\(\int \frac {\sqrt [3]{c^2-d^2 x^2}}{c+d x} \, dx\) [246]

Optimal result

Integrand size = 24, antiderivative size = 310 \[ \int \frac {\sqrt [3]{c^2-d^2 x^2}}{c+d x} \, dx=\frac {3 \sqrt [3]{c^2-d^2 x^2}}{2 d}+\frac {3^{3/4} \sqrt {2-\sqrt {3}} c \left (c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right ) \sqrt {\frac {c^{4/3}+c^{2/3} \sqrt [3]{c^2-d^2 x^2}+\left (c^2-d^2 x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}{\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}\right ),-7+4 \sqrt {3}\right )}{d^2 x \sqrt {-\frac {c^{2/3} \left (c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )}{\left (\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )^2}}} \] Output:

3/2*(-d^2*x^2+c^2)^(1/3)/d+3^(3/4)*(1/2*6^(1/2)-1/2*2^(1/2))*c*(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 = 5.41 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.25 \[ \int \frac {\sqrt [3]{c^2-d^2 x^2}}{c+d x} \, dx=-\frac {3 (c-d x) \left (1+\frac {d x}{c}\right )^{2/3} \sqrt [3]{c^2-d^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {2}{3},\frac {4}{3},\frac {7}{3},\frac {c-d x}{2 c}\right )}{4\ 2^{2/3} d (c+d x)} \] Input:

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

Output:

(-3*(c - d*x)*(1 + (d*x)/c)^(2/3)*(c^2 - d^2*x^2)^(1/3)*Hypergeometric2F1[ 
2/3, 4/3, 7/3, (c - d*x)/(2*c)])/(4*2^(2/3)*d*(c + d*x))
 

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {504, 234, 241, 760}

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.

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

\(\Big \downarrow \) 504

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

\(\Big \downarrow \) 234

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

\(\Big \downarrow \) 241

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

\(\Big \downarrow \) 760

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

Input:

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

Output:

(3*(c^2 - d^2*x^2)^(1/3))/(2*d) + (3^(3/4)*Sqrt[2 - Sqrt[3]]*c*(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]*Ell 
ipticF[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]])/(d^2*x*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)])
 

Defintions of rubi rules used

rule 234

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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