\(\int \frac {\sqrt [3]{c^2-d^2 x^2}}{(c+d x)^{4/3}} \, dx\) [276]

Optimal result

Integrand size = 26, antiderivative size = 251 \[ \int \frac {\sqrt [3]{c^2-d^2 x^2}}{(c+d x)^{4/3}} \, dx=\frac {3 \sqrt [3]{c^2-d^2 x^2}}{d \sqrt [3]{c+d x}}-\frac {\sqrt [3]{2} \sqrt {3} \sqrt [3]{c} \sqrt [3]{c^2-d^2 x^2} \arctan \left (\frac {\sqrt [3]{c}+2^{2/3} \sqrt [3]{c-d x}}{\sqrt {3} \sqrt [3]{c}}\right )}{d \sqrt [3]{c-d x} \sqrt [3]{c+d x}}-\frac {\sqrt [3]{c} \sqrt [3]{c^2-d^2 x^2} \log (c+d x)}{2^{2/3} d \sqrt [3]{c-d x} \sqrt [3]{c+d x}}+\frac {3 \sqrt [3]{c} \sqrt [3]{c^2-d^2 x^2} \log \left (\sqrt [3]{2} \sqrt [3]{c}-\sqrt [3]{c-d x}\right )}{2^{2/3} d \sqrt [3]{c-d x} \sqrt [3]{c+d x}} \] Output:

3*(-d^2*x^2+c^2)^(1/3)/d/(d*x+c)^(1/3)-2^(1/3)*3^(1/2)*c^(1/3)*(-d^2*x^2+c 
^2)^(1/3)*arctan(1/3*(c^(1/3)+2^(2/3)*(-d*x+c)^(1/3))*3^(1/2)/c^(1/3))/d/( 
-d*x+c)^(1/3)/(d*x+c)^(1/3)-1/2*c^(1/3)*(-d^2*x^2+c^2)^(1/3)*ln(d*x+c)*2^( 
1/3)/d/(-d*x+c)^(1/3)/(d*x+c)^(1/3)+3/2*c^(1/3)*(-d^2*x^2+c^2)^(1/3)*ln(2^ 
(1/3)*c^(1/3)-(-d*x+c)^(1/3))*2^(1/3)/d/(-d*x+c)^(1/3)/(d*x+c)^(1/3)
 

Mathematica [A] (verified)

Time = 0.53 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.01 \[ \int \frac {\sqrt [3]{c^2-d^2 x^2}}{(c+d x)^{4/3}} \, dx=\frac {\frac {6 \sqrt [3]{c^2-d^2 x^2}}{\sqrt [3]{c+d x}}+2 \sqrt [3]{2} \sqrt {3} \sqrt [3]{c} \arctan \left (\frac {\sqrt {3} \sqrt [3]{c} \sqrt [3]{c+d x}}{\sqrt [3]{c} \sqrt [3]{c+d x}+2^{2/3} \sqrt [3]{c^2-d^2 x^2}}\right )+2 \sqrt [3]{2} \sqrt [3]{c} \log \left (-2 \sqrt [3]{c} \sqrt [3]{c+d x}+2^{2/3} \sqrt [3]{c^2-d^2 x^2}\right )-\sqrt [3]{2} \sqrt [3]{c} \log \left (2 c^{2/3} (c+d x)^{2/3}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c+d x} \sqrt [3]{c^2-d^2 x^2}+\sqrt [3]{2} \left (c^2-d^2 x^2\right )^{2/3}\right )}{2 d} \] Input:

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

Output:

((6*(c^2 - d^2*x^2)^(1/3))/(c + d*x)^(1/3) + 2*2^(1/3)*Sqrt[3]*c^(1/3)*Arc 
Tan[(Sqrt[3]*c^(1/3)*(c + d*x)^(1/3))/(c^(1/3)*(c + d*x)^(1/3) + 2^(2/3)*( 
c^2 - d^2*x^2)^(1/3))] + 2*2^(1/3)*c^(1/3)*Log[-2*c^(1/3)*(c + d*x)^(1/3) 
+ 2^(2/3)*(c^2 - d^2*x^2)^(1/3)] - 2^(1/3)*c^(1/3)*Log[2*c^(2/3)*(c + d*x) 
^(2/3) + 2^(2/3)*c^(1/3)*(c + d*x)^(1/3)*(c^2 - d^2*x^2)^(1/3) + 2^(1/3)*( 
c^2 - d^2*x^2)^(2/3)])/(2*d)
 

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.78, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {474, 473, 27, 60, 69, 16, 1082, 217}

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.

