\(\int (A+B x) (c+d x)^3 (c^2-d^2 x^2)^p \, dx\) [114]

Optimal result

Integrand size = 27, antiderivative size = 126 \[ \int (A+B x) (c+d x)^3 \left (c^2-d^2 x^2\right )^p \, dx=-\frac {B (c+d x)^3 \left (c^2-d^2 x^2\right )^{1+p}}{d^2 (5+2 p)}+\frac {2^p (3 B c+A d (5+2 p)) \left (\frac {c-d x}{c}\right )^{-4-p} \left (c^2-d^2 x^2\right )^{4+p} \operatorname {Hypergeometric2F1}\left (-p,4+p,5+p,\frac {c+d x}{2 c}\right )}{c^4 d^2 (4+p) (5+2 p)} \] Output:

-B*(d*x+c)^3*(-d^2*x^2+c^2)^(p+1)/d^2/(5+2*p)+2^p*(3*B*c+A*d*(5+2*p))*((-d 
*x+c)/c)^(-4-p)*(-d^2*x^2+c^2)^(4+p)*hypergeom([-p, 4+p],[5+p],1/2*(d*x+c) 
/c)/c^4/d^2/(4+p)/(5+2*p)
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(302\) vs. \(2(126)=252\).

Time = 1.13 (sec) , antiderivative size = 302, normalized size of antiderivative = 2.40 \[ \int (A+B x) (c+d x)^3 \left (c^2-d^2 x^2\right )^p \, dx=\frac {1}{10} \left (c^2-d^2 x^2\right )^p \left (-\frac {20 B c^5}{d^2 (1+p)}+\frac {15 B c^5}{d^2 (2+p)}+\frac {5 A c^4}{d (2+p)}-\frac {20 A c^4}{d+d p}+\frac {20 B c^3 x^2}{1+p}+\frac {20 A c^2 d x^2}{1+p}-\frac {30 B c^3 x^2}{2+p}-\frac {10 A c^2 d x^2}{2+p}+\frac {15 B c d^2 x^4}{2+p}+\frac {5 A d^3 x^4}{2+p}+10 A c^3 x \left (1-\frac {d^2 x^2}{c^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},\frac {d^2 x^2}{c^2}\right )+10 c d (B c+A d) x^3 \left (1-\frac {d^2 x^2}{c^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},-p,\frac {5}{2},\frac {d^2 x^2}{c^2}\right )+2 B d^3 x^5 \left (1-\frac {d^2 x^2}{c^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},-p,\frac {7}{2},\frac {d^2 x^2}{c^2}\right )\right ) \] Input:

Integrate[(A + B*x)*(c + d*x)^3*(c^2 - d^2*x^2)^p,x]
 

Output:

((c^2 - d^2*x^2)^p*((-20*B*c^5)/(d^2*(1 + p)) + (15*B*c^5)/(d^2*(2 + p)) + 
 (5*A*c^4)/(d*(2 + p)) - (20*A*c^4)/(d + d*p) + (20*B*c^3*x^2)/(1 + p) + ( 
20*A*c^2*d*x^2)/(1 + p) - (30*B*c^3*x^2)/(2 + p) - (10*A*c^2*d*x^2)/(2 + p 
) + (15*B*c*d^2*x^4)/(2 + p) + (5*A*d^3*x^4)/(2 + p) + (10*A*c^3*x*Hyperge 
ometric2F1[1/2, -p, 3/2, (d^2*x^2)/c^2])/(1 - (d^2*x^2)/c^2)^p + (10*c*d*( 
B*c + A*d)*x^3*Hypergeometric2F1[3/2, -p, 5/2, (d^2*x^2)/c^2])/(1 - (d^2*x 
^2)/c^2)^p + (2*B*d^3*x^5*Hypergeometric2F1[5/2, -p, 7/2, (d^2*x^2)/c^2])/ 
(1 - (d^2*x^2)/c^2)^p))/10
 

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {672, 473, 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.

