3.1044 \(\int \frac{(a+b x^4)^{3/4}}{x^6} \, dx\)

Optimal. Leaf size=99 \[ \frac{3 b^{3/2} x \sqrt [4]{\frac{a}{b x^4}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{5 \sqrt{a} \sqrt [4]{a+b x^4}}-\frac{3 b}{5 x \sqrt [4]{a+b x^4}}-\frac{\left (a+b x^4\right )^{3/4}}{5 x^5} \]

[Out]

(-3*b)/(5*x*(a + b*x^4)^(1/4)) - (a + b*x^4)^(3/4)/(5*x^5) + (3*b^(3/2)*(1 + a/(b*x^4))^(1/4)*x*EllipticE[ArcC
ot[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(5*Sqrt[a]*(a + b*x^4)^(1/4))

________________________________________________________________________________________

Rubi [A]  time = 0.0470486, antiderivative size = 99, 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 = {277, 312, 281, 335, 275, 196} \[ \frac{3 b^{3/2} x \sqrt [4]{\frac{a}{b x^4}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{5 \sqrt{a} \sqrt [4]{a+b x^4}}-\frac{3 b}{5 x \sqrt [4]{a+b x^4}}-\frac{\left (a+b x^4\right )^{3/4}}{5 x^5} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^4)^(3/4)/x^6,x]

[Out]

(-3*b)/(5*x*(a + b*x^4)^(1/4)) - (a + b*x^4)^(3/4)/(5*x^5) + (3*b^(3/2)*(1 + a/(b*x^4))^(1/4)*x*EllipticE[ArcC
ot[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(5*Sqrt[a]*(a + b*x^4)^(1/4))

Rule 277

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

Rule 312

Int[1/((x_)^2*((a_) + (b_.)*(x_)^4)^(1/4)), x_Symbol] :> -Simp[(x*(a + b*x^4)^(1/4))^(-1), x] - Dist[b, 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{\left (a+b x^4\right )^{3/4}}{x^6} \, dx &=-\frac{\left (a+b x^4\right )^{3/4}}{5 x^5}+\frac{1}{5} (3 b) \int \frac{1}{x^2 \sqrt [4]{a+b x^4}} \, dx\\ &=-\frac{3 b}{5 x \sqrt [4]{a+b x^4}}-\frac{\left (a+b x^4\right )^{3/4}}{5 x^5}-\frac{1}{5} \left (3 b^2\right ) \int \frac{x^2}{\left (a+b x^4\right )^{5/4}} \, dx\\ &=-\frac{3 b}{5 x \sqrt [4]{a+b x^4}}-\frac{\left (a+b x^4\right )^{3/4}}{5 x^5}-\frac{\left (3 b \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}{5 \sqrt [4]{a+b x^4}}\\ &=-\frac{3 b}{5 x \sqrt [4]{a+b x^4}}-\frac{\left (a+b x^4\right )^{3/4}}{5 x^5}+\frac{\left (3 b \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 )}{5 \sqrt [4]{a+b x^4}}\\ &=-\frac{3 b}{5 x \sqrt [4]{a+b x^4}}-\frac{\left (a+b x^4\right )^{3/4}}{5 x^5}+\frac{\left (3 b \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 )}{10 \sqrt [4]{a+b x^4}}\\ &=-\frac{3 b}{5 x \sqrt [4]{a+b x^4}}-\frac{\left (a+b x^4\right )^{3/4}}{5 x^5}+\frac{3 b^{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 )}{5 \sqrt{a} \sqrt [4]{a+b x^4}}\\ \end{align*}

Mathematica [C]  time = 0.0100199, size = 51, normalized size = 0.52 \[ -\frac{\left (a+b x^4\right )^{3/4} \, _2F_1\left (-\frac{5}{4},-\frac{3}{4};-\frac{1}{4};-\frac{b x^4}{a}\right )}{5 x^5 \left (\frac{b x^4}{a}+1\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^4)^(3/4)/x^6,x]

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.029, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{6}} \left ( b{x}^{4}+a \right ) ^{{\frac{3}{4}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^4+a)^(3/4)/x^6,x)

[Out]

int((b*x^4+a)^(3/4)/x^6,x)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{x^{6}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^(3/4)/x^6,x, algorithm="maxima")

[Out]

integrate((b*x^4 + a)^(3/4)/x^6, x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{x^{6}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^(3/4)/x^6,x, algorithm="fricas")

[Out]

integral((b*x^4 + a)^(3/4)/x^6, x)

________________________________________________________________________________________

Sympy [C]  time = 1.32333, size = 31, normalized size = 0.31 \begin{align*} - \frac{b^{\frac{3}{4}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{4}}} \right )}}{2 x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**4+a)**(3/4)/x**6,x)

[Out]

-b**(3/4)*hyper((-3/4, 1/2), (3/2,), a*exp_polar(I*pi)/(b*x**4))/(2*x**2)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{x^{6}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^(3/4)/x^6,x, algorithm="giac")

[Out]

integrate((b*x^4 + a)^(3/4)/x^6, x)