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

\(\Big \downarrow \) 474

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

\(\Big \downarrow \) 473

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 69

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

\(\Big \downarrow \) 16

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

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

Input:

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

Output:

((c^2 - d^2*x^2)^(4/3)*((3*(c^2 - c*d*x)^(1/3))/d + 2*c^2*(-((Sqrt[3]*ArcT 
an[(1 + (2^(2/3)*(c^2 - c*d*x)^(1/3))/c^(2/3))/Sqrt[3]])/(2^(2/3)*c^(4/3)* 
d)) - Log[c + d*x]/(2*2^(2/3)*c^(4/3)*d) + (3*Log[2^(1/3)*c^(2/3) - (c^2 - 
 c*d*x)^(1/3)])/(2*2^(2/3)*c^(4/3)*d))))/((c + d*x)^(1/3)*(1 + (d*x)/c)*(c 
^2 - c*d*x)^(4/3))
 

Defintions of rubi rules used

rule 16

Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + 
b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 60

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( 
b*(m + n + 1)))   Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, 
 c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[a, b, c, d, m, n, x]
 

rule 69

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[ 
{q = Rt[(b*c - a*d)/b, 3]}, Simp[-Log[RemoveContent[a + b*x, x]]/(2*b*q^2), 
 x] + (-Simp[3/(2*b*q)   Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1 
/3)], x] - Simp[3/(2*b*q^2)   Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)], 
 x])] /; FreeQ[{a, b, c, d}, x] && PosQ[(b*c - a*d)/b]
 

rule 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 473

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
c^(n - 1)*((a + b*x^2)^(p + 1)/((1 + d*(x/c))^(p + 1)*(a/c + (b*x)/d)^(p + 
1)))   Int[(1 + d*(x/c))^(n + p)*(a/c + (b/d)*x)^p, x], x] /; FreeQ[{a, b, 
c, d, n}, x] && EqQ[b*c^2 + a*d^2, 0] && (IntegerQ[n] || GtQ[c, 0]) &&  !Gt 
Q[a, 0] &&  !(IntegerQ[n] && (IntegerQ[3*p] || IntegerQ[4*p]))
 

rule 474

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
c^IntPart[n]*((c + d*x)^FracPart[n]/(1 + d*(x/c))^FracPart[n])   Int[(1 + d 
*(x/c))^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c^2 + 
 a*d^2, 0] &&  !(IntegerQ[n] || GtQ[c, 0])
 

rule 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

Maple [F]

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

Input:

int((-d^2*x^2+c^2)^(1/3)/(d*x+c)^(4/3),x)
 

Output:

int((-d^2*x^2+c^2)^(1/3)/(d*x+c)^(4/3),x)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.01 \[ \int \frac {\sqrt [3]{c^2-d^2 x^2}}{(c+d x)^{4/3}} \, dx=-\frac {2 \, \sqrt {3} 2^{\frac {1}{3}} {\left (d x + c\right )} c^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} 2^{\frac {2}{3}} {\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} c^{\frac {2}{3}} + \sqrt {3} {\left (c d x + c^{2}\right )}}{3 \, {\left (c d x + c^{2}\right )}}\right ) + 2^{\frac {1}{3}} {\left (d x + c\right )} c^{\frac {1}{3}} \log \left (\frac {2^{\frac {2}{3}} {\left (d x + c\right )} c^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} c^{\frac {1}{3}} + {\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}}}{d x + c}\right ) - 2 \cdot 2^{\frac {1}{3}} {\left (d x + c\right )} c^{\frac {1}{3}} \log \left (-\frac {2^{\frac {1}{3}} {\left (d x + c\right )} c^{\frac {1}{3}} - {\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}}}{d x + c}\right ) - 6 \, {\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}}}{2 \, {\left (d^{2} x + c d\right )}} \] Input:

integrate((-d^2*x^2+c^2)^(1/3)/(d*x+c)^(4/3),x, algorithm="fricas")
                                                                                    
