3.1054 \(\int x^5 \left (a+b x^4\right )^{5/4} \, dx\)

Optimal. Leaf size=122 \[ -\frac{10 a^{7/2} \left (\frac{b x^4}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{231 b^{3/2} \left (a+b x^4\right )^{3/4}}+\frac{5 a^2 x^2 \sqrt [4]{a+b x^4}}{231 b}+\frac{1}{11} x^6 \left (a+b x^4\right )^{5/4}+\frac{5}{77} a x^6 \sqrt [4]{a+b x^4} \]

[Out]

(5*a^2*x^2*(a + b*x^4)^(1/4))/(231*b) + (5*a*x^6*(a + b*x^4)^(1/4))/77 + (x^6*(a
+ b*x^4)^(5/4))/11 - (10*a^(7/2)*(1 + (b*x^4)/a)^(3/4)*EllipticF[ArcTan[(Sqrt[b
]*x^2)/Sqrt[a]]/2, 2])/(231*b^(3/2)*(a + b*x^4)^(3/4))

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Rubi [A]  time = 0.185824, antiderivative size = 122, 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 \[ -\frac{10 a^{7/2} \left (\frac{b x^4}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{231 b^{3/2} \left (a+b x^4\right )^{3/4}}+\frac{5 a^2 x^2 \sqrt [4]{a+b x^4}}{231 b}+\frac{1}{11} x^6 \left (a+b x^4\right )^{5/4}+\frac{5}{77} a x^6 \sqrt [4]{a+b x^4} \]

Antiderivative was successfully verified.

[In]  Int[x^5*(a + b*x^4)^(5/4),x]

[Out]

(5*a^2*x^2*(a + b*x^4)^(1/4))/(231*b) + (5*a*x^6*(a + b*x^4)^(1/4))/77 + (x^6*(a
+ b*x^4)^(5/4))/11 - (10*a^(7/2)*(1 + (b*x^4)/a)^(3/4)*EllipticF[ArcTan[(Sqrt[b
]*x^2)/Sqrt[a]]/2, 2])/(231*b^(3/2)*(a + b*x^4)^(3/4))

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Rubi in Sympy [A]  time = 19.1123, size = 109, normalized size = 0.89 \[ - \frac{10 a^{\frac{7}{2}} \left (1 + \frac{b x^{4}}{a}\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{2}\middle | 2\right )}{231 b^{\frac{3}{2}} \left (a + b x^{4}\right )^{\frac{3}{4}}} + \frac{5 a^{2} x^{2} \sqrt [4]{a + b x^{4}}}{231 b} + \frac{5 a x^{6} \sqrt [4]{a + b x^{4}}}{77} + \frac{x^{6} \left (a + b x^{4}\right )^{\frac{5}{4}}}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-10*a**(7/2)*(1 + b*x**4/a)**(3/4)*elliptic_f(atan(sqrt(b)*x**2/sqrt(a))/2, 2)/(
231*b**(3/2)*(a + b*x**4)**(3/4)) + 5*a**2*x**2*(a + b*x**4)**(1/4)/(231*b) + 5*
a*x**6*(a + b*x**4)**(1/4)/77 + x**6*(a + b*x**4)**(5/4)/11

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Mathematica [C]  time = 0.0722902, size = 91, normalized size = 0.75 \[ \frac{x^2 \left (-5 a^3 \left (\frac{b x^4}{a}+1\right )^{3/4} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{3}{2};-\frac{b x^4}{a}\right )+5 a^3+41 a^2 b x^4+57 a b^2 x^8+21 b^3 x^{12}\right )}{231 b \left (a+b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

(x^2*(5*a^3 + 41*a^2*b*x^4 + 57*a*b^2*x^8 + 21*b^3*x^12 - 5*a^3*(1 + (b*x^4)/a)^
(3/4)*Hypergeometric2F1[1/2, 3/4, 3/2, -((b*x^4)/a)]))/(231*b*(a + b*x^4)^(3/4))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5*(b*x^4+a)^(5/4),x)

[Out]

int(x^5*(b*x^4+a)^(5/4),x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^4 + a)^(5/4)*x^5, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b x^{9} + a x^{5}\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b*x^9 + a*x^5)*(b*x^4 + a)^(1/4), x)

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Sympy [A]  time = 12.2386, size = 29, normalized size = 0.24 \[ \frac{a^{\frac{5}{4}} x^{6}{{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{4}, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**(5/4)*x**6*hyper((-5/4, 3/2), (5/2,), b*x**4*exp_polar(I*pi)/a)/6

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{4} + a\right )}^{\frac{5}{4}} x^{5}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^4 + a)^(5/4)*x^5, x)