Integrand size = 26, antiderivative size = 136 \[ \int \frac {(c+d x)^{11/3}}{\left (c^2-d^2 x^2\right )^{2/3}} \, dx=-\frac {486 c^3 \sqrt [3]{c^2-d^2 x^2}}{35 d \sqrt [3]{c+d x}}-\frac {81 c^2 (c+d x)^{2/3} \sqrt [3]{c^2-d^2 x^2}}{35 d}-\frac {27 c (c+d x)^{5/3} \sqrt [3]{c^2-d^2 x^2}}{35 d}-\frac {3 (c+d x)^{8/3} \sqrt [3]{c^2-d^2 x^2}}{10 d} \] Output:
-486/35*c^3*(-d^2*x^2+c^2)^(1/3)/d/(d*x+c)^(1/3)-81/35*c^2*(d*x+c)^(2/3)*( -d^2*x^2+c^2)^(1/3)/d-27/35*c*(d*x+c)^(5/3)*(-d^2*x^2+c^2)^(1/3)/d-3/10*(d *x+c)^(8/3)*(-d^2*x^2+c^2)^(1/3)/d
Time = 0.38 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.46 \[ \int \frac {(c+d x)^{11/3}}{\left (c^2-d^2 x^2\right )^{2/3}} \, dx=-\frac {3 \sqrt [3]{c^2-d^2 x^2} \left (403 c^3+111 c^2 d x+39 c d^2 x^2+7 d^3 x^3\right )}{70 d \sqrt [3]{c+d x}} \] Input:
Integrate[(c + d*x)^(11/3)/(c^2 - d^2*x^2)^(2/3),x]
Output:
(-3*(c^2 - d^2*x^2)^(1/3)*(403*c^3 + 111*c^2*d*x + 39*c*d^2*x^2 + 7*d^3*x^ 3))/(70*d*(c + d*x)^(1/3))
Time = 0.39 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.04, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {459, 459, 459, 458}
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.
\(\displaystyle \int \frac {(c+d x)^{11/3}}{\left (c^2-d^2 x^2\right )^{2/3}} \, dx\) |
\(\Big \downarrow \) 459 |
\(\displaystyle \frac {9}{5} c \int \frac {(c+d x)^{8/3}}{\left (c^2-d^2 x^2\right )^{2/3}}dx-\frac {3 (c+d x)^{8/3} \sqrt [3]{c^2-d^2 x^2}}{10 d}\) |
\(\Big \downarrow \) 459 |
\(\displaystyle \frac {9}{5} c \left (\frac {12}{7} c \int \frac {(c+d x)^{5/3}}{\left (c^2-d^2 x^2\right )^{2/3}}dx-\frac {3 (c+d x)^{5/3} \sqrt [3]{c^2-d^2 x^2}}{7 d}\right )-\frac {3 (c+d x)^{8/3} \sqrt [3]{c^2-d^2 x^2}}{10 d}\) |
\(\Big \downarrow \) 459 |
\(\displaystyle \frac {9}{5} c \left (\frac {12}{7} c \left (\frac {3}{2} c \int \frac {(c+d x)^{2/3}}{\left (c^2-d^2 x^2\right )^{2/3}}dx-\frac {3 (c+d x)^{2/3} \sqrt [3]{c^2-d^2 x^2}}{4 d}\right )-\frac {3 (c+d x)^{5/3} \sqrt [3]{c^2-d^2 x^2}}{7 d}\right )-\frac {3 (c+d x)^{8/3} \sqrt [3]{c^2-d^2 x^2}}{10 d}\) |
\(\Big \downarrow \) 458 |
\(\displaystyle \frac {9}{5} c \left (\frac {12}{7} c \left (-\frac {9 c \sqrt [3]{c^2-d^2 x^2}}{2 d \sqrt [3]{c+d x}}-\frac {3 (c+d x)^{2/3} \sqrt [3]{c^2-d^2 x^2}}{4 d}\right )-\frac {3 (c+d x)^{5/3} \sqrt [3]{c^2-d^2 x^2}}{7 d}\right )-\frac {3 (c+d x)^{8/3} \sqrt [3]{c^2-d^2 x^2}}{10 d}\) |
Input:
Int[(c + d*x)^(11/3)/(c^2 - d^2*x^2)^(2/3),x]
Output:
(-3*(c + d*x)^(8/3)*(c^2 - d^2*x^2)^(1/3))/(10*d) + (9*c*((-3*(c + d*x)^(5 /3)*(c^2 - d^2*x^2)^(1/3))/(7*d) + (12*c*((-9*c*(c^2 - d^2*x^2)^(1/3))/(2* d*(c + d*x)^(1/3)) - (3*(c + d*x)^(2/3)*(c^2 - d^2*x^2)^(1/3))/(4*d)))/7)) /5
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ d*(c + d*x)^(n - 1)*((a + b*x^2)^(p + 1)/(b*(p + 1))), x] /; FreeQ[{a, b, c , d, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] && EqQ[n + p, 0]
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ d*(c + d*x)^(n - 1)*((a + b*x^2)^(p + 1)/(b*(n + 2*p + 1))), x] + Simp[2*c* (Simplify[n + p]/(n + 2*p + 1)) Int[(c + d*x)^(n - 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] && IGtQ[Simplif y[n + p], 0]
Time = 0.21 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.46
method | result | size |
gosper | \(-\frac {3 \left (-d x +c \right ) \left (7 d^{3} x^{3}+39 c \,d^{2} x^{2}+111 c^{2} d x +403 c^{3}\right ) \left (d x +c \right )^{\frac {2}{3}}}{70 d \left (-d^{2} x^{2}+c^{2}\right )^{\frac {2}{3}}}\) | \(63\) |
orering | \(-\frac {3 \left (-d x +c \right ) \left (7 d^{3} x^{3}+39 c \,d^{2} x^{2}+111 c^{2} d x +403 c^{3}\right ) \left (d x +c \right )^{\frac {2}{3}}}{70 d \left (-d^{2} x^{2}+c^{2}\right )^{\frac {2}{3}}}\) | \(63\) |
risch | \(-\frac {3 {\left (\left (-d^{2} x^{2}+c^{2}\right )^{2}\right )}^{\frac {1}{3}} {\left (\frac {\left (d^{2} x^{2}-c^{2}\right )^{2}}{\left (d x +c \right )^{2}}\right )}^{\frac {1}{3}} \left (d x +c \right )^{\frac {2}{3}} \left (7 d^{3} x^{3}+39 c \,d^{2} x^{2}+111 c^{2} d x +403 c^{3}\right ) \left (-d x +c \right )}{70 \left (-d^{2} x^{2}+c^{2}\right )^{\frac {2}{3}} {\left (\left (d^{2} x^{2}-c^{2}\right )^{2}\right )}^{\frac {1}{3}} d \left (\left (d x -c \right )^{2}\right )^{\frac {1}{3}}}\) | \(132\) |
Input:
int((d*x+c)^(11/3)/(-d^2*x^2+c^2)^(2/3),x,method=_RETURNVERBOSE)
Output:
-3/70*(-d*x+c)*(7*d^3*x^3+39*c*d^2*x^2+111*c^2*d*x+403*c^3)*(d*x+c)^(2/3)/ d/(-d^2*x^2+c^2)^(2/3)
Time = 0.08 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.47 \[ \int \frac {(c+d x)^{11/3}}{\left (c^2-d^2 x^2\right )^{2/3}} \, dx=-\frac {3 \, {\left (7 \, d^{3} x^{3} + 39 \, c d^{2} x^{2} + 111 \, c^{2} d x + 403 \, c^{3}\right )} {\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}}}{70 \, {\left (d^{2} x + c d\right )}} \] Input:
integrate((d*x+c)^(11/3)/(-d^2*x^2+c^2)^(2/3),x, algorithm="fricas")
Output:
-3/70*(7*d^3*x^3 + 39*c*d^2*x^2 + 111*c^2*d*x + 403*c^3)*(-d^2*x^2 + c^2)^ (1/3)*(d*x + c)^(2/3)/(d^2*x + c*d)
\[ \int \frac {(c+d x)^{11/3}}{\left (c^2-d^2 x^2\right )^{2/3}} \, dx=\int \frac {\left (c + d x\right )^{\frac {11}{3}}}{\left (- \left (- c + d x\right ) \left (c + d x\right )\right )^{\frac {2}{3}}}\, dx \] Input:
integrate((d*x+c)**(11/3)/(-d**2*x**2+c**2)**(2/3),x)
Output:
Integral((c + d*x)**(11/3)/(-(-c + d*x)*(c + d*x))**(2/3), x)
Time = 0.05 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.40 \[ \int \frac {(c+d x)^{11/3}}{\left (c^2-d^2 x^2\right )^{2/3}} \, dx=\frac {3 \, {\left (7 \, d^{4} x^{4} + 32 \, c d^{3} x^{3} + 72 \, c^{2} d^{2} x^{2} + 292 \, c^{3} d x - 403 \, c^{4}\right )}}{70 \, {\left (-d x + c\right )}^{\frac {2}{3}} d} \] Input:
integrate((d*x+c)^(11/3)/(-d^2*x^2+c^2)^(2/3),x, algorithm="maxima")
Output:
3/70*(7*d^4*x^4 + 32*c*d^3*x^3 + 72*c^2*d^2*x^2 + 292*c^3*d*x - 403*c^4)/( (-d*x + c)^(2/3)*d)
\[ \int \frac {(c+d x)^{11/3}}{\left (c^2-d^2 x^2\right )^{2/3}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {11}{3}}}{{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {2}{3}}} \,d x } \] Input:
integrate((d*x+c)^(11/3)/(-d^2*x^2+c^2)^(2/3),x, algorithm="giac")
Output:
integrate((d*x + c)^(11/3)/(-d^2*x^2 + c^2)^(2/3), x)
Time = 7.13 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.61 \[ \int \frac {(c+d x)^{11/3}}{\left (c^2-d^2 x^2\right )^{2/3}} \, dx=-\frac {{\left (c^2-d^2\,x^2\right )}^{1/3}\,\left (\frac {1209\,c^3\,{\left (c+d\,x\right )}^{2/3}}{70\,d^2}+\frac {117\,c\,x^2\,{\left (c+d\,x\right )}^{2/3}}{70}+\frac {3\,d\,x^3\,{\left (c+d\,x\right )}^{2/3}}{10}+\frac {333\,c^2\,x\,{\left (c+d\,x\right )}^{2/3}}{70\,d}\right )}{x+\frac {c}{d}} \] Input:
int((c + d*x)^(11/3)/(c^2 - d^2*x^2)^(2/3),x)
Output:
-((c^2 - d^2*x^2)^(1/3)*((1209*c^3*(c + d*x)^(2/3))/(70*d^2) + (117*c*x^2* (c + d*x)^(2/3))/70 + (3*d*x^3*(c + d*x)^(2/3))/10 + (333*c^2*x*(c + d*x)^ (2/3))/(70*d)))/(x + c/d)
\[ \int \frac {(c+d x)^{11/3}}{\left (c^2-d^2 x^2\right )^{2/3}} \, dx=\frac {-3 \left (-d x +c \right )^{\frac {1}{3}} c^{3}+\left (\int \frac {\left (d x +c \right )^{\frac {2}{3}} x^{3}}{\left (-d^{2} x^{2}+c^{2}\right )^{\frac {2}{3}}}d x \right ) d^{4}+3 \left (\int \frac {\left (d x +c \right )^{\frac {2}{3}} x^{2}}{\left (-d^{2} x^{2}+c^{2}\right )^{\frac {2}{3}}}d x \right ) c \,d^{3}+3 \left (\int \frac {\left (d x +c \right )^{\frac {2}{3}} x}{\left (-d^{2} x^{2}+c^{2}\right )^{\frac {2}{3}}}d x \right ) c^{2} d^{2}}{d} \] Input:
int((d*x+c)^(11/3)/(-d^2*x^2+c^2)^(2/3),x)
Output:
( - 3*(c - d*x)**(1/3)*c**3 + int(((c + d*x)**(2/3)*x**3)/(c**2 - d**2*x** 2)**(2/3),x)*d**4 + 3*int(((c + d*x)**(2/3)*x**2)/(c**2 - d**2*x**2)**(2/3 ),x)*c*d**3 + 3*int(((c + d*x)**(2/3)*x)/(c**2 - d**2*x**2)**(2/3),x)*c**2 *d**2)/d