3.1093 \(\int \frac {1}{x^7 \sqrt [4]{a+b x^4}} \, dx\)

Optimal. Leaf size=128 \[ \frac {b^{3/2} \sqrt [4]{\frac {b x^4}{a}+1} E\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{4 a^{3/2} \sqrt [4]{a+b x^4}}-\frac {b^2 x^2}{4 a^2 \sqrt [4]{a+b x^4}}+\frac {b \left (a+b x^4\right )^{3/4}}{4 a^2 x^2}-\frac {\left (a+b x^4\right )^{3/4}}{6 a x^6} \]

[Out]

-1/4*b^2*x^2/a^2/(b*x^4+a)^(1/4)-1/6*(b*x^4+a)^(3/4)/a/x^6+1/4*b*(b*x^4+a)^(3/4)/a^2/x^2+1/4*b^(3/2)*(1+b*x^4/
a)^(1/4)*(cos(1/2*arctan(x^2*b^(1/2)/a^(1/2)))^2)^(1/2)/cos(1/2*arctan(x^2*b^(1/2)/a^(1/2)))*EllipticE(sin(1/2
*arctan(x^2*b^(1/2)/a^(1/2))),2^(1/2))/a^(3/2)/(b*x^4+a)^(1/4)

________________________________________________________________________________________

Rubi [A]  time = 0.07, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {275, 325, 229, 227, 196} \[ -\frac {b^2 x^2}{4 a^2 \sqrt [4]{a+b x^4}}+\frac {b^{3/2} \sqrt [4]{\frac {b x^4}{a}+1} E\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{4 a^{3/2} \sqrt [4]{a+b x^4}}+\frac {b \left (a+b x^4\right )^{3/4}}{4 a^2 x^2}-\frac {\left (a+b x^4\right )^{3/4}}{6 a x^6} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^7*(a + b*x^4)^(1/4)),x]

[Out]

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

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]

Rule 227

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

Rule 229

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

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 325

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

Rubi steps

\begin {align*} \int \frac {1}{x^7 \sqrt [4]{a+b x^4}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x^4 \sqrt [4]{a+b x^2}} \, dx,x,x^2\right )\\ &=-\frac {\left (a+b x^4\right )^{3/4}}{6 a x^6}-\frac {b \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt [4]{a+b x^2}} \, dx,x,x^2\right )}{4 a}\\ &=-\frac {\left (a+b x^4\right )^{3/4}}{6 a x^6}+\frac {b \left (a+b x^4\right )^{3/4}}{4 a^2 x^2}-\frac {b^2 \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{a+b x^2}} \, dx,x,x^2\right )}{8 a^2}\\ &=-\frac {\left (a+b x^4\right )^{3/4}}{6 a x^6}+\frac {b \left (a+b x^4\right )^{3/4}}{4 a^2 x^2}-\frac {\left (b^2 \sqrt [4]{1+\frac {b x^4}{a}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{1+\frac {b x^2}{a}}} \, dx,x,x^2\right )}{8 a^2 \sqrt [4]{a+b x^4}}\\ &=-\frac {b^2 x^2}{4 a^2 \sqrt [4]{a+b x^4}}-\frac {\left (a+b x^4\right )^{3/4}}{6 a x^6}+\frac {b \left (a+b x^4\right )^{3/4}}{4 a^2 x^2}+\frac {\left (b^2 \sqrt [4]{1+\frac {b x^4}{a}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+\frac {b x^2}{a}\right )^{5/4}} \, dx,x,x^2\right )}{8 a^2 \sqrt [4]{a+b x^4}}\\ &=-\frac {b^2 x^2}{4 a^2 \sqrt [4]{a+b x^4}}-\frac {\left (a+b x^4\right )^{3/4}}{6 a x^6}+\frac {b \left (a+b x^4\right )^{3/4}}{4 a^2 x^2}+\frac {b^{3/2} \sqrt [4]{1+\frac {b x^4}{a}} E\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{4 a^{3/2} \sqrt [4]{a+b x^4}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.01, size = 51, normalized size = 0.40 \[ -\frac {\sqrt [4]{\frac {b x^4}{a}+1} \, _2F_1\left (-\frac {3}{2},\frac {1}{4};-\frac {1}{2};-\frac {b x^4}{a}\right )}{6 x^6 \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^7*(a + b*x^4)^(1/4)),x]

[Out]

-1/6*((1 + (b*x^4)/a)^(1/4)*Hypergeometric2F1[-3/2, 1/4, -1/2, -((b*x^4)/a)])/(x^6*(a + b*x^4)^(1/4))

________________________________________________________________________________________

fricas [F]  time = 0.82, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x^{4} + a\right )}^{\frac {3}{4}}}{b x^{11} + a x^{7}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(b*x^4+a)^(1/4),x, algorithm="fricas")

[Out]

integral((b*x^4 + a)^(3/4)/(b*x^11 + a*x^7), x)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{4} + a\right )}^{\frac {1}{4}} x^{7}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(b*x^4+a)^(1/4),x, algorithm="giac")

[Out]

integrate(1/((b*x^4 + a)^(1/4)*x^7), x)

________________________________________________________________________________________

maple [F]  time = 0.16, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \,x^{4}+a \right )^{\frac {1}{4}} x^{7}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^7/(b*x^4+a)^(1/4),x)

[Out]

int(1/x^7/(b*x^4+a)^(1/4),x)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{4} + a\right )}^{\frac {1}{4}} x^{7}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(b*x^4+a)^(1/4),x, algorithm="maxima")

[Out]

integrate(1/((b*x^4 + a)^(1/4)*x^7), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{x^7\,{\left (b\,x^4+a\right )}^{1/4}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^7*(a + b*x^4)^(1/4)),x)

[Out]

int(1/(x^7*(a + b*x^4)^(1/4)), x)

________________________________________________________________________________________

sympy [C]  time = 2.48, size = 32, normalized size = 0.25 \[ - \frac {{{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{2}, \frac {1}{4} \\ - \frac {1}{2} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{6 \sqrt [4]{a} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**7/(b*x**4+a)**(1/4),x)

[Out]

-hyper((-3/2, 1/4), (-1/2,), b*x**4*exp_polar(I*pi)/a)/(6*a**(1/4)*x**6)

________________________________________________________________________________________