frac{1}{8} x^3 \left (a+b x^4\right )^{5/4}+\frac{5}{32} a x^3 \sqrt [4]{a+b x^4} \]

[Out]

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

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Rubi [A]  time = 0.104091, antiderivative size = 100, 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 \[ -\frac{5 a^2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{3/4}}+\frac{5 a^2 \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{3/4}}+\frac{1}{8} x^3 \left (a+b x^4\right )^{5/4}+\frac{5}{32} a x^3 \sqrt [4]{a+b x^4} \]

Antiderivative was successfully verified.

[In]  Int[x^2*(a + b*x^4)^(5/4),x]

[Out]

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

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Rubi in Sympy [A]  time = 13.5919, size = 94, normalized size = 0.94 \[ - \frac{5 a^{2} \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a + b x^{4}}} \right )}}{64 b^{\frac{3}{4}}} + \frac{5 a^{2} \operatorname{atanh}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a + b x^{4}}} \right )}}{64 b^{\frac{3}{4}}} + \frac{5 a x^{3} \sqrt [4]{a + b x^{4}}}{32} + \frac{x^{3} \left (a + b x^{4}\right )^{\frac{5}{4}}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.0528986, size = 77, normalized size = 0.77 \[ \frac{x^3 \left (5 a^2 \left (\frac{b x^4}{a}+1\right )^{3/4} \, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};-\frac{b x^4}{a}\right )+27 a^2+39 a b x^4+12 b^2 x^8\right )}{96 \left (a+b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

(x^3*(27*a^2 + 39*a*b*x^4 + 12*b^2*x^8 + 5*a^2*(1 + (b*x^4)/a)^(3/4)*Hypergeomet
ric2F1[3/4, 3/4, 7/4, -((b*x^4)/a)]))/(96*(a + b*x^4)^(3/4))

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Maple [F]  time = 0.039, size = 0, normalized size = 0. \[ \int{x}^{2} \left ( b{x}^{4}+a \right ) ^{{\frac{5}{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(b*x^4+a)^(5/4),x)

[Out]

int(x^2*(b*x^4+a)^(5/4),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)^(5/4)*x^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.267101, size = 257, normalized size = 2.57 \[ \frac{1}{32} \,{\left (4 \, b x^{7} + 9 \, a x^{3}\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}} - \frac{5}{32} \, \left (\frac{a^{8}}{b^{3}}\right )^{\frac{1}{4}} \arctan \left (\frac{\left (\frac{a^{8}}{b^{3}}\right )^{\frac{1}{4}} b x}{{\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{2} + x \sqrt{\frac{\sqrt{b x^{4} + a} a^{4} + \sqrt{\frac{a^{8}}{b^{3}}} b^{2} x^{2}}{x^{2}}}}\right ) + \frac{5}{128} \, \left (\frac{a^{8}}{b^{3}}\right )^{\frac{1}{4}} \log \left (\frac{5 \,{\left ({\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{2} + \left (\frac{a^{8}}{b^{3}}\right )^{\frac{1}{4}} b x\right )}}{x}\right ) - \frac{5}{128} \, \left (\frac{a^{8}}{b^{3}}\right )^{\frac{1}{4}} \log \left (\frac{5 \,{\left ({\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{2} - \left (\frac{a^{8}}{b^{3}}\right )^{\frac{1}{4}} b x\right )}}{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/32*(4*b*x^7 + 9*a*x^3)*(b*x^4 + a)^(1/4) - 5/32*(a^8/b^3)^(1/4)*arctan((a^8/b^
3)^(1/4)*b*x/((b*x^4 + a)^(1/4)*a^2 + x*sqrt((sqrt(b*x^4 + a)*a^4 + sqrt(a^8/b^3
)*b^2*x^2)/x^2))) + 5/128*(a^8/b^3)^(1/4)*log(5*((b*x^4 + a)^(1/4)*a^2 + (a^8/b^
3)^(1/4)*b*x)/x) - 5/128*(a^8/b^3)^(1/4)*log(5*((b*x^4 + a)^(1/4)*a^2 - (a^8/b^3
)^(1/4)*b*x)/x)

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Sympy [A]  time = 9.40525, size = 39, normalized size = 0.39 \[ \frac{a^{\frac{5}{4}} x^{3} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{4}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{7}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.241536, size = 346, normalized size = 3.46 \[ \frac{1}{256} \,{\left (\frac{8 \, x^{8}{\left (\frac{9 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}{\left (b + \frac{a}{x^{4}}\right )}}{x} - \frac{5 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} b}{x}\right )}}{a^{2}} + \frac{10 \, \sqrt{2} \left (-b\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (-b\right )^{\frac{1}{4}} + \frac{2 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{x}\right )}}{2 \, \left (-b\right )^{\frac{1}{4}}}\right )}{b} + \frac{10 \, \sqrt{2} \left (-b\right )^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (-b\right )^{\frac{1}{4}} - \frac{2 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{x}\right )}}{2 \, \left (-b\right )^{\frac{1}{4}}}\right )}{b} + \frac{5 \, \sqrt{2} \left (-b\right )^{\frac{1}{4}}{\rm ln}\left (\sqrt{-b} + \frac{\sqrt{2}{\left (b x^{4} + a\right )}^{\frac{1}{4}} \left (-b\right )^{\frac{1}{4}}}{x} + \frac{\sqrt{b x^{4} + a}}{x^{2}}\right )}{b} - \frac{5 \, \sqrt{2} \left (-b\right )^{\frac{1}{4}}{\rm ln}\left (\sqrt{-b} - \frac{\sqrt{2}{\left (b x^{4} + a\right )}^{\frac{1}{4}} \left (-b\right )^{\frac{1}{4}}}{x} + \frac{\sqrt{b x^{4} + a}}{x^{2}}\right )}{b}\right )} a^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/256*(8*x^8*(9*(b*x^4 + a)^(1/4)*(b + a/x^4)/x - 5*(b*x^4 + a)^(1/4)*b/x)/a^2 +
10*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 + 10*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 + 5*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 - 5*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)*a^2