Integrand size = 24, antiderivative size = 65 \[ \int \frac {\left (c^2-d^2 x^2\right )^{2/5}}{(c+d x)^2} \, dx=-\frac {5\ 2^{2/5} c \left (\frac {c-d x}{c}\right )^{3/5} \operatorname {Hypergeometric2F1}\left (-\frac {3}{5},-\frac {2}{5},\frac {2}{5},\frac {c+d x}{2 c}\right )}{3 d \left (c^2-d^2 x^2\right )^{3/5}} \] Output:
-5/3*2^(2/5)*c*((-d*x+c)/c)^(3/5)*hypergeom([-3/5, -2/5],[2/5],1/2*(d*x+c) /c)/d/(-d^2*x^2+c^2)^(3/5)
Time = 8.41 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.22 \[ \int \frac {\left (c^2-d^2 x^2\right )^{2/5}}{(c+d x)^2} \, dx=-\frac {5 (c-d x) \left (1+\frac {d x}{c}\right )^{3/5} \left (c^2-d^2 x^2\right )^{2/5} \operatorname {Hypergeometric2F1}\left (\frac {7}{5},\frac {8}{5},\frac {12}{5},\frac {c-d x}{2 c}\right )}{14\ 2^{3/5} c d (c+d x)} \] Input:
Integrate[(c^2 - d^2*x^2)^(2/5)/(c + d*x)^2,x]
Output:
(-5*(c - d*x)*(1 + (d*x)/c)^(3/5)*(c^2 - d^2*x^2)^(2/5)*Hypergeometric2F1[ 7/5, 8/5, 12/5, (c - d*x)/(2*c)])/(14*2^(3/5)*c*d*(c + d*x))
Time = 0.29 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.02, 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.
\(\displaystyle \int \frac {\left (c^2-d^2 x^2\right )^{2/5}}{(c+d x)^2} \, dx\) |
\(\Big \downarrow \) 473 |
\(\displaystyle \frac {\left (c^2-d^2 x^2\right )^{7/5} \int \frac {(c-d x)^{2/5}}{\left (\frac {d x}{c}+1\right )^{8/5}}dx}{c^3 (c-d x)^{7/5} \left (\frac {d x}{c}+1\right )^{7/5}}\) |
\(\Big \downarrow \) 79 |
\(\displaystyle -\frac {5 \left (c^2-d^2 x^2\right )^{7/5} \operatorname {Hypergeometric2F1}\left (\frac {7}{5},\frac {8}{5},\frac {12}{5},\frac {c-d x}{2 c}\right )}{14\ 2^{3/5} c^3 d \left (\frac {d x}{c}+1\right )^{7/5}}\) |
Input:
Int[(c^2 - d^2*x^2)^(2/5)/(c + d*x)^2,x]
Output:
(-5*(c^2 - d^2*x^2)^(7/5)*Hypergeometric2F1[7/5, 8/5, 12/5, (c - d*x)/(2*c )])/(14*2^(3/5)*c^3*d*(1 + (d*x)/c)^(7/5))
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]))
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]))
\[\int \frac {\left (-d^{2} x^{2}+c^{2}\right )^{\frac {2}{5}}}{\left (d x +c \right )^{2}}d x\]
Input:
int((-d^2*x^2+c^2)^(2/5)/(d*x+c)^2,x)
Output:
int((-d^2*x^2+c^2)^(2/5)/(d*x+c)^2,x)
\[ \int \frac {\left (c^2-d^2 x^2\right )^{2/5}}{(c+d x)^2} \, dx=\int { \frac {{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {2}{5}}}{{\left (d x + c\right )}^{2}} \,d x } \] Input:
integrate((-d^2*x^2+c^2)^(2/5)/(d*x+c)^2,x, algorithm="fricas")
Output:
integral((-d^2*x^2 + c^2)^(2/5)/(d^2*x^2 + 2*c*d*x + c^2), x)
\[ \int \frac {\left (c^2-d^2 x^2\right )^{2/5}}{(c+d x)^2} \, dx=\int \frac {\left (- \left (- c + d x\right ) \left (c + d x\right )\right )^{\frac {2}{5}}}{\left (c + d x\right )^{2}}\, dx \] Input:
integrate((-d**2*x**2+c**2)**(2/5)/(d*x+c)**2,x)
Output:
Integral((-(-c + d*x)*(c + d*x))**(2/5)/(c + d*x)**2, x)
\[ \int \frac {\left (c^2-d^2 x^2\right )^{2/5}}{(c+d x)^2} \, dx=\int { \frac {{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {2}{5}}}{{\left (d x + c\right )}^{2}} \,d x } \] Input:
integrate((-d^2*x^2+c^2)^(2/5)/(d*x+c)^2,x, algorithm="maxima")
Output:
integrate((-d^2*x^2 + c^2)^(2/5)/(d*x + c)^2, x)
\[ \int \frac {\left (c^2-d^2 x^2\right )^{2/5}}{(c+d x)^2} \, dx=\int { \frac {{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {2}{5}}}{{\left (d x + c\right )}^{2}} \,d x } \] Input:
integrate((-d^2*x^2+c^2)^(2/5)/(d*x+c)^2,x, algorithm="giac")
Output:
integrate((-d^2*x^2 + c^2)^(2/5)/(d*x + c)^2, x)
Timed out. \[ \int \frac {\left (c^2-d^2 x^2\right )^{2/5}}{(c+d x)^2} \, dx=\int \frac {{\left (c^2-d^2\,x^2\right )}^{2/5}}{{\left (c+d\,x\right )}^2} \,d x \] 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)
\[ \int \frac {\left (c^2-d^2 x^2\right )^{2/5}}{(c+d x)^2} \, dx=\frac {-4 \left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{5}} \left (\int \frac {\left (-d^{2} x^{2}+c^{2}\right )^{\frac {2}{5}} x}{-d^{3} x^{3}-c \,d^{2} x^{2}+c^{2} d x +c^{3}}d x \right ) d^{2}-5 c +5 d x}{5 \left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{5}} d} \] Input:
int((-d^2*x^2+c^2)^(2/5)/(d*x+c)^2,x)
Output:
( - 4*(c**2 - d**2*x**2)**(3/5)*int(((c**2 - d**2*x**2)**(2/5)*x)/(c**3 + c**2*d*x - c*d**2*x**2 - d**3*x**3),x)*d**2 - 5*c + 5*d*x)/(5*(c**2 - d**2 *x**2)**(3/5)*d)