\(\int x^4 (c+d x)^n (a+b x^2)^p \, dx\) [354]

Optimal result

Integrand size = 20, antiderivative size = 670 \[ \int x^4 (c+d x)^n \left (a+b x^2\right )^p \, dx=-\frac {\left (a d^2 (3+n) (4+n+2 p)-b c^2 \left (n^2+n (11+6 p)+6 \left (6+7 p+2 p^2\right )\right )\right ) (c+d x)^{1+n} \left (a+b x^2\right )^{1+p}}{b^2 d^3 (3+n+2 p) (4+n+2 p) (5+n+2 p)}-\frac {2 c (6+n+3 p) (c+d x)^{2+n} \left (a+b x^2\right )^{1+p}}{b d^3 (4+n+2 p) (5+n+2 p)}+\frac {(c+d x)^{3+n} \left (a+b x^2\right )^{1+p}}{b d^3 (5+n+2 p)}+\frac {\left (a d^2 \left (a d^2 \left (3+4 n+n^2\right ) (4+n+2 p)-4 b c^2 n \left (2+3 p+p^2\right )\right )-4 b c^2 (1+p) \left (a d^2 \left (6+5 n+n^2+3 p+2 n p\right )-b c^2 \left (6+7 p+2 p^2\right )\right )\right ) (c+d x)^{1+n} \left (a+b x^2\right )^p \left (1-\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^{-p} \left (1-\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^{-p} \operatorname {AppellF1}\left (1+n,-p,-p,2+n,\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}},\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )}{b^2 d^5 (1+n) (3+n+2 p) (4+n+2 p) (5+n+2 p)}+\frac {4 c (1+p) \left (a d^2 \left (6+5 n+n^2+3 p+2 n p\right )-b c^2 \left (6+7 p+2 p^2\right )\right ) (c+d x)^{2+n} \left (a+b x^2\right )^p \left (1-\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^{-p} \left (1-\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^{-p} \operatorname {AppellF1}\left (2+n,-p,-p,3+n,\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}},\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )}{b d^5 (2+n) (3+n+2 p) (4+n+2 p) (5+n+2 p)} \] Output:

-(a*d^2*(3+n)*(4+n+2*p)-b*c^2*(n^2+n*(11+6*p)+12*p^2+42*p+36))*(d*x+c)^(1+ 
n)*(b*x^2+a)^(p+1)/b^2/d^3/(3+n+2*p)/(4+n+2*p)/(5+n+2*p)-2*c*(6+n+3*p)*(d* 
x+c)^(2+n)*(b*x^2+a)^(p+1)/b/d^3/(4+n+2*p)/(5+n+2*p)+(d*x+c)^(3+n)*(b*x^2+ 
a)^(p+1)/b/d^3/(5+n+2*p)+(a*d^2*(a*d^2*(n^2+4*n+3)*(4+n+2*p)-4*b*c^2*n*(p^ 
2+3*p+2))-4*b*c^2*(p+1)*(a*d^2*(n^2+2*n*p+5*n+3*p+6)-b*c^2*(2*p^2+7*p+6))) 
*(d*x+c)^(1+n)*(b*x^2+a)^p*AppellF1(1+n,-p,-p,2+n,(d*x+c)/(c-(-a)^(1/2)*d/ 
b^(1/2)),(d*x+c)/(c+(-a)^(1/2)*d/b^(1/2)))/b^2/d^5/(1+n)/(3+n+2*p)/(4+n+2* 
p)/(5+n+2*p)/((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)+4*c*(p+1)*(a*d^2*(n^2+2*n*p+5*n+3*p+6)-b*c^2*(2*p^2+7 
*p+6))*(d*x+c)^(2+n)*(b*x^2+a)^p*AppellF1(2+n,-p,-p,3+n,(d*x+c)/(c-(-a)^(1 
/2)*d/b^(1/2)),(d*x+c)/(c+(-a)^(1/2)*d/b^(1/2)))/b/d^5/(2+n)/(3+n+2*p)/(4+ 
n+2*p)/(5+n+2*p)/((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]

\[ \int x^4 (c+d x)^n \left (a+b x^2\right )^p \, dx=\int x^4 (c+d x)^n \left (a+b x^2\right )^p \, dx \] Input:

Integrate[x^4*(c + d*x)^n*(a + b*x^2)^p,x]
 

Output:

Integrate[x^4*(c + d*x)^n*(a + b*x^2)^p, x]
 

Rubi [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(2533\) vs. \(2(670)=1340\).

