\(\int (A+B x) (c+d x)^{3/2} (c^2-d^2 x^2)^p \, dx\) [122]

Optimal result

Integrand size = 29, antiderivative size = 144 \[ \int (A+B x) (c+d x)^{3/2} \left (c^2-d^2 x^2\right )^p \, dx=-\frac {2 B (c+d x)^{3/2} \left (c^2-d^2 x^2\right )^{1+p}}{d^2 (7+4 p)}-\frac {2^{\frac {3}{2}+p} (3 B c+A d (7+4 p)) \sqrt {c+d x} \left (1+\frac {d x}{c}\right )^{-\frac {3}{2}-p} \left (c^2-d^2 x^2\right )^{1+p} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2}-p,1+p,2+p,\frac {c-d x}{2 c}\right )}{d^2 (1+p) (7+4 p)} \] Output:

-2*B*(d*x+c)^(3/2)*(-d^2*x^2+c^2)^(p+1)/d^2/(7+4*p)-2^(3/2+p)*(3*B*c+A*d*( 
7+4*p))*(d*x+c)^(1/2)*(1+d*x/c)^(-3/2-p)*(-d^2*x^2+c^2)^(p+1)*hypergeom([p 
+1, -3/2-p],[2+p],1/2*(-d*x+c)/c)/d^2/(p+1)/(7+4*p)
 

Mathematica [A] (verified)

Time = 1.91 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.96 \[ \int (A+B x) (c+d x)^{3/2} \left (c^2-d^2 x^2\right )^p \, dx=-\frac {2 (c-d x) \sqrt {c+d x} \left (1+\frac {d x}{c}\right )^{-\frac {1}{2}-p} \left (c^2-d^2 x^2\right )^p \left (B (1+p) (c+d x)^2 \left (1+\frac {d x}{c}\right )^{\frac {1}{2}+p}+2^{\frac {1}{2}+p} c (3 B c+A d (7+4 p)) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2}-p,1+p,2+p,\frac {c-d x}{2 c}\right )\right )}{d^2 (1+p) (7+4 p)} \] Input:

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

Output:

(-2*(c - d*x)*Sqrt[c + d*x]*(1 + (d*x)/c)^(-1/2 - p)*(c^2 - d^2*x^2)^p*(B* 
(1 + p)*(c + d*x)^2*(1 + (d*x)/c)^(1/2 + p) + 2^(1/2 + p)*c*(3*B*c + A*d*( 
7 + 4*p))*Hypergeometric2F1[-3/2 - p, 1 + p, 2 + p, (c - d*x)/(2*c)]))/(d^ 
2*(1 + p)*(7 + 4*p))
 

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.97, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {672, 474, 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.

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

\(\Big \downarrow \) 672

\(\displaystyle \left (A+\frac {3 B c}{4 d p+7 d}\right ) \int (c+d x)^{3/2} \left (c^2-d^2 x^2\right )^pdx-\frac {2 B (c+d x)^{3/2} \left (c^2-d^2 x^2\right )^{p+1}}{d^2 (4 p+7)}\)

\(\Big \downarrow \) 474

\(\displaystyle \frac {c \sqrt {c+d x} \left (A+\frac {3 B c}{4 d p+7 d}\right ) \int \left (\frac {d x}{c}+1\right )^{3/2} \left (c^2-d^2 x^2\right )^pdx}{\sqrt {\frac {d x}{c}+1}}-\frac {2 B (c+d x)^{3/2} \left (c^2-d^2 x^2\right )^{p+1}}{d^2 (4 p+7)}\)

\(\Big \downarrow \) 473

\(\displaystyle c \sqrt {c+d x} \left (\frac {d x}{c}+1\right )^{-p-\frac {3}{2}} \left (c^2-c d x\right )^{-p-1} \left (c^2-d^2 x^2\right )^{p+1} \left (A+\frac {3 B c}{4 d p+7 d}\right ) \int \left (\frac {d x}{c}+1\right )^{p+\frac {3}{2}} \left (c^2-c d x\right )^pdx-\frac {2 B (c+d x)^{3/2} \left (c^2-d^2 x^2\right )^{p+1}}{d^2 (4 p+7)}\)

