\(\int \frac {x^5 (a+b x^2)^p}{c+d x} \, dx\) [29]

Optimal result

Integrand size = 20, antiderivative size = 202 \[ \int \frac {x^5 \left (a+b x^2\right )^p}{c+d x} \, dx=-\frac {c \left (b c^2-a d^2\right ) \left (a+b x^2\right )^{1+p}}{2 b^2 d^4 (1+p)}-\frac {c \left (a+b x^2\right )^{2+p}}{2 b^2 d^2 (2+p)}-\frac {d x^7 \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {7}{2},-p,1,\frac {9}{2},-\frac {b x^2}{a},\frac {d^2 x^2}{c^2}\right )}{7 c^2}+\frac {c^5 \left (a+b x^2\right )^{1+p} \operatorname {Hypergeometric2F1}\left (1,1+p,2+p,\frac {d^2 \left (a+b x^2\right )}{b c^2+a d^2}\right )}{2 d^4 \left (b c^2+a d^2\right ) (1+p)} \] Output:

-1/2*c*(-a*d^2+b*c^2)*(b*x^2+a)^(p+1)/b^2/d^4/(p+1)-1/2*c*(b*x^2+a)^(2+p)/ 
b^2/d^2/(2+p)-1/7*d*x^7*(b*x^2+a)^p*AppellF1(7/2,1,-p,9/2,d^2*x^2/c^2,-b*x 
^2/a)/c^2/((1+b*x^2/a)^p)+1/2*c^5*(b*x^2+a)^(p+1)*hypergeom([1, p+1],[2+p] 
,d^2*(b*x^2+a)/(a*d^2+b*c^2))/d^4/(a*d^2+b*c^2)/(p+1)
 

Mathematica [F]

\[ \int \frac {x^5 \left (a+b x^2\right )^p}{c+d x} \, dx=\int \frac {x^5 \left (a+b x^2\right )^p}{c+d x} \, dx \] Input:

Integrate[(x^5*(a + b*x^2)^p)/(c + d*x),x]
 

Output:

Integrate[(x^5*(a + b*x^2)^p)/(c + d*x), x]
 

Rubi [A] (verified)

Time = 0.67 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.99, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {621, 354, 99, 395, 394, 2009}

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.

\(\displaystyle \int \frac {x^5 \left (a+b x^2\right )^p}{c+d x} \, dx\)

\(\Big \downarrow \) 621

\(\displaystyle c \int \frac {x^5 \left (b x^2+a\right )^p}{c^2-d^2 x^2}dx-d \int \frac {x^6 \left (b x^2+a\right )^p}{c^2-d^2 x^2}dx\)

\(\Big \downarrow \) 354

\(\displaystyle \frac {1}{2} c \int \frac {x^4 \left (b x^2+a\right )^p}{c^2-d^2 x^2}dx^2-d \int \frac {x^6 \left (b x^2+a\right )^p}{c^2-d^2 x^2}dx\)

\(\Big \downarrow \) 99

\(\displaystyle \frac {1}{2} c \int \left (\frac {\left (a d^2-b c^2\right ) \left (b x^2+a\right )^p}{b d^4}+\frac {c^4 \left (b x^2+a\right )^p}{d^4 \left (c^2-d^2 x^2\right )}-\frac {\left (b x^2+a\right )^{p+1}}{b d^2}\right )dx^2-d \int \frac {x^6 \left (b x^2+a\right )^p}{c^2-d^2 x^2}dx\)

\(\Big \downarrow \) 395

\(\displaystyle \frac {1}{2} c \int \left (\frac {\left (a d^2-b c^2\right ) \left (b x^2+a\right )^p}{b d^4}+\frac {c^4 \left (b x^2+a\right )^p}{d^4 \left (c^2-d^2 x^2\right )}-\frac {\left (b x^2+a\right )^{p+1}}{b d^2}\right )dx^2-d \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \int \frac {x^6 \left (\frac {b x^2}{a}+1\right )^p}{c^2-d^2 x^2}dx\)