Time = 2.08 (sec) , antiderivative size = 2533, normalized size of antiderivative = 3.78, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {624, 624, 624, 624, 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.

\(\displaystyle \int x^4 \left (a+b x^2\right )^p (c+d x)^n \, dx\)

\(\Big \downarrow \) 624

\(\displaystyle \frac {\int x^3 (c+d x)^{n+1} \left (b x^2+a\right )^pdx}{d}-\frac {c \int x^3 (c+d x)^n \left (b x^2+a\right )^pdx}{d}\)

\(\Big \downarrow \) 624

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

\(\Big \downarrow \) 624

\(\displaystyle \frac {\frac {\frac {\int x (c+d x)^{n+3} \left (b x^2+a\right )^pdx}{d}-\frac {c \int x (c+d x)^{n+2} \left (b x^2+a\right )^pdx}{d}}{d}-\frac {c \left (\frac {\int x (c+d x)^{n+2} \left (b x^2+a\right )^pdx}{d}-\frac {c \int x (c+d x)^{n+1} \left (b x^2+a\right )^pdx}{d}\right )}{d}}{d}-\frac {c \left (\frac {\frac {\int x (c+d x)^{n+2} \left (b x^2+a\right )^pdx}{d}-\frac {c \int x (c+d x)^{n+1} \left (b x^2+a\right )^pdx}{d}}{d}-\frac {c \left (\frac {\int x (c+d x)^{n+1} \left (b x^2+a\right )^pdx}{d}-\frac {c \int x (c+d x)^n \left (b x^2+a\right )^pdx}{d}\right )}{d}\right )}{d}\)

\(\Big \downarrow \) 624

\(\displaystyle \frac {\frac {\frac {\frac {\int (c+d x)^{n+4} \left (b x^2+a\right )^pdx}{d}-\frac {c \int (c+d x)^{n+3} \left (b x^2+a\right )^pdx}{d}}{d}-\frac {c \left (\frac {\int (c+d x)^{n+3} \left (b x^2+a\right )^pdx}{d}-\frac {c \int (c+d x)^{n+2} \left (b x^2+a\right )^pdx}{d}\right )}{d}}{d}-\frac {c \left (\frac {\frac {\int (c+d x)^{n+3} \left (b x^2+a\right )^pdx}{d}-\frac {c \int (c+d x)^{n+2} \left (b x^2+a\right )^pdx}{d}}{d}-\frac {c \left (\frac {\int (c+d x)^{n+2} \left (b x^2+a\right )^pdx}{d}-\frac {c \int (c+d x)^{n+1} \left (b x^2+a\right )^pdx}{d}\right )}{d}\right )}{d}}{d}-\frac {c \left (\frac {\frac {\frac {\int (c+d x)^{n+3} \left (b x^2+a\right )^pdx}{d}-\frac {c \int (c+d x)^{n+2} \left (b x^2+a\right )^pdx}{d}}{d}-\frac {c \left (\frac {\int (c+d x)^{n+2} \left (b x^2+a\right )^pdx}{d}-\frac {c \int (c+d x)^{n+1} \left (b x^2+a\right )^pdx}{d}\right )}{d}}{d}-\frac {c \left (\frac {\frac {\int (c+d x)^{n+2} \left (b x^2+a\right )^pdx}{d}-\frac {c \int (c+d x)^{n+1} \left (b x^2+a\right )^pdx}{d}}{d}-\frac {c \left (\frac {\int (c+d x)^{n+1} \left (b x^2+a\right )^pdx}{d}-\frac {c \int (c+d x)^n \left (b x^2+a\right )^pdx}{d}\right )}{d}\right )}{d}\right )}{d}\)

