\(\int (c+d x) (c^2-d^2 x^2)^{2/3} \, dx\) [251]

Optimal result

Integrand size = 22, antiderivative size = 666 \[ \int (c+d x) \left (c^2-d^2 x^2\right )^{2/3} \, dx=\frac {3}{7} c x \left (c^2-d^2 x^2\right )^{2/3}-\frac {3 \left (c^2-d^2 x^2\right )^{5/3}}{10 d}-\frac {12 c^3 x}{7 \left (\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )}-\frac {6 \sqrt [4]{3} \sqrt {2+\sqrt {3}} c^{11/3} \left (c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right ) \sqrt {\frac {c^{4/3}+c^{2/3} \sqrt [3]{c^2-d^2 x^2}+\left (c^2-d^2 x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )^2}} E\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}{\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}\right )|-7+4 \sqrt {3}\right )}{7 d^2 x \sqrt {-\frac {c^{2/3} \left (c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )}{\left (\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )^2}}}+\frac {4 \sqrt {2} 3^{3/4} c^{11/3} \left (c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right ) \sqrt {\frac {c^{4/3}+c^{2/3} \sqrt [3]{c^2-d^2 x^2}+\left (c^2-d^2 x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}{\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}\right ),-7+4 \sqrt {3}\right )}{7 d^2 x \sqrt {-\frac {c^{2/3} \left (c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )}{\left (\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )^2}}} \] Output:

3/7*c*x*(-d^2*x^2+c^2)^(2/3)-3/10*(-d^2*x^2+c^2)^(5/3)/d-12*c^3*x/(7*(1-3^ 
(1/2))*c^(2/3)-7*(-d^2*x^2+c^2)^(1/3))-6/7*3^(1/4)*(1/2*6^(1/2)+1/2*2^(1/2 
))*c^(11/3)*(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)+4/7* 
2^(1/2)*3^(3/4)*c^(11/3)*(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 = 6.10 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.12 \[ \int (c+d x) \left (c^2-d^2 x^2\right )^{2/3} \, dx=-\frac {3 \left (c^2-d^2 x^2\right )^{5/3}}{10 d}+\frac {c x \left (c^2-d^2 x^2\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},\frac {1}{2},\frac {3}{2},\frac {d^2 x^2}{c^2}\right )}{\left (1-\frac {d^2 x^2}{c^2}\right )^{2/3}} \] Input:

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

Output:

(-3*(c^2 - d^2*x^2)^(5/3))/(10*d) + (c*x*(c^2 - d^2*x^2)^(2/3)*Hypergeomet 
ric2F1[-2/3, 1/2, 3/2, (d^2*x^2)/c^2])/(1 - (d^2*x^2)/c^2)^(2/3)
 

Rubi [A] (warning: unable to verify)

Time = 0.88 (sec) , antiderivative size = 722, normalized size of antiderivative = 1.08, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {455, 211, 233, 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.

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

\(\Big \downarrow \) 455

\(\displaystyle c \int \left (c^2-d^2 x^2\right )^{2/3}dx-\frac {3 \left (c^2-d^2 x^2\right )^{5/3}}{10 d}\)

\(\Big \downarrow \) 211

\(\displaystyle c \left (\frac {4}{7} c^2 \int \frac {1}{\sqrt [3]{c^2-d^2 x^2}}dx+\frac {3}{7} x \left (c^2-d^2 x^2\right )^{2/3}\right )-\frac {3 \left (c^2-d^2 x^2\right )^{5/3}}{10 d}\)

\(\Big \downarrow \) 233

\(\displaystyle c \left (\frac {3}{7} x \left (c^2-d^2 x^2\right )^{2/3}-\frac {6 c^2 \sqrt {-d^2 x^2} \int \frac {\sqrt [3]{c^2-d^2 x^2}}{\sqrt {-d^2 x^2}}d\sqrt [3]{c^2-d^2 x^2}}{7 d^2 x}\right )-\frac {3 \left (c^2-d^2 x^2\right )^{5/3}}{10 d}\)

\(\Big \downarrow \) 833

\(\displaystyle c \left (\frac {3}{7} x \left (c^2-d^2 x^2\right )^{2/3}-\frac {6 c^2 \sqrt {-d^2 x^2} \left (\left (1+\sqrt {3}\right ) c^{2/3} \int \frac {1}{\sqrt {-d^2 x^2}}d\sqrt [3]{c^2-d^2 x^2}-\int \frac {\left (1+\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}{\sqrt {-d^2 x^2}}d\sqrt [3]{c^2-d^2 x^2}\right )}{7 d^2 x}\right )-\frac {3 \left (c^2-d^2 x^2\right )^{5/3}}{10 d}\)

\(\Big \downarrow \) 760

\(\displaystyle c \left (\frac {3}{7} x \left (c^2-d^2 x^2\right )^{2/3}-\frac {6 c^2 \sqrt {-d^2 x^2} \left (-\int \frac {\left (1+\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}{\sqrt {-d^2 x^2}}d\sqrt [3]{c^2-d^2 x^2}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) c^{2/3} \left (c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right ) \sqrt {\frac {c^{4/3}+\left (c^2-d^2 x^2\right )^{2/3}+c^{2/3} \sqrt [3]{c^2-d^2 x^2}}{\left (\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}{\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-d^2 x^2} \sqrt {-\frac {c^{2/3} \left (c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )}{\left (\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )^2}}}\right )}{7 d^2 x}\right )-\frac {3 \left (c^2-d^2 x^2\right )^{5/3}}{10 d}\)

