\(\int \frac {1}{(a d e+(c d^2+a e^2) x+c d e x^2)^{5/4}} \, dx\) [379]

Optimal result

Integrand size = 29, antiderivative size = 198 \[ \int \frac {1}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/4}} \, dx=-\frac {4 \left (c d^2+a e^2+2 c d e x\right )}{\left (c d^2-a e^2\right )^2 \sqrt [4]{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {4 \sqrt {2} \sqrt [4]{-\frac {c d e \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )}{\left (c d^2-a e^2\right )^2}} E\left (\left .\frac {1}{2} \arcsin \left (\frac {a e^2+c d (d+2 e x)}{c d^2-a e^2}\right )\right |2\right )}{\left (c d^2-a e^2\right ) \sqrt [4]{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \] Output:

(-8*c*d*e*x-4*a*e^2-4*c*d^2)/(-a*e^2+c*d^2)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e 
*x^2)^(1/4)+4*2^(1/2)*(-c*d*e*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)/(-a*e^2+c* 
d^2)^2)^(1/4)*EllipticE(sin(1/2*arcsin((a*e^2+c*d*(2*e*x+d))/(-a*e^2+c*d^2 
))),2^(1/2))/(-a*e^2+c*d^2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/4)
 

Mathematica [C] (verified)

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

Time = 0.03 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.47 \[ \int \frac {1}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/4}} \, dx=-\frac {4 \sqrt [4]{\frac {c d (d+e x)}{c d^2-a e^2}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {5}{4},\frac {3}{4},\frac {e (a e+c d x)}{-c d^2+a e^2}\right )}{\left (c d^2-a e^2\right ) \sqrt [4]{(a e+c d x) (d+e x)}} \] Input:

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(-5/4),x]
 

Output:

(-4*((c*d*(d + e*x))/(c*d^2 - a*e^2))^(1/4)*Hypergeometric2F1[-1/4, 5/4, 3 
/4, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)])/((c*d^2 - a*e^2)*((a*e + c*d*x) 
*(d + e*x))^(1/4))
 

Rubi [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(1016\) vs. \(2(198)=396\).

Time = 0.67 (sec) , antiderivative size = 1016, normalized size of antiderivative = 5.13, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {1089, 1094, 834, 761, 1510}

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.

\(\displaystyle \int \frac {1}{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/4}} \, dx\)

\(\Big \downarrow \) 1089

\(\displaystyle \frac {4 c d e \int \frac {1}{\sqrt [4]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{\left (c d^2-a e^2\right )^2}-\frac {4 \left (a e^2+c d^2+2 c d e x\right )}{\left (c d^2-a e^2\right )^2 \sqrt [4]{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 1094

\(\displaystyle \frac {16 c d e \sqrt {\left (a e^2+c d^2+2 c d e x\right )^2} \int \frac {\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{\sqrt {\left (c d^2-a e^2\right )^2+4 c d e \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )}}d\sqrt [4]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{\left (c d^2-a e^2\right )^2 \left (a e^2+c d^2+2 c d e x\right )}-\frac {4 \left (a e^2+c d^2+2 c d e x\right )}{\left (c d^2-a e^2\right )^2 \sqrt [4]{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 834

\(\displaystyle \frac {16 c d e \sqrt {\left (a e^2+c d^2+2 c d e x\right )^2} \left (\frac {\left (c d^2-a e^2\right ) \int \frac {1}{\sqrt {\left (c d^2-a e^2\right )^2+4 c d e \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )}}d\sqrt [4]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{2 \sqrt {c} \sqrt {d} \sqrt {e}}-\frac {\left (c d^2-a e^2\right ) \int \frac {1-\frac {2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{c d^2-a e^2}}{\sqrt {\left (c d^2-a e^2\right )^2+4 c d e \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )}}d\sqrt [4]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{2 \sqrt {c} \sqrt {d} \sqrt {e}}\right )}{\left (c d^2-a e^2\right )^2 \left (a e^2+c d^2+2 c d e x\right )}-\frac {4 \left (a e^2+c d^2+2 c d e x\right )}{\left (c d^2-a e^2\right )^2 \sqrt [4]{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 761