\(\Big \downarrow \) 514

\(\displaystyle \frac {\frac {\frac {\frac {\left (b x^2+a\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} \int (c+d x)^{n+4} \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 {c \left (b x^2+a\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} \int (c+d x)^{n+3} \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}}{d}-\frac {c \left (\frac {\left (b x^2+a\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} \int (c+d x)^{n+3} \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 {c \left (b x^2+a\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} \int (c+d x)^{n+2} \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}\right )}{d}}{d}-\frac {c \left (\frac {\frac {\left (b x^2+a\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} \int (c+d x)^{n+3} \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 {c \left (b x^2+a\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} \int (c+d x)^{n+2} \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}}{d}-\frac {c \left (\frac {\left (b x^2+a\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} \int (c+d x)^{n+2} \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 {c \left (b x^2+a\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} \int (c+d x)^{n+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}\right )}{d}\right )}{d}}{d}-\frac {c \left (\frac {\frac {\frac {\left (b x^2+a\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} \int (c+d x)^{n+3} \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 {c \left (b x^2+a\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} \int (c+d x)^{n+2} \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}}{d}-\frac {c \left (\frac {\left (b x^2+a\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} \int (c+d x)^{n+2} \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 {c \left (b x^2+a\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} \int (c+d x)^{n+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}\right )}{d}}{d}-\frac {c \left (\frac {\frac {\left (b x^2+a\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} \int (c+d x)^{n+2} \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 {c \left (b x^2+a\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} \int (c+d x)^{n+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}}{d}-\frac {c \left (\frac {\left (b x^2+a\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} \int (c+d x)^{n+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 {c \left (b x^2+a\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} \int (c+d x)^n \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}\right )}{d}\right )}{d}\right )}{d}\)

\(\Big \downarrow \) 150

\(\displaystyle \frac {\frac {\frac {\frac {(c+d x)^{n+5} \left (b x^2+a\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 (n+5,-p,-p,n+6,\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 (n+5)}-\frac {c (c+d x)^{n+4} \left (b x^2+a\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 (n+4,-p,-p,n+5,\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 (n+4)}}{d}-\frac {c \left (\frac {(c+d x)^{n+4} \left (b x^2+a\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 (n+4,-p,-p,n+5,\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 (n+4)}-\frac {c (c+d x)^{n+3} \left (b x^2+a\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 (n+3,-p,-p,n+4,\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 (n+3)}\right )}{d}}{d}-\frac {c \left (\frac {\frac {(c+d x)^{n+4} \left (b x^2+a\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 (n+4,-p,-p,n+5,\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 (n+4)}-\frac {c (c+d x)^{n+3} \left (b x^2+a\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 (n+3,-p,-p,n+4,\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 (n+3)}}{d}-\frac {c \left (\frac {(c+d x)^{n+3} \left (b x^2+a\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 (n+3,-p,-p,n+4,\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 (n+3)}-\frac {c (c+d x)^{n+2} \left (b x^2+a\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 (n+2,-p,-p,n+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 (n+2)}\right )}{d}\right )}{d}}{d}-\frac {c \left (\frac {\frac {\frac {(c+d x)^{n+4} \left (b x^2+a\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 (n+4,-p,-p,n+5,\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 (n+4)}-\frac {c (c+d x)^{n+3} \left (b x^2+a\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 (n+3,-p,-p,n+4,\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 (n+3)}}{d}-\frac {c \left (\frac {(c+d x)^{n+3} \left (b x^2+a\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 (n+3,-p,-p,n+4,\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 (n+3)}-\frac {c (c+d x)^{n+2} \left (b x^2+a\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 (n+2,-p,-p,n+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 (n+2)}\right )}{d}}{d}-\frac {c \left (\frac {\frac {(c+d x)^{n+3} \left (b x^2+a\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 (n+3,-p,-p,n+4,\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 (n+3)}-\frac {c (c+d x)^{n+2} \left (b x^2+a\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 (n+2,-p,-p,n+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 (n+2)}}{d}-\frac {c \left (\frac {(c+d x)^{n+2} \left (b x^2+a\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 (n+2,-p,-p,n+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 (n+2)}-\frac {c (c+d x)^{n+1} \left (b x^2+a\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 (n+1,-p,-p,n+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 (n+1)}\right )}{d}\right )}{d}\right )}{d}\)

Input:

Int[x^4*(c + d*x)^n*(a + b*x^2)^p,x]
 

Output:

