\(\int (d+e x)^m (f+g x)^n (a+c x^2) \, dx\) [172]

Optimal result

Integrand size = 22, antiderivative size = 233 \[ \int (d+e x)^m (f+g x)^n \left (a+c x^2\right ) \, dx=-\frac {c (e f (2+m)+d g (4+m+2 n)) (d+e x)^{1+m} (f+g x)^{1+n}}{e^2 g^2 (2+m+n) (3+m+n)}+\frac {c (d+e x)^{2+m} (f+g x)^{1+n}}{e^2 g (3+m+n)}+\frac {\left (c (e f (1+m)+d g (1+n)) (e f (2+m)+d g (4+m+2 n))+g (2+m+n) \left (a e^2 g (3+m+n)-c d (e f (2+m)+d g (1+n))\right )\right ) (d+e x)^{1+m} (f+g x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,2+m+n,2+m,-\frac {g (d+e x)}{e f-d g}\right )}{e^2 g^2 (e f-d g) (1+m) (2+m+n) (3+m+n)} \] Output:

-c*(e*f*(2+m)+d*g*(4+m+2*n))*(e*x+d)^(1+m)*(g*x+f)^(1+n)/e^2/g^2/(2+m+n)/( 
3+m+n)+c*(e*x+d)^(2+m)*(g*x+f)^(1+n)/e^2/g/(3+m+n)+(c*(e*f*(1+m)+d*g*(1+n) 
)*(e*f*(2+m)+d*g*(4+m+2*n))+g*(2+m+n)*(a*e^2*g*(3+m+n)-c*d*(e*f*(2+m)+d*g* 
(1+n))))*(e*x+d)^(1+m)*(g*x+f)^(1+n)*hypergeom([1, 2+m+n],[2+m],-g*(e*x+d) 
/(-d*g+e*f))/e^2/g^2/(-d*g+e*f)/(1+m)/(2+m+n)/(3+m+n)
 

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.75 \[ \int (d+e x)^m (f+g x)^n \left (a+c x^2\right ) \, dx=\frac {(d+e x)^{1+m} (f+g x)^n \left (\frac {e (f+g x)}{e f-d g}\right )^{-n} \left (c (e f-d g)^2 \operatorname {Hypergeometric2F1}\left (1+m,-2-n,2+m,\frac {g (d+e x)}{-e f+d g}\right )+e \left (2 c f (-e f+d g) \operatorname {Hypergeometric2F1}\left (1+m,-1-n,2+m,\frac {g (d+e x)}{-e f+d g}\right )+e \left (c f^2+a g^2\right ) \operatorname {Hypergeometric2F1}\left (1+m,-n,2+m,\frac {g (d+e x)}{-e f+d g}\right )\right )\right )}{e^3 g^2 (1+m)} \] Input:

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

Output:

((d + e*x)^(1 + m)*(f + g*x)^n*(c*(e*f - d*g)^2*Hypergeometric2F1[1 + m, - 
2 - n, 2 + m, (g*(d + e*x))/(-(e*f) + d*g)] + e*(2*c*f*(-(e*f) + d*g)*Hype 
rgeometric2F1[1 + m, -1 - n, 2 + m, (g*(d + e*x))/(-(e*f) + d*g)] + e*(c*f 
^2 + a*g^2)*Hypergeometric2F1[1 + m, -n, 2 + m, (g*(d + e*x))/(-(e*f) + d* 
g)])))/(e^3*g^2*(1 + m)*((e*(f + g*x))/(e*f - d*g))^n)
 

Rubi [A] (warning: unable to verify)

Time = 0.72 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.06, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {651, 90, 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.

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

\(\Big \downarrow \) 651

\(\displaystyle \frac {\int (d+e x)^m (f+g x)^n \left (a g (m+n+3) e^2-c (e f (m+2)+d g (m+2 n+4)) x e-c d (e f (m+2)+d g (n+1))\right )dx}{e^2 g (m+n+3)}+\frac {c (d+e x)^{m+2} (f+g x)^{n+1}}{e^2 g (m+n+3)}\)

