Integrand size = 12, antiderivative size = 69 \[ \int \frac {1}{\left (6-5 x+x^2\right )^{7/4}} \, dx=\frac {4 (5-2 x)}{3 \left (6-5 x+x^2\right )^{3/4}}+\frac {8 \sqrt {2} \left (-6+5 x-x^2\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arcsin (5-2 x),2\right )}{3 \left (6-5 x+x^2\right )^{3/4}} \] Output:
4/3*(5-2*x)/(x^2-5*x+6)^(3/4)-8/3*(-x^2+5*x-6)^(3/4)*InverseJacobiAM(1/2*a rcsin(-5+2*x),2^(1/2))*2^(1/2)/(x^2-5*x+6)^(3/4)
Time = 7.85 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.78 \[ \int \frac {1}{\left (6-5 x+x^2\right )^{7/4}} \, dx=\frac {20-8 x+8 \sqrt {2} \left (-6+5 x-x^2\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arcsin (5-2 x),2\right )}{3 \left (6-5 x+x^2\right )^{3/4}} \] Input:
Integrate[(6 - 5*x + x^2)^(-7/4),x]
Output:
(20 - 8*x + 8*Sqrt[2]*(-6 + 5*x - x^2)^(3/4)*EllipticF[ArcSin[5 - 2*x]/2, 2])/(3*(6 - 5*x + x^2)^(3/4))
Leaf count is larger than twice the leaf count of optimal. \(141\) vs. \(2(69)=138\).
Time = 0.36 (sec) , antiderivative size = 141, normalized size of antiderivative = 2.04, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1089, 1094, 761}
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.
\(\displaystyle \int \frac {1}{\left (x^2-5 x+6\right )^{7/4}} \, dx\) |
\(\Big \downarrow \) 1089 |
\(\displaystyle \frac {4 (5-2 x)}{3 \left (x^2-5 x+6\right )^{3/4}}-\frac {4}{3} \int \frac {1}{\left (x^2-5 x+6\right )^{3/4}}dx\) |
\(\Big \downarrow \) 1094 |
\(\displaystyle \frac {16 \sqrt {(2 x-5)^2} \int \frac {1}{\sqrt {4 \left (x^2-5 x+6\right )+1}}d\sqrt [4]{x^2-5 x+6}}{3 (5-2 x)}+\frac {4 (5-2 x)}{3 \left (x^2-5 x+6\right )^{3/4}}\) |
\(\Big \downarrow \) 761 |
\(\displaystyle \frac {4 \sqrt {2} \sqrt {(2 x-5)^2} \left (2 \sqrt {x^2-5 x+6}+1\right ) \sqrt {\frac {4 \left (x^2-5 x+6\right )+1}{\left (2 \sqrt {x^2-5 x+6}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {2} \sqrt [4]{x^2-5 x+6}\right ),\frac {1}{2}\right )}{3 \sqrt {4 \left (x^2-5 x+6\right )+1} (5-2 x)}+\frac {4 (5-2 x)}{3 \left (x^2-5 x+6\right )^{3/4}}\) |
Input:
Int[(6 - 5*x + x^2)^(-7/4),x]
Output:
(4*(5 - 2*x))/(3*(6 - 5*x + x^2)^(3/4)) + (4*Sqrt[2]*Sqrt[(-5 + 2*x)^2]*(1 + 2*Sqrt[6 - 5*x + x^2])*Sqrt[(1 + 4*(6 - 5*x + x^2))/(1 + 2*Sqrt[6 - 5*x + x^2])^2]*EllipticF[2*ArcTan[Sqrt[2]*(6 - 5*x + x^2)^(1/4)], 1/2])/(3*(5 - 2*x)*Sqrt[1 + 4*(6 - 5*x + x^2)])
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[( 1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))* EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) *((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] - Simp[2*c*((2*p + 3)/((p + 1)*(b^2 - 4*a*c))) Int[(a + b*x + c*x^2)^(p + 1), x], x] /; Fre eQ[{a, b, c}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[3*p])
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[4*(Sqrt[(b + 2*c*x)^2]/(b + 2*c*x)) Subst[Int[x^(4*(p + 1) - 1)/Sqrt[b^2 - 4*a*c + 4 *c*x^4], x], x, (a + b*x + c*x^2)^(1/4)], x] /; FreeQ[{a, b, c}, x] && Inte gerQ[4*p]
\[\int \frac {1}{\left (x^{2}-5 x +6\right )^{\frac {7}{4}}}d x\]
Input:
int(1/(x^2-5*x+6)^(7/4),x)
Output:
int(1/(x^2-5*x+6)^(7/4),x)
\[ \int \frac {1}{\left (6-5 x+x^2\right )^{7/4}} \, dx=\int { \frac {1}{{\left (x^{2} - 5 \, x + 6\right )}^{\frac {7}{4}}} \,d x } \] Input:
integrate(1/(x^2-5*x+6)^(7/4),x, algorithm="fricas")
Output:
integral((x^2 - 5*x + 6)^(1/4)/(x^4 - 10*x^3 + 37*x^2 - 60*x + 36), x)
\[ \int \frac {1}{\left (6-5 x+x^2\right )^{7/4}} \, dx=\int \frac {1}{\left (x^{2} - 5 x + 6\right )^{\frac {7}{4}}}\, dx \] Input:
integrate(1/(x**2-5*x+6)**(7/4),x)
Output:
Integral((x**2 - 5*x + 6)**(-7/4), x)
\[ \int \frac {1}{\left (6-5 x+x^2\right )^{7/4}} \, dx=\int { \frac {1}{{\left (x^{2} - 5 \, x + 6\right )}^{\frac {7}{4}}} \,d x } \] Input:
integrate(1/(x^2-5*x+6)^(7/4),x, algorithm="maxima")
Output:
integrate((x^2 - 5*x + 6)^(-7/4), x)
\[ \int \frac {1}{\left (6-5 x+x^2\right )^{7/4}} \, dx=\int { \frac {1}{{\left (x^{2} - 5 \, x + 6\right )}^{\frac {7}{4}}} \,d x } \] Input:
integrate(1/(x^2-5*x+6)^(7/4),x, algorithm="giac")
Output:
integrate((x^2 - 5*x + 6)^(-7/4), x)
Timed out. \[ \int \frac {1}{\left (6-5 x+x^2\right )^{7/4}} \, dx=\int \frac {1}{{\left (x^2-5\,x+6\right )}^{7/4}} \,d x \] Input:
int(1/(x^2 - 5*x + 6)^(7/4),x)
Output:
int(1/(x^2 - 5*x + 6)^(7/4), x)
\[ \int \frac {1}{\left (6-5 x+x^2\right )^{7/4}} \, dx=\int \frac {1}{\left (x^{2}-5 x +6\right )^{\frac {3}{4}} x^{2}-5 \left (x^{2}-5 x +6\right )^{\frac {3}{4}} x +6 \left (x^{2}-5 x +6\right )^{\frac {3}{4}}}d x \] Input:
int(1/(x^2-5*x+6)^(7/4),x)
Output:
int(1/((x**2 - 5*x + 6)**(3/4)*x**2 - 5*(x**2 - 5*x + 6)**(3/4)*x + 6*(x** 2 - 5*x + 6)**(3/4)),x)