\(\displaystyle \int (A+B x) (c+d x)^3 \left (c^2-d^2 x^2\right )^p \, dx\)

\(\Big \downarrow \) 672

\(\displaystyle \left (A+\frac {3 B c}{2 d p+5 d}\right ) \int (c+d x)^3 \left (c^2-d^2 x^2\right )^pdx-\frac {B (c+d x)^3 \left (c^2-d^2 x^2\right )^{p+1}}{d^2 (2 p+5)}\)

\(\Big \downarrow \) 473

\(\displaystyle c^2 (c-d x)^{-p-1} \left (\frac {d x}{c}+1\right )^{-p-1} \left (c^2-d^2 x^2\right )^{p+1} \left (A+\frac {3 B c}{2 d p+5 d}\right ) \int (c-d x)^p \left (\frac {d x}{c}+1\right )^{p+3}dx-\frac {B (c+d x)^3 \left (c^2-d^2 x^2\right )^{p+1}}{d^2 (2 p+5)}\)

\(\Big \downarrow \) 79

\(\displaystyle -\frac {c^2 2^{p+3} \left (c^2-d^2 x^2\right )^{p+1} \left (\frac {d x}{c}+1\right )^{-p-1} \left (A+\frac {3 B c}{2 d p+5 d}\right ) \operatorname {Hypergeometric2F1}\left (-p-3,p+1,p+2,\frac {c-d x}{2 c}\right )}{d (p+1)}-\frac {B (c+d x)^3 \left (c^2-d^2 x^2\right )^{p+1}}{d^2 (2 p+5)}\)

Input:

Int[(A + B*x)*(c + d*x)^3*(c^2 - d^2*x^2)^p,x]
 

Output:

-((B*(c + d*x)^3*(c^2 - d^2*x^2)^(1 + p))/(d^2*(5 + 2*p))) - (2^(3 + p)*c^ 
2*(A + (3*B*c)/(5*d + 2*d*p))*(1 + (d*x)/c)^(-1 - p)*(c^2 - d^2*x^2)^(1 + 
p)*Hypergeometric2F1[-3 - p, 1 + p, 2 + p, (c - d*x)/(2*c)])/(d*(1 + p))
 

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 473

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
c^(n - 1)*((a + b*x^2)^(p + 1)/((1 + d*(x/c))^(p + 1)*(a/c + (b*x)/d)^(p + 
1)))   Int[(1 + d*(x/c))^(n + p)*(a/c + (b/d)*x)^p, x], x] /; FreeQ[{a, b, 
c, d, n}, x] && EqQ[b*c^2 + a*d^2, 0] && (IntegerQ[n] || GtQ[c, 0]) &&  !Gt 
Q[a, 0] &&  !(IntegerQ[n] && (IntegerQ[3*p] || IntegerQ[4*p]))
 

rule 672

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

Maple [F]

\[\int \left (B x +A \right ) \left (d x +c \right )^{3} \left (-d^{2} x^{2}+c^{2}\right )^{p}d x\]

Input:

int((B*x+A)*(d*x+c)^3*(-d^2*x^2+c^2)^p,x)
 

Output:

int((B*x+A)*(d*x+c)^3*(-d^2*x^2+c^2)^p,x)
 

Fricas [F]

\[ \int (A+B x) (c+d x)^3 \left (c^2-d^2 x^2\right )^p \, dx=\int { {\left (B x + A\right )} {\left (d x + c\right )}^{3} {\left (-d^{2} x^{2} + c^{2}\right )}^{p} \,d x } \] Input:

integrate((B*x+A)*(d*x+c)^3*(-d^2*x^2+c^2)^p,x, algorithm="fricas")
 

Output:

integral((B*d^3*x^4 + A*c^3 + (3*B*c*d^2 + A*d^3)*x^3 + 3*(B*c^2*d + A*c*d 
^2)*x^2 + (B*c^3 + 3*A*c^2*d)*x)*(-d^2*x^2 + c^2)^p, x)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 350 vs. \(2 (104) = 208\).

