Integrand size = 19, antiderivative size = 515 \[ \int \frac {c+d x}{\sqrt {-a+b x^3}} \, dx=-\frac {2 d \sqrt {-a+b x^3}}{b^{2/3} \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )}+\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \sqrt [3]{a} d \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} E\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right )|-7+4 \sqrt {3}\right )}{b^{2/3} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt {-a+b x^3}}-\frac {2 \sqrt {2-\sqrt {3}} \left (\sqrt [3]{b} c+\left (1+\sqrt {3}\right ) \sqrt [3]{a} d\right ) \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt {-a+b x^3}} \] Output:
-2*d*(b*x^3-a)^(1/2)/b^(2/3)/((1-3^(1/2))*a^(1/3)-b^(1/3)*x)+3^(1/4)*(1/2* 6^(1/2)+1/2*2^(1/2))*a^(1/3)*d*(a^(1/3)-b^(1/3)*x)*((a^(2/3)+a^(1/3)*b^(1/ 3)*x+b^(2/3)*x^2)/((1-3^(1/2))*a^(1/3)-b^(1/3)*x)^2)^(1/2)*EllipticE(((1+3 ^(1/2))*a^(1/3)-b^(1/3)*x)/((1-3^(1/2))*a^(1/3)-b^(1/3)*x),2*I-I*3^(1/2))/ b^(2/3)/(-a^(1/3)*(a^(1/3)-b^(1/3)*x)/((1-3^(1/2))*a^(1/3)-b^(1/3)*x)^2)^( 1/2)/(b*x^3-a)^(1/2)-2/3*(1/2*6^(1/2)-1/2*2^(1/2))*(b^(1/3)*c+(1+3^(1/2))* a^(1/3)*d)*(a^(1/3)-b^(1/3)*x)*((a^(2/3)+a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(( 1-3^(1/2))*a^(1/3)-b^(1/3)*x)^2)^(1/2)*EllipticF(((1+3^(1/2))*a^(1/3)-b^(1 /3)*x)/((1-3^(1/2))*a^(1/3)-b^(1/3)*x),2*I-I*3^(1/2))*3^(3/4)/b^(2/3)/(-a^ (1/3)*(a^(1/3)-b^(1/3)*x)/((1-3^(1/2))*a^(1/3)-b^(1/3)*x)^2)^(1/2)/(b*x^3- a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.02 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.15 \[ \int \frac {c+d x}{\sqrt {-a+b x^3}} \, dx=\frac {x \sqrt {1-\frac {b x^3}{a}} \left (2 c \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},\frac {b x^3}{a}\right )+d x \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2}{3},\frac {5}{3},\frac {b x^3}{a}\right )\right )}{2 \sqrt {-a+b x^3}} \] Input:
Integrate[(c + d*x)/Sqrt[-a + b*x^3],x]
Output:
(x*Sqrt[1 - (b*x^3)/a]*(2*c*Hypergeometric2F1[1/3, 1/2, 4/3, (b*x^3)/a] + d*x*Hypergeometric2F1[1/2, 2/3, 5/3, (b*x^3)/a]))/(2*Sqrt[-a + b*x^3])
Time = 0.91 (sec) , antiderivative size = 522, normalized size of antiderivative = 1.01, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2419, 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.
