Integrand size = 19, antiderivative size = 233 \[ \int \frac {\sqrt {a+b x^3}}{\sqrt {c x}} \, dx=\frac {\sqrt {c x} \sqrt {a+b x^3}}{2 c}+\frac {3^{3/4} a^{2/3} \sqrt {c x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{a}+\left (1-\sqrt {3}\right ) \sqrt [3]{b} x}{\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{4 c \sqrt {\frac {\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \] Output:
1/2*(c*x)^(1/2)*(b*x^3+a)^(1/2)/c+1/4*3^(3/4)*a^(2/3)*(c*x)^(1/2)*(a^(1/3) +b^(1/3)*x)*((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(1/3)+(1+3^(1/2))* b^(1/3)*x)^2)^(1/2)*InverseJacobiAM(arccos((a^(1/3)+(1-3^(1/2))*b^(1/3)*x) /(a^(1/3)+(1+3^(1/2))*b^(1/3)*x)),1/4*6^(1/2)+1/4*2^(1/2))/c/(b^(1/3)*x*(a ^(1/3)+b^(1/3)*x)/(a^(1/3)+(1+3^(1/2))*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 = 10.01 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.23 \[ \int \frac {\sqrt {a+b x^3}}{\sqrt {c x}} \, dx=\frac {2 x \sqrt {a+b x^3} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{6},\frac {7}{6},-\frac {b x^3}{a}\right )}{\sqrt {c x} \sqrt {1+\frac {b x^3}{a}}} \] Input:
Integrate[Sqrt[a + b*x^3]/Sqrt[c*x],x]
Output:
(2*x*Sqrt[a + b*x^3]*Hypergeometric2F1[-1/2, 1/6, 7/6, -((b*x^3)/a)])/(Sqr t[c*x]*Sqrt[1 + (b*x^3)/a])
Time = 0.51 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.12, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {811, 851, 766}
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.
| \(\displaystyle \int \frac {\sqrt {a+b x^3}}{\sqrt {c x}} \, dx\) |
| \(\Big \downarrow \) 811 |
| \(\displaystyle \frac {3}{4} a \int \frac {1}{\sqrt {c x} \sqrt {b x^3+a}}dx+\frac {\sqrt {c x} \sqrt {a+b x^3}}{2 c}\) |
| \(\Big \downarrow \) 851 |
| \(\displaystyle \frac {3 a \int \frac {1}{\sqrt {b x^3+a}}d\sqrt {c x}}{2 c}+\frac {\sqrt {c x} \sqrt {a+b x^3}}{2 c}\) |
| \(\Big \downarrow \) 766 |
| \(\displaystyle \frac {3^{3/4} a^{2/3} \sqrt {c x} \left (\sqrt [3]{a} c+\sqrt [3]{b} c x\right ) \sqrt {\frac {a^{2/3} c^2-\sqrt [3]{a} \sqrt [3]{b} c^2 x+b^{