\(\int \frac {(a+b \sqrt {c+d x^n})^p}{e x} \, dx\) [372]

Optimal result

Integrand size = 24, antiderivative size = 159 \[ \int \frac {\left (a+b \sqrt {c+d x^n}\right )^p}{e x} \, dx=-\frac {\left (a+b \sqrt {c+d x^n}\right )^{1+p} \operatorname {Hypergeometric2F1}\left (1,1+p,2+p,\frac {a+b \sqrt {c+d x^n}}{a-b \sqrt {c}}\right )}{\left (a-b \sqrt {c}\right ) e n (1+p)}-\frac {\left (a+b \sqrt {c+d x^n}\right )^{1+p} \operatorname {Hypergeometric2F1}\left (1,1+p,2+p,\frac {a+b \sqrt {c+d x^n}}{a+b \sqrt {c}}\right )}{\left (a+b \sqrt {c}\right ) e n (1+p)} \] Output:

-(a+b*(c+d*x^n)^(1/2))^(p+1)*hypergeom([1, p+1],[2+p],(a+b*(c+d*x^n)^(1/2) 
)/(a-b*c^(1/2)))/(a-b*c^(1/2))/e/n/(p+1)-(a+b*(c+d*x^n)^(1/2))^(p+1)*hyper 
geom([1, p+1],[2+p],(a+b*(c+d*x^n)^(1/2))/(a+b*c^(1/2)))/(a+b*c^(1/2))/e/n 
/(p+1)
 

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.93 \[ \int \frac {\left (a+b \sqrt {c+d x^n}\right )^p}{e x} \, dx=-\frac {\left (a+b \sqrt {c+d x^n}\right )^{1+p} \left (\left (a+b \sqrt {c}\right ) \operatorname {Hypergeometric2F1}\left (1,1+p,2+p,\frac {a+b \sqrt {c+d x^n}}{a-b \sqrt {c}}\right )+\left (a-b \sqrt {c}\right ) \operatorname {Hypergeometric2F1}\left (1,1+p,2+p,\frac {a+b \sqrt {c+d x^n}}{a+b \sqrt {c}}\right )\right )}{\left (a-b \sqrt {c}\right ) \left (a+b \sqrt {c}\right ) e n (1+p)} \] Input:

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

Output:

-(((a + b*Sqrt[c + d*x^n])^(1 + p)*((a + b*Sqrt[c])*Hypergeometric2F1[1, 1 
 + p, 2 + p, (a + b*Sqrt[c + d*x^n])/(a - b*Sqrt[c])] + (a - b*Sqrt[c])*Hy 
pergeometric2F1[1, 1 + p, 2 + p, (a + b*Sqrt[c + d*x^n])/(a + b*Sqrt[c])]) 
)/((a - b*Sqrt[c])*(a + b*Sqrt[c])*e*n*(1 + p)))
 

Rubi [A] (verified)

Time = 0.86 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {27, 7282, 896, 25, 1732, 615, 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.

\(\displaystyle \int \frac {\left (a+b \sqrt {c+d x^n}\right )^p}{e x} \, dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {\left (a+b \sqrt {d x^n+c}\right )^p}{x}dx}{e}\)

\(\Big \downarrow \) 7282

\(\displaystyle \frac {\int x^{-n} \left (a+b \sqrt {d x^n+c}\right )^pdx^n}{e n}\)

\(\Big \downarrow \) 896

\(\displaystyle \frac {\int \frac {x^{-n} \left (a+b \sqrt {d x^n+c}\right )^p}{d}d\left (d x^n+c\right )}{e n}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\int -\frac {x^{-n} \left (a+b \sqrt {d x^n+c}\right )^p}{d}d\left (d x^n+c\right )}{e n}\)

\(\Big \downarrow \) 1732

\(\displaystyle -\frac {2 \int \frac {\sqrt {d x^n+c} \left (a+b \sqrt {d x^n+c}\right )^p}{c-x^{2 n}}d\sqrt {d x^n+c}}{e n}\)

