\(\int (A+B x) (c+d x)^m (a+b x^2)^p \, dx\) [302]

Optimal result

Integrand size = 22, antiderivative size = 315 \[ \int (A+B x) (c+d x)^m \left (a+b x^2\right )^p \, dx=-\frac {(B c-A d) (c+d x)^{1+m} \left (a+b x^2\right )^p \left (1-\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^{-p} \left (1-\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^{-p} \operatorname {AppellF1}\left (1+m,-p,-p,2+m,\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}},\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )}{d^2 (1+m)}+\frac {B (c+d x)^{2+m} \left (a+b x^2\right )^p \left (1-\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^{-p} \left (1-\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^{-p} \operatorname {AppellF1}\left (2+m,-p,-p,3+m,\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}},\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )}{d^2 (2+m)} \] Output:

-(-A*d+B*c)*(d*x+c)^(1+m)*(b*x^2+a)^p*AppellF1(1+m,-p,-p,2+m,(d*x+c)/(c-(- 
a)^(1/2)*d/b^(1/2)),(d*x+c)/(c+(-a)^(1/2)*d/b^(1/2)))/d^2/(1+m)/((1-(d*x+c 
)/(c-(-a)^(1/2)*d/b^(1/2)))^p)/((1-(d*x+c)/(c+(-a)^(1/2)*d/b^(1/2)))^p)+B* 
(d*x+c)^(2+m)*(b*x^2+a)^p*AppellF1(2+m,-p,-p,3+m,(d*x+c)/(c-(-a)^(1/2)*d/b 
^(1/2)),(d*x+c)/(c+(-a)^(1/2)*d/b^(1/2)))/d^2/(2+m)/((1-(d*x+c)/(c-(-a)^(1 
/2)*d/b^(1/2)))^p)/((1-(d*x+c)/(c+(-a)^(1/2)*d/b^(1/2)))^p)
 

Mathematica [F]

\[ \int (A+B x) (c+d x)^m \left (a+b x^2\right )^p \, dx=\int (A+B x) (c+d x)^m \left (a+b x^2\right )^p \, dx \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {719, 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.

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

\(\Big \downarrow \) 719

\(\displaystyle \frac {B \int (c+d x)^{m+1} \left (b x^2+a\right )^pdx}{d}-\frac {(B c-A d) \int (c+d x)^m \left (b x^2+a\right )^pdx}{d}\)

\(\Big \downarrow \) 514

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

\(\Big \downarrow \) 150

\(\displaystyle \frac {B \left (a+b x^2\right )^p (c+d x)^{m+2} \left (1-\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^{-p} \left (1-\frac {c+d x}{\frac {\sqrt {-a} d}{\sqrt {b}}+c}\right )^{-p} \operatorname {AppellF1}\left (m+2,-p,-p,m+3,\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}},\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )}{d^2 (m+2)}-\frac {\left (a+b x^2\right )^p (B c-A d) (c+d x)^{m+1} \left (1-\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^{-p} \left (1-\frac {c+d x}{\frac {\sqrt {-a} d}{\sqrt {b}}+c}\right )^{-p} \operatorname {AppellF1}\left (m+1,-p,-p,m+2,\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}},\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )}{d^2 (m+1)}\)

Input:

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

Output:

-(((B*c - A*d)*(c + d*x)^(1 + m)*(a + b*x^2)^p*AppellF1[1 + m, -p, -p, 2 + 
 m, (c + d*x)/(c - (Sqrt[-a]*d)/Sqrt[b]), (c + d*x)/(c + (Sqrt[-a]*d)/Sqrt 
[b])])/(d^2*(1 + m)*(1 - (c + d*x)/(c - (Sqrt[-a]*d)/Sqrt[b]))^p*(1 - (c + 
 d*x)/(c + (Sqrt[-a]*d)/Sqrt[b]))^p)) + (B*(c + d*x)^(2 + m)*(a + b*x^2)^p 
*AppellF1[2 + m, -p, -p, 3 + m, (c + d*x)/(c - (Sqrt[-a]*d)/Sqrt[b]), (c + 
 d*x)/(c + (Sqrt[-a]*d)/Sqrt[b])])/(d^2*(2 + m)*(1 - (c + d*x)/(c - (Sqrt[ 
-a]*d)/Sqrt[b]))^p*(1 - (c + d*x)/(c + (Sqrt[-a]*d)/Sqrt[b]))^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]
 

rule 719

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

Maple [F]