\(\Big \downarrow \) 79

\(\displaystyle -\frac {2^{p+\frac {3}{2}} \sqrt {c+d x} \left (c^2-d^2 x^2\right )^{p+1} \left (\frac {d x}{c}+1\right )^{-p-\frac {3}{2}} \left (A+\frac {3 B c}{4 d p+7 d}\right ) \operatorname {Hypergeometric2F1}\left (-p-\frac {3}{2},p+1,p+2,\frac {c-d x}{2 c}\right )}{d (p+1)}-\frac {2 B (c+d x)^{3/2} \left (c^2-d^2 x^2\right )^{p+1}}{d^2 (4 p+7)}\)

Input:

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

Output:

(-2*B*(c + d*x)^(3/2)*(c^2 - d^2*x^2)^(1 + p))/(d^2*(7 + 4*p)) - (2^(3/2 + 
 p)*(A + (3*B*c)/(7*d + 4*d*p))*Sqrt[c + d*x]*(1 + (d*x)/c)^(-3/2 - p)*(c^ 
2 - d^2*x^2)^(1 + p)*Hypergeometric2F1[-3/2 - 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 474

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

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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(2*(64*sqrt(c + d*x)*(c**2 - d**2*x**2)**p*a*c**2*d*p**3 + 208*sqrt(c + d* 
x)*(c**2 - d**2*x**2)**p*a*c**2*d*p**2 + 180*sqrt(c + d*x)*(c**2 - d**2*x* 
*2)**p*a*c**2*d*p + 21*sqrt(c + d*x)*(c**2 - d**2*x**2)**p*a*c**2*d + 16*s 
qrt(c + d*x)*(c**2 - d**2*x**2)**p*a*c*d**2*p**2*x + 52*sqrt(c + d*x)*(c** 
2 - d**2*x**2)**p*a*c*d**2*p*x + 42*sqrt(c + d*x)*(c**2 - d**2*x**2)**p*a* 
c*d**2*x + 16*sqrt(c + d*x)*(c**2 - d**2*x**2)**p*a*d**3*p**2*x**2 + 40*sq 
rt(c + d*x)*(c**2 - d**2*x**2)**p*a*d**3*p*x**2 + 21*sqrt(c + d*x)*(c**2 - 
 d**2*x**2)**p*a*d**3*x**2 + 32*sqrt(c + d*x)*(c**2 - d**2*x**2)**p*b*c**3 
*p**2 + 40*sqrt(c + d*x)*(c**2 - d**2*x**2)**p*b*c**3*p - 6*sqrt(c + d*x)* 
(c**2 - d**2*x**2)**p*b*c**3 - 16*sqrt(c + d*x)*(c**2 - d**2*x**2)**p*b*c* 
*2*d*p**2*x - 20*sqrt(c + d*x)*(c**2 - d**2*x**2)**p*b*c**2*d*p*x + 3*sqrt 
(c + d*x)*(c**2 - d**2*x**2)**p*b*c**2*d*x + 16*sqrt(c + d*x)*(c**2 - d**2 
*x**2)**p*b*c*d**2*p**2*x**2 + 44*sqrt(c + d*x)*(c**2 - d**2*x**2)**p*b*c* 
d**2*p*x**2 + 24*sqrt(c + d*x)*(c**2 - d**2*x**2)**p*b*c*d**2*x**2 + 16*sq 
rt(c + d*x)*(c**2 - d**2*x**2)**p*b*d**3*p**2*x**3 + 32*sqrt(c + d*x)*(c** 
2 - d**2*x**2)**p*b*d**3*p*x**3 + 15*sqrt(c + d*x)*(c**2 - d**2*x**2)**p*b 
*d**3*x**3 + 8192*int((sqrt(c + d*x)*(c**2 - d**2*x**2)**p*x)/(64*c**2*p** 
3 + 240*c**2*p**2 + 284*c**2*p + 105*c**2 - 64*d**2*p**3*x**2 - 240*d**2*p 
**2*x**2 - 284*d**2*p*x**2 - 105*d**2*x**2),x)*a*c**2*d**3*p**7 + 61440*in 
t((sqrt(c + d*x)*(c**2 - d**2*x**2)**p*x)/(64*c**2*p**3 + 240*c**2*p**2...