\(\int \frac {(a+b x^4)^{3/4}}{x^{11}} \, dx\) [1030]

Optimal result

Integrand size = 15, antiderivative size = 149 \[ \int \frac {\left (a+b x^4\right )^{3/4}}{x^{11}} \, dx=-\frac {3 b^3 x^2}{40 a^2 \sqrt [4]{a+b x^4}}-\frac {\left (a+b x^4\right )^{3/4}}{10 x^{10}}-\frac {b \left (a+b x^4\right )^{3/4}}{20 a x^6}+\frac {3 b^2 \left (a+b x^4\right )^{3/4}}{40 a^2 x^2}+\frac {3 b^{5/2} \sqrt [4]{1+\frac {b x^4}{a}} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{40 a^{3/2} \sqrt [4]{a+b x^4}} \]

[Out]

-3/40*b^3*x^2/a^2/(b*x^4+a)^(1/4)-1/10*(b*x^4+a)^(3/4)/x^10-1/20*b*(b*x^4+a)^(3/4)/a/x^6+3/40*b^2*(b*x^4+a)^(3
/4)/a^2/x^2+3/40*b^(5/2)*(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] (verified)

Time = 0.08 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {281, 283, 331, 235, 233, 202} \[ \int \frac {\left (a+b x^4\right )^{3/4}}{x^{11}} \, dx=\frac {3 b^{5/2} \sqrt [4]{\frac {b x^4}{a}+1} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{40 a^{3/2} \sqrt [4]{a+b x^4}}-\frac {3 b^3 x^2}{40 a^2 \sqrt [4]{a+b x^4}}+\frac {3 b^2 \left (a+b x^4\right )^{3/4}}{40 a^2 x^2}-\frac {\left (a+b x^4\right )^{3/4}}{10 x^{10}}-\frac {b \left (a+b x^4\right )^{3/4}}{20 a x^6} \]

[In]

Int[(a + b*x^4)^(3/4)/x^11,x]

[Out]

(-3*b^3*x^2)/(40*a^2*(a + b*x^4)^(1/4)) - (a + b*x^4)^(3/4)/(10*x^10) - (b*(a + b*x^4)^(3/4))/(20*a*x^6) + (3*
b^2*(a + b*x^4)^(3/4))/(40*a^2*x^2) + (3*b^(5/2)*(1 + (b*x^4)/a)^(1/4)*EllipticE[ArcTan[(Sqrt[b]*x^2)/Sqrt[a]]
/2, 2])/(40*a^(3/2)*(a + b*x^4)^(1/4))

Rule 202

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

Rule 233

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 235

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 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 283

