3.12.34 \(\int \frac {x^4}{(a+b x^4)^{3/4}} \, dx\) [1134]

Optimal. Leaf size=83 \[ \frac {x \sqrt [4]{a+b x^4}}{2 b}+\frac {\sqrt {a} \left (1+\frac {a}{b x^4}\right )^{3/4} x^3 F\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{2 \sqrt {b} \left (a+b x^4\right )^{3/4}} \]

[Out]

1/2*x*(b*x^4+a)^(1/4)/b+1/2*(1+a/b/x^4)^(3/4)*x^3*(cos(1/2*arccot(x^2*b^(1/2)/a^(1/2)))^2)^(1/2)/cos(1/2*arcco
t(x^2*b^(1/2)/a^(1/2)))*EllipticF(sin(1/2*arccot(x^2*b^(1/2)/a^(1/2))),2^(1/2))*a^(1/2)/(b*x^4+a)^(3/4)/b^(1/2
)

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Rubi [A]
time = 0.03, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {327, 243, 342, 281, 237} \begin {gather*} \frac {x \sqrt [4]{a+b x^4}}{2 b}+\frac {\sqrt {a} x^3 \left (\frac {a}{b x^4}+1\right )^{3/4} F\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{2 \sqrt {b} \left (a+b x^4\right )^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^4/(a + b*x^4)^(3/4),x]

[Out]

(x*(a + b*x^4)^(1/4))/(2*b) + (Sqrt[a]*(1 + a/(b*x^4))^(3/4)*x^3*EllipticF[ArcCot[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2]
)/(2*Sqrt[b]*(a + b*x^4)^(3/4))

Rule 237

Int[((a_) + (b_.)*(x_)^2)^(-3/4), x_Symbol] :> Simp[(2/(a^(3/4)*Rt[b/a, 2]))*EllipticF[(1/2)*ArcTan[Rt[b/a, 2]
*x], 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b/a]

Rule 243

Int[((a_) + (b_.)*(x_)^4)^(-3/4), x_Symbol] :> Dist[x^3*((1 + a/(b*x^4))^(3/4)/(a + b*x^4)^(3/4)), Int[1/(x^3*
(1 + a/(b*x^4))^(3/4)), x], x] /; FreeQ[{a, b}, x]

Rule 281

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 342

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^(m + 2), x], x, 1/x] /;
FreeQ[{a, b, p}, x] && ILtQ[n, 0] && IntegerQ[m]

Rubi steps

\begin {align*} \int \frac {x^4}{\left (a+b x^4\right )^{3/4}} \, dx &=\frac {x \sqrt [4]{a+b x^4}}{2 b}-\frac {a \int \frac {1}{\left (a+b x^4\right )^{3/4}} \, dx}{2 b}\\ &=\frac {x \sqrt [4]{a+b x^4}}{2 b}-\frac {\left (a \left (1+\frac {a}{b x^4}\right )^{3/4} x^3\right ) \int \frac {1}{\left (1+\frac {a}{b x^4}\right )^{3/4} x^3} \, dx}{2 b \left (a+b x^4\right )^{3/4}}\\ &=\frac {x \sqrt [4]{a+b x^4}}{2 b}+\frac {\left (a \left (1+\frac {a}{b x^4}\right )^{3/4} x^3\right ) \text {Subst}\left (\int \frac {x}{\left (1+\frac {a x^4}{b}\right )^{3/4}} \, dx,x,\frac {1}{x}\right )}{2 b \left (a+b x^4\right )^{3/4}}\\ &=\frac {x \sqrt [4]{a+b x^4}}{2 b}+\frac {\left (a \left (1+\frac {a}{b x^4}\right )^{3/4} x^3\right ) \text {Subst}\left (\int \frac {1}{\left (1+\frac {a x^2}{b}\right )^{3/4}} \, dx,x,\frac {1}{x^2}\right )}{4 b \left (a+b x^4\right )^{3/4}}\\ &=\frac {x \sqrt [4]{a+b x^4}}{2 b}+\frac {\sqrt {a} \left (1+\frac {a}{b x^4}\right )^{3/4} x^3 F\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{2 \sqrt {b} \left (a+b x^4\right )^{3/4}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 6.88, size = 62, normalized size = 0.75 \begin {gather*} \frac {x \left (a+b x^4-a \left (1+\frac {b x^4}{a}\right )^{3/4} \, _2F_1\left (\frac {1}{4},\frac {3}{4};\frac {5}{4};-\frac {b x^4}{a}\right )\right )}{2 b \left (a+b x^4\right )^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^4/(a + b*x^4)^(3/4),x]

[Out]

(x*(a + b*x^4 - a*(1 + (b*x^4)/a)^(3/4)*Hypergeometric2F1[1/4, 3/4, 5/4, -((b*x^4)/a)]))/(2*b*(a + b*x^4)^(3/4
))

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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {x^{4}}{\left (b \,x^{4}+a \right )^{\frac {3}{4}}}\, dx\]

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)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^4/(b*x^4 + a)^(3/4), x)

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Fricas [F]
time = 0.08, size = 15, normalized size = 0.18 \begin {gather*} {\rm integral}\left (\frac {x^{4}}{{\left (b x^{4} + a\right )}^{\frac {3}{4}}}, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x^4/(b*x^4 + a)^(3/4), x)

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Sympy [C] Result contains complex when optimal does not.
time = 0.40, size = 37, normalized size = 0.45 \begin {gather*} \frac {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 a^{\frac {3}{4}} \Gamma \left (\frac {9}{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^4/(b*x^4 + a)^(3/4), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^4}{{\left (b\,x^4+a\right )}^{3/4}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(a + b*x^4)^(3/4),x)

[Out]

int(x^4/(a + b*x^4)^(3/4), x)

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