Integrand size = 24, antiderivative size = 640 \[ \int \frac {\left (c^2-d^2 x^2\right )^{2/3}}{(c+d x)^2} \, dx=-\frac {3 \left (c^2-d^2 x^2\right )^{2/3}}{d (c+d x)}+\frac {6 x}{\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}+\frac {3 \sqrt [4]{3} \sqrt {2+\sqrt {3}} c^{2/3} \left (c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right ) \sqrt {\frac {c^{4/3}+c^{2/3} \sqrt [3]{c^2-d^2 x^2}+\left (c^2-d^2 x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )^2}} E\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}{\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}\right )|-7+4 \sqrt {3}\right )}{d^2 x \sqrt {-\frac {c^{2/3} \left (c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )}{\left (\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )^2}}}-\frac {2 \sqrt {2} 3^{3/4} c^{2/3} \left (c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right ) \sqrt {\frac {c^{4/3}+c^{2/3} \sqrt [3]{c^2-d^2 x^2}+\left (c^2-d^2 x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}{\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}\right ),-7+4 \sqrt {3}\right )}{d^2 x \sqrt {-\frac {c^{2/3} \left (c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )}{\left (\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )^2}}} \] Output:
-3*(-d^2*x^2+c^2)^(2/3)/d/(d*x+c)+6*x/((1-3^(1/2))*c^(2/3)-(-d^2*x^2+c^2)^ (1/3))+3*3^(1/4)*(1/2*6^(1/2)+1/2*2^(1/2))*c^(2/3)*(c^(2/3)-(-d^2*x^2+c^2) ^(1/3))*((c^(4/3)+c^(2/3)*(-d^2*x^2+c^2)^(1/3)+(-d^2*x^2+c^2)^(2/3))/((1-3 ^(1/2))*c^(2/3)-(-d^2*x^2+c^2)^(1/3))^2)^(1/2)*EllipticE(((1+3^(1/2))*c^(2 /3)-(-d^2*x^2+c^2)^(1/3))/((1-3^(1/2))*c^(2/3)-(-d^2*x^2+c^2)^(1/3)),2*I-I *3^(1/2))/d^2/x/(-c^(2/3)*(c^(2/3)-(-d^2*x^2+c^2)^(1/3))/((1-3^(1/2))*c^(2 /3)-(-d^2*x^2+c^2)^(1/3))^2)^(1/2)-2*2^(1/2)*3^(3/4)*c^(2/3)*(c^(2/3)-(-d^ 2*x^2+c^2)^(1/3))*((c^(4/3)+c^(2/3)*(-d^2*x^2+c^2)^(1/3)+(-d^2*x^2+c^2)^(2 /3))/((1-3^(1/2))*c^(2/3)-(-d^2*x^2+c^2)^(1/3))^2)^(1/2)*EllipticF(((1+3^( 1/2))*c^(2/3)-(-d^2*x^2+c^2)^(1/3))/((1-3^(1/2))*c^(2/3)-(-d^2*x^2+c^2)^(1 /3)),2*I-I*3^(1/2))/d^2/x/(-c^(2/3)*(c^(2/3)-(-d^2*x^2+c^2)^(1/3))/((1-3^( 1/2))*c^(2/3)-(-d^2*x^2+c^2)^(1/3))^2)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 6.58 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.12 \[ \int \frac {\left (c^2-d^2 x^2\right )^{2/3}}{(c+d x)^2} \, dx=-\frac {3 (c-d x) \sqrt [3]{1+\frac {d x}{c}} \left (c^2-d^2 x^2\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {4}{3},\frac {5}{3},\frac {8}{3},\frac {c-d x}{2 c}\right )}{10 \sqrt [3]{2} c d (c+d x)} \] Input:
Integrate[(c^2 - d^2*x^2)^(2/3)/(c + d*x)^2,x]
Output:
(-3*(c - d*x)*(1 + (d*x)/c)^(1/3)*(c^2 - d^2*x^2)^(2/3)*Hypergeometric2F1[ 4/3, 5/3, 8/3, (c - d*x)/(2*c)])/(10*2^(1/3)*c*d*(c + d*x))
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.29 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.10, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {506, 74}
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.
