\(\int \frac {(c^2-d^2 x^2)^{2/3}}{c+d x} \, dx\) [253]

Optimal result

Integrand size = 24, antiderivative size = 637 \[ \int \frac {\left (c^2-d^2 x^2\right )^{2/3}}{c+d x} \, dx=\frac {3 \left (c^2-d^2 x^2\right )^{2/3}}{4 d}-\frac {3 c x}{\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}-\frac {3 \sqrt [4]{3} \sqrt {2+\sqrt {3}} c^{5/3} \left (c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right ) \sqrt {\frac {c^{4/3}+c^{2/3} \sqrt [3]{c^2-d^2 x^2}+\left (c^2-d^2 x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )^2}} E\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}{\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}\right )|-7+4 \sqrt {3}\right )}{2 d^2 x \sqrt {-\frac {c^{2/3} \left (c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )}{\left (\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )^2}}}+\frac {\sqrt {2} 3^{3/4} c^{5/3} \left (c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right ) \sqrt {\frac {c^{4/3}+c^{2/3} \sqrt [3]{c^2-d^2 x^2}+\left (c^2-d^2 x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}{\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}\right ),-7+4 \sqrt {3}\right )}{d^2 x \sqrt {-\frac {c^{2/3} \left (c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )}{\left (\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )^2}}} \] Output:

3/4*(-d^2*x^2+c^2)^(2/3)/d-3*c*x/((1-3^(1/2))*c^(2/3)-(-d^2*x^2+c^2)^(1/3) 
)-3/2*3^(1/4)*(1/2*6^(1/2)+1/2*2^(1/2))*c^(5/3)*(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^(1/2)*3^(3/4)*c^(5/3)*(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)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 6.02 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.12 \[ \int \frac {\left (c^2-d^2 x^2\right )^{2/3}}{c+d x} \, dx=-\frac {3 (c-d x) \sqrt [3]{1+\frac {d x}{c}} \left (c^2-d^2 x^2\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {5}{3},\frac {8}{3},\frac {c-d x}{2 c}\right )}{5 \sqrt [3]{2} d (c+d x)} \] Input:

Integrate[(c^2 - d^2*x^2)^(2/3)/(c + d*x),x]
 

Output:

(-3*(c - d*x)*(1 + (d*x)/c)^(1/3)*(c^2 - d^2*x^2)^(2/3)*Hypergeometric2F1[ 
1/3, 5/3, 8/3, (c - d*x)/(2*c)])/(5*2^(1/3)*d*(c + d*x))
 

Rubi [A] (warning: unable to verify)

Time = 0.84 (sec) , antiderivative size = 696, normalized size of antiderivative = 1.09, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {504, 233, 241, 833, 760, 2418}

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.

\(\displaystyle \int \frac {\left (c^2-d^2 x^2\right )^{2/3}}{c+d x} \, dx\)

\(\Big \downarrow \) 504

\(\displaystyle c \int \frac {1}{\sqrt [3]{c^2-d^2 x^2}}dx-d \int \frac {x}{\sqrt [3]{c^2-d^2 x^2}}dx\)

\(\Big \downarrow \) 233

\(\displaystyle -d \int \frac {x}{\sqrt [3]{c^2-d^2 x^2}}dx-\frac {3 c \sqrt {-d^2 x^2} \int \frac {\sqrt [3]{c^2-d^2 x^2}}{\sqrt {-d^2 x^2}}d\sqrt [3]{c^2-d^2 x^2}}{2 d^2 x}\)

\(\Big \downarrow \) 241

\(\displaystyle \frac {3 \left (c^2-d^2 x^2\right )^{2/3}}{4 d}-\frac {3 c \sqrt {-d^2 x^2} \int \frac {\sqrt [3]{c^2-d^2 x^2}}{\sqrt {-d^2 x^2}}d\sqrt [3]{c^2-d^2 x^2}}{2 d^2 x}\)

\(\Big \downarrow \) 833

\(\displaystyle \frac {3 \left (c^2-d^2 x^2\right )^{2/3}}{4 d}-\frac {3 c \sqrt {-d^2 x^2} \left (\left (1+\sqrt {3}\right ) c^{2/3} \int \frac {1}{\sqrt {-d^2 x^2}}d\sqrt [3]{c^2-d^2 x^2}-\int \frac {\left (1+\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}{\sqrt {-d^2 x^2}}d\sqrt [3]{c^2-d^2 x^2}\right )}{2 d^2 x}\)

\(\Big \downarrow \) 760

\(\displaystyle \frac {3 \left (c^2-d^2 x^2\right )^{2/3}}{4 d}-\frac {3 c \sqrt {-d^2 x^2} \left (-\int \frac {\left (1+\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}{\sqrt {-d^2 x^2}}d\sqrt [3]{c^2-d^2 x^2}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) c^{2/3} \left (c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right ) \sqrt {\frac {c^{4/3}+\left (c^2-d^2 x^2\right )^{2/3}+c^{2/3} \sqrt [3]{c^2-d^2 x^2}}{\left (\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}{\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-d^2 x^2} \sqrt {-\frac {c^{2/3} \left (c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )}{\left (\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )^2}}}\right )}{2 d^2 x}\)

