\(\int \frac {(e x)^{2/3}}{(c+d x)^2 \sqrt {a+b x^2}} \, dx\) [1611]

Optimal result

Integrand size = 26, antiderivative size = 221 \[ \int \frac {(e x)^{2/3}}{(c+d x)^2 \sqrt {a+b x^2}} \, dx=\frac {3 (e x)^{5/3} \sqrt {1+\frac {b x^2}{a}} \operatorname {AppellF1}\left (\frac {5}{6},\frac {1}{2},2,\frac {11}{6},-\frac {b x^2}{a},\frac {d^2 x^2}{c^2}\right )}{5 c^2 e \sqrt {a+b x^2}}-\frac {3 d (e x)^{8/3} \sqrt {1+\frac {b x^2}{a}} \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},2,\frac {7}{3},-\frac {b x^2}{a},\frac {d^2 x^2}{c^2}\right )}{4 c^3 e^2 \sqrt {a+b x^2}}+\frac {3 d^2 (e x)^{11/3} \sqrt {1+\frac {b x^2}{a}} \operatorname {AppellF1}\left (\frac {11}{6},\frac {1}{2},2,\frac {17}{6},-\frac {b x^2}{a},\frac {d^2 x^2}{c^2}\right )}{11 c^4 e^3 \sqrt {a+b x^2}} \] Output:

3/5*(e*x)^(5/3)*(1+b*x^2/a)^(1/2)*AppellF1(5/6,2,1/2,11/6,d^2*x^2/c^2,-b*x 
^2/a)/c^2/e/(b*x^2+a)^(1/2)-3/4*d*(e*x)^(8/3)*(1+b*x^2/a)^(1/2)*AppellF1(4 
/3,2,1/2,7/3,d^2*x^2/c^2,-b*x^2/a)/c^3/e^2/(b*x^2+a)^(1/2)+3/11*d^2*(e*x)^ 
(11/3)*(1+b*x^2/a)^(1/2)*AppellF1(11/6,2,1/2,17/6,d^2*x^2/c^2,-b*x^2/a)/c^ 
4/e^3/(b*x^2+a)^(1/2)
 

Mathematica [F]

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

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

Output:

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

Rubi [F]

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.

\(\displaystyle \int \frac {(e x)^{2/3}}{\sqrt {a+b x^2} (c+d x)^2} \, dx\)

\(\Big \downarrow \) 616

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1888

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

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 616

Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), 
x_Symbol] :> With[{k = Denominator[m]}, Simp[k/e   Subst[Int[x^(k*(m + 1) - 
 1)*(c + d*(x^k/e))^n*(a + b*(x^(2*k)/e^2))^p, x], x, (e*x)^(1/k)], x]] /; 
FreeQ[{a, b, c, d, e, p}, x] && ILtQ[n, 0] && FractionQ[m]
 

rule 1888

Int[((f_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^ 
(n_))^(q_.), x_Symbol] :> Unintegrable[(f*x)^m*(d + e*x^n)^q*(a + c*x^(2*n) 
)^p, x] /; FreeQ[{a, c, d, e, f, m, n, p, q}, x] && EqQ[n2, 2*n]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

Integral((e*x)**(2/3)/(sqrt(a + b*x**2)*(c + d*x)**2), x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(e x)^{2/3}}{(c+d x)^2 \sqrt {a+b x^2}} \, dx=e^{\frac {2}{3}} \left (\int \frac {x^{\frac {2}{3}} \sqrt {b \,x^{2}+a}}{b \,d^{2} x^{4}+2 b c d \,x^{3}+a \,d^{2} x^{2}+b \,c^{2} x^{2}+2 a c d x +a \,c^{2}}d x \right ) \] Input:

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

Output:

e**(2/3)*int((x**(2/3)*sqrt(a + b*x**2))/(a*c**2 + 2*a*c*d*x + a*d**2*x**2 
 + b*c**2*x**2 + 2*b*c*d*x**3 + b*d**2*x**4),x)