\(\int \frac {1}{(d+e x) \sqrt [3]{a d e+(c d^2+a e^2) x+c d e x^2}} \, dx\) [335]

Optimal result

Integrand size = 37, antiderivative size = 82 \[ \int \frac {1}{(d+e x) \sqrt [3]{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},1,\frac {2}{3},\frac {c d (d+e x)}{c d^2-a e^2}\right )}{\left (c d^2-a e^2\right ) (d+e x)} \] Output:

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.20 \[ \int \frac {1}{(d+e x) \sqrt [3]{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {3 c d ((a e+c d x) (d+e x))^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {2}{3},\frac {4}{3},\frac {5}{3},\frac {e (a e+c d x)}{-c d^2+a e^2}\right )}{2 \left (c d^2-a e^2\right )^2 \left (\frac {c d (d+e x)}{c d^2-a e^2}\right )^{2/3}} \] Input:

Integrate[1/((d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(1/3)),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.41, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.081, Rules used = {1138, 80, 79}

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.

\(\displaystyle \int \frac {1}{(d+e x) \sqrt [3]{x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \, dx\)

\(\Big \downarrow \) 1138

\(\displaystyle \frac {\sqrt [3]{\frac {e x}{d}+1} \sqrt [3]{a e+c d x} \int \frac {1}{\sqrt [3]{a e+c d x} \left (\frac {e x}{d}+1\right )^{4/3}}dx}{d \sqrt [3]{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 80

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

\(\Big \downarrow \) 79

\(\displaystyle \frac {3 (a e+c d x) \sqrt [3]{\frac {c d (d+e x)}{c d^2-a e^2}} \operatorname {Hypergeometric2F1}\left (\frac {2}{3},\frac {4}{3},\frac {5}{3},-\frac {e (a e+c d x)}{c d^2-a e^2}\right )}{2 \left (c d^2-a e^2\right ) \sqrt [3]{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

Input:

Int[1/((d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(1/3)),x]
 

Output:

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

Defintions of rubi rules used

rule 79

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(( 
a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n))*Hypergeometric2F1[-n, m + 1 
, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}, x] 
&&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] 
 ||  !(RationalQ[n] && GtQ[-d/(b*c - a*d), 0]))
 

rule 80

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(c 
 + d*x)^FracPart[n]/((b/(b*c - a*d))^IntPart[n]*(b*((c + d*x)/(b*c - a*d))) 
^FracPart[n])   Int[(a + b*x)^m*Simp[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d) 
), x]^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] &&  !IntegerQ[m] &&  !Integ 
erQ[n] && (RationalQ[m] ||  !SimplerQ[n + 1, m + 1])
 

rule 1138

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Sy 
mbol] :> Simp[d^m*((a + b*x + c*x^2)^FracPart[p]/((1 + e*(x/d))^FracPart[p] 
*(a/d + (c*x)/e)^FracPart[p]))   Int[(1 + e*(x/d))^(m + p)*(a/d + (c/e)*x)^ 
p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] 
&& (IntegerQ[m] || GtQ[d, 0]) &&  !(IGtQ[m, 0] && (IntegerQ[3*p] || Integer 
Q[4*p]))
 

Maple [F]

\[\int \frac {1}{\left (e x +d \right ) {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{\frac {1}{3}}}d x\]

Input:

int(1/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/3),x)
 

Output:

int(1/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/3),x)
 

Fricas [F]

\[ \int \frac {1}{(d+e x) \sqrt [3]{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int { \frac {1}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {1}{3}} {\left (e x + d\right )}} \,d x } \] Input:

integrate(1/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/3),x, algorithm=" 
fricas")
 

Output:

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

Sympy [F]

\[ \int \frac {1}{(d+e x) \sqrt [3]{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int \frac {1}{\sqrt [3]{\left (d + e x\right ) \left (a e + c d x\right )} \left (d + e x\right )}\, dx \] Input:

integrate(1/(e*x+d)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/3),x)
 

Output:

Integral(1/(((d + e*x)*(a*e + c*d*x))**(1/3)*(d + e*x)), x)
 

Maxima [F]

\[ \int \frac {1}{(d+e x) \sqrt [3]{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int { \frac {1}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {1}{3}} {\left (e x + d\right )}} \,d x } \] Input:

integrate(1/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/3),x, algorithm=" 
maxima")
 

Output:

integrate(1/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(1/3)*(e*x + d)), x)
 

Giac [F]

\[ \int \frac {1}{(d+e x) \sqrt [3]{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int { \frac {1}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {1}{3}} {\left (e x + d\right )}} \,d x } \] Input:

integrate(1/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/3),x, algorithm=" 
giac")
 

Output:

integrate(1/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(1/3)*(e*x + d)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x) \sqrt [3]{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int \frac {1}{\left (d+e\,x\right )\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{1/3}} \,d x \] Input:

int(1/((d + e*x)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/3)),x)
 

Output:

int(1/((d + e*x)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/3)), x)
 

Reduce [F]

\[ \int \frac {1}{(d+e x) \sqrt [3]{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int \frac {1}{\left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {1}{3}} d +\left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {1}{3}} e x}d x \] Input:

int(1/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/3),x)
 

Output:

int(1/((a*d*e + a*e**2*x + c*d**2*x + c*d*e*x**2)**(1/3)*d + (a*d*e + a*e* 
*2*x + c*d**2*x + c*d*e*x**2)**(1/3)*e*x),x)