Integrand size = 22, antiderivative size = 637 \[ \int \frac {c+d x}{\sqrt [3]{c^2-d^2 x^2}} \, dx=-\frac {3 \left (c^2-d^2 x^2\right )^{2/3}}{4 d}-\frac {3 c x}{\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}-\frac {3 \sqrt [4]{3} \sqrt {2+\sqrt {3}} c^{5/3} \left (c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right ) \sqrt {\frac {c^{4/3}+c^{2/3} \sqrt [3]{c^2-d^2 x^2}+\left (c^2-d^2 x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )^2}} E\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}{\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}\right )|-7+4 \sqrt {3}\right )}{2 d^2 x \sqrt {-\frac {c^{2/3} \left (c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )}{\left (\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )^2}}}+\frac {\sqrt {2} 3^{3/4} c^{5/3} \left (c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right ) \sqrt {\frac {c^{4/3}+c^{2/3} \sqrt [3]{c^2-d^2 x^2}+\left (c^2-d^2 x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}{\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}\right ),-7+4 \sqrt {3}\right )}{d^2 x \sqrt {-\frac {c^{2/3} \left (c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )}{\left (\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )^2}}} \] 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)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 9.62 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.13 \[ \int \frac {c+d x}{\sqrt [3]{c^2-d^2 x^2}} \, dx=-\frac {3 \left (c^2-d^2 x^2\right )^{2/3}}{4 d}+\frac {c x \sqrt [3]{1-\frac {d^2 x^2}{c^2}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {3}{2},\frac {d^2 x^2}{c^2}\right )}{\sqrt [3]{c^2-d^2 x^2}} \] Input:
Integrate[(c + d*x)/(c^2 - d^2*x^2)^(1/3),x]
Output:
(-3*(c^2 - d^2*x^2)^(2/3))/(4*d) + (c*x*(1 - (d^2*x^2)/c^2)^(1/3)*Hypergeo metric2F1[1/3, 1/2, 3/2, (d^2*x^2)/c^2])/(c^2 - d^2*x^2)^(1/3)
Time = 0.84 (sec) , antiderivative size = 696, normalized size of antiderivative = 1.09, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {455, 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.
\(\displaystyle \int \frac {c+d x}{\sqrt [3]{c^2-d^2 x^2}} \, dx\) |
\(\Big \downarrow \) 455 |
\(\displaystyle c \int \frac {1}{\sqrt [3]{c^2-d^2 x^2}}dx-\frac {3 \left (c^2-d^2 x^2\right )^{2/3}}{4 d}\) |
\(\Big \downarrow \) 233 |
\(\displaystyle -\frac {3 c \sqrt {-d^2 x^2} \int \frac {\sqrt [3]{c^2-d^2 x^2}}{\sqrt {-d^2 x^2}}d\sqrt [3]{c^2-d^2 x^2}}{2 d^2 x}-\frac {3 \left (c^2-d^2 x^2\right )^{2/3}}{4 d}\) |
\(\Big \downarrow \) 833 |
\(\displaystyle -\frac {3 c \sqrt {-d^2 x^2} \left (\left (1+\sqrt {3}\right ) c^{2/3} \int \frac {1}{\sqrt {-d^2 x^2}}d\sqrt [3]{c^2-d^2 x^2}-\int \frac {\left (1+\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}{\sqrt {-d^2 x^2}}d\sqrt [3]{c^2-d^2 x^2}\right )}{2 d^2 x}-\frac {3 \left (c^2-d^2 x^2\right )^{2/3}}{4 d}\) |
\(\Big \downarrow \) 760 |
\(\displaystyle -\frac {3 c \sqrt {-d^2 x^2} \left (-\int \frac {\left (1+\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}{\sqrt {-d^2 x^2}}d\sqrt [3]{c^2-d^2 x^2}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) c^{2/3} \left (c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right ) \sqrt {\frac {c^{4/3}+\left (c^2-d^2 x^2\right )^{2/3}+c^{2/3} \sqrt [3]{c^2-d^2 x^2}}{\left (\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}{\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-d^2 x^2} \sqrt {-\frac {c^{2/3} \left (c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )}{\left (\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )^2}}}\right )}{2 d^2 x}-\frac {3 \left (c^2-d^2 x^2\right )^{2/3}}{4 d}\) |
\(\Big \downarrow \) 2418 |