-((c*(-((c*(-((c*(-((c*(c + d*x)^(1 + n)*(a + b*x^2)^p*AppellF1[1 + n, -p, 
 -p, 2 + n, (c + d*x)/(c - (Sqrt[-a]*d)/Sqrt[b]), (c + d*x)/(c + (Sqrt[-a] 
*d)/Sqrt[b])])/(d^2*(1 + n)*(1 - (c + d*x)/(c - (Sqrt[-a]*d)/Sqrt[b]))^p*( 
1 - (c + d*x)/(c + (Sqrt[-a]*d)/Sqrt[b]))^p)) + ((c + d*x)^(2 + n)*(a + b* 
x^2)^p*AppellF1[2 + n, -p, -p, 3 + n, (c + d*x)/(c - (Sqrt[-a]*d)/Sqrt[b]) 
, (c + d*x)/(c + (Sqrt[-a]*d)/Sqrt[b])])/(d^2*(2 + n)*(1 - (c + d*x)/(c - 
(Sqrt[-a]*d)/Sqrt[b]))^p*(1 - (c + d*x)/(c + (Sqrt[-a]*d)/Sqrt[b]))^p)))/d 
) + (-((c*(c + d*x)^(2 + n)*(a + b*x^2)^p*AppellF1[2 + n, -p, -p, 3 + n, ( 
c + d*x)/(c - (Sqrt[-a]*d)/Sqrt[b]), (c + d*x)/(c + (Sqrt[-a]*d)/Sqrt[b])] 
)/(d^2*(2 + n)*(1 - (c + d*x)/(c - (Sqrt[-a]*d)/Sqrt[b]))^p*(1 - (c + d*x) 
/(c + (Sqrt[-a]*d)/Sqrt[b]))^p)) + ((c + d*x)^(3 + n)*(a + b*x^2)^p*Appell 
F1[3 + n, -p, -p, 4 + n, (c + d*x)/(c - (Sqrt[-a]*d)/Sqrt[b]), (c + d*x)/( 
c + (Sqrt[-a]*d)/Sqrt[b])])/(d^2*(3 + n)*(1 - (c + d*x)/(c - (Sqrt[-a]*d)/ 
Sqrt[b]))^p*(1 - (c + d*x)/(c + (Sqrt[-a]*d)/Sqrt[b]))^p))/d))/d) + (-((c* 
(-((c*(c + d*x)^(2 + n)*(a + b*x^2)^p*AppellF1[2 + n, -p, -p, 3 + n, (c + 
d*x)/(c - (Sqrt[-a]*d)/Sqrt[b]), (c + d*x)/(c + (Sqrt[-a]*d)/Sqrt[b])])/(d 
^2*(2 + n)*(1 - (c + d*x)/(c - (Sqrt[-a]*d)/Sqrt[b]))^p*(1 - (c + d*x)/(c 
+ (Sqrt[-a]*d)/Sqrt[b]))^p)) + ((c + d*x)^(3 + n)*(a + b*x^2)^p*AppellF1[3 
 + n, -p, -p, 4 + n, (c + d*x)/(c - (Sqrt[-a]*d)/Sqrt[b]), (c + d*x)/(c + 
(Sqrt[-a]*d)/Sqrt[b])])/(d^2*(3 + n)*(1 - (c + d*x)/(c - (Sqrt[-a]*d)/S...
 

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 624

Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> Simp[1/d   Int[x^(m - 1)*(c + d*x)^(n + 1)*(a + b*x^2)^p, x], x] - Si 
mp[c/d   Int[x^(m - 1)*(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, 
 d, n, p}, x] && IGtQ[m, 0]
 

Maple [F]

\[\int x^{4} \left (d x +c \right )^{n} \left (b \,x^{2}+a \right )^{p}d x\]

Input:

int(x^4*(d*x+c)^n*(b*x^2+a)^p,x)
 

Output:

int(x^4*(d*x+c)^n*(b*x^2+a)^p,x)
 

Fricas [F]

\[ \int x^4 (c+d x)^n \left (a+b x^2\right )^p \, dx=\int { {\left (b x^{2} + a\right )}^{p} {\left (d x + c\right )}^{n} x^{4} \,d x } \] Input:

integrate(x^4*(d*x+c)^n*(b*x^2+a)^p,x, algorithm="fricas")
 

Output:

integral((b*x^2 + a)^p*(d*x + c)^n*x^4, x)
 

Sympy [F(-1)]

Timed out. \[ \int x^4 (c+d x)^n \left (a+b x^2\right )^p \, dx=\text {Timed out} \] Input:

integrate(x**4*(d*x+c)**n*(b*x**2+a)**p,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int x^4 (c+d x)^n \left (a+b x^2\right )^p \, dx=\int { {\left (b x^{2} + a\right )}^{p} {\left (d x + c\right )}^{n} x^{4} \,d x } \] Input:

integrate(x^4*(d*x+c)^n*(b*x^2+a)^p,x, algorithm="maxima")
 

