\(\int (c+d x)^n (c^2-d^2 x^2)^{3/2} \, dx\) [342]

Optimal result

Integrand size = 24, antiderivative size = 83 \[ \int (c+d x)^n \left (c^2-d^2 x^2\right )^{3/2} \, dx=-\frac {2^{\frac {5}{2}+n} (c+d x)^n \left (1+\frac {d x}{c}\right )^{-\frac {5}{2}-n} \left (c^2-d^2 x^2\right )^{5/2} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},-\frac {3}{2}-n,\frac {7}{2},\frac {c-d x}{2 c}\right )}{5 c d} \] Output:

-1/5*2^(5/2+n)*(d*x+c)^n*(1+d*x/c)^(-5/2-n)*(-d^2*x^2+c^2)^(5/2)*hypergeom 
([5/2, -3/2-n],[7/2],1/2*(-d*x+c)/c)/c/d
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.

Time = 0.54 (sec) , antiderivative size = 191, normalized size of antiderivative = 2.30 \[ \int (c+d x)^n \left (c^2-d^2 x^2\right )^{3/2} \, dx=-\frac {2^n (c+d x)^n \left (1+\frac {d x}{c}\right )^{-\frac {1}{2}-2 n} \left (d^3 x^3 \sqrt {c-d x} \sqrt {c+d x} \left (\frac {1}{2}+\frac {d x}{2 c}\right )^n \operatorname {AppellF1}\left (3,-\frac {1}{2},-\frac {1}{2}-n,4,\frac {d x}{c},-\frac {d x}{c}\right )+2 c^2 (c-d x) \sqrt {2-\frac {2 d x}{c}} \left (1+\frac {d x}{c}\right )^n \sqrt {c^2-d^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},-\frac {1}{2}-n,\frac {5}{2},\frac {c-d x}{2 c}\right )\right )}{3 d \sqrt {1-\frac {d x}{c}}} \] Input:

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

Output:

-1/3*(2^n*(c + d*x)^n*(1 + (d*x)/c)^(-1/2 - 2*n)*(d^3*x^3*Sqrt[c - d*x]*Sq 
rt[c + d*x]*(1/2 + (d*x)/(2*c))^n*AppellF1[3, -1/2, -1/2 - n, 4, (d*x)/c, 
-((d*x)/c)] + 2*c^2*(c - d*x)*Sqrt[2 - (2*d*x)/c]*(1 + (d*x)/c)^n*Sqrt[c^2 
 - d^2*x^2]*Hypergeometric2F1[3/2, -1/2 - n, 5/2, (c - d*x)/(2*c)]))/(d*Sq 
rt[1 - (d*x)/c])
 

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {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.

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

\(\Big \downarrow \) 474

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

\(\Big \downarrow \) 473

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

\(\Big \downarrow \) 79

\(\displaystyle -\frac {2^{n+\frac {5}{2}} \left (c^2-d^2 x^2\right )^{5/2} (c+d x)^n \left (\frac {d x}{c}+1\right )^{-n-\frac {5}{2}} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},-n-\frac {3}{2},\frac {7}{2},\frac {c-d x}{2 c}\right )}{5 c d}\)

Input:

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

Output:

-1/5*(2^(5/2 + n)*(c + d*x)^n*(1 + (d*x)/c)^(-5/2 - n)*(c^2 - d^2*x^2)^(5/ 
2)*Hypergeometric2F1[5/2, -3/2 - n, 7/2, (c - d*x)/(2*c)])/(c*d)
 

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])
 

Maple [F]

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

Input:

int((d*x+c)^n*(-d^2*x^2+c^2)^(3/2),x)
 

Output:

int((d*x+c)^n*(-d^2*x^2+c^2)^(3/2),x)
 

Fricas [F]

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

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

Output:

integral(-(d^2*x^2 - c^2)*sqrt(-d^2*x^2 + c^2)*(d*x + c)^n, x)
 

Sympy [F]

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

integrate((d*x+c)**n*(-d**2*x**2+c**2)**(3/2),x)
 

Output:

Integral((-(-c + d*x)*(c + d*x))**(3/2)*(c + d*x)**n, x)
 

Maxima [F]

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

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

Output:

integrate((-d^2*x^2 + c^2)^(3/2)*(d*x + c)^n, x)
 

Giac [F]

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

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

Output:

integrate((-d^2*x^2 + c^2)^(3/2)*(d*x + c)^n, x)
 

Mupad [F(-1)]

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

int((c^2 - d^2*x^2)^(3/2)*(c + d*x)^n,x)
 

Output:

int((c^2 - d^2*x^2)^(3/2)*(c + d*x)^n, x)
 

Reduce [F]

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

int((d*x+c)^n*(-d^2*x^2+c^2)^(3/2),x)
 

Output:

 - int((c + d*x)**n*sqrt(c**2 - d**2*x**2)*x**2,x)*d**2 + int((c + d*x)**n 
*sqrt(c**2 - d**2*x**2),x)*c**2