\(\int \frac {(a+b (c+d x^n)^2)^p}{e x} \, dx\) [343]

Optimal result

Integrand size = 22, antiderivative size = 195 \[ \int \frac {\left (a+b \left (c+d x^n\right )^2\right )^p}{e x} \, dx=\frac {\left (-\frac {x^{-n} \left (\sqrt {-a}-\sqrt {b} c-\sqrt {b} d x^n\right )}{\sqrt {b} d}\right )^{-p} \left (\frac {x^{-n} \left (\sqrt {-a}+\sqrt {b} c+\sqrt {b} d x^n\right )}{\sqrt {b} d}\right )^{-p} \left (a+b c^2+2 b c d x^n+b d^2 x^{2 n}\right )^p \operatorname {AppellF1}\left (-2 p,-p,-p,1-2 p,\frac {\left (\frac {\sqrt {-a}}{\sqrt {b}}-c\right ) x^{-n}}{d},-\frac {\left (\frac {\sqrt {-a}}{\sqrt {b}}+c\right ) x^{-n}}{d}\right )}{2 e n p} \] Output:

1/2*(a+b*c^2+2*b*c*d*x^n+b*d^2*x^(2*n))^p*AppellF1(-2*p,-p,-p,1-2*p,((-a)^ 
(1/2)/b^(1/2)-c)/d/(x^n),-((-a)^(1/2)/b^(1/2)+c)/d/(x^n))/e/n/p/((-((-a)^( 
1/2)-b^(1/2)*c-b^(1/2)*d*x^n)/b^(1/2)/d/(x^n))^p)/((((-a)^(1/2)+b^(1/2)*c+ 
b^(1/2)*d*x^n)/b^(1/2)/d/(x^n))^p)
 

Mathematica [A] (verified)

Time = 0.39 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.92 \[ \int \frac {\left (a+b \left (c+d x^n\right )^2\right )^p}{e x} \, dx=\frac {\left (\frac {x^{-n} \left (-\sqrt {-a b d^2}+b d \left (c+d x^n\right )\right )}{b d^2}\right )^{-p} \left (\frac {x^{-n} \left (\sqrt {-a b d^2}+b d \left (c+d x^n\right )\right )}{b d^2}\right )^{-p} \left (a+b \left (c+d x^n\right )^2\right )^p \operatorname {AppellF1}\left (-2 p,-p,-p,1-2 p,-\frac {\left (b c d+\sqrt {-a b d^2}\right ) x^{-n}}{b d^2},\frac {\left (-b c d+\sqrt {-a b d^2}\right ) x^{-n}}{b d^2}\right )}{2 e n p} \] Input:

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

Output:

((a + b*(c + d*x^n)^2)^p*AppellF1[-2*p, -p, -p, 1 - 2*p, -((b*c*d + Sqrt[- 
(a*b*d^2)])/(b*d^2*x^n)), (-(b*c*d) + Sqrt[-(a*b*d^2)])/(b*d^2*x^n)])/(2*e 
*n*p*((-Sqrt[-(a*b*d^2)] + b*d*(c + d*x^n))/(b*d^2*x^n))^p*((Sqrt[-(a*b*d^ 
2)] + b*d*(c + d*x^n))/(b*d^2*x^n))^p)
 

Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {27, 2086, 1693, 1178, 150}

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.

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2086

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

\(\Big \downarrow \) 1693

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

\(\Big \downarrow \) 1178

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

\(\Big \downarrow \) 150

\(\displaystyle \frac {\left (-\frac {x^{-n} \left (\sqrt {-a}-\sqrt {b} c-\sqrt {b} d x^n\right )}{\sqrt {b} d}\right )^{-p} \left (\frac {x^{-n} \left (\sqrt {-a}+\sqrt {b} c+\sqrt {b} d x^n\right )}{\sqrt {b} d}\right )^{-p} \left (a+b c^2+2 b c d x^n+b d^2 x^{2 n}\right )^p \operatorname {AppellF1}\left (-2 p,-p,-p,1-2 p,\frac {\left (\frac {\sqrt {-a}}{\sqrt {b}}-c\right ) x^{-n}}{d},-\frac {\left (c+\frac {\sqrt {-a}}{\sqrt {b}}\right ) x^{-n}}{d}\right )}{2 e n p}\)

Input:

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

Output:

((a + b*c^2 + 2*b*c*d*x^n + b*d^2*x^(2*n))^p*AppellF1[-2*p, -p, -p, 1 - 2* 
p, (Sqrt[-a]/Sqrt[b] - c)/(d*x^n), -((Sqrt[-a]/Sqrt[b] + c)/(d*x^n))])/(2* 
e*n*p*(-((Sqrt[-a] - Sqrt[b]*c - Sqrt[b]*d*x^n)/(Sqrt[b]*d*x^n)))^p*((Sqrt 
[-a] + Sqrt[b]*c + Sqrt[b]*d*x^n)/(Sqrt[b]*d*x^n))^p)
 

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 150

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_ 
] :> Simp[c^n*e^p*((b*x)^(m + 1)/(b*(m + 1)))*AppellF1[m + 1, -n, -p, m + 2 
, (-d)*(x/c), (-f)*(x/e)], x] /; FreeQ[{b, c, d, e, f, m, n, p}, x] &&  !In 
tegerQ[m] &&  !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])
 

rule 1178

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(-(1/(d + e*x))^(2*p))*((a + 
b*x + c*x^2)^p/(e*(e*((b - q + 2*c*x)/(2*c*(d + e*x))))^p*(e*((b + q + 2*c* 
x)/(2*c*(d + e*x))))^p))   Subst[Int[x^(-m - 2*(p + 1))*Simp[1 - (d - e*((b 
 - q)/(2*c)))*x, x]^p*Simp[1 - (d - e*((b + q)/(2*c)))*x, x]^p, x], x, 1/(d 
 + e*x)], x]] /; FreeQ[{a, b, c, d, e, p}, x] && ILtQ[m, 0]
 

rule 1693

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol 
] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x + c*x^2)^p, 
x], x, x^n], x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[n2, 2*n] && IntegerQ 
[Simplify[(m + 1)/n]]
 

rule 2086

Int[(u_)^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Int[(d*x)^m*ExpandToSum[u, 
x]^p, x] /; FreeQ[{d, m, p}, x] && TrinomialQ[u, x] &&  !TrinomialMatchQ[u, 
 x]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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