\(\int \frac {(c-d x)^3}{(c^2-d^2 x^2)^{13/5}} \, dx\) [326]

Optimal result

Integrand size = 25, antiderivative size = 67 \[ \int \frac {(c-d x)^3}{\left (c^2-d^2 x^2\right )^{13/5}} \, dx=-\frac {5 \left (c^2-d^2 x^2\right )^{7/5} \operatorname {Hypergeometric2F1}\left (\frac {7}{5},\frac {13}{5},\frac {12}{5},\frac {c-d x}{2 c}\right )}{28\ 2^{3/5} c^4 d \left (\frac {c+d x}{c}\right )^{7/5}} \] Output:

-5/56*(-d^2*x^2+c^2)^(7/5)*hypergeom([7/5, 13/5],[12/5],1/2*(-d*x+c)/c)*2^ 
(2/5)/c^4/d/((d*x+c)/c)^(7/5)
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(157\) vs. \(2(67)=134\).

Time = 10.10 (sec) , antiderivative size = 157, normalized size of antiderivative = 2.34 \[ \int \frac {(c-d x)^3}{\left (c^2-d^2 x^2\right )^{13/5}} \, dx=\frac {-5 \left (c^5+2 c^3 d^2 x^2\right )+12 c^2 d x \left (c^2-d^2 x^2\right ) \left (1-\frac {d^2 x^2}{c^2}\right )^{3/5} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {13}{5},\frac {3}{2},\frac {d^2 x^2}{c^2}\right )+12 d^3 x^3 \left (c^2-d^2 x^2\right ) \left (1-\frac {d^2 x^2}{c^2}\right )^{3/5} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {13}{5},\frac {5}{2},\frac {d^2 x^2}{c^2}\right )}{12 c^3 d \left (c^2-d^2 x^2\right )^{8/5}} \] Input:

Integrate[(c - d*x)^3/(c^2 - d^2*x^2)^(13/5),x]
 

Output:

(-5*(c^5 + 2*c^3*d^2*x^2) + 12*c^2*d*x*(c^2 - d^2*x^2)*(1 - (d^2*x^2)/c^2) 
^(3/5)*Hypergeometric2F1[1/2, 13/5, 3/2, (d^2*x^2)/c^2] + 12*d^3*x^3*(c^2 
- d^2*x^2)*(1 - (d^2*x^2)/c^2)^(3/5)*Hypergeometric2F1[3/2, 13/5, 5/2, (d^ 
2*x^2)/c^2])/(12*c^3*d*(c^2 - d^2*x^2)^(8/5))
 

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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, 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.

\(\displaystyle \int \frac {(c-d x)^3}{\left (c^2-d^2 x^2\right )^{13/5}} \, dx\)

\(\Big \downarrow \) 473

\(\displaystyle \frac {c^2 (c+d x)^{8/5} \left (1-\frac {d x}{c}\right )^{8/5} \int \frac {\left (1-\frac {d x}{c}\right )^{2/5}}{(c+d x)^{13/5}}dx}{\left (c^2-d^2 x^2\right )^{8/5}}\)

\(\Big \downarrow \) 79

\(\displaystyle -\frac {5 c^2 \left (1-\frac {d x}{c}\right )^{8/5} \operatorname {Hypergeometric2F1}\left (-\frac {8}{5},-\frac {2}{5},-\frac {3}{5},\frac {c+d x}{2 c}\right )}{4\ 2^{3/5} d \left (c^2-d^2 x^2\right )^{8/5}}\)

Input:

Int[(c - d*x)^3/(c^2 - d^2*x^2)^(13/5),x]
 

Output:

(-5*c^2*(1 - (d*x)/c)^(8/5)*Hypergeometric2F1[-8/5, -2/5, -3/5, (c + d*x)/ 
(2*c)])/(4*2^(3/5)*d*(c^2 - d^2*x^2)^(8/5))
 

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]

\[\int \frac {\left (-d x +c \right )^{3}}{\left (-d^{2} x^{2}+c^{2}\right )^{\frac {13}{5}}}d x\]

Input:

int((-d*x+c)^3/(-d^2*x^2+c^2)^(13/5),x)
 

Output:

int((-d*x+c)^3/(-d^2*x^2+c^2)^(13/5),x)
 