Output:

integrate((b*x^2 + a)^p*(d*x + c)^n*x^4, x)
 

Giac [F]

\[ \int x^4 (c+d x)^n \left (a+b x^2\right )^p \, dx=\int { {\left (b x^{2} + a\right )}^{p} {\left (d x + c\right )}^{n} x^{4} \,d x } \] Input:

integrate(x^4*(d*x+c)^n*(b*x^2+a)^p,x, algorithm="giac")
 

Output:

integrate((b*x^2 + a)^p*(d*x + c)^n*x^4, x)
 

Mupad [F(-1)]

Timed out. \[ \int x^4 (c+d x)^n \left (a+b x^2\right )^p \, dx=\int x^4\,{\left (b\,x^2+a\right )}^p\,{\left (c+d\,x\right )}^n \,d x \] Input:

int(x^4*(a + b*x^2)^p*(c + d*x)^n,x)
 

Output:

int(x^4*(a + b*x^2)^p*(c + d*x)^n, x)
 

Reduce [F]

\[ \int x^4 (c+d x)^n \left (a+b x^2\right )^p \, dx=\text {too large to display} \] Input:

int(x^4*(d*x+c)^n*(b*x^2+a)^p,x)
 

Output:

((c + d*x)**n*(a + b*x**2)**p*a**3*d**5*n**4 + 4*(c + d*x)**n*(a + b*x**2) 
**p*a**3*d**5*n**3*p + 10*(c + d*x)**n*(a + b*x**2)**p*a**3*d**5*n**3 + 4* 
(c + d*x)**n*(a + b*x**2)**p*a**3*d**5*n**2*p**2 + 28*(c + d*x)**n*(a + b* 
x**2)**p*a**3*d**5*n**2*p + 35*(c + d*x)**n*(a + b*x**2)**p*a**3*d**5*n**2 
 + 16*(c + d*x)**n*(a + b*x**2)**p*a**3*d**5*n*p**2 + 60*(c + d*x)**n*(a + 
 b*x**2)**p*a**3*d**5*n*p + 50*(c + d*x)**n*(a + b*x**2)**p*a**3*d**5*n + 
12*(c + d*x)**n*(a + b*x**2)**p*a**3*d**5*p**2 + 36*(c + d*x)**n*(a + b*x* 
*2)**p*a**3*d**5*p + 24*(c + d*x)**n*(a + b*x**2)**p*a**3*d**5 - (c + d*x) 
**n*(a + b*x**2)**p*a**2*b*c**2*d**3*n**4 + 2*(c + d*x)**n*(a + b*x**2)**p 
*a**2*b*c**2*d**3*n**3*p - 2*(c + d*x)**n*(a + b*x**2)**p*a**2*b*c**2*d**3 
*n**3 + 8*(c + d*x)**n*(a + b*x**2)**p*a**2*b*c**2*d**3*n**2*p**2 + 10*(c 
+ d*x)**n*(a + b*x**2)**p*a**2*b*c**2*d**3*n**2*p + (c + d*x)**n*(a + b*x* 
*2)**p*a**2*b*c**2*d**3*n**2 - 8*(c + d*x)**n*(a + b*x**2)**p*a**2*b*c**2* 
d**3*n*p**2 - 12*(c + d*x)**n*(a + b*x**2)**p*a**2*b*c**2*d**3*n*p + 2*(c 
+ d*x)**n*(a + b*x**2)**p*a**2*b*c**2*d**3*n - 2*(c + d*x)**n*(a + b*x**2) 
**p*a**2*b*c*d**4*n**3*p*x - 8*(c + d*x)**n*(a + b*x**2)**p*a**2*b*c*d**4* 
n**2*p**2*x - 18*(c + d*x)**n*(a + b*x**2)**p*a**2*b*c*d**4*n**2*p*x - 8*( 
c + d*x)**n*(a + b*x**2)**p*a**2*b*c*d**4*n*p**3*x - 48*(c + d*x)**n*(a + 
b*x**2)**p*a**2*b*c*d**4*n*p**2*x - 52*(c + d*x)**n*(a + b*x**2)**p*a**2*b 
*c*d**4*n*p*x - 24*(c + d*x)**n*(a + b*x**2)**p*a**2*b*c*d**4*p**3*x - ...