\(\int \frac {1}{(c+d x) \sqrt [3]{c^2-d^2 x^2}} \, dx\) [260]

Optimal result

Integrand size = 24, antiderivative size = 652 \[ \int \frac {1}{(c+d x) \sqrt [3]{c^2-d^2 x^2}} \, dx=-\frac {3 \left (c^2-d^2 x^2\right )^{2/3}}{2 c d (c+d x)}+\frac {3 x}{2 c \left (\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )}+\frac {3 \sqrt [4]{3} \sqrt {2+\sqrt {3}} \left (c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right ) \sqrt {\frac {c^{4/3}+c^{2/3} \sqrt [3]{c^2-d^2 x^2}+\left (c^2-d^2 x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )^2}} E\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}{\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}\right )|-7+4 \sqrt {3}\right )}{4 \sqrt [3]{c} d^2 x \sqrt {-\frac {c^{2/3} \left (c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )}{\left (\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )^2}}}-\frac {3^{3/4} \left (c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right ) \sqrt {\frac {c^{4/3}+c^{2/3} \sqrt [3]{c^2-d^2 x^2}+\left (c^2-d^2 x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}{\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}\right ),-7+4 \sqrt {3}\right )}{\sqrt {2} \sqrt [3]{c} d^2 x \sqrt {-\frac {c^{2/3} \left (c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )}{\left (\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )^2}}} \] Output:

-3/2*(-d^2*x^2+c^2)^(2/3)/c/d/(d*x+c)+3/2*x/c/((1-3^(1/2))*c^(2/3)-(-d^2*x 
^2+c^2)^(1/3))+3/4*3^(1/4)*(1/2*6^(1/2)+1/2*2^(1/2))*(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))/c^(1/3)/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)-1/2*3^(3/4)*(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))*2^(1/2)/c^(1/3)/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 = 5.30 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.11 \[ \int \frac {1}{(c+d x) \sqrt [3]{c^2-d^2 x^2}} \, dx=-\frac {3 (c-d x) \sqrt [3]{1+\frac {d x}{c}} \operatorname {Hypergeometric2F1}\left (\frac {2}{3},\frac {4}{3},\frac {5}{3},\frac {c-d x}{2 c}\right )}{4 \sqrt [3]{2} c d \sqrt [3]{c^2-d^2 x^2}} \] Input:

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

Output:

(-3*(c - d*x)*(1 + (d*x)/c)^(1/3)*Hypergeometric2F1[2/3, 4/3, 5/3, (c - d* 
x)/(2*c)])/(4*2^(1/3)*c*d*(c^2 - d^2*x^2)^(1/3))
 

Rubi [A] (warning: unable to verify)

Time = 0.87 (sec) , antiderivative size = 725, normalized size of antiderivative = 1.11, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {504, 215, 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.

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

\(\Big \downarrow \) 504

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

\(\Big \downarrow \) 215

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

\(\Big \downarrow \) 233

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

\(\Big \downarrow \) 241

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

\(\Big \downarrow \) 833

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

\(\Big \downarrow \) 760

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

\(\Big \downarrow \) 2418

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

Input:

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

Output:

-3/(2*d*(c^2 - d^2*x^2)^(1/3)) + c*((3*x)/(2*c^2*(c^2 - d^2*x^2)^(1/3)) + 
(3*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 + 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]*EllipticE[Ar 
cSin[((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]])/(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]*E 
llipticF[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)])))/(4*c^2*d^2*x))
 

Defintions of rubi rules used

rule 215

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) 
/(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1))   Int[(a + b*x^2)^(p + 1 
), x], x] /; FreeQ[{a, b}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[6 
*p])
 

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]

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

Input:

int(1/(d*x+c)/(-d^2*x^2+c^2)^(1/3),x)
 

Output:

int(1/(d*x+c)/(-d^2*x^2+c^2)^(1/3),x)
 

Fricas [F]

\[ \int \frac {1}{(c+d x) \sqrt [3]{c^2-d^2 x^2}} \, dx=\int { \frac {1}{{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {1}{3}} {\left (d x + c\right )}} \,d x } \] Input:

integrate(1/(d*x+c)/(-d^2*x^2+c^2)^(1/3),x, algorithm="fricas")
 

Output:

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

Sympy [F]

\[ \int \frac {1}{(c+d x) \sqrt [3]{c^2-d^2 x^2}} \, dx=\int \frac {1}{\sqrt [3]{- \left (- c + d x\right ) \left (c + d x\right )} \left (c + d x\right )}\, dx \] Input:

integrate(1/(d*x+c)/(-d**2*x**2+c**2)**(1/3),x)
 

Output:

Integral(1/((-(-c + d*x)*(c + d*x))**(1/3)*(c + d*x)), x)
 

Maxima [F]

\[ \int \frac {1}{(c+d x) \sqrt [3]{c^2-d^2 x^2}} \, dx=\int { \frac {1}{{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {1}{3}} {\left (d x + c\right )}} \,d x } \] Input:

integrate(1/(d*x+c)/(-d^2*x^2+c^2)^(1/3),x, algorithm="maxima")
 

Output:

integrate(1/((-d^2*x^2 + c^2)^(1/3)*(d*x + c)), x)
 

Giac [F]

\[ \int \frac {1}{(c+d x) \sqrt [3]{c^2-d^2 x^2}} \, dx=\int { \frac {1}{{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {1}{3}} {\left (d x + c\right )}} \,d x } \] Input:

integrate(1/(d*x+c)/(-d^2*x^2+c^2)^(1/3),x, algorithm="giac")
 

Output:

integrate(1/((-d^2*x^2 + c^2)^(1/3)*(d*x + c)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(c+d x) \sqrt [3]{c^2-d^2 x^2}} \, dx=\int \frac {1}{{\left (c^2-d^2\,x^2\right )}^{1/3}\,\left (c+d\,x\right )} \,d x \] Input:

int(1/((c^2 - d^2*x^2)^(1/3)*(c + d*x)),x)
 

Output:

int(1/((c^2 - d^2*x^2)^(1/3)*(c + d*x)), x)
 

Reduce [F]

\[ \int \frac {1}{(c+d x) \sqrt [3]{c^2-d^2 x^2}} \, dx=\int \frac {1}{\left (-d^{2} x^{2}+c^{2}\right )^{\frac {1}{3}} c +\left (-d^{2} x^{2}+c^{2}\right )^{\frac {1}{3}} d x}d x \] Input:

int(1/(d*x+c)/(-d^2*x^2+c^2)^(1/3),x)
 

Output:

int(1/((c**2 - d**2*x**2)**(1/3)*c + (c**2 - d**2*x**2)**(1/3)*d*x),x)