\(\Big \downarrow \) 2418

\(\displaystyle \frac {3 \left (c^2-d^2 x^2\right )^{2/3}}{4 d}-\frac {3 c \sqrt {-d^2 x^2} \left (-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) c^{2/3} \left (c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right ) \sqrt {\frac {c^{4/3}+\left (c^2-d^2 x^2\right )^{2/3}+c^{2/3} \sqrt [3]{c^2-d^2 x^2}}{\left (\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}{\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-d^2 x^2} \sqrt {-\frac {c^{2/3} \left (c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )}{\left (\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )^2}}}+\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} c^{2/3} \left (c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right ) \sqrt {\frac {c^{4/3}+\left (c^2-d^2 x^2\right )^{2/3}+c^{2/3} \sqrt [3]{c^2-d^2 x^2}}{\left (\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )^2}} E\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}{\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}\right )|-7+4 \sqrt {3}\right )}{\sqrt {-d^2 x^2} \sqrt {-\frac {c^{2/3} \left (c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )}{\left (\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )^2}}}-\frac {2 \sqrt {-d^2 x^2}}{\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}\right )}{2 d^2 x}\)

Input:

Int[(c^2 - d^2*x^2)^(2/3)/(c + d*x),x]
 

Output:

(3*(c^2 - d^2*x^2)^(2/3))/(4*d) - (3*c*Sqrt[-(d^2*x^2)]*((-2*Sqrt[-(d^2*x^ 
2)])/((1 - Sqrt[3])*c^(2/3) - (c^2 - d^2*x^2)^(1/3)) + (3^(1/4)*Sqrt[2 + S 
qrt[3]]*c^(2/3)*(c^(2/3) - (c^2 - d^2*x^2)^(1/3))*Sqrt[(c^(4/3) + c^(2/3)* 
(c^2 - d^2*x^2)^(1/3) + (c^2 - d^2*x^2)^(2/3))/((1 - Sqrt[3])*c^(2/3) - (c 
^2 - d^2*x^2)^(1/3))^2]*EllipticE[ArcSin[((1 + Sqrt[3])*c^(2/3) - (c^2 - d 
^2*x^2)^(1/3))/((1 - Sqrt[3])*c^(2/3) - (c^2 - d^2*x^2)^(1/3))], -7 + 4*Sq 
rt[3]])/(Sqrt[-(d^2*x^2)]*Sqrt[-((c^(2/3)*(c^(2/3) - (c^2 - d^2*x^2)^(1/3) 
))/((1 - Sqrt[3])*c^(2/3) - (c^2 - d^2*x^2)^(1/3))^2)]) - (2*Sqrt[2 - Sqrt 
[3]]*(1 + Sqrt[3])*c^(2/3)*(c^(2/3) - (c^2 - d^2*x^2)^(1/3))*Sqrt[(c^(4/3) 
 + c^(2/3)*(c^2 - d^2*x^2)^(1/3) + (c^2 - d^2*x^2)^(2/3))/((1 - Sqrt[3])*c 
^(2/3) - (c^2 - d^2*x^2)^(1/3))^2]*EllipticF[ArcSin[((1 + Sqrt[3])*c^(2/3) 
 - (c^2 - d^2*x^2)^(1/3))/((1 - Sqrt[3])*c^(2/3) - (c^2 - d^2*x^2)^(1/3))] 
, -7 + 4*Sqrt[3]])/(3^(1/4)*Sqrt[-(d^2*x^2)]*Sqrt[-((c^(2/3)*(c^(2/3) - (c 
^2 - d^2*x^2)^(1/3)))/((1 - Sqrt[3])*c^(2/3) - (c^2 - d^2*x^2)^(1/3))^2)]) 
))/(2*d^2*x)
 

Defintions of rubi rules used

rule 233

Int[((a_) + (b_.)*(x_)^2)^(-1/3), x_Symbol] :> Simp[3*(Sqrt[b*x^2]/(2*b*x)) 
   Subst[Int[x/Sqrt[-a + x^3], x], x, (a + b*x^2)^(1/3)], x] /; FreeQ[{a, b 
}, x]
 

rule 241

Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x^2)^(p + 1)/ 
(2*b*(p + 1)), x] /; FreeQ[{a, b, p}, x] && NeQ[p, -1]
 

rule 504

Int[((a_) + (b_.)*(x_)^2)^(p_)/((c_) + (d_.)*(x_)), x_Symbol] :> Simp[c   I 
nt[(a + b*x^2)^p/(c^2 - d^2*x^2), x], x] - Simp[d   Int[x*((a + b*x^2)^p/(c 
^2 - d^2*x^2)), x], x] /; FreeQ[{a, b, c, d, p}, x]
 