Fricas [F]

\[ \int \frac {(c-d x)^3}{\left (c^2-d^2 x^2\right )^{13/5}} \, dx=\int { -\frac {{\left (d x - c\right )}^{3}}{{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {13}{5}}} \,d x } \] Input:

integrate((-d*x+c)^3/(-d^2*x^2+c^2)^(13/5),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]

\[ \int \frac {(c-d x)^3}{\left (c^2-d^2 x^2\right )^{13/5}} \, dx=- \int \left (- \frac {c^{3}}{c^{4} \left (c^{2} - d^{2} x^{2}\right )^{\frac {3}{5}} - 2 c^{2} d^{2} x^{2} \left (c^{2} - d^{2} x^{2}\right )^{\frac {3}{5}} + d^{4} x^{4} \left (c^{2} - d^{2} x^{2}\right )^{\frac {3}{5}}}\right )\, dx - \int \frac {d^{3} x^{3}}{c^{4} \left (c^{2} - d^{2} x^{2}\right )^{\frac {3}{5}} - 2 c^{2} d^{2} x^{2} \left (c^{2} - d^{2} x^{2}\right )^{\frac {3}{5}} + d^{4} x^{4} \left (c^{2} - d^{2} x^{2}\right )^{\frac {3}{5}}}\, dx - \int \left (- \frac {3 c d^{2} x^{2}}{c^{4} \left (c^{2} - d^{2} x^{2}\right )^{\frac {3}{5}} - 2 c^{2} d^{2} x^{2} \left (c^{2} - d^{2} x^{2}\right )^{\frac {3}{5}} + d^{4} x^{4} \left (c^{2} - d^{2} x^{2}\right )^{\frac {3}{5}}}\right )\, dx - \int \frac {3 c^{2} d x}{c^{4} \left (c^{2} - d^{2} x^{2}\right )^{\frac {3}{5}} - 2 c^{2} d^{2} x^{2} \left (c^{2} - d^{2} x^{2}\right )^{\frac {3}{5}} + d^{4} x^{4} \left (c^{2} - d^{2} x^{2}\right )^{\frac {3}{5}}}\, dx \] Input:

integrate((-d*x+c)**3/(-d**2*x**2+c**2)**(13/5),x)
 

Output:

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

Maxima [F]

\[ \int \frac {(c-d x)^3}{\left (c^2-d^2 x^2\right )^{13/5}} \, dx=\int { -\frac {{\left (d x - c\right )}^{3}}{{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {13}{5}}} \,d x } \] Input:

integrate((-d*x+c)^3/(-d^2*x^2+c^2)^(13/5),x, algorithm="maxima")
 

Output:

-integrate((d*x - c)^3/(-d^2*x^2 + c^2)^(13/5), x)
 

Giac [F]

\[ \int \frac {(c-d x)^3}{\left (c^2-d^2 x^2\right )^{13/5}} \, dx=\int { -\frac {{\left (d x - c\right )}^{3}}{{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {13}{5}}} \,d x } \] Input:

integrate((-d*x+c)^3/(-d^2*x^2+c^2)^(13/5),x, algorithm="giac")
 

Output:

integrate(-(d*x - c)^3/(-d^2*x^2 + c^2)^(13/5), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(c-d x)^3}{\left (c^2-d^2 x^2\right )^{13/5}} \, dx=\int \frac {{\left (c-d\,x\right )}^3}{{\left (c^2-d^2\,x^2\right )}^{13/5}} \,d x \] Input:

int((c - d*x)^3/(c^2 - d^2*x^2)^(13/5),x)
 

Output:

int((c - d*x)^3/(c^2 - d^2*x^2)^(13/5), x)
 

Reduce [F]

\[ \int \frac {(c-d x)^3}{\left (c^2-d^2 x^2\right )^{13/5}} \, dx=\frac {-4 \left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{5}} \left (\int \frac {\left (-d^{2} x^{2}+c^{2}\right )^{\frac {2}{5}}}{-d^{4} x^{4}-2 c \,d^{3} x^{3}+2 c^{3} d x +c^{4}}d x \right ) c^{2} d -4 \left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{5}} \left (\int \frac {\left (-d^{2} x^{2}+c^{2}\right )^{\frac {2}{5}}}{-d^{4} x^{4}-2 c \,d^{3} x^{3}+2 c^{3} d x +c^{4}}d x \right ) c \,d^{2} x -5 c +5 d x}{6 \left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{5}} d \left (d x +c \right )} \] Input:

int((-d*x+c)^3/(-d^2*x^2+c^2)^(13/5),x)
 

Output:

( - 4*(c**2 - d**2*x**2)**(3/5)*int((c**2 - d**2*x**2)**(2/5)/(c**4 + 2*c* 
*3*d*x - 2*c*d**3*x**3 - d**4*x**4),x)*c**2*d - 4*(c**2 - d**2*x**2)**(3/5 
)*int((c**2 - d**2*x**2)**(2/5)/(c**4 + 2*c**3*d*x - 2*c*d**3*x**3 - d**4* 
x**4),x)*c*d**2*x - 5*c + 5*d*x)/(6*(c**2 - d**2*x**2)**(3/5)*d*(c + d*x))