Time = 3.71 (sec) , antiderivative size = 971, normalized size of antiderivative = 7.71 \[ \int (A+B x) (c+d x)^3 \left (c^2-d^2 x^2\right )^p \, dx=\text {Too large to display} \] Input:

integrate((B*x+A)*(d*x+c)**3*(-d**2*x**2+c**2)**p,x)
 

Output:

A*c**3*c**(2*p)*x*hyper((1/2, -p), (3/2,), d**2*x**2*exp_polar(2*I*pi)/c** 
2) + 3*A*c**2*d*Piecewise((x**2*(c**2)**p/2, Eq(d**2, 0)), (-Piecewise(((c 
**2 - d**2*x**2)**(p + 1)/(p + 1), Ne(p, -1)), (log(c**2 - d**2*x**2), Tru 
e))/(2*d**2), True)) + A*c*c**(2*p)*d**2*x**3*hyper((3/2, -p), (5/2,), d** 
2*x**2*exp_polar(2*I*pi)/c**2) + A*d**3*Piecewise((x**4*(c**2)**p/4, Eq(d, 
 0)), (-c**2*log(-c/d + x)/(-2*c**2*d**4 + 2*d**6*x**2) - c**2*log(c/d + x 
)/(-2*c**2*d**4 + 2*d**6*x**2) - c**2/(-2*c**2*d**4 + 2*d**6*x**2) + d**2* 
x**2*log(-c/d + x)/(-2*c**2*d**4 + 2*d**6*x**2) + d**2*x**2*log(c/d + x)/( 
-2*c**2*d**4 + 2*d**6*x**2), Eq(p, -2)), (-c**2*log(-c/d + x)/(2*d**4) - c 
**2*log(c/d + x)/(2*d**4) - x**2/(2*d**2), Eq(p, -1)), (-c**4*(c**2 - d**2 
*x**2)**p/(2*d**4*p**2 + 6*d**4*p + 4*d**4) - c**2*d**2*p*x**2*(c**2 - d** 
2*x**2)**p/(2*d**4*p**2 + 6*d**4*p + 4*d**4) + d**4*p*x**4*(c**2 - d**2*x* 
*2)**p/(2*d**4*p**2 + 6*d**4*p + 4*d**4) + d**4*x**4*(c**2 - d**2*x**2)**p 
/(2*d**4*p**2 + 6*d**4*p + 4*d**4), True)) + B*c**3*Piecewise((x**2*(c**2) 
**p/2, Eq(d**2, 0)), (-Piecewise(((c**2 - d**2*x**2)**(p + 1)/(p + 1), Ne( 
p, -1)), (log(c**2 - d**2*x**2), True))/(2*d**2), True)) + B*c**2*c**(2*p) 
*d*x**3*hyper((3/2, -p), (5/2,), d**2*x**2*exp_polar(2*I*pi)/c**2) + 3*B*c 
*d**2*Piecewise((x**4*(c**2)**p/4, Eq(d, 0)), (-c**2*log(-c/d + x)/(-2*c** 
2*d**4 + 2*d**6*x**2) - c**2*log(c/d + x)/(-2*c**2*d**4 + 2*d**6*x**2) - c 
**2/(-2*c**2*d**4 + 2*d**6*x**2) + d**2*x**2*log(-c/d + x)/(-2*c**2*d**...
                                                                                    
                                                                                    
 

Maxima [F]

\[ \int (A+B x) (c+d x)^3 \left (c^2-d^2 x^2\right )^p \, dx=\int { {\left (B x + A\right )} {\left (d x + c\right )}^{3} {\left (-d^{2} x^{2} + c^{2}\right )}^{p} \,d x } \] Input:

integrate((B*x+A)*(d*x+c)^3*(-d^2*x^2+c^2)^p,x, algorithm="maxima")
 

Output:

integrate((B*x + A)*(d*x + c)^3*(-d^2*x^2 + c^2)^p, x)
 

Giac [F]

\[ \int (A+B x) (c+d x)^3 \left (c^2-d^2 x^2\right )^p \, dx=\int { {\left (B x + A\right )} {\left (d x + c\right )}^{3} {\left (-d^{2} x^{2} + c^{2}\right )}^{p} \,d x } \] Input:

integrate((B*x+A)*(d*x+c)^3*(-d^2*x^2+c^2)^p,x, algorithm="giac")
 