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 331

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*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {\left (a+b x^2\right )^{3/4}}{x^6} \, dx,x,x^2\right ) \\ & = -\frac {\left (a+b x^4\right )^{3/4}}{10 x^{10}}+\frac {1}{20} (3 b) \text {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}}{10 x^{10}}-\frac {b \left (a+b x^4\right )^{3/4}}{20 a x^6}-\frac {\left (3 b^2\right ) \text {Subst}\left (\int \frac {1}{x^2 \sqrt [4]{a+b x^2}} \, dx,x,x^2\right )}{40 a} \\ & = -\frac {\left (a+b x^4\right )^{3/4}}{10 x^{10}}-\frac {b \left (a+b x^4\right )^{3/4}}{20 a x^6}+\frac {3 b^2 \left (a+b x^4\right )^{3/4}}{40 a^2 x^2}-\frac {\left (3 b^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{a+b x^2}} \, dx,x,x^2\right )}{80 a^2} \\ & = -\frac {\left (a+b x^4\right )^{3/4}}{10 x^{10}}-\frac {b \left (a+b x^4\right )^{3/4}}{20 a x^6}+\frac {3 b^2 \left (a+b x^4\right )^{3/4}}{40 a^2 x^2}-\frac {\left (3 b^3 \sqrt [4]{1+\frac {b x^4}{a}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{1+\frac {b x^2}{a}}} \, dx,x,x^2\right )}{80 a^2 \sqrt [4]{a+b x^4}} \\ & = -\frac {3 b^3 x^2}{40 a^2 \sqrt [4]{a+b x^4}}-\frac {\left (a+b x^4\right )^{3/4}}{10 x^{10}}-\frac {b \left (a+b x^4\right )^{3/4}}{20 a x^6}+\frac {3 b^2 \left (a+b x^4\right )^{3/4}}{40 a^2 x^2}+\frac {\left (3 b^3 \sqrt [4]{1+\frac {b x^4}{a}}\right ) \text {Subst}\left (\int \frac {1}{\left (1+\frac {b x^2}{a}\right )^{5/4}} \, dx,x,x^2\right )}{80 a^2 \sqrt [4]{a+b x^4}} \\ & = -\frac {3 b^3 x^2}{40 a^2 \sqrt [4]{a+b x^4}}-\frac {\left (a+b x^4\right )^{3/4}}{10 x^{10}}-\frac {b \left (a+b x^4\right )^{3/4}}{20 a x^6}+\frac {3 b^2 \left (a+b x^4\right )^{3/4}}{40 a^2 x^2}+\frac {3 b^{5/2} \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 )}{40 a^{3/2} \sqrt [4]{a+b x^4}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 10.02 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.34 \[ \int \frac {\left (a+b x^4\right )^{3/4}}{x^{11}} \, dx=-\frac {\left (a+b x^4\right )^{3/4} \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},-\frac {3}{4},-\frac {3}{2},-\frac {b x^4}{a}\right )}{10 x^{10} \left (1+\frac {b x^4}{a}\right )^{3/4}} \]

[In]

Integrate[(a + b*x^4)^(3/4)/x^11,x]

[Out]

-1/10*((a + b*x^4)^(3/4)*Hypergeometric2F1[-5/2, -3/4, -3/2, -((b*x^4)/a)])/(x^10*(1 + (b*x^4)/a)^(3/4))

Maple [F]

\[\int \frac {\left (b \,x^{4}+a \right )^{\frac {3}{4}}}{x^{11}}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int \frac {\left (a+b x^4\right )^{3/4}}{x^{11}} \, dx=\int { \frac {{\left (b x^{4} + a\right )}^{\frac {3}{4}}}{x^{11}} \,d x } \]

[In]

integrate((b*x^4+a)^(3/4)/x^11,x, algorithm="fricas")

[Out]

integral((b*x^4 + a)^(3/4)/x^11, x)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.83 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.23 \[ \int \frac {\left (a+b x^4\right )^{3/4}}{x^{11}} \, dx=- \frac {a^{\frac {3}{4}} {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{2}, - \frac {3}{4} \\ - \frac {3}{2} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{10 x^{10}} \]

[In]

integrate((b*x**4+a)**(3/4)/x**11,x)

[Out]

-a**(3/4)*hyper((-5/2, -3/4), (-3/2,), b*x**4*exp_polar(I*pi)/a)/(10*x**10)

Maxima [F]

\[ \int \frac {\left (a+b x^4\right )^{3/4}}{x^{11}} \, dx=\int { \frac {{\left (b x^{4} + a\right )}^{\frac {3}{4}}}{x^{11}} \,d x } \]

[In]

integrate((b*x^4+a)^(3/4)/x^11,x, algorithm="maxima")

[Out]

integrate((b*x^4 + a)^(3/4)/x^11, x)

Giac [F]

\[ \int \frac {\left (a+b x^4\right )^{3/4}}{x^{11}} \, dx=\int { \frac {{\left (b x^{4} + a\right )}^{\frac {3}{4}}}{x^{11}} \,d x } \]

[In]

integrate((b*x^4+a)^(3/4)/x^11,x, algorithm="giac")

[Out]

integrate((b*x^4 + a)^(3/4)/x^11, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^4\right )^{3/4}}{x^{11}} \, dx=\int \frac {{\left (b\,x^4+a\right )}^{3/4}}{x^{11}} \,d x \]

[In]

int((a + b*x^4)^(3/4)/x^11,x)

[Out]

int((a + b*x^4)^(3/4)/x^11, x)