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

Input:

int((B*x+A)*(d*x+c)^m*(b*x^2+a)^p,x)
 

Output:

int((B*x+A)*(d*x+c)^m*(b*x^2+a)^p,x)
 

Fricas [F]

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

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

Output:

integral((B*x + A)*(b*x^2 + a)^p*(d*x + c)^m, x)
 

Sympy [F(-1)]

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

integrate((B*x+A)*(d*x+c)**m*(b*x**2+a)**p,x)
 

Output:

Timed out
 

Maxima [F]

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

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

Output:

integrate((B*x + A)*(b*x^2 + a)^p*(d*x + c)^m, x)
 

Giac [F]

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

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

Output:

integrate((B*x + A)*(b*x^2 + a)^p*(d*x + c)^m, x)
 

Mupad [F(-1)]

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

int((a + b*x^2)^p*(A + B*x)*(c + d*x)^m,x)
 

Output:

int((a + b*x^2)^p*(A + B*x)*(c + d*x)^m, x)
 

Reduce [F]

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

int((B*x+A)*(d*x+c)^m*(b*x^2+a)^p,x)
 

Output:

((c + d*x)**m*(a + b*x**2)**p*a**2*d**2*m + 2*(c + d*x)**m*(a + b*x**2)**p 
*a**2*d**2*p + 2*(c + d*x)**m*(a + b*x**2)**p*a**2*d**2 + (c + d*x)**m*(a 
+ b*x**2)**p*a*b*c*d*m*x + 2*(c + d*x)**m*(a + b*x**2)**p*a*b*c*d*m + 2*(c 
 + d*x)**m*(a + b*x**2)**p*a*b*c*d*p*x + 2*(c + d*x)**m*(a + b*x**2)**p*a* 
b*c*d*p + 2*(c + d*x)**m*(a + b*x**2)**p*a*b*c*d*x + (c + d*x)**m*(a + b*x 
**2)**p*a*b*c*d + (c + d*x)**m*(a + b*x**2)**p*b**2*c**2*m*x + (c + d*x)** 
m*(a + b*x**2)**p*b**2*c*d*m*x**2 + 2*(c + d*x)**m*(a + b*x**2)**p*b**2*c* 
d*p*x**2 + (c + d*x)**m*(a + b*x**2)**p*b**2*c*d*x**2 - int(((c + d*x)**m* 
(a + b*x**2)**p*x**2)/(a*c*m**2 + 4*a*c*m*p + 3*a*c*m + 4*a*c*p**2 + 6*a*c 
*p + 2*a*c + a*d*m**2*x + 4*a*d*m*p*x + 3*a*d*m*x + 4*a*d*p**2*x + 6*a*d*p 
*x + 2*a*d*x + b*c*m**2*x**2 + 4*b*c*m*p*x**2 + 3*b*c*m*x**2 + 4*b*c*p**2* 
x**2 + 6*b*c*p*x**2 + 2*b*c*x**2 + b*d*m**2*x**3 + 4*b*d*m*p*x**3 + 3*b*d* 
m*x**3 + 4*b*d*p**2*x**3 + 6*b*d*p*x**3 + 2*b*d*x**3),x)*a**2*b*d**3*m**4 
- 8*int(((c + d*x)**m*(a + b*x**2)**p*x**2)/(a*c*m**2 + 4*a*c*m*p + 3*a*c* 
m + 4*a*c*p**2 + 6*a*c*p + 2*a*c + a*d*m**2*x + 4*a*d*m*p*x + 3*a*d*m*x + 
4*a*d*p**2*x + 6*a*d*p*x + 2*a*d*x + b*c*m**2*x**2 + 4*b*c*m*p*x**2 + 3*b* 
c*m*x**2 + 4*b*c*p**2*x**2 + 6*b*c*p*x**2 + 2*b*c*x**2 + b*d*m**2*x**3 + 4 
*b*d*m*p*x**3 + 3*b*d*m*x**3 + 4*b*d*p**2*x**3 + 6*b*d*p*x**3 + 2*b*d*x**3 
),x)*a**2*b*d**3*m**3*p - 5*int(((c + d*x)**m*(a + b*x**2)**p*x**2)/(a*c*m 
**2 + 4*a*c*m*p + 3*a*c*m + 4*a*c*p**2 + 6*a*c*p + 2*a*c + a*d*m**2*x +...