Output:

integrate((B*x + A)*(d*x + c)^3*(-d^2*x^2 + c^2)^p, x)
 

Mupad [F(-1)]

Timed out. \[ \int (A+B x) (c+d x)^3 \left (c^2-d^2 x^2\right )^p \, dx=\int {\left (c^2-d^2\,x^2\right )}^p\,\left (A+B\,x\right )\,{\left (c+d\,x\right )}^3 \,d x \] Input:

int((c^2 - d^2*x^2)^p*(A + B*x)*(c + d*x)^3,x)
 

Output:

int((c^2 - d^2*x^2)^p*(A + B*x)*(c + d*x)^3, x)
 

Reduce [F]

\[ \int (A+B x) (c+d x)^3 \left (c^2-d^2 x^2\right )^p \, dx=\text {too large to display} \] Input:

int((B*x+A)*(d*x+c)^3*(-d^2*x^2+c^2)^p,x)
 

Output:

( - 24*(c**2 - d**2*x**2)**p*a*c**4*d*p**4 - 164*(c**2 - d**2*x**2)**p*a*c 
**4*d*p**3 - 390*(c**2 - d**2*x**2)**p*a*c**4*d*p**2 - 367*(c**2 - d**2*x* 
*2)**p*a*c**4*d*p - 105*(c**2 - d**2*x**2)**p*a*c**4*d - 16*(c**2 - d**2*x 
**2)**p*a*c**3*d**2*p**4*x - 76*(c**2 - d**2*x**2)**p*a*c**3*d**2*p**3*x - 
 86*(c**2 - d**2*x**2)**p*a*c**3*d**2*p**2*x + 34*(c**2 - d**2*x**2)**p*a* 
c**3*d**2*p*x + 60*(c**2 - d**2*x**2)**p*a*c**3*d**2*x + 16*(c**2 - d**2*x 
**2)**p*a*c**2*d**3*p**4*x**2 + 120*(c**2 - d**2*x**2)**p*a*c**2*d**3*p**3 
*x**2 + 308*(c**2 - d**2*x**2)**p*a*c**2*d**3*p**2*x**2 + 306*(c**2 - d**2 
*x**2)**p*a*c**2*d**3*p*x**2 + 90*(c**2 - d**2*x**2)**p*a*c**2*d**3*x**2 + 
 24*(c**2 - d**2*x**2)**p*a*c*d**4*p**4*x**3 + 144*(c**2 - d**2*x**2)**p*a 
*c*d**4*p**3*x**3 + 294*(c**2 - d**2*x**2)**p*a*c*d**4*p**2*x**3 + 234*(c* 
*2 - d**2*x**2)**p*a*c*d**4*p*x**3 + 60*(c**2 - d**2*x**2)**p*a*c*d**4*x** 
3 + 8*(c**2 - d**2*x**2)**p*a*d**5*p**4*x**4 + 44*(c**2 - d**2*x**2)**p*a* 
d**5*p**3*x**4 + 82*(c**2 - d**2*x**2)**p*a*d**5*p**2*x**4 + 61*(c**2 - d* 
*2*x**2)**p*a*d**5*p*x**4 + 15*(c**2 - d**2*x**2)**p*a*d**5*x**4 - 8*(c**2 
 - d**2*x**2)**p*b*c**5*p**4 - 76*(c**2 - d**2*x**2)**p*b*c**5*p**3 - 226* 
(c**2 - d**2*x**2)**p*b*c**5*p**2 - 245*(c**2 - d**2*x**2)**p*b*c**5*p - 7 
5*(c**2 - d**2*x**2)**p*b*c**5 - 24*(c**2 - d**2*x**2)**p*b*c**4*d*p**4*x 
- 144*(c**2 - d**2*x**2)**p*b*c**4*d*p**3*x - 264*(c**2 - d**2*x**2)**p*b* 
c**4*d*p**2*x - 144*(c**2 - d**2*x**2)**p*b*c**4*d*p*x - 16*(c**2 - d**...