| \(\displaystyle \int \frac {\left (c^2-d^2 x^2\right )^{2/3}}{(c+d x)^2} \, dx\) |
| \(\Big \downarrow \) 506 |
| \(\displaystyle -\frac {\left (\frac {1}{c+d x}\right )^{4/3} \left (c^2-d^2 x^2\right )^{2/3} \int \frac {\left (1-\frac {2 c}{c+d x}\right )^{2/3}}{\left (\frac {1}{c+d x}\right )^{4/3}}d\frac {1}{c+d x}}{d \left (1-\frac {2 c}{c+d x}\right )^{2/3}}\) |
| \(\Big \downarrow \) 74 |
| \(\displaystyle \frac {3 \left (c^2-d^2 x^2\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{3},\frac {2}{3},\frac {2 c}{c+d x}\right )}{d (c+d x) \left (1-\frac {2 c}{c+d x}\right )^{2/3}}\) |
Input:
Int[(c^2 - d^2*x^2)^(2/3)/(c + d*x)^2,x]
Output:
(3*(c^2 - d^2*x^2)^(2/3)*Hypergeometric2F1[-2/3, -1/3, 2/3, (2*c)/(c + d*x )])/(d*(c + d*x)*(1 - (2*c)/(c + d*x))^(2/3))
Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[c^n*((b*x )^(m + 1)/(b*(m + 1)))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] && !IntegerQ[m] && (IntegerQ[n] || (GtQ[c, 0] && !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0] && GtQ[-d/(b*c), 0])))
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ {q = Rt[-a/b, 2]}, Simp[(-(a + b*x^2)^p)*((1/(c + d*x))^(2*p)/(d*(1 - (c - d*q)/(c + d*x))^p*(1 - (c + d*q)/(c + d*x))^p)) Subst[Int[(1 - (c - d*q)* x)^p*((1 - (c + d*q)*x)^p/x^(n + 2*p + 2)), x], x, 1/(c + d*x)], x]] /; Fre eQ[{a, b, c, d, p}, x] && ILtQ[n, -1] && NegQ[a/b]
\[\int \frac {\left (-d^{2} x^{2}+c^{2}\right )^{\frac {2}{3}}}{\left (d x +c \right )^{2}}d x\]
Input:
int((-d^2*x^2+c^2)^(2/3)/(d*x+c)^2,x)
Output:
int((-d^2*x^2+c^2)^(2/3)/(d*x+c)^2,x)
\[ \int \frac {\left (c^2-d^2 x^2\right )^{2/3}}{(c+d x)^2} \, dx=\int { \frac {{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {2}{3}}}{{\left (d x + c\right )}^{2}} \,d x } \] Input:
integrate((-d^2*x^2+c^2)^(2/3)/(d*x+c)^2,x, algorithm="fricas")
Output:
integral((-d^2*x^2 + c^2)^(2/3)/(d^2*x^2 + 2*c*d*x + c^2), x)
\[ \int \frac {\left (c^2-d^2 x^2\right )^{2/3}}{(c+d x)^2} \, dx=\int \frac {\left (- \left (- c + d x\right ) \left (c + d x\right )\right )^{\frac {2}{3}}}{\left (c + d x\right )^{2}}\, dx \] Input:
integrate((-d**2*x**2+c**2)**(2/3)/(d*x+c)**2,x)
Output:
Integral((-(-c + d*x)*(c + d*x))**(2/3)/(c + d*x)**2, x)
\[ \int \frac {\left (c^2-d^2 x^2\right )^{2/3}}{(c+d x)^2} \, dx=\int { \frac {{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {2}{3}}}{{\left (d x + c\right )}^{2}} \,d x } \] Input:
integrate((-d^2*x^2+c^2)^(2/3)/(d*x+c)^2,x, algorithm="maxima")
Output:
integrate((-d^2*x^2 + c^2)^(2/3)/(d*x + c)^2, x)
\[ \int \frac {\left (c^2-d^2 x^2\right )^{2/3}}{(c+d x)^2} \, dx=\int { \frac {{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {2}{3}}}{{\left (d x + c\right )}^{2}} \,d x } \] Input:
integrate((-d^2*x^2+c^2)^(2/3)/(d*x+c)^2,x, algorithm="giac")
Output:
integrate((-d^2*x^2 + c^2)^(2/3)/(d*x + c)^2, x)
Timed out. \[ \int \frac {\left (c^2-d^2 x^2\right )^{2/3}}{(c+d x)^2} \, dx=\int \frac {{\left (c^2-d^2\,x^2\right )}^{2/3}}{{\left (c+d\,x\right )}^2} \,d x \] Input:
int((c^2 - d^2*x^2)^(2/3)/(c + d*x)^2,x)
Output:
int((c^2 - d^2*x^2)^(2/3)/(c + d*x)^2, x)
\[ \int \frac {\left (c^2-d^2 x^2\right )^{2/3}}{(c+d x)^2} \, dx=\frac {-3 \left (-d^{2} x^{2}+c^{2}\right )^{\frac {2}{3}}-4 \left (\int \frac {\left (-d^{2} x^{2}+c^{2}\right )^{\frac {2}{3}} x}{-d^{3} x^{3}-c \,d^{2} x^{2}+c^{2} d x +c^{3}}d x \right ) c \,d^{2}-4 \left (\int \frac {\left (-d^{2} x^{2}+c^{2}\right )^{\frac {2}{3}} x}{-d^{3} x^{3}-c \,d^{2} x^{2}+c^{2} d x +c^{3}}d x \right ) d^{3} x}{3 d \left (d x +c \right )} \] Input:
int((-d^2*x^2+c^2)^(2/3)/(d*x+c)^2,x)
Output:
( - 3*(c**2 - d**2*x**2)**(2/3) - 4*int(((c**2 - d**2*x**2)**(2/3)*x)/(c** 3 + c**2*d*x - c*d**2*x**2 - d**3*x**3),x)*c*d**2 - 4*int(((c**2 - d**2*x* *2)**(2/3)*x)/(c**3 + c**2*d*x - c*d**2*x**2 - d**3*x**3),x)*d**3*x)/(3*d* (c + d*x))