\(\displaystyle -\frac {3 c \sqrt {-d^2 x^2} \left (-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) c^{2/3} \left (c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right ) \sqrt {\frac {c^{4/3}+\left (c^2-d^2 x^2\right )^{2/3}+c^{2/3} \sqrt [3]{c^2-d^2 x^2}}{\left (\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}{\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-d^2 x^2} \sqrt {-\frac {c^{2/3} \left (c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )}{\left (\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )^2}}}+\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} c^{2/3} \left (c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right ) \sqrt {\frac {c^{4/3}+\left (c^2-d^2 x^2\right )^{2/3}+c^{2/3} \sqrt [3]{c^2-d^2 x^2}}{\left (\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )^2}} E\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}{\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}\right )|-7+4 \sqrt {3}\right )}{\sqrt {-d^2 x^2} \sqrt {-\frac {c^{2/3} \left (c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )}{\left (\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )^2}}}-\frac {2 \sqrt {-d^2 x^2}}{\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}\right )}{2 d^2 x}-\frac {3 \left (c^2-d^2 x^2\right )^{2/3}}{4 d}\) |
Input:
Int[(c + d*x)/(c^2 - d^2*x^2)^(1/3),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 + 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[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*S qrt[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 - Sqr t[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)
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]
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]
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]
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]
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]
\[\int \frac {d x +c}{\left (-d^{2} x^{2}+c^{2}\right )^{\frac {1}{3}}}d x\]
Input:
int((d*x+c)/(-d^2*x^2+c^2)^(1/3),x)
Output:
int((d*x+c)/(-d^2*x^2+c^2)^(1/3),x)
\[ \int \frac {c+d x}{\sqrt [3]{c^2-d^2 x^2}} \, dx=\int { \frac {d x + c}{{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {1}{3}}} \,d x } \] Input:
integrate((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*x - c), x)
Time = 0.96 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.10 \[ \int \frac {c+d x}{\sqrt [3]{c^2-d^2 x^2}} \, dx=\sqrt [3]{c} x {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{2} \\ \frac {3}{2} \end {matrix}\middle | {\frac {d^{2} x^{2} e^{2 i \pi }}{c^{2}}} \right )} + d \left (\begin {cases} \frac {x^{2}}{2 \sqrt [3]{c^{2}}} & \text {for}\: d^{2} = 0 \\- \frac {3 \left (c^{2} - d^{2} x^{2}\right )^{\frac {2}{3}}}{4 d^{2}} & \text {otherwise} \end {cases}\right ) \] Input:
integrate((d*x+c)/(-d**2*x**2+c**2)**(1/3),x)
Output:
c**(1/3)*x*hyper((1/3, 1/2), (3/2,), d**2*x**2*exp_polar(2*I*pi)/c**2) + d *Piecewise((x**2/(2*(c**2)**(1/3)), Eq(d**2, 0)), (-3*(c**2 - d**2*x**2)** (2/3)/(4*d**2), True))
\[ \int \frac {c+d x}{\sqrt [3]{c^2-d^2 x^2}} \, dx=\int { \frac {d x + c}{{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {1}{3}}} \,d x } \] Input:
integrate((d*x+c)/(-d^2*x^2+c^2)^(1/3),x, algorithm="maxima")
Output:
integrate((d*x + c)/(-d^2*x^2 + c^2)^(1/3), x)
\[ \int \frac {c+d x}{\sqrt [3]{c^2-d^2 x^2}} \, dx=\int { \frac {d x + c}{{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {1}{3}}} \,d x } \] Input:
integrate((d*x+c)/(-d^2*x^2+c^2)^(1/3),x, algorithm="giac")
Output:
integrate((d*x + c)/(-d^2*x^2 + c^2)^(1/3), x)
Time = 6.72 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.11 \[ \int \frac {c+d x}{\sqrt [3]{c^2-d^2 x^2}} \, dx=\frac {c\,x\,{\left (1-\frac {d^2\,x^2}{c^2}\right )}^{1/3}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{3},\frac {1}{2};\ \frac {3}{2};\ \frac {d^2\,x^2}{c^2}\right )}{{\left (c^2-d^2\,x^2\right )}^{1/3}}-\frac {3\,{\left (c^2-d^2\,x^2\right )}^{2/3}}{4\,d} \] Input:
int((c + d*x)/(c^2 - d^2*x^2)^(1/3),x)
Output:
(c*x*(1 - (d^2*x^2)/c^2)^(1/3)*hypergeom([1/3, 1/2], 3/2, (d^2*x^2)/c^2))/ (c^2 - d^2*x^2)^(1/3) - (3*(c^2 - d^2*x^2)^(2/3))/(4*d)
\[ \int \frac {c+d x}{\sqrt [3]{c^2-d^2 x^2}} \, dx=\left (\int \frac {x}{\left (-d^{2} x^{2}+c^{2}\right )^{\frac {1}{3}}}d x \right ) d +\left (\int \frac {1}{\left (-d^{2} x^{2}+c^{2}\right )^{\frac {1}{3}}}d x \right ) c \] Input:
int((d*x+c)/(-d^2*x^2+c^2)^(1/3),x)
Output:
int(x/(c**2 - d**2*x**2)**(1/3),x)*d + int(1/(c**2 - d**2*x**2)**(1/3),x)* c