\(\displaystyle \frac {16 c d e \sqrt {\left (a e^2+c d^2+2 c d e x\right )^2} \left (\frac {\left (c d^2-a e^2\right )^{3/2} \left (\frac {2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c d^2-a e^2}+1\right ) \sqrt {\frac {4 c d e \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )+\left (c d^2-a e^2\right )^2}{\left (c d^2-a e^2\right )^2 \left (\frac {2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c d^2-a e^2}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt [4]{e} \sqrt [4]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{\sqrt {c d^2-a e^2}}\right ),\frac {1}{2}\right )}{4 \sqrt {2} c^{3/4} d^{3/4} e^{3/4} \sqrt {4 c d e \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )+\left (c d^2-a e^2\right )^2}}-\frac {\left (c d^2-a e^2\right ) \int \frac {1-\frac {2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{c d^2-a e^2}}{\sqrt {\left (c d^2-a e^2\right )^2+4 c d e \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )}}d\sqrt [4]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{2 \sqrt {c} \sqrt {d} \sqrt {e}}\right )}{\left (c d^2-a e^2\right )^2 \left (a e^2+c d^2+2 c d e x\right )}-\frac {4 \left (a e^2+c d^2+2 c d e x\right )}{\left (c d^2-a e^2\right )^2 \sqrt [4]{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 1510

\(\displaystyle \frac {16 c d e \sqrt {\left (c d^2+2 c e x d+a e^2\right )^2} \left (\frac {\left (c d^2-a e^2\right )^{3/2} \left (\frac {2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{c d^2-a e^2}+1\right ) \sqrt {\frac {\left (c d^2-a e^2\right )^2+4 c d e \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )}{\left (c d^2-a e^2\right )^2 \left (\frac {2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{c d^2-a e^2}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt [4]{e} \sqrt [4]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{\sqrt {c d^2-a e^2}}\right ),\frac {1}{2}\right )}{4 \sqrt {2} c^{3/4} d^{3/4} e^{3/4} \sqrt {\left (c d^2-a e^2\right )^2+4 c d e \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )}}-\frac {\left (c d^2-a e^2\right ) \left (\frac {\sqrt {c d^2-a e^2} \left (\frac {2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{c d^2-a e^2}+1\right ) \sqrt {\frac {\left (c d^2-a e^2\right )^2+4 c d e \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )}{\left (c d^2-a e^2\right )^2 \left (\frac {2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{c d^2-a e^2}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt [4]{e} \sqrt [4]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{\sqrt {c d^2-a e^2}}\right )|\frac {1}{2}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt [4]{e} \sqrt {\left (c d^2-a e^2\right )^2+4 c d e \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )}}-\frac {\sqrt [4]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e} \sqrt {\left (c d^2-a e^2\right )^2+4 c d e \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )}}{\left (c d^2-a e^2\right )^2 \left (\frac {2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{c d^2-a e^2}+1\right )}\right )}{2 \sqrt {c} \sqrt {d} \sqrt {e}}\right )}{\left (c d^2-a e^2\right )^2 \left (c d^2+2 c e x d+a e^2\right )}-\frac {4 \left (c d^2+2 c e x d+a e^2\right )}{\left (c d^2-a e^2\right )^2 \sqrt [4]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}\)

Input:

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(-5/4),x]
 

Output:

(-4*(c*d^2 + a*e^2 + 2*c*d*e*x))/((c*d^2 - a*e^2)^2*(a*d*e + (c*d^2 + a*e^ 
2)*x + c*d*e*x^2)^(1/4)) + (16*c*d*e*Sqrt[(c*d^2 + a*e^2 + 2*c*d*e*x)^2]*( 
-1/2*((c*d^2 - a*e^2)*(-(((a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(1/4)*Sq 
rt[(c*d^2 - a*e^2)^2 + 4*c*d*e*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)])/( 
(c*d^2 - a*e^2)^2*(1 + (2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a* 
e^2)*x + c*d*e*x^2])/(c*d^2 - a*e^2)))) + (Sqrt[c*d^2 - a*e^2]*(1 + (2*Sqr 
t[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(c*d^2 - 
 a*e^2))*Sqrt[((c*d^2 - a*e^2)^2 + 4*c*d*e*(a*d*e + (c*d^2 + a*e^2)*x + c* 
d*e*x^2))/((c*d^2 - a*e^2)^2*(1 + (2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + 
(c*d^2 + a*e^2)*x + c*d*e*x^2])/(c*d^2 - a*e^2))^2)]*EllipticE[2*ArcTan[(S 
qrt[2]*c^(1/4)*d^(1/4)*e^(1/4)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(1/ 
4))/Sqrt[c*d^2 - a*e^2]], 1/2])/(Sqrt[2]*c^(1/4)*d^(1/4)*e^(1/4)*Sqrt[(c*d 
^2 - a*e^2)^2 + 4*c*d*e*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)])))/(Sqrt[ 
c]*Sqrt[d]*Sqrt[e]) + ((c*d^2 - a*e^2)^(3/2)*(1 + (2*Sqrt[c]*Sqrt[d]*Sqrt[ 
e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(c*d^2 - a*e^2))*Sqrt[((c* 
d^2 - a*e^2)^2 + 4*c*d*e*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/((c*d^2 
- a*e^2)^2*(1 + (2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x 
+ c*d*e*x^2])/(c*d^2 - a*e^2))^2)]*EllipticF[2*ArcTan[(Sqrt[2]*c^(1/4)*d^( 
1/4)*e^(1/4)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(1/4))/Sqrt[c*d^2 - a 
*e^2]], 1/2])/(4*Sqrt[2]*c^(3/4)*d^(3/4)*e^(3/4)*Sqrt[(c*d^2 - a*e^2)^2...
 

