3.1104 \(\int \frac{x^6}{\sqrt [4]{a+b x^4}} \, dx\)

Optimal. Leaf size=105 \[ -\frac{a^{3/2} x \sqrt [4]{\frac{a}{b x^4}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{4 b^{3/2} \sqrt [4]{a+b x^4}}+\frac{x^3 \left (a+b x^4\right )^{3/4}}{6 b}-\frac{a x^3}{4 b \sqrt [4]{a+b x^4}} \]

[Out]

-(a*x^3)/(4*b*(a + b*x^4)^(1/4)) + (x^3*(a + b*x^4)^(3/4))/(6*b) - (a^(3/2)*(1 + a/(b*x^4))^(1/4)*x*EllipticE[
ArcCot[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(4*b^(3/2)*(a + b*x^4)^(1/4))

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Rubi [A]  time = 0.0457805, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {321, 310, 281, 335, 275, 196} \[ -\frac{a^{3/2} x \sqrt [4]{\frac{a}{b x^4}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{4 b^{3/2} \sqrt [4]{a+b x^4}}+\frac{x^3 \left (a+b x^4\right )^{3/4}}{6 b}-\frac{a x^3}{4 b \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]

Int[x^6/(a + b*x^4)^(1/4),x]

[Out]

-(a*x^3)/(4*b*(a + b*x^4)^(1/4)) + (x^3*(a + b*x^4)^(3/4))/(6*b) - (a^(3/2)*(1 + a/(b*x^4))^(1/4)*x*EllipticE[
ArcCot[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(4*b^(3/2)*(a + b*x^4)^(1/4))

Rule 321

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 310

Int[(x_)^2/((a_) + (b_.)*(x_)^4)^(1/4), x_Symbol] :> Simp[x^3/(2*(a + b*x^4)^(1/4)), x] - Dist[a/2, Int[x^2/(a
+ b*x^4)^(5/4), x], x] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 281

Int[(x_)^2/((a_) + (b_.)*(x_)^4)^(5/4), x_Symbol] :> Dist[(x*(1 + a/(b*x^4))^(1/4))/(b*(a + b*x^4)^(1/4)), Int
[1/(x^3*(1 + a/(b*x^4))^(5/4)), x], x] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 335

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]

Rule 275

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 196

Int[((a_) + (b_.)*(x_)^2)^(-5/4), x_Symbol] :> Simp[(2*EllipticE[(1*ArcTan[Rt[b/a, 2]*x])/2, 2])/(a^(5/4)*Rt[b
/a, 2]), x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b/a]

Rubi steps

\begin{align*} \int \frac{x^6}{\sqrt [4]{a+b x^4}} \, dx &=\frac{x^3 \left (a+b x^4\right )^{3/4}}{6 b}-\frac{a \int \frac{x^2}{\sqrt [4]{a+b x^4}} \, dx}{2 b}\\ &=-\frac{a x^3}{4 b \sqrt [4]{a+b x^4}}+\frac{x^3 \left (a+b x^4\right )^{3/4}}{6 b}+\frac{a^2 \int \frac{x^2}{\left (a+b x^4\right )^{5/4}} \, dx}{4 b}\\ &=-\frac{a x^3}{4 b \sqrt [4]{a+b x^4}}+\frac{x^3 \left (a+b x^4\right )^{3/4}}{6 b}+\frac{\left (a^2 \sqrt [4]{1+\frac{a}{b x^4}} x\right ) \int \frac{1}{\left (1+\frac{a}{b x^4}\right )^{5/4} x^3} \, dx}{4 b^2 \sqrt [4]{a+b x^4}}\\ &=-\frac{a x^3}{4 b \sqrt [4]{a+b x^4}}+\frac{x^3 \left (a+b x^4\right )^{3/4}}{6 b}-\frac{\left (a^2 \sqrt [4]{1+\frac{a}{b x^4}} x\right ) \operatorname{Subst}\left (\int \frac{x}{\left (1+\frac{a x^4}{b}\right )^{5/4}} \, dx,x,\frac{1}{x}\right )}{4 b^2 \sqrt [4]{a+b x^4}}\\ &=-\frac{a x^3}{4 b \sqrt [4]{a+b x^4}}+\frac{x^3 \left (a+b x^4\right )^{3/4}}{6 b}-\frac{\left (a^2 \sqrt [4]{1+\frac{a}{b x^4}} x\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1+\frac{a x^2}{b}\right )^{5/4}} \, dx,x,\frac{1}{x^2}\right )}{8 b^2 \sqrt [4]{a+b x^4}}\\ &=-\frac{a x^3}{4 b \sqrt [4]{a+b x^4}}+\frac{x^3 \left (a+b x^4\right )^{3/4}}{6 b}-\frac{a^{3/2} \sqrt [4]{1+\frac{a}{b x^4}} x E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{4 b^{3/2} \sqrt [4]{a+b x^4}}\\ \end{align*}

Mathematica [C]  time = 0.020441, size = 64, normalized size = 0.61 \[ \frac{x^3 \left (-a \sqrt [4]{\frac{b x^4}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};-\frac{b x^4}{a}\right )+a+b x^4\right )}{6 b \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^6/(a + b*x^4)^(1/4),x]

[Out]

(x^3*(a + b*x^4 - a*(1 + (b*x^4)/a)^(1/4)*Hypergeometric2F1[1/4, 3/4, 7/4, -((b*x^4)/a)]))/(6*b*(a + b*x^4)^(1
/4))

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Maple [F]  time = 0.028, size = 0, normalized size = 0. \begin{align*} \int{{x}^{6}{\frac{1}{\sqrt [4]{b{x}^{4}+a}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/(b*x^4+a)^(1/4),x)

[Out]

int(x^6/(b*x^4+a)^(1/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{6}}{{\left (b x^{4} + a\right )}^{\frac{1}{4}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(b*x^4+a)^(1/4),x, algorithm="maxima")

[Out]

integrate(x^6/(b*x^4 + a)^(1/4), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{6}}{{\left (b x^{4} + a\right )}^{\frac{1}{4}}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(b*x^4+a)^(1/4),x, algorithm="fricas")

[Out]

integral(x^6/(b*x^4 + a)^(1/4), x)

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Sympy [C]  time = 1.12518, size = 37, normalized size = 0.35 \begin{align*} \frac{x^{7} \Gamma \left (\frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt [4]{a} \Gamma \left (\frac{11}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6/(b*x**4+a)**(1/4),x)

[Out]

x**7*gamma(7/4)*hyper((1/4, 7/4), (11/4,), b*x**4*exp_polar(I*pi)/a)/(4*a**(1/4)*gamma(11/4))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{6}}{{\left (b x^{4} + a\right )}^{\frac{1}{4}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(b*x^4+a)^(1/4),x, algorithm="giac")

[Out]

integrate(x^6/(b*x^4 + a)^(1/4), x)