| \(\displaystyle \int \frac {c+d x}{\sqrt {b x^3-a}} \, dx\) |
| \(\Big \downarrow \) 2419 |
| \(\displaystyle \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \int \frac {1}{\sqrt {b x^3-a}}dx-\frac {d \int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\sqrt {b x^3-a}}dx}{\sqrt [3]{b}}\) |
| \(\Big \downarrow \) 760 |
| \(\displaystyle -\frac {d \int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\sqrt {b x^3-a}}dx}{\sqrt [3]{b}}-\frac {2 \sqrt {2-\sqrt {3}} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt [3]{b} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt {b x^3-a}}\) |
| \(\Big \downarrow \) 2418 |
| \(\displaystyle -\frac {2 \sqrt {2-\sqrt {3}} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt [3]{b} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt {b x^3-a}}-\frac {d \left (\frac {2 \sqrt {b x^3-a}}{\sqrt [3]{b} \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )}-\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} E\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right )|-7+4 \sqrt {3}\right )}{\sqrt [3]{b} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt {b x^3-a}}\right )}{\sqrt [3]{b}}\) |
Input:
Int[(c + d*x)/Sqrt[-a + b*x^3],x]
Output:
-((d*((2*Sqrt[-a + b*x^3])/(b^(1/3)*((1 - Sqrt[3])*a^(1/3) - b^(1/3)*x)) - (3^(1/4)*Sqrt[2 + Sqrt[3]]*a^(1/3)*(a^(1/3) - b^(1/3)*x)*Sqrt[(a^(2/3) + a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/((1 - Sqrt[3])*a^(1/3) - b^(1/3)*x)^2]*El lipticE[ArcSin[((1 + Sqrt[3])*a^(1/3) - b^(1/3)*x)/((1 - Sqrt[3])*a^(1/3) - b^(1/3)*x)], -7 + 4*Sqrt[3]])/(b^(1/3)*Sqrt[-((a^(1/3)*(a^(1/3) - b^(1/3 )*x))/((1 - Sqrt[3])*a^(1/3) - b^(1/3)*x)^2)]*Sqrt[-a + b*x^3])))/b^(1/3)) - (2*Sqrt[2 - Sqrt[3]]*(c + ((1 + Sqrt[3])*a^(1/3)*d)/b^(1/3))*(a^(1/3) - b^(1/3)*x)*Sqrt[(a^(2/3) + a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/((1 - Sqrt[3] )*a^(1/3) - b^(1/3)*x)^2]*EllipticF[ArcSin[((1 + Sqrt[3])*a^(1/3) - b^(1/3 )*x)/((1 - Sqrt[3])*a^(1/3) - b^(1/3)*x)], -7 + 4*Sqrt[3]])/(3^(1/4)*b^(1/ 3)*Sqrt[-((a^(1/3)*(a^(1/3) - b^(1/3)*x))/((1 - Sqrt[3])*a^(1/3) - b^(1/3) *x)^2)]*Sqrt[-a + b*x^3])
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[((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[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = N umer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(c*r - (1 + Sqrt[3])*d*s)/r Int[1/Sqrt[a + b*x^3], x], x] + Simp[d/r Int[((1 + Sqrt[3])*s + r*x)/Sq rt[a + b*x^3], x], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[a] && NeQ[b*c^3 - 2*(5 + 3*Sqrt[3])*a*d^3, 0]
Time = 0.25 (sec) , antiderivative size = 683, normalized size of antiderivative = 1.33
| method | result | size |
| default | \(\frac {2 i c \sqrt {3}\, \left (a \,b^{2}\right )^{\frac {1}{3}} \sqrt {-\frac {i \left (x +\frac {\left (a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (a \,b^{2}\right )^{\frac {1}{3}}}{b}}{-\frac {3 \left (a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (a \,b^{2}\right )^{\frac {1}{3}}}{2 b}}}\, \sqrt {\frac {i \left (x +\frac {\left (a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (a \,b^{2}\right )^{\frac {1}{3}}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {-\frac {i \left (x +\frac {\left (a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {-\frac {i \sqrt {3}\, \left (a \,b^{2}\right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )}{3 