Defintions of rubi rules used

rule 761

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]
 

rule 834

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 2]}, S 
imp[1/q   Int[1/Sqrt[a + b*x^4], x], x] - Simp[1/q   Int[(1 - q*x^2)/Sqrt[a 
 + b*x^4], x], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
 

rule 1089

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])
 

rule 1094

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]
 

rule 1510

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = 
 Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + c*x^4]/(a*(1 + q^2*x^2))), x] + Simp[d* 
(1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]/(q*Sqrt[a + c*x^4]))*E 
llipticE[2*ArcTan[q*x], 1/2], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e 
}, x] && PosQ[c/a]
 

Maple [F]

\[\int \frac {1}{{\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{\frac {5}{4}}}d x\]

Input:

int(1/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(5/4),x)
 

Output:

int(1/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(5/4),x)
 

Fricas [F]

\[ \int \frac {1}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/4}} \, dx=\int { \frac {1}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {5}{4}}} \,d x } \] Input:

integrate(1/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/4),x, algorithm="fricas")
 

Output:

integral((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/4)/(c^2*d^2*e^2*x^4 + 
a^2*d^2*e^2 + 2*(c^2*d^3*e + a*c*d*e^3)*x^3 + (c^2*d^4 + 4*a*c*d^2*e^2 + a 
^2*e^4)*x^2 + 2*(a*c*d^3*e + a^2*d*e^3)*x), x)
 

Sympy [F]

\[ \int \frac {1}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/4}} \, dx=\int \frac {1}{\left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac {5}{4}}}\, dx \] Input:

integrate(1/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/4),x)
 

Output:

Integral((a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(-5/4), x)
 

Maxima [F]

\[ \int \frac {1}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/4}} \, dx=\int { \frac {1}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {5}{4}}} \,d x } \] Input:

integrate(1/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/4),x, algorithm="maxima")
 

Output:

integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(-5/4), x)
 

Giac [F]

\[ \int \frac {1}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/4}} \, dx=\int { \frac {1}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {5}{4}}} \,d x } \] Input:

integrate(1/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/4),x, algorithm="giac")
 

Output:

integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(-5/4), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/4}} \, dx=\int \frac {1}{{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/4}} \,d x \] Input:

int(1/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/4),x)
 

Output:

int(1/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/4), x)
 

Reduce [F]

\[ \int \frac {1}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/4}} \, dx=\int \frac {1}{\left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {1}{4}} a d e +\left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {1}{4}} a \,e^{2} x +\left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {1}{4}} c \,d^{2} x +\left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {1}{4}} c d e \,x^{2}}d x \] Input:

int(1/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/4),x)
 

Output:

int(1/((a*d*e + a*e**2*x + c*d**2*x + c*d*e*x**2)**(1/4)*a*d*e + (a*d*e + 
a*e**2*x + c*d**2*x + c*d*e*x**2)**(1/4)*a*e**2*x + (a*d*e + a*e**2*x + c* 
d**2*x + c*d*e*x**2)**(1/4)*c*d**2*x + (a*d*e + a*e**2*x + c*d**2*x + c*d* 
e*x**2)**(1/4)*c*d*e*x**2),x)