\(\Big \downarrow \) 90

\(\displaystyle \frac {\frac {\left (a e^2 g^2 \left (m^2+m (2 n+5)+n^2+5 n+6\right )+c \left (d^2 g^2 \left (n^2+3 n+2\right )+2 d e f g (m+1) (n+1)+e^2 f^2 \left (m^2+3 m+2\right )\right )\right ) \int (d+e x)^m (f+g x)^ndx}{g (m+n+2)}-\frac {c (d+e x)^{m+1} (f+g x)^{n+1} (d g (m+2 n+4)+e f (m+2))}{g (m+n+2)}}{e^2 g (m+n+3)}+\frac {c (d+e x)^{m+2} (f+g x)^{n+1}}{e^2 g (m+n+3)}\)

\(\Big \downarrow \) 80

\(\displaystyle \frac {\frac {(f+g x)^n \left (\frac {e (f+g x)}{e f-d g}\right )^{-n} \left (a e^2 g^2 \left (m^2+m (2 n+5)+n^2+5 n+6\right )+c \left (d^2 g^2 \left (n^2+3 n+2\right )+2 d e f g (m+1) (n+1)+e^2 f^2 \left (m^2+3 m+2\right )\right )\right ) \int (d+e x)^m \left (\frac {e f}{e f-d g}+\frac {e g x}{e f-d g}\right )^ndx}{g (m+n+2)}-\frac {c (d+e x)^{m+1} (f+g x)^{n+1} (d g (m+2 n+4)+e f (m+2))}{g (m+n+2)}}{e^2 g (m+n+3)}+\frac {c (d+e x)^{m+2} (f+g x)^{n+1}}{e^2 g (m+n+3)}\)

\(\Big \downarrow \) 79

\(\displaystyle \frac {\frac {(d+e x)^{m+1} (f+g x)^n \left (\frac {e (f+g x)}{e f-d g}\right )^{-n} \left (a e^2 g^2 \left (m^2+m (2 n+5)+n^2+5 n+6\right )+c \left (d^2 g^2 \left (n^2+3 n+2\right )+2 d e f g (m+1) (n+1)+e^2 f^2 \left (m^2+3 m+2\right )\right )\right ) \operatorname {Hypergeometric2F1}\left (m+1,-n,m+2,-\frac {g (d+e x)}{e f-d g}\right )}{e g (m+1) (m+n+2)}-\frac {c (d+e x)^{m+1} (f+g x)^{n+1} (d g (m+2 n+4)+e f (m+2))}{g (m+n+2)}}{e^2 g (m+n+3)}+\frac {c (d+e x)^{m+2} (f+g x)^{n+1}}{e^2 g (m+n+3)}\)

Input:

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

Output:

(c*(d + e*x)^(2 + m)*(f + g*x)^(1 + n))/(e^2*g*(3 + m + n)) + (-((c*(e*f*( 
2 + m) + d*g*(4 + m + 2*n))*(d + e*x)^(1 + m)*(f + g*x)^(1 + n))/(g*(2 + m 
 + n))) + ((a*e^2*g^2*(6 + m^2 + 5*n + n^2 + m*(5 + 2*n)) + c*(e^2*f^2*(2 
+ 3*m + m^2) + 2*d*e*f*g*(1 + m)*(1 + n) + d^2*g^2*(2 + 3*n + n^2)))*(d + 
e*x)^(1 + m)*(f + g*x)^n*Hypergeometric2F1[1 + m, -n, 2 + m, -((g*(d + e*x 
))/(e*f - d*g))])/(e*g*(1 + m)*(2 + m + n)*((e*(f + g*x))/(e*f - d*g))^n)) 
/(e^2*g*(3 + m + n))
 

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 90

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), 
 x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p 
+ 2))   Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, 
p}, x] && NeQ[n + p + 2, 0]
 

rule 651

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^ 
2)^(p_.), x_Symbol] :> Simp[c^p*(d + e*x)^(m + 2*p)*((f + g*x)^(n + 1)/(g*e 
^(2*p)*(m + n + 2*p + 1))), x] + Simp[1/(g*e^(2*p)*(m + n + 2*p + 1))   Int 
[(d + e*x)^m*(f + g*x)^n*ExpandToSum[g*(m + n + 2*p + 1)*(e^(2*p)*(a + c*x^ 
2)^p - c^p*(d + e*x)^(2*p)) - c^p*(e*f - d*g)*(m + 2*p)*(d + e*x)^(2*p - 1) 
, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && IGtQ[p, 0] &&  !IntegerQ[m] 
 &&  !IntegerQ[n] && NeQ[m + n + 2*p + 1, 0]
 

Maple [F]