\(\Big \downarrow \) 2418

\(\displaystyle c \left (\frac {3}{7} x \left (c^2-d^2 x^2\right )^{2/3}-\frac {6 c^2 \sqrt {-d^2 x^2} \left (-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) c^{2/3} \left (c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right ) \sqrt {\frac {c^{4/3}+\left (c^2-d^2 x^2\right )^{2/3}+c^{2/3} \sqrt [3]{c^2-d^2 x^2}}{\left (\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}{\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-d^2 x^2} \sqrt {-\frac {c^{2/3} \left (c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )}{\left (\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )^2}}}+\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} c^{2/3} \left (c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right ) \sqrt {\frac {c^{4/3}+\left (c^2-d^2 x^2\right )^{2/3}+c^{2/3} \sqrt [3]{c^2-d^2 x^2}}{\left (\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )^2}} E\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}{\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}\right )|-7+4 \sqrt {3}\right )}{\sqrt {-d^2 x^2} \sqrt {-\frac {c^{2/3} \left (c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )}{\left (\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )^2}}}-\frac {2 \sqrt {-d^2 x^2}}{\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}\right )}{7 d^2 x}\right )-\frac {3 \left (c^2-d^2 x^2\right )^{5/3}}{10 d}\)

Input:

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

Output:

(-3*(c^2 - d^2*x^2)^(5/3))/(10*d) + c*((3*x*(c^2 - d^2*x^2)^(2/3))/7 - (6* 
c^2*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[A 
rcSin[((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]* 
EllipticF[ArcSin[((1 + Sqrt[3])*c^(2/3) - (c^2 - d^2*x^2)^(1/3))/((1 - Sqr 
t[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)])))/(7*d^2*x))
 

Defintions of rubi rules used

rule 211

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 
)), x] + Simp[2*a*(p/(2*p + 1))   Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ 
{a, b}, x] && GtQ[p, 0] && (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 455

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

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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [A] (verification not implemented)

Time = 1.08 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.10 \[ \int (c+d x) \left (c^2-d^2 x^2\right )^{2/3} \, dx=c^{\frac {7}{3}} x {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {1}{2} \\ \frac {3}{2} \end {matrix}\middle | {\frac {d^{2} x^{2} e^{2 i \pi }}{c^{2}}} \right )} + d \left (\begin {cases} \frac {x^{2} \left (c^{2}\right )^{\frac {2}{3}}}{2} & \text {for}\: d^{2} = 0 \\- \frac {3 \left (c^{2} - d^{2} x^{2}\right )^{\frac {5}{3}}}{10 d^{2}} & \text {otherwise} \end {cases}\right ) \] Input:

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

Output:

c**(7/3)*x*hyper((-2/3, 1/2), (3/2,), d**2*x**2*exp_polar(2*I*pi)/c**2) + 
d*Piecewise((x**2*(c**2)**(2/3)/2, Eq(d**2, 0)), (-3*(c**2 - d**2*x**2)**( 
5/3)/(10*d**2), True))
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [B] (verification not implemented)

Time = 6.74 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.10 \[ \int (c+d x) \left (c^2-d^2 x^2\right )^{2/3} \, dx=\frac {c\,x\,{\left (c^2-d^2\,x^2\right )}^{2/3}\,{{}}_2{\mathrm {F}}_1\left (-\frac {2}{3},\frac {1}{2};\ \frac {3}{2};\ \frac {d^2\,x^2}{c^2}\right )}{{\left (1-\frac {d^2\,x^2}{c^2}\right )}^{2/3}}-\frac {3\,{\left (c^2-d^2\,x^2\right )}^{5/3}}{10\,d} \] Input:

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

Output:

(c*x*(c^2 - d^2*x^2)^(2/3)*hypergeom([-2/3, 1/2], 3/2, (d^2*x^2)/c^2))/(1 
- (d^2*x^2)/c^2)^(2/3) - (3*(c^2 - d^2*x^2)^(5/3))/(10*d)
 

Reduce [F]

\[ \int (c+d x) \left (c^2-d^2 x^2\right )^{2/3} \, dx=\frac {-21 \left (-d^{2} x^{2}+c^{2}\right )^{\frac {2}{3}} c^{2}+30 \left (-d^{2} x^{2}+c^{2}\right )^{\frac {2}{3}} c d x +21 \left (-d^{2} x^{2}+c^{2}\right )^{\frac {2}{3}} d^{2} x^{2}+40 \left (\int \frac {1}{\left (-d^{2} x^{2}+c^{2}\right )^{\frac {1}{3}}}d x \right ) c^{3} d}{70 d} \] Input:

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

Output:

( - 21*(c**2 - d**2*x**2)**(2/3)*c**2 + 30*(c**2 - d**2*x**2)**(2/3)*c*d*x 
 + 21*(c**2 - d**2*x**2)**(2/3)*d**2*x**2 + 40*int((c**2 - d**2*x**2)**(2/ 
3)/(c**2 - d**2*x**2),x)*c**3*d)/(70*d)