\(\Big \downarrow \) 394

\(\displaystyle \frac {1}{2} c \int \left (\frac {\left (a d^2-b c^2\right ) \left (b x^2+a\right )^p}{b d^4}+\frac {c^4 \left (b x^2+a\right )^p}{d^4 \left (c^2-d^2 x^2\right )}-\frac {\left (b x^2+a\right )^{p+1}}{b d^2}\right )dx^2-\frac {d x^7 \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {7}{2},-p,1,\frac {9}{2},-\frac {b x^2}{a},\frac {d^2 x^2}{c^2}\right )}{7 c^2}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{2} c \left (-\frac {\left (b c^2-a d^2\right ) \left (a+b x^2\right )^{p+1}}{b^2 d^4 (p+1)}-\frac {\left (a+b x^2\right )^{p+2}}{b^2 d^2 (p+2)}+\frac {c^4 \left (a+b x^2\right )^{p+1} \operatorname {Hypergeometric2F1}\left (1,p+1,p+2,\frac {d^2 \left (b x^2+a\right )}{b c^2+a d^2}\right )}{d^4 (p+1) \left (a d^2+b c^2\right )}\right )-\frac {d x^7 \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {7}{2},-p,1,\frac {9}{2},-\frac {b x^2}{a},\frac {d^2 x^2}{c^2}\right )}{7 c^2}\)

Input:

Int[(x^5*(a + b*x^2)^p)/(c + d*x),x]
 

Output:

-1/7*(d*x^7*(a + b*x^2)^p*AppellF1[7/2, -p, 1, 9/2, -((b*x^2)/a), (d^2*x^2 
)/c^2])/(c^2*(1 + (b*x^2)/a)^p) + (c*(-(((b*c^2 - a*d^2)*(a + b*x^2)^(1 + 
p))/(b^2*d^4*(1 + p))) - (a + b*x^2)^(2 + p)/(b^2*d^2*(2 + p)) + (c^4*(a + 
 b*x^2)^(1 + p)*Hypergeometric2F1[1, 1 + p, 2 + p, (d^2*(a + b*x^2))/(b*c^ 
2 + a*d^2)])/(d^4*(b*c^2 + a*d^2)*(1 + p))))/2
 

Defintions of rubi rules used

rule 99

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], 
 x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | 
| (GtQ[m, 0] && GeQ[n, -1]))
 

rule 354

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

rule 394

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ 
), x_Symbol] :> Simp[a^p*c^q*((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m + 1)/2 
, -p, -q, 1 + (m + 1)/2, (-b)*(x^2/a), (-d)*(x^2/c)], x] /; FreeQ[{a, b, c, 
 d, e, m, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, 1] && (Int 
egerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
 

rule 395

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ 
), x_Symbol] :> Simp[a^IntPart[p]*((a + b*x^2)^FracPart[p]/(1 + b*(x^2/a))^ 
FracPart[p])   Int[(e*x)^m*(1 + b*(x^2/a))^p*(c + d*x^2)^q, x], x] /; FreeQ 
[{a, b, c, d, e, m, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, 
1] &&  !(IntegerQ[p] || GtQ[a, 0])
 

rule 621

Int[((x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_))/((c_) + (d_.)*(x_)), x_Symbol] 
:> Simp[c   Int[x^m*((a + b*x^2)^p/(c^2 - d^2*x^2)), x], x] - Simp[d   Int[ 
x^(m + 1)*((a + b*x^2)^p/(c^2 - d^2*x^2)), x], x] /; FreeQ[{a, b, c, d, m, 
p}, x]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [F]

\[\int \frac {x^{5} \left (b \,x^{2}+a \right )^{p}}{d x +c}d x\]

Input:

int(x^5*(b*x^2+a)^p/(d*x+c),x)
 

Output:

int(x^5*(b*x^2+a)^p/(d*x+c),x)
 

Fricas [F]

\[ \int \frac {x^5 \left (a+b x^2\right )^p}{c+d x} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{p} x^{5}}{d x + c} \,d x } \] Input:

integrate(x^5*(b*x^2+a)^p/(d*x+c),x, algorithm="fricas")
 

Output:

integral((b*x^2 + a)^p*x^5/(d*x + c), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^5 \left (a+b x^2\right )^p}{c+d x} \, dx=\text {Timed out} \] Input:

integrate(x**5*(b*x**2+a)**p/(d*x+c),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {x^5 \left (a+b x^2\right )^p}{c+d x} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{p} x^{5}}{d x + c} \,d x } \] Input:

integrate(x^5*(b*x^2+a)^p/(d*x+c),x, algorithm="maxima")
 