rule 760

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], 
s = Denom[Rt[b/a, 3]]}, Simp[2*Sqrt[2 - Sqrt[3]]*(s + r*x)*(Sqrt[(s^2 - r*s 
*x + r^2*x^2)/((1 - Sqrt[3])*s + r*x)^2]/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[(- 
s)*((s + r*x)/((1 - Sqrt[3])*s + r*x)^2)]))*EllipticF[ArcSin[((1 + Sqrt[3]) 
*s + r*x)/((1 - Sqrt[3])*s + r*x)], -7 + 4*Sqrt[3]], x]] /; FreeQ[{a, b}, x 
] && NegQ[a]
 

rule 833

Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3] 
], s = Denom[Rt[b/a, 3]]}, Simp[(-(1 + Sqrt[3]))*(s/r)   Int[1/Sqrt[a + b*x 
^3], x], x] + Simp[1/r   Int[((1 + Sqrt[3])*s + r*x)/Sqrt[a + b*x^3], x], x 
]] /; FreeQ[{a, b}, x] && NegQ[a]
 

rule 2418

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = N 
umer[Simplify[(1 + Sqrt[3])*(d/c)]], s = Denom[Simplify[(1 + Sqrt[3])*(d/c) 
]]}, Simp[2*d*s^3*(Sqrt[a + b*x^3]/(a*r^2*((1 - Sqrt[3])*s + r*x))), x] + S 
imp[3^(1/4)*Sqrt[2 + Sqrt[3]]*d*s*(s + r*x)*(Sqrt[(s^2 - r*s*x + r^2*x^2)/( 
(1 - Sqrt[3])*s + r*x)^2]/(r^2*Sqrt[a + b*x^3]*Sqrt[(-s)*((s + r*x)/((1 - S 
qrt[3])*s + r*x)^2)]))*EllipticE[ArcSin[((1 + Sqrt[3])*s + r*x)/((1 - Sqrt[ 
3])*s + r*x)], -7 + 4*Sqrt[3]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[a] && 
 EqQ[b*c^3 - 2*(5 + 3*Sqrt[3])*a*d^3, 0]
 

Maple [F]

\[\int \frac {\left (-d^{2} x^{2}+c^{2}\right )^{\frac {2}{3}}}{d x +c}d x\]

Input:

int((-d^2*x^2+c^2)^(2/3)/(d*x+c),x)
 

Output:

int((-d^2*x^2+c^2)^(2/3)/(d*x+c),x)
 

Fricas [F]

\[ \int \frac {\left (c^2-d^2 x^2\right )^{2/3}}{c+d x} \, dx=\int { \frac {{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {2}{3}}}{d x + c} \,d x } \] Input:

integrate((-d^2*x^2+c^2)^(2/3)/(d*x+c),x, algorithm="fricas")
 

Output:

integral((-d^2*x^2 + c^2)^(2/3)/(d*x + c), x)
 

Sympy [F]

\[ \int \frac {\left (c^2-d^2 x^2\right )^{2/3}}{c+d x} \, dx=\int \frac {\left (- \left (- c + d x\right ) \left (c + d x\right )\right )^{\frac {2}{3}}}{c + d x}\, dx \] Input:

integrate((-d**2*x**2+c**2)**(2/3)/(d*x+c),x)
 

Output:

Integral((-(-c + d*x)*(c + d*x))**(2/3)/(c + d*x), x)
 

Maxima [F]

\[ \int \frac {\left (c^2-d^2 x^2\right )^{2/3}}{c+d x} \, dx=\int { \frac {{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {2}{3}}}{d x + c} \,d x } \] Input:

integrate((-d^2*x^2+c^2)^(2/3)/(d*x+c),x, algorithm="maxima")
 

Output:

integrate((-d^2*x^2 + c^2)^(2/3)/(d*x + c), x)
 

Giac [F]

\[ \int \frac {\left (c^2-d^2 x^2\right )^{2/3}}{c+d x} \, dx=\int { \frac {{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {2}{3}}}{d x + c} \,d x } \] Input:

integrate((-d^2*x^2+c^2)^(2/3)/(d*x+c),x, algorithm="giac")
 

Output:

integrate((-d^2*x^2 + c^2)^(2/3)/(d*x + c), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (c^2-d^2 x^2\right )^{2/3}}{c+d x} \, dx=\int \frac {{\left (c^2-d^2\,x^2\right )}^{2/3}}{c+d\,x} \,d x \] Input:

int((c^2 - d^2*x^2)^(2/3)/(c + d*x),x)
 

Output:

int((c^2 - d^2*x^2)^(2/3)/(c + d*x), x)
 

Reduce [F]

\[ \int \frac {\left (c^2-d^2 x^2\right )^{2/3}}{c+d x} \, dx=\int \frac {\left (-d^{2} x^{2}+c^{2}\right )^{\frac {2}{3}}}{d x +c}d x \] Input:

int((-d^2*x^2+c^2)^(2/3)/(d*x+c),x)
                                                                                    
                                                                                    
 

Output:

int((c**2 - d**2*x**2)**(2/3)/(c + d*x),x)