Integrand size = 38, antiderivative size = 116 \[ \int \frac {b+a x^4}{\left (-b+x^2+a x^4\right ) \sqrt [4]{-b x^2+a x^6}} \, dx=-\frac {\arctan \left (\frac {\sqrt {2} x \sqrt [4]{-b x^2+a x^6}}{-x^2+\sqrt {-b x^2+a x^6}}\right )}{\sqrt {2}}-\frac {\text {arctanh}\left (\frac {\frac {x^2}{\sqrt {2}}+\frac {\sqrt {-b x^2+a x^6}}{\sqrt {2}}}{x \sqrt [4]{-b x^2+a x^6}}\right )}{\sqrt {2}} \] Output:
-1/2*arctan(2^(1/2)*x*(a*x^6-b*x^2)^(1/4)/(-x^2+(a*x^6-b*x^2)^(1/2)))*2^(1 /2)-1/2*arctanh((1/2*2^(1/2)*x^2+1/2*(a*x^6-b*x^2)^(1/2)*2^(1/2))/x/(a*x^6 -b*x^2)^(1/4))*2^(1/2)
\[ \int \frac {b+a x^4}{\left (-b+x^2+a x^4\right ) \sqrt [4]{-b x^2+a x^6}} \, dx=\int \frac {b+a x^4}{\left (-b+x^2+a x^4\right ) \sqrt [4]{-b x^2+a x^6}} \, dx \] Input:
Integrate[(b + a*x^4)/((-b + x^2 + a*x^4)*(-(b*x^2) + a*x^6)^(1/4)),x]
Output:
Integrate[(b + a*x^4)/((-b + x^2 + a*x^4)*(-(b*x^2) + a*x^6)^(1/4)), x]
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 1.57 (sec) , antiderivative size = 482, normalized size of antiderivative = 4.16, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.132, Rules used = {2467, 25, 2035, 7279, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {a x^4+b}{\left (a x^4-b+x^2\right ) \sqrt [4]{a x^6-b x^2}} \, dx\) |
\(\Big \downarrow \) 2467 |
\(\displaystyle \frac {\sqrt {x} \sqrt [4]{a x^4-b} \int -\frac {a x^4+b}{\sqrt {x} \left (-a x^4-x^2+b\right ) \sqrt [4]{a x^4-b}}dx}{\sqrt [4]{a x^6-b x^2}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\sqrt {x} \sqrt [4]{a x^4-b} \int \frac {a x^4+b}{\sqrt {x} \left (-a x^4-x^2+b\right ) \sqrt [4]{a x^4-b}}dx}{\sqrt [4]{a x^6-b x^2}}\) |
\(\Big \downarrow \) 2035 |
\(\displaystyle -\frac {2 \sqrt {x} \sqrt [4]{a x^4-b} \int \frac {a x^4+b}{\left (-a x^4-x^2+b\right ) \sqrt [4]{a x^4-b}}d\sqrt {x}}{\sqrt [4]{a x^6-b x^2}}\) |
\(\Big \downarrow \) 7279 |
\(\displaystyle -\frac {2 \sqrt {x} \sqrt [4]{a x^4-b} \int \left (\frac {2 b-x^2}{\left (-a x^4-x^2+b\right ) \sqrt [4]{a x^4-b}}-\frac {1}{\sqrt [4]{a x^4-b}}\right )d\sqrt {x}}{\sqrt [4]{a x^6-b x^2}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2 \sqrt {x} \sqrt [4]{a x^4-b} \left (\frac {\sqrt {x} \sqrt [4]{1-\frac {a x^4}{b}} \operatorname {AppellF1}\left (\frac {1}{8},1,\frac {1}{4},\frac {9}{8},\frac {2 a^2 x^4}{2 a b-\sqrt {4 a b+1}+1},\frac {a x^4}{b}\right )}{\sqrt [4]{a x^4-b}}+\frac {\sqrt {x} \sqrt [4]{1-\frac {a x^4}{b}} \operatorname {AppellF1}\left (\frac {1}{8},1,\frac {1}{4},\frac {9}{8},\frac {2 a^2 x^4}{2 a b+\sqrt {4 a b+1}+1},\frac {a x^4}{b}\right )}{\sqrt [4]{a x^4-b}}-\frac {a x^{5/2} \left (1-\sqrt {4 a b+1}\right ) \sqrt [4]{1-\frac {a x^4}{b}} \operatorname {AppellF1}\left (\frac {5}{8},1,\frac {1}{4},\frac {13}{8},\frac {2 a^2 x^4}{2 a b-\sqrt {4 a b+1}+1},\frac {a x^4}{b}\right )}{5 \left (2 a b-\sqrt {4 a b+1}+1\right ) \sqrt [4]{a x^4-b}}-\frac {a x^{5/2} \left (\sqrt {4 a b+1}+1\right ) \sqrt [4]{1-\frac {a x^4}{b}} \operatorname {AppellF1}\left (\frac {5}{8},1,\frac {1}{4},\frac {13}{8},\frac {2 a^2 x^4}{2 a b+\sqrt {4 a b+1}+1},\frac {a x^4}{b}\right )}{5 \left (2 a b+\sqrt {4 a b+1}+1\right ) \sqrt [4]{a x^4-b}}-\frac {\sqrt {x} \sqrt [4]{1-\frac {a x^4}{b}} \operatorname {Hypergeometric2F1}\left (\frac {1}{8},\frac {1}{4},\frac {9}{8},\frac {a x^4}{b}\right )}{\sqrt [4]{a x^4-b}}\right )}{\sqrt [4]{a x^6-b x^2}}\) |
Input:
Int[(b + a*x^4)/((-b + x^2 + a*x^4)*(-(b*x^2) + a*x^6)^(1/4)),x]
Output:
(-2*Sqrt[x]*(-b + a*x^4)^(1/4)*((Sqrt[x]*(1 - (a*x^4)/b)^(1/4)*AppellF1[1/ 8, 1, 1/4, 9/8, (2*a^2*x^4)/(1 + 2*a*b - Sqrt[1 + 4*a*b]), (a*x^4)/b])/(-b + a*x^4)^(1/4) + (Sqrt[x]*(1 - (a*x^4)/b)^(1/4)*AppellF1[1/8, 1, 1/4, 9/8 , (2*a^2*x^4)/(1 + 2*a*b + Sqrt[1 + 4*a*b]), (a*x^4)/b])/(-b + a*x^4)^(1/4 ) - (a*(1 - Sqrt[1 + 4*a*b])*x^(5/2)*(1 - (a*x^4)/b)^(1/4)*AppellF1[5/8, 1 , 1/4, 13/8, (2*a^2*x^4)/(1 + 2*a*b - Sqrt[1 + 4*a*b]), (a*x^4)/b])/(5*(1 + 2*a*b - Sqrt[1 + 4*a*b])*(-b + a*x^4)^(1/4)) - (a*(1 + Sqrt[1 + 4*a*b])* x^(5/2)*(1 - (a*x^4)/b)^(1/4)*AppellF1[5/8, 1, 1/4, 13/8, (2*a^2*x^4)/(1 + 2*a*b + Sqrt[1 + 4*a*b]), (a*x^4)/b])/(5*(1 + 2*a*b + Sqrt[1 + 4*a*b])*(- b + a*x^4)^(1/4)) - (Sqrt[x]*(1 - (a*x^4)/b)^(1/4)*Hypergeometric2F1[1/8, 1/4, 9/8, (a*x^4)/b])/(-b + a*x^4)^(1/4)))/(-(b*x^2) + a*x^6)^(1/4)
Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k Subst [Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti onQ[m] && AlgebraicFunctionQ[Fx, x]
Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p]) Int[x^( p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P olyQ[Px, x] && !IntegerQ[p] && !MonomialQ[Px, x] && !PolyQ[Fx, x]
Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ {v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Time = 1.49 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.28
method | result | size |
pseudoelliptic | \(\frac {\sqrt {2}\, \left (\ln \left (\frac {-\left (x^{2} \left (a \,x^{4}-b \right )\right )^{\frac {1}{4}} \sqrt {2}\, x +x^{2}+\sqrt {x^{2} \left (a \,x^{4}-b \right )}}{\left (x^{2} \left (a \,x^{4}-b \right )\right )^{\frac {1}{4}} \sqrt {2}\, x +x^{2}+\sqrt {x^{2} \left (a \,x^{4}-b \right )}}\right )+2 \arctan \left (\frac {\left (x^{2} \left (a \,x^{4}-b \right )\right )^{\frac {1}{4}} \sqrt {2}+x}{x}\right )+2 \arctan \left (\frac {\left (x^{2} \left (a \,x^{4}-b \right )\right )^{\frac {1}{4}} \sqrt {2}-x}{x}\right )\right )}{4}\) | \(148\) |
Input:
int((a*x^4+b)/(a*x^4+x^2-b)/(a*x^6-b*x^2)^(1/4),x,method=_RETURNVERBOSE)
Output:
1/4*2^(1/2)*(ln((-(x^2*(a*x^4-b))^(1/4)*2^(1/2)*x+x^2+(x^2*(a*x^4-b))^(1/2 ))/((x^2*(a*x^4-b))^(1/4)*2^(1/2)*x+x^2+(x^2*(a*x^4-b))^(1/2)))+2*arctan(( (x^2*(a*x^4-b))^(1/4)*2^(1/2)+x)/x)+2*arctan(((x^2*(a*x^4-b))^(1/4)*2^(1/2 )-x)/x))
Timed out. \[ \int \frac {b+a x^4}{\left (-b+x^2+a x^4\right ) \sqrt [4]{-b x^2+a x^6}} \, dx=\text {Timed out} \] Input:
integrate((a*x^4+b)/(a*x^4+x^2-b)/(a*x^6-b*x^2)^(1/4),x, algorithm="fricas ")
Output:
Timed out
\[ \int \frac {b+a x^4}{\left (-b+x^2+a x^4\right ) \sqrt [4]{-b x^2+a x^6}} \, dx=\int \frac {a x^{4} + b}{\sqrt [4]{x^{2} \left (a x^{4} - b\right )} \left (a x^{4} - b + x^{2}\right )}\, dx \] Input:
integrate((a*x**4+b)/(a*x**4+x**2-b)/(a*x**6-b*x**2)**(1/4),x)
Output:
Integral((a*x**4 + b)/((x**2*(a*x**4 - b))**(1/4)*(a*x**4 - b + x**2)), x)
\[ \int \frac {b+a x^4}{\left (-b+x^2+a x^4\right ) \sqrt [4]{-b x^2+a x^6}} \, dx=\int { \frac {a x^{4} + b}{{\left (a x^{6} - b x^{2}\right )}^{\frac {1}{4}} {\left (a x^{4} + x^{2} - b\right )}} \,d x } \] Input:
integrate((a*x^4+b)/(a*x^4+x^2-b)/(a*x^6-b*x^2)^(1/4),x, algorithm="maxima ")
Output:
integrate((a*x^4 + b)/((a*x^6 - b*x^2)^(1/4)*(a*x^4 + x^2 - b)), x)
\[ \int \frac {b+a x^4}{\left (-b+x^2+a x^4\right ) \sqrt [4]{-b x^2+a x^6}} \, dx=\int { \frac {a x^{4} + b}{{\left (a x^{6} - b x^{2}\right )}^{\frac {1}{4}} {\left (a x^{4} + x^{2} - b\right )}} \,d x } \] Input:
integrate((a*x^4+b)/(a*x^4+x^2-b)/(a*x^6-b*x^2)^(1/4),x, algorithm="giac")
Output:
integrate((a*x^4 + b)/((a*x^6 - b*x^2)^(1/4)*(a*x^4 + x^2 - b)), x)
Timed out. \[ \int \frac {b+a x^4}{\left (-b+x^2+a x^4\right ) \sqrt [4]{-b x^2+a x^6}} \, dx=\int \frac {a\,x^4+b}{{\left (a\,x^6-b\,x^2\right )}^{1/4}\,\left (a\,x^4+x^2-b\right )} \,d x \] Input:
int((b + a*x^4)/((a*x^6 - b*x^2)^(1/4)*(a*x^4 - b + x^2)),x)
Output:
int((b + a*x^4)/((a*x^6 - b*x^2)^(1/4)*(a*x^4 - b + x^2)), x)
\[ \int \frac {b+a x^4}{\left (-b+x^2+a x^4\right ) \sqrt [4]{-b x^2+a x^6}} \, dx =\text {Too large to display} \] Input:
int((a*x^4+b)/(a*x^4+x^2-b)/(a*x^6-b*x^2)^(1/4),x)
Output:
(6*sqrt(x)*(a*x**4 - b)**(5/4) + 2*sqrt(x)*(a*x**4 - b)**(1/4)*a*x**4 - 2* sqrt(x)*(a*x**4 - b)**(1/4)*b - 7*sqrt(a*x**4 - b)*int((a*x**4 - b)**(3/4) /(sqrt(x)*a**3*x**12 - 3*sqrt(x)*a**2*b*x**8 + sqrt(x)*a**2*x**10 + 3*sqrt (x)*a*b**2*x**4 - 2*sqrt(x)*a*b*x**6 - sqrt(x)*b**3 + sqrt(x)*b**2*x**2),x )*a*b**2*x**4 + 7*sqrt(a*x**4 - b)*int((a*x**4 - b)**(3/4)/(sqrt(x)*a**3*x **12 - 3*sqrt(x)*a**2*b*x**8 + sqrt(x)*a**2*x**10 + 3*sqrt(x)*a*b**2*x**4 - 2*sqrt(x)*a*b*x**6 - sqrt(x)*b**3 + sqrt(x)*b**2*x**2),x)*b**3 + 3*sqrt( a*x**4 - b)*int((sqrt(x)*(a*x**4 - b)**(3/4)*x**7)/(a**3*x**12 - 3*a**2*b* x**8 + a**2*x**10 + 3*a*b**2*x**4 - 2*a*b*x**6 - b**3 + b**2*x**2),x)*a**3 *x**4 - 3*sqrt(a*x**4 - b)*int((sqrt(x)*(a*x**4 - b)**(3/4)*x**7)/(a**3*x* *12 - 3*a**2*b*x**8 + a**2*x**10 + 3*a*b**2*x**4 - 2*a*b*x**6 - b**3 + b** 2*x**2),x)*a**2*b + 4*sqrt(a*x**4 - b)*int((sqrt(x)*(a*x**4 - b)**(3/4)*x* *3)/(a**3*x**12 - 3*a**2*b*x**8 + a**2*x**10 + 3*a*b**2*x**4 - 2*a*b*x**6 - b**3 + b**2*x**2),x)*a**2*b*x**4 - 4*sqrt(a*x**4 - b)*int((sqrt(x)*(a*x* *4 - b)**(3/4)*x**3)/(a**3*x**12 - 3*a**2*b*x**8 + a**2*x**10 + 3*a*b**2*x **4 - 2*a*b*x**6 - b**3 + b**2*x**2),x)*a*b**2 + 4*sqrt(a*x**4 - b)*int((s qrt(x)*(a*x**4 - b)**(3/4)*x)/(a**3*x**12 - 3*a**2*b*x**8 + a**2*x**10 + 3 *a*b**2*x**4 - 2*a*b*x**6 - b**3 + b**2*x**2),x)*a*b*x**4 - 4*sqrt(a*x**4 - b)*int((sqrt(x)*(a*x**4 - b)**(3/4)*x)/(a**3*x**12 - 3*a**2*b*x**8 + a** 2*x**10 + 3*a*b**2*x**4 - 2*a*b*x**6 - b**3 + b**2*x**2),x)*b**2)/(3*sq...