b \sqrt {b \,x^{3}-a}}+\frac {2 i d \sqrt {3}\, \left (a \,b^{2}\right )^{\frac {1}{3}} \sqrt {-\frac {i \left (x +\frac {\left (a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (a \,b^{2}\right )^{\frac {1}{3}}}{b}}{-\frac {3 \left (a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (a \,b^{2}\right )^{\frac {1}{3}}}{2 b}}}\, \sqrt {\frac {i \left (x +\frac {\left (a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (a \,b^{2}\right )^{\frac {1}{3}}}}\, \left (\left (-\frac {3 \left (a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \operatorname {EllipticE}\left (\frac {\sqrt {3}\, \sqrt {-\frac {i \left (x +\frac {\left (a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {-\frac {i \sqrt {3}\, \left (a \,b^{2}\right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )+\frac {\left (a \,b^{2}\right )^{\frac {1}{3}} \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {-\frac {i \left (x +\frac {\left (a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {-\frac {i \sqrt {3}\, \left (a \,b^{2}\right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )}{b}\right )}{3 b \sqrt {b \,x^{3}-a}}\) | \(683\) |
| elliptic | \(\frac {2 i c \sqrt {3}\, \left (a \,b^{2}\right )^{\frac {1}{3}} \sqrt {-\frac {i \left (x +\frac {\left (a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (a \,b^{2}\right )^{\frac {1}{3}}}{b}}{-\frac {3 \left (a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (a \,b^{2}\right )^{\frac {1}{3}}}{2 b}}}\, \sqrt {\frac {i \left (x +\frac {\left (a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (a \,b^{2}\right )^{\frac {1}{3}}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {-\frac {i \left (x +\frac {\left (a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {-\frac {i \sqrt {3}\, \left (a \,b^{2}\right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )}{3 b \sqrt {b \,x^{3}-a}}+\frac {2 i d \sqrt {3}\, \left (a \,b^{2}\right )^{\frac {1}{3}} \sqrt {-\frac {i \left (x +\frac {\left (a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (a \,b^{2}\right )^{\frac {1}{3}}}{b}}{-\frac {3 \left (a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (a \,b^{2}\right )^{\frac {1}{3}}}{2 b}}}\, \sqrt {\frac {i \left (x +\frac {\left (a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (a \,b^{2}\right )^{\frac {1}{3}}}}\, \left (\left (-\frac {3 \left (a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \operatorname {EllipticE}\left (\frac {\sqrt {3}\, \sqrt {-\frac {i \left (x +\frac {\left (a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {-\frac {i \sqrt {3}\, \left (a \,b^{2}\right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )+\frac {\left (a \,b^{2}\right )^{\frac {1}{3}} \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {-\frac {i \left (x +\frac {\left (a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {-\frac {i \sqrt {3}\, \left (a \,b^{2}\right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )}{b}\right )}{3 b \sqrt {b \,x^{3}-a}}\) | \(683\) |
Input:
int((d*x+c)/(b*x^3-a)^(1/2),x,method=_RETURNVERBOSE)
Output:
