Integrand size = 24, antiderivative size = 62 \[ \int \frac {\sqrt [8]{c^2-d^2 x^2}}{c+d x} \, dx=\frac {8 \sqrt [8]{2} \sqrt [8]{c^2-d^2 x^2} \operatorname {Hypergeometric2F1}\left (-\frac {1}{8},\frac {1}{8},\frac {9}{8},\frac {c+d x}{2 c}\right )}{d \sqrt [8]{\frac {c-d x}{c}}} \] Output:
8*2^(1/8)*(-d^2*x^2+c^2)^(1/8)*hypergeom([-1/8, 1/8],[9/8],1/2*(d*x+c)/c)/ d/((-d*x+c)/c)^(1/8)
Time = 8.94 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.23 \[ \int \frac {\sqrt [8]{c^2-d^2 x^2}}{c+d x} \, dx=-\frac {4 \sqrt [8]{2} (c-d x) \left (1+\frac {d x}{c}\right )^{7/8} \sqrt [8]{c^2-d^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {7}{8},\frac {9}{8},\frac {17}{8},\frac {c-d x}{2 c}\right )}{9 d (c+d x)} \] Input:
Integrate[(c^2 - d^2*x^2)^(1/8)/(c + d*x),x]
Output:
(-4*2^(1/8)*(c - d*x)*(1 + (d*x)/c)^(7/8)*(c^2 - d^2*x^2)^(1/8)*Hypergeome tric2F1[7/8, 9/8, 17/8, (c - d*x)/(2*c)])/(9*d*(c + d*x))
Time = 0.28 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.06, 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.
\(\displaystyle \int \frac {\sqrt [8]{c^2-d^2 x^2}}{c+d x} \, dx\) |
\(\Big \downarrow \) 473 |
\(\displaystyle \frac {\left (c^2-d^2 x^2\right )^{9/8} \int \frac {\sqrt [8]{c-d x}}{\left (\frac {d x}{c}+1\right )^{7/8}}dx}{c^2 (c-d x)^{9/8} \left (\frac {d x}{c}+1\right )^{9/8}}\) |
\(\Big \downarrow \) 79 |
\(\displaystyle -\frac {4 \sqrt [8]{2} \left (c^2-d^2 x^2\right )^{9/8} \operatorname {Hypergeometric2F1}\left (\frac {7}{8},\frac {9}{8},\frac {17}{8},\frac {c-d x}{2 c}\right )}{9 c^2 d \left (\frac {d x}{c}+1\right )^{9/8}}\) |
Input:
Int[(c^2 - d^2*x^2)^(1/8)/(c + d*x),x]
Output:
(-4*2^(1/8)*(c^2 - d^2*x^2)^(9/8)*Hypergeometric2F1[7/8, 9/8, 17/8, (c - d *x)/(2*c)])/(9*c^2*d*(1 + (d*x)/c)^(9/8))
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]))
\[\int \frac {\left (-d^{2} x^{2}+c^{2}\right )^{\frac {1}{8}}}{d x +c}d x\]
Input:
int((-d^2*x^2+c^2)^(1/8)/(d*x+c),x)
Output:
int((-d^2*x^2+c^2)^(1/8)/(d*x+c),x)
\[ \int \frac {\sqrt [8]{c^2-d^2 x^2}}{c+d x} \, dx=\int { \frac {{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {1}{8}}}{d x + c} \,d x } \] Input:
integrate((-d^2*x^2+c^2)^(1/8)/(d*x+c),x, algorithm="fricas")
Output:
integral((-d^2*x^2 + c^2)^(1/8)/(d*x + c), x)
\[ \int \frac {\sqrt [8]{c^2-d^2 x^2}}{c+d x} \, dx=\int \frac {\sqrt [8]{- \left (- c + d x\right ) \left (c + d x\right )}}{c + d x}\, dx \] Input:
integrate((-d**2*x**2+c**2)**(1/8)/(d*x+c),x)
Output:
Integral((-(-c + d*x)*(c + d*x))**(1/8)/(c + d*x), x)
\[ \int \frac {\sqrt [8]{c^2-d^2 x^2}}{c+d x} \, dx=\int { \frac {{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {1}{8}}}{d x + c} \,d x } \] Input:
integrate((-d^2*x^2+c^2)^(1/8)/(d*x+c),x, algorithm="maxima")
Output:
integrate((-d^2*x^2 + c^2)^(1/8)/(d*x + c), x)
\[ \int \frac {\sqrt [8]{c^2-d^2 x^2}}{c+d x} \, dx=\int { \frac {{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {1}{8}}}{d x + c} \,d x } \] Input:
integrate((-d^2*x^2+c^2)^(1/8)/(d*x+c),x, algorithm="giac")
Output:
integrate((-d^2*x^2 + c^2)^(1/8)/(d*x + c), x)
Timed out. \[ \int \frac {\sqrt [8]{c^2-d^2 x^2}}{c+d x} \, dx=\int \frac {{\left (c^2-d^2\,x^2\right )}^{1/8}}{c+d\,x} \,d x \] Input:
int((c^2 - d^2*x^2)^(1/8)/(c + d*x),x)
Output:
int((c^2 - d^2*x^2)^(1/8)/(c + d*x), x)
\[ \int \frac {\sqrt [8]{c^2-d^2 x^2}}{c+d x} \, dx=\frac {-4 \left (-d^{2} x^{2}+c^{2}\right )^{\frac {7}{8}}+5 \left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{4}} \left (\int \frac {\left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{4}}}{\left (-d^{2} x^{2}+c^{2}\right )^{\frac {5}{8}} c^{2}-\left (-d^{2} x^{2}+c^{2}\right )^{\frac {5}{8}} d^{2} x^{2}}d x \right ) c d}{5 \left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{4}} d} \] Input:
int((-d^2*x^2+c^2)^(1/8)/(d*x+c),x)
Output:
( - 4*(c**2 - d**2*x**2)**(7/8) + 5*(c**2 - d**2*x**2)**(3/4)*int((c**2 - d**2*x**2)**(3/4)/((c**2 - d**2*x**2)**(5/8)*c**2 - (c**2 - d**2*x**2)**(5 /8)*d**2*x**2),x)*c*d)/(5*(c**2 - d**2*x**2)**(3/4)*d)