\(\int (c+d x)^2 (c^2-d^2 x^2)^{2/5} \, dx\) [320]

Optimal result

Integrand size = 24, antiderivative size = 67 \[ \int (c+d x)^2 \left (c^2-d^2 x^2\right )^{2/5} \, dx=\frac {5\ 2^{2/5} \left (c^2-d^2 x^2\right )^{17/5} \operatorname {Hypergeometric2F1}\left (-\frac {2}{5},\frac {17}{5},\frac {22}{5},\frac {c+d x}{2 c}\right )}{17 c^3 d \left (\frac {c-d x}{c}\right )^{17/5}} \] Output:

5/17*2^(2/5)*(-d^2*x^2+c^2)^(17/5)*hypergeom([-2/5, 17/5],[22/5],1/2*(d*x+ 
c)/c)/c^3/d/((-d*x+c)/c)^(17/5)
 

Mathematica [A] (verified)

Time = 8.33 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.91 \[ \int (c+d x)^2 \left (c^2-d^2 x^2\right )^{2/5} \, dx=\frac {\left (c^2-d^2 x^2\right )^{2/5} \left (-15 c \left (c^2-d^2 x^2\right ) \left (1-\frac {d^2 x^2}{c^2}\right )^{2/5}+21 c^2 d x \operatorname {Hypergeometric2F1}\left (-\frac {2}{5},\frac {1}{2},\frac {3}{2},\frac {d^2 x^2}{c^2}\right )+7 d^3 x^3 \operatorname {Hypergeometric2F1}\left (-\frac {2}{5},\frac {3}{2},\frac {5}{2},\frac {d^2 x^2}{c^2}\right )\right )}{21 d \left (1-\frac {d^2 x^2}{c^2}\right )^{2/5}} \] Input:

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

Output:

((c^2 - d^2*x^2)^(2/5)*(-15*c*(c^2 - d^2*x^2)*(1 - (d^2*x^2)/c^2)^(2/5) + 
21*c^2*d*x*Hypergeometric2F1[-2/5, 1/2, 3/2, (d^2*x^2)/c^2] + 7*d^3*x^3*Hy 
pergeometric2F1[-2/5, 3/2, 5/2, (d^2*x^2)/c^2]))/(21*d*(1 - (d^2*x^2)/c^2) 
^(2/5))
 

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.96, 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.

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

\(\Big \downarrow \) 473

\(\displaystyle \frac {c \left (c^2-d^2 x^2\right )^{7/5} \int (c-d x)^{2/5} \left (\frac {d x}{c}+1\right )^{12/5}dx}{(c-d x)^{7/5} \left (\frac {d x}{c}+1\right )^{7/5}}\)

\(\Big \downarrow \) 79

\(\displaystyle -\frac {20\ 2^{2/5} c \left (c^2-d^2 x^2\right )^{7/5} \operatorname {Hypergeometric2F1}\left (-\frac {12}{5},\frac {7}{5},\frac {12}{5},\frac {c-d x}{2 c}\right )}{7 d \left (\frac {d x}{c}+1\right )^{7/5}}\)

Input:

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

Output:

(-20*2^(2/5)*c*(c^2 - d^2*x^2)^(7/5)*Hypergeometric2F1[-12/5, 7/5, 12/5, ( 
c - d*x)/(2*c)])/(7*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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.43 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.64 \[ \int (c+d x)^2 \left (c^2-d^2 x^2\right )^{2/5} \, dx=c^{\frac {14}{5}} x {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{5}, \frac {1}{2} \\ \frac {3}{2} \end {matrix}\middle | {\frac {d^{2} x^{2} e^{2 i \pi }}{c^{2}}} \right )} + \frac {c^{\frac {4}{5}} d^{2} x^{3} {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{5}, \frac {3}{2} \\ \frac {5}{2} \end {matrix}\middle | {\frac {d^{2} x^{2} e^{2 i \pi }}{c^{2}}} \right )}}{3} + 2 c d \left (\begin {cases} \frac {x^{2} \left (c^{2}\right )^{\frac {2}{5}}}{2} & \text {for}\: d^{2} = 0 \\- \frac {5 \left (c^{2} - d^{2} x^{2}\right )^{\frac {7}{5}}}{14 d^{2}} & \text {otherwise} \end {cases}\right ) \] Input:

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

Output:

c**(14/5)*x*hyper((-2/5, 1/2), (3/2,), d**2*x**2*exp_polar(2*I*pi)/c**2) + 
 c**(4/5)*d**2*x**3*hyper((-2/5, 3/2), (5/2,), d**2*x**2*exp_polar(2*I*pi) 
/c**2)/3 + 2*c*d*Piecewise((x**2*(c**2)**(2/5)/2, Eq(d**2, 0)), (-5*(c**2 
- d**2*x**2)**(7/5)/(14*d**2), True))
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (c+d x)^2 \left (c^2-d^2 x^2\right )^{2/5} \, dx=\frac {224 \left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{5}} \left (\int \frac {1}{\left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{5}}}d x \right ) c^{4} d -285 c^{5}+175 c^{4} d x +570 c^{3} d^{2} x^{2}-70 c^{2} d^{3} x^{3}-285 c \,d^{4} x^{4}-105 d^{5} x^{5}}{399 \left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{5}} d} \] Input:

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

Output:

(224*(c**2 - d**2*x**2)**(3/5)*int((c**2 - d**2*x**2)**(2/5)/(c**2 - d**2* 
x**2),x)*c**4*d - 285*c**5 + 175*c**4*d*x + 570*c**3*d**2*x**2 - 70*c**2*d 
**3*x**3 - 285*c*d**4*x**4 - 105*d**5*x**5)/(399*(c**2 - d**2*x**2)**(3/5) 
*d)