Output:

integrate((b*x^2 + a)^p*x^5/(d*x + c), x)
 

Giac [F]

\[ \int \frac {x^5 \left (a+b x^2\right )^p}{c+d x} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{p} x^{5}}{d x + c} \,d x } \] Input:

integrate(x^5*(b*x^2+a)^p/(d*x+c),x, algorithm="giac")
 

Output:

integrate((b*x^2 + a)^p*x^5/(d*x + c), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^5 \left (a+b x^2\right )^p}{c+d x} \, dx=\int \frac {x^5\,{\left (b\,x^2+a\right )}^p}{c+d\,x} \,d x \] Input:

int((x^5*(a + b*x^2)^p)/(c + d*x),x)
 

Output:

int((x^5*(a + b*x^2)^p)/(c + d*x), x)
 

Reduce [F]

\[ \int \frac {x^5 \left (a+b x^2\right )^p}{c+d x} \, dx=\text {too large to display} \] Input:

int(x^5*(b*x^2+a)^p/(d*x+c),x)
                                                                                    
                                                                                    
 

Output:

(6*(a + b*x**2)**p*a**3*d**5*p**2 + 18*(a + b*x**2)**p*a**3*d**5*p + 12*(a 
 + b*x**2)**p*a**3*d**5 + 4*(a + b*x**2)**p*a**2*b*c**2*d**3*p**3 + 14*(a 
+ b*x**2)**p*a**2*b*c**2*d**3*p**2 + 8*(a + b*x**2)**p*a**2*b*c**2*d**3*p 
- 5*(a + b*x**2)**p*a**2*b*c**2*d**3 - 12*(a + b*x**2)**p*a**2*b*c*d**4*p* 
*3*x - 36*(a + b*x**2)**p*a**2*b*c*d**4*p**2*x - 24*(a + b*x**2)**p*a**2*b 
*c*d**4*p*x + 4*(a + b*x**2)**p*a*b**2*c**4*d*p**3 + 24*(a + b*x**2)**p*a* 
b**2*c**4*d*p**2 + 47*(a + b*x**2)**p*a*b**2*c**4*d*p + 30*(a + b*x**2)**p 
*a*b**2*c**4*d + 8*(a + b*x**2)**p*a*b**2*c**3*d**2*p**4*x + 44*(a + b*x** 
2)**p*a*b**2*c**3*d**2*p**3*x + 76*(a + b*x**2)**p*a*b**2*c**3*d**2*p**2*x 
 + 40*(a + b*x**2)**p*a*b**2*c**3*d**2*p*x - 8*(a + b*x**2)**p*a*b**2*c**2 
*d**3*p**4*x**2 - 36*(a + b*x**2)**p*a*b**2*c**2*d**3*p**3*x**2 - 46*(a + 
b*x**2)**p*a*b**2*c**2*d**3*p**2*x**2 - 15*(a + b*x**2)**p*a*b**2*c**2*d** 
3*p*x**2 + 8*(a + b*x**2)**p*a*b**2*c*d**4*p**4*x**3 + 28*(a + b*x**2)**p* 
a*b**2*c*d**4*p**3*x**3 + 28*(a + b*x**2)**p*a*b**2*c*d**4*p**2*x**3 + 8*( 
a + b*x**2)**p*a*b**2*c*d**4*p*x**3 + 8*(a + b*x**2)**p*b**3*c**5*p**4*x + 
 56*(a + b*x**2)**p*b**3*c**5*p**3*x + 142*(a + b*x**2)**p*b**3*c**5*p**2* 
x + 154*(a + b*x**2)**p*b**3*c**5*p*x + 60*(a + b*x**2)**p*b**3*c**5*x - 8 
*(a + b*x**2)**p*b**3*c**4*d*p**4*x**2 - 52*(a + b*x**2)**p*b**3*c**4*d*p* 
*3*x**2 - 118*(a + b*x**2)**p*b**3*c**4*d*p**2*x**2 - 107*(a + b*x**2)**p* 
b**3*c**4*d*p*x**2 - 30*(a + b*x**2)**p*b**3*c**4*d*x**2 + 8*(a + b*x**...