2/3*I*c*3^(1/2)/b*(a*b^2)^(1/3)*(-I*(x+1/2/b*(a*b^2)^(1/3)+1/2*I*3^(1/2)/b *(a*b^2)^(1/3))*3^(1/2)*b/(a*b^2)^(1/3))^(1/2)*((x-1/b*(a*b^2)^(1/3))/(-3/ 2/b*(a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(a*b^2)^(1/3)))^(1/2)*(I*(x+1/2/b*(a*b^2 )^(1/3)-1/2*I*3^(1/2)/b*(a*b^2)^(1/3))*3^(1/2)*b/(a*b^2)^(1/3))^(1/2)/(b*x ^3-a)^(1/2)*EllipticF(1/3*3^(1/2)*(-I*(x+1/2/b*(a*b^2)^(1/3)+1/2*I*3^(1/2) /b*(a*b^2)^(1/3))*3^(1/2)*b/(a*b^2)^(1/3))^(1/2),(-I*3^(1/2)/b*(a*b^2)^(1/ 3)/(-3/2/b*(a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(a*b^2)^(1/3)))^(1/2))+2/3*I*d*3^ (1/2)/b*(a*b^2)^(1/3)*(-I*(x+1/2/b*(a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(a*b^2)^( 1/3))*3^(1/2)*b/(a*b^2)^(1/3))^(1/2)*((x-1/b*(a*b^2)^(1/3))/(-3/2/b*(a*b^2 )^(1/3)-1/2*I*3^(1/2)/b*(a*b^2)^(1/3)))^(1/2)*(I*(x+1/2/b*(a*b^2)^(1/3)-1/ 2*I*3^(1/2)/b*(a*b^2)^(1/3))*3^(1/2)*b/(a*b^2)^(1/3))^(1/2)/(b*x^3-a)^(1/2 )*((-3/2/b*(a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(a*b^2)^(1/3))*EllipticE(1/3*3^(1 /2)*(-I*(x+1/2/b*(a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(a*b^2)^(1/3))*3^(1/2)*b/(a *b^2)^(1/3))^(1/2),(-I*3^(1/2)/b*(a*b^2)^(1/3)/(-3/2/b*(a*b^2)^(1/3)-1/2*I *3^(1/2)/b*(a*b^2)^(1/3)))^(1/2))+1/b*(a*b^2)^(1/3)*EllipticF(1/3*3^(1/2)* (-I*(x+1/2/b*(a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(a*b^2)^(1/3))*3^(1/2)*b/(a*b^2 )^(1/3))^(1/2),(-I*3^(1/2)/b*(a*b^2)^(1/3)/(-3/2/b*(a*b^2)^(1/3)-1/2*I*3^( 1/2)/b*(a*b^2)^(1/3)))^(1/2)))
Time = 0.08 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.08 \[ \int \frac {c+d x}{\sqrt {-a+b x^3}} \, dx=\frac {2 \, {\left (\sqrt {b} c {\rm weierstrassPInverse}\left (0, \frac {4 \, a}{b}, x\right ) - \sqrt {b} d {\rm weierstrassZeta}\left (0, \frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, \frac {4 \, a}{b}, x\right )\right )\right )}}{b} \] Input:
integrate((d*x+c)/(b*x^3-a)^(1/2),x, algorithm="fricas")
Output:
2*(sqrt(b)*c*weierstrassPInverse(0, 4*a/b, x) - sqrt(b)*d*weierstrassZeta( 0, 4*a/b, weierstrassPInverse(0, 4*a/b, x)))/b
Time = 1.11 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.14 \[ \int \frac {c+d x}{\sqrt {-a+b x^3}} \, dx=- \frac {i c x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{2} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3}}{a}} \right )}}{3 \sqrt {a} \Gamma \left (\frac {4}{3}\right )} - \frac {i d x^{2} \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {\frac {b x^{3}}{a}} \right )}}{3 \sqrt {a} \Gamma \left (\frac {5}{3}\right )} \] Input:
integrate((d*x+c)/(b*x**3-a)**(1/2),x)
Output:
-I*c*x*gamma(1/3)*hyper((1/3, 1/2), (4/3,), b*x**3/a)/(3*sqrt(a)*gamma(4/3 )) - I*d*x**2*gamma(2/3)*hyper((1/2, 2/3), (5/3,), b*x**3/a)/(3*sqrt(a)*ga mma(5/3))
\[ \int \frac {c+d x}{\sqrt {-a+b x^3}} \, dx=\int { \frac {d x + c}{\sqrt {b x^{3} - a}} \,d x } \] Input:
integrate((d*x+c)/(b*x^3-a)^(1/2),x, algorithm="maxima")
Output:
integrate((d*x + c)/sqrt(b*x^3 - a), x)
\[ \int \frac {c+d x}{\sqrt {-a+b x^3}} \, dx=\int { \frac {d x + c}{\sqrt {b x^{3} - a}} \,d x } \] Input:
integrate((d*x+c)/(b*x^3-a)^(1/2),x, algorithm="giac")
Output:
integrate((d*x + c)/sqrt(b*x^3 - a), x)
Timed out. \[ \int \frac {c+d x}{\sqrt {-a+b x^3}} \, dx=\int \frac {c+d\,x}{\sqrt {b\,x^3-a}} \,d x \] Input:
int((c + d*x)/(b*x^3 - a)^(1/2),x)
Output:
int((c + d*x)/(b*x^3 - a)^(1/2), x)
\[ \int \frac {c+d x}{\sqrt {-a+b x^3}} \, dx=-\left (\int \frac {\sqrt {b \,x^{3}-a}}{-b \,x^{3}+a}d x \right ) c -\left (\int \frac {\sqrt {b \,x^{3}-a}\, x}{-b \,x^{3}+a}d x \right ) d \] Input:
int((d*x+c)/(b*x^3-a)^(1/2),x)
Output:
- (int(sqrt( - a + b*x**3)/(a - b*x**3),x)*c + int((sqrt( - a + b*x**3)*x )/(a - b*x**3),x)*d)