\(\int (d+e x)^m (a+c x^2)^p \, dx\) [175]

Optimal result

Integrand size = 17, antiderivative size = 152 \[ \int (d+e x)^m \left (a+c x^2\right )^p \, dx=\frac {(d+e x)^{1+m} \left (a+c x^2\right )^p \left (1-\frac {d+e x}{d-\frac {\sqrt {-a} e}{\sqrt {c}}}\right )^{-p} \left (1-\frac {d+e x}{d+\frac {\sqrt {-a} e}{\sqrt {c}}}\right )^{-p} \operatorname {AppellF1}\left (1+m,-p,-p,2+m,\frac {d+e x}{d-\frac {\sqrt {-a} e}{\sqrt {c}}},\frac {d+e x}{d+\frac {\sqrt {-a} e}{\sqrt {c}}}\right )}{e (1+m)} \] Output:

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

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.03 \[ \int (d+e x)^m \left (a+c x^2\right )^p \, dx=\frac {\left (\frac {e \left (\sqrt {-\frac {a}{c}}-x\right )}{d+\sqrt {-\frac {a}{c}} e}\right )^{-p} \left (\frac {e \left (\sqrt {-\frac {a}{c}}+x\right )}{-d+\sqrt {-\frac {a}{c}} e}\right )^{-p} (d+e x)^{1+m} \left (a+c x^2\right )^p \operatorname {AppellF1}\left (1+m,-p,-p,2+m,\frac {d+e x}{d-\sqrt {-\frac {a}{c}} e},\frac {d+e x}{d+\sqrt {-\frac {a}{c}} e}\right )}{e (1+m)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {514, 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.

\(\displaystyle \int \left (a+c x^2\right )^p (d+e x)^m \, dx\)

\(\Big \downarrow \) 514

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

\(\Big \downarrow \) 150

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

Input:

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

Output:

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

Defintions of rubi rules used

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 514

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ 
{q = Rt[-a/b, 2]}, Simp[(a + b*x^2)^p/(d*(1 - (c + d*x)/(c - d*q))^p*(1 - ( 
c + d*x)/(c + d*q))^p)   Subst[Int[x^n*Simp[1 - x/(c + d*q), x]^p*Simp[1 - 
x/(c - d*q), x]^p, x], x, c + d*x], x]] /; FreeQ[{a, b, c, d, n, p}, x] && 
NeQ[b*c^2 + a*d^2, 0]
 

Maple [F]

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

Input:

int((e*x+d)^m*(c*x^2+a)^p,x)
 

Output:

int((e*x+d)^m*(c*x^2+a)^p,x)
 

Fricas [F]

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

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

Output:

integral((c*x^2 + a)^p*(e*x + d)^m, x)
 

Sympy [F(-1)]

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

integrate((e*x+d)**m*(c*x**2+a)**p,x)
 

Output:

Timed out
 

Maxima [F]

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

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

Output:

integrate((c*x^2 + a)^p*(e*x + d)^m, x)
 

Giac [F]

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

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

Output:

integrate((c*x^2 + a)^p*(e*x + d)^m, x)
 

Mupad [F(-1)]

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

int((a + c*x^2)^p*(d + e*x)^m,x)
 

Output:

int((a + c*x^2)^p*(d + e*x)^m, x)
 

Reduce [F]

\[ \int (d+e x)^m \left (a+c x^2\right )^p \, dx=\text {too large to display} \] Input:

int((e*x+d)^m*(c*x^2+a)^p,x)
 

Output:

((d + e*x)**m*(a + c*x**2)**p*a*e + (d + e*x)**m*(a + c*x**2)**p*c*d*x - i 
nt(((d + e*x)**m*(a + c*x**2)**p*x**2)/(a*d*m + 2*a*d*p + a*d + a*e*m*x + 
2*a*e*p*x + a*e*x + c*d*m*x**2 + 2*c*d*p*x**2 + c*d*x**2 + c*e*m*x**3 + 2* 
c*e*p*x**3 + c*e*x**3),x)*a*c*e**2*m**2 - 4*int(((d + e*x)**m*(a + c*x**2) 
**p*x**2)/(a*d*m + 2*a*d*p + a*d + a*e*m*x + 2*a*e*p*x + a*e*x + c*d*m*x** 
2 + 2*c*d*p*x**2 + c*d*x**2 + c*e*m*x**3 + 2*c*e*p*x**3 + c*e*x**3),x)*a*c 
*e**2*m*p - int(((d + e*x)**m*(a + c*x**2)**p*x**2)/(a*d*m + 2*a*d*p + a*d 
 + a*e*m*x + 2*a*e*p*x + a*e*x + c*d*m*x**2 + 2*c*d*p*x**2 + c*d*x**2 + c* 
e*m*x**3 + 2*c*e*p*x**3 + c*e*x**3),x)*a*c*e**2*m - 4*int(((d + e*x)**m*(a 
 + c*x**2)**p*x**2)/(a*d*m + 2*a*d*p + a*d + a*e*m*x + 2*a*e*p*x + a*e*x + 
 c*d*m*x**2 + 2*c*d*p*x**2 + c*d*x**2 + c*e*m*x**3 + 2*c*e*p*x**3 + c*e*x* 
*3),x)*a*c*e**2*p**2 - 2*int(((d + e*x)**m*(a + c*x**2)**p*x**2)/(a*d*m + 
2*a*d*p + a*d + a*e*m*x + 2*a*e*p*x + a*e*x + c*d*m*x**2 + 2*c*d*p*x**2 + 
c*d*x**2 + c*e*m*x**3 + 2*c*e*p*x**3 + c*e*x**3),x)*a*c*e**2*p + int(((d + 
 e*x)**m*(a + c*x**2)**p*x**2)/(a*d*m + 2*a*d*p + a*d + a*e*m*x + 2*a*e*p* 
x + a*e*x + c*d*m*x**2 + 2*c*d*p*x**2 + c*d*x**2 + c*e*m*x**3 + 2*c*e*p*x* 
*3 + c*e*x**3),x)*c**2*d**2*m**2 + 2*int(((d + e*x)**m*(a + c*x**2)**p*x** 
2)/(a*d*m + 2*a*d*p + a*d + a*e*m*x + 2*a*e*p*x + a*e*x + c*d*m*x**2 + 2*c 
*d*p*x**2 + c*d*x**2 + c*e*m*x**3 + 2*c*e*p*x**3 + c*e*x**3),x)*c**2*d**2* 
m*p + int(((d + e*x)**m*(a + c*x**2)**p*x**2)/(a*d*m + 2*a*d*p + a*d + ...