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

Input:

int((e*x+d)^m*(g*x+f)^n*(c*x^2+a),x)
 

Output:

int((e*x+d)^m*(g*x+f)^n*(c*x^2+a),x)
 

Fricas [F]

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

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

Output:

integral((c*x^2 + a)*(e*x + d)^m*(g*x + f)^n, x)
 

Sympy [F(-2)]

Exception generated. \[ \int (d+e x)^m (f+g x)^n \left (a+c x^2\right ) \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:

integrate((e*x+d)**m*(g*x+f)**n*(c*x**2+a),x)
 

Output:

Exception raised: HeuristicGCDFailed >> no luck
 

Maxima [F]

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

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

Output:

integrate((c*x^2 + a)*(e*x + d)^m*(g*x + f)^n, x)
 

Giac [F]

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

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

Output:

integrate((c*x^2 + a)*(e*x + d)^m*(g*x + f)^n, x)
 

Mupad [F(-1)]

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

int((f + g*x)^n*(a + c*x^2)*(d + e*x)^m,x)
 

Output:

int((f + g*x)^n*(a + c*x^2)*(d + e*x)^m, x)
 

Reduce [F]

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

int((e*x+d)^m*(g*x+f)^n*(c*x^2+a),x)
 

Output:

((f + g*x)**n*(d + e*x)**m*a*d*e**2*f*g**2*m**3 + 3*(f + g*x)**n*(d + e*x) 
**m*a*d*e**2*f*g**2*m**2*n + 5*(f + g*x)**n*(d + e*x)**m*a*d*e**2*f*g**2*m 
**2 + 3*(f + g*x)**n*(d + e*x)**m*a*d*e**2*f*g**2*m*n**2 + 10*(f + g*x)**n 
*(d + e*x)**m*a*d*e**2*f*g**2*m*n + 6*(f + g*x)**n*(d + e*x)**m*a*d*e**2*f 
*g**2*m + (f + g*x)**n*(d + e*x)**m*a*d*e**2*f*g**2*n**3 + 5*(f + g*x)**n* 
(d + e*x)**m*a*d*e**2*f*g**2*n**2 + 6*(f + g*x)**n*(d + e*x)**m*a*d*e**2*f 
*g**2*n + (f + g*x)**n*(d + e*x)**m*a*d*e**2*g**3*m**2*n*x + 2*(f + g*x)** 
n*(d + e*x)**m*a*d*e**2*g**3*m*n**2*x + 5*(f + g*x)**n*(d + e*x)**m*a*d*e* 
*2*g**3*m*n*x + (f + g*x)**n*(d + e*x)**m*a*d*e**2*g**3*n**3*x + 5*(f + g* 
x)**n*(d + e*x)**m*a*d*e**2*g**3*n**2*x + 6*(f + g*x)**n*(d + e*x)**m*a*d* 
e**2*g**3*n*x + (f + g*x)**n*(d + e*x)**m*a*e**3*f*g**2*m**3*x + 2*(f + g* 
x)**n*(d + e*x)**m*a*e**3*f*g**2*m**2*n*x + 5*(f + g*x)**n*(d + e*x)**m*a* 
e**3*f*g**2*m**2*x + (f + g*x)**n*(d + e*x)**m*a*e**3*f*g**2*m*n**2*x + 5* 
(f + g*x)**n*(d + e*x)**m*a*e**3*f*g**2*m*n*x + 6*(f + g*x)**n*(d + e*x)** 
m*a*e**3*f*g**2*m*x + (f + g*x)**n*(d + e*x)**m*c*d**3*f*g**2*m*n + 2*(f + 
 g*x)**n*(d + e*x)**m*c*d**3*f*g**2*m - (f + g*x)**n*(d + e*x)**m*c*d**3*g 
**3*m*n**2*x - 2*(f + g*x)**n*(d + e*x)**m*c*d**3*g**3*m*n*x - 2*(f + g*x) 
**n*(d + e*x)**m*c*d**2*e*f**2*g*m*n - (f + g*x)**n*(d + e*x)**m*c*d**2*e* 
f*g**2*m**2*n*x - 2*(f + g*x)**n*(d + e*x)**m*c*d**2*e*f*g**2*m**2*x + 2*( 
f + g*x)**n*(d + e*x)**m*c*d**2*e*f*g**2*m*n**2*x + (f + g*x)**n*(d + e...