\(\int \frac {(c^2-d^2 x^2)^{2/5}}{(c+d x)^3} \, dx\) [325]

Optimal result

Integrand size = 24, antiderivative size = 67 \[ \int \frac {\left (c^2-d^2 x^2\right )^{2/5}}{(c+d x)^3} \, dx=-\frac {5 c^2 \left (\frac {c-d x}{c}\right )^{8/5} \operatorname {Hypergeometric2F1}\left (-\frac {8}{5},-\frac {2}{5},-\frac {3}{5},\frac {c+d x}{2 c}\right )}{4\ 2^{3/5} d \left (c^2-d^2 x^2\right )^{8/5}} \] Output:

-5/8*c^2*((-d*x+c)/c)^(8/5)*hypergeom([-8/5, -2/5],[-3/5],1/2*(d*x+c)/c)*2 
^(2/5)/d/(-d^2*x^2+c^2)^(8/5)
 

Mathematica [A] (verified)

Time = 10.13 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.18 \[ \int \frac {\left (c^2-d^2 x^2\right )^{2/5}}{(c+d x)^3} \, dx=-\frac {5 (c-d x) \left (1+\frac {d x}{c}\right )^{3/5} \left (c^2-d^2 x^2\right )^{2/5} \operatorname {Hypergeometric2F1}\left (\frac {7}{5},\frac {13}{5},\frac {12}{5},\frac {c-d x}{2 c}\right )}{28\ 2^{3/5} c^2 d (c+d x)} \] Input:

Integrate[(c^2 - d^2*x^2)^(2/5)/(c + d*x)^3,x]
 

Output:

(-5*(c - d*x)*(1 + (d*x)/c)^(3/5)*(c^2 - d^2*x^2)^(2/5)*Hypergeometric2F1[ 
7/5, 13/5, 12/5, (c - d*x)/(2*c)])/(28*2^(3/5)*c^2*d*(c + d*x))
 

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.99, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {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.

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

\(\Big \downarrow \) 473

\(\displaystyle \frac {\left (c^2-d^2 x^2\right )^{7/5} \int \frac {(c-d x)^{2/5}}{\left (\frac {d x}{c}+1\right )^{13/5}}dx}{c^4 (c-d x)^{7/5} \left (\frac {d x}{c}+1\right )^{7/5}}\)

\(\Big \downarrow \) 79

\(\displaystyle -\frac {5 \left (c^2-d^2 x^2\right )^{7/5} \operatorname {Hypergeometric2F1}\left (\frac {7}{5},\frac {13}{5},\frac {12}{5},\frac {c-d x}{2 c}\right )}{28\ 2^{3/5} c^4 d \left (\frac {d x}{c}+1\right )^{7/5}}\)

Input:

Int[(c^2 - d^2*x^2)^(2/5)/(c + d*x)^3,x]
 

Output:

(-5*(c^2 - d^2*x^2)^(7/5)*Hypergeometric2F1[7/5, 13/5, 12/5, (c - d*x)/(2* 
c)])/(28*2^(3/5)*c^4*d*(1 + (d*x)/c)^(7/5))
 

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]))
 

Maple [F]

\[\int \frac {\left (-d^{2} x^{2}+c^{2}\right )^{\frac {2}{5}}}{\left (d x +c \right )^{3}}d x\]

Input:

int((-d^2*x^2+c^2)^(2/5)/(d*x+c)^3,x)
 

Output:

int((-d^2*x^2+c^2)^(2/5)/(d*x+c)^3,x)
 

Fricas [F]

\[ \int \frac {\left (c^2-d^2 x^2\right )^{2/5}}{(c+d x)^3} \, dx=\int { \frac {{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {2}{5}}}{{\left (d x + c\right )}^{3}} \,d x } \] Input:

integrate((-d^2*x^2+c^2)^(2/5)/(d*x+c)^3,x, algorithm="fricas")
 

Output:

integral((-d^2*x^2 + c^2)^(2/5)/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3), 
 x)
 

Sympy [F]

\[ \int \frac {\left (c^2-d^2 x^2\right )^{2/5}}{(c+d x)^3} \, dx=\int \frac {\left (- \left (- c + d x\right ) \left (c + d x\right )\right )^{\frac {2}{5}}}{\left (c + d x\right )^{3}}\, dx \] Input:

integrate((-d**2*x**2+c**2)**(2/5)/(d*x+c)**3,x)
 

Output:

Integral((-(-c + d*x)*(c + d*x))**(2/5)/(c + d*x)**3, x)
 

Maxima [F]

\[ \int \frac {\left (c^2-d^2 x^2\right )^{2/5}}{(c+d x)^3} \, dx=\int { \frac {{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {2}{5}}}{{\left (d x + c\right )}^{3}} \,d x } \] Input:

integrate((-d^2*x^2+c^2)^(2/5)/(d*x+c)^3,x, algorithm="maxima")
 

Output:

integrate((-d^2*x^2 + c^2)^(2/5)/(d*x + c)^3, x)
 

Giac [F]

\[ \int \frac {\left (c^2-d^2 x^2\right )^{2/5}}{(c+d x)^3} \, dx=\int { \frac {{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {2}{5}}}{{\left (d x + c\right )}^{3}} \,d x } \] Input:

integrate((-d^2*x^2+c^2)^(2/5)/(d*x+c)^3,x, algorithm="giac")
 

Output:

integrate((-d^2*x^2 + c^2)^(2/5)/(d*x + c)^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (c^2-d^2 x^2\right )^{2/5}}{(c+d x)^3} \, dx=\int \frac {{\left (c^2-d^2\,x^2\right )}^{2/5}}{{\left (c+d\,x\right )}^3} \,d x \] Input:

int((c^2 - d^2*x^2)^(2/5)/(c + d*x)^3,x)
 

Output:

int((c^2 - d^2*x^2)^(2/5)/(c + d*x)^3, x)
 

Reduce [F]

\[ \int \frac {\left (c^2-d^2 x^2\right )^{2/5}}{(c+d x)^3} \, dx=\frac {-4 \left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{5}} \left (\int \frac {\left (-d^{2} x^{2}+c^{2}\right )^{\frac {2}{5}} x}{-d^{4} x^{4}-2 c \,d^{3} x^{3}+2 c^{3} d x +c^{4}}d x \right ) c \,d^{2}-4 \left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{5}} \left (\int \frac {\left (-d^{2} x^{2}+c^{2}\right )^{\frac {2}{5}} x}{-d^{4} x^{4}-2 c \,d^{3} x^{3}+2 c^{3} d x +c^{4}}d x \right ) d^{3} x -5 c +5 d x}{10 \left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{5}} d \left (d x +c \right )} \] Input:

int((-d^2*x^2+c^2)^(2/5)/(d*x+c)^3,x)
 

Output:

( - 4*(c**2 - d**2*x**2)**(3/5)*int(((c**2 - d**2*x**2)**(2/5)*x)/(c**4 + 
2*c**3*d*x - 2*c*d**3*x**3 - d**4*x**4),x)*c*d**2 - 4*(c**2 - d**2*x**2)** 
(3/5)*int(((c**2 - d**2*x**2)**(2/5)*x)/(c**4 + 2*c**3*d*x - 2*c*d**3*x**3 
 - d**4*x**4),x)*d**3*x - 5*c + 5*d*x)/(10*(c**2 - d**2*x**2)**(3/5)*d*(c 
+ d*x))