\(\Big \downarrow \) 615

\(\displaystyle -\frac {2 \int \left (\frac {\left (a+b \sqrt {d x^n+c}\right )^p}{2 \left (-d x^n-c+\sqrt {c}\right )}-\frac {\left (a+b \sqrt {d x^n+c}\right )^p}{2 \left (\sqrt {c}+\sqrt {d x^n+c}\right )}\right )d\sqrt {d x^n+c}}{e n}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2 \left (\frac {\left (a+b \sqrt {c+d x^n}\right )^{p+1} \operatorname {Hypergeometric2F1}\left (1,p+1,p+2,\frac {a+b \sqrt {d x^n+c}}{a-b \sqrt {c}}\right )}{2 (p+1) \left (a-b \sqrt {c}\right )}+\frac {\left (a+b \sqrt {c+d x^n}\right )^{p+1} \operatorname {Hypergeometric2F1}\left (1,p+1,p+2,\frac {a+b \sqrt {d x^n+c}}{a+b \sqrt {c}}\right )}{2 (p+1) \left (a+b \sqrt {c}\right )}\right )}{e n}\)

Input:

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

Output:

(-2*(((a + b*Sqrt[c + d*x^n])^(1 + p)*Hypergeometric2F1[1, 1 + p, 2 + p, ( 
a + b*Sqrt[c + d*x^n])/(a - b*Sqrt[c])])/(2*(a - b*Sqrt[c])*(1 + p)) + ((a 
 + b*Sqrt[c + d*x^n])^(1 + p)*Hypergeometric2F1[1, 1 + p, 2 + p, (a + b*Sq 
rt[c + d*x^n])/(a + b*Sqrt[c])])/(2*(a + b*Sqrt[c])*(1 + p))))/(e*n)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 615

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

rule 896

Int[((a_) + (b_.)*(v_)^(n_))^(p_.)*(x_)^(m_.), x_Symbol] :> With[{c = Coeff 
icient[v, x, 0], d = Coefficient[v, x, 1]}, Simp[1/d^(m + 1)   Subst[Int[Si 
mplifyIntegrand[(x - c)^m*(a + b*x^n)^p, x], x], x, v], x] /; NeQ[c, 0]] /; 
 FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && IntegerQ[m]
 

rule 1732

Int[((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symb 
ol] :> With[{g = Denominator[n]}, Simp[g   Subst[Int[x^(g - 1)*(d + e*x^(g* 
n))^q*(a + c*x^(2*g*n))^p, x], x, x^(1/g)], x]] /; FreeQ[{a, c, d, e, p, q} 
, x] && EqQ[n2, 2*n] && FractionQ[n]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7282

Int[(u_)/(x_), x_Symbol] :> With[{lst = PowerVariableExpn[u, 0, x]}, Simp[1 
/lst[[2]]   Subst[Int[NormalizeIntegrand[Simplify[lst[[1]]/x], x], x], x, ( 
lst[[3]]*x)^lst[[2]]], x] /;  !FalseQ[lst] && NeQ[lst[[2]], 0]] /; NonsumQ[ 
u] &&  !RationalFunctionQ[u, x]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-2)]

Exception generated. \[ \int \frac {\left (a+b \sqrt {c+d x^n}\right )^p}{e x} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >>  Error detected within library code:   do_a 
lg_rde: unimplemented kernel
 

Sympy [F]

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

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

Output:

Integral((a + b*sqrt(c + d*x**n))**p/x, x)/e
 

Maxima [F]

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

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

Output:

integrate((sqrt(d*x^n + c)*b + a)^p/x, x)/e
 

Giac [F]

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

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

Output:

integrate((sqrt(d*x^n + c)*b + a)^p/(e*x), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int((sqrt(x**n*d + c)*b + a)**p/x,x)/e