                                                                                    
 

Output:

-1/2*(2*sqrt(3)*2^(1/3)*(d*x + c)*c^(1/3)*arctan(1/3*(sqrt(3)*2^(2/3)*(-d^ 
2*x^2 + c^2)^(1/3)*(d*x + c)^(2/3)*c^(2/3) + sqrt(3)*(c*d*x + c^2))/(c*d*x 
 + c^2)) + 2^(1/3)*(d*x + c)*c^(1/3)*log((2^(2/3)*(d*x + c)*c^(2/3) + 2^(1 
/3)*(-d^2*x^2 + c^2)^(1/3)*(d*x + c)^(2/3)*c^(1/3) + (-d^2*x^2 + c^2)^(2/3 
)*(d*x + c)^(1/3))/(d*x + c)) - 2*2^(1/3)*(d*x + c)*c^(1/3)*log(-(2^(1/3)* 
(d*x + c)*c^(1/3) - (-d^2*x^2 + c^2)^(1/3)*(d*x + c)^(2/3))/(d*x + c)) - 6 
*(-d^2*x^2 + c^2)^(1/3)*(d*x + c)^(2/3))/(d^2*x + c*d)
 

Sympy [F]

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

integrate((-d**2*x**2+c**2)**(1/3)/(d*x+c)**(4/3),x)
 

Output:

Integral((-(-c + d*x)*(c + d*x))**(1/3)/(c + d*x)**(4/3), x)
 

Maxima [F]

\[ \int \frac {\sqrt [3]{c^2-d^2 x^2}}{(c+d x)^{4/3}} \, dx=\int { \frac {{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {1}{3}}}{{\left (d x + c\right )}^{\frac {4}{3}}} \,d x } \] Input:

integrate((-d^2*x^2+c^2)^(1/3)/(d*x+c)^(4/3),x, algorithm="maxima")
 

Output:

integrate((-d^2*x^2 + c^2)^(1/3)/(d*x + c)^(4/3), x)
 

Giac [F]

\[ \int \frac {\sqrt [3]{c^2-d^2 x^2}}{(c+d x)^{4/3}} \, dx=\int { \frac {{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {1}{3}}}{{\left (d x + c\right )}^{\frac {4}{3}}} \,d x } \] Input:

integrate((-d^2*x^2+c^2)^(1/3)/(d*x+c)^(4/3),x, algorithm="giac")
 

Output:

integrate((-d^2*x^2 + c^2)^(1/3)/(d*x + c)^(4/3), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt [3]{c^2-d^2 x^2}}{(c+d x)^{4/3}} \, dx=\int \frac {{\left (c^2-d^2\,x^2\right )}^{1/3}}{{\left (c+d\,x\right )}^{4/3}} \,d x \] Input:

int((c^2 - d^2*x^2)^(1/3)/(c + d*x)^(4/3),x)
 

Output:

int((c^2 - d^2*x^2)^(1/3)/(c + d*x)^(4/3), x)
 

Reduce [F]

\[ \int \frac {\sqrt [3]{c^2-d^2 x^2}}{(c+d x)^{4/3}} \, dx=\int \frac {\left (-d^{2} x^{2}+c^{2}\right )^{\frac {1}{3}}}{\left (d x +c \right )^{\frac {1}{3}} c +\left (d x +c \right )^{\frac {1}{3}} d x}d x \] Input:

int((-d^2*x^2+c^2)^(1/3)/(d*x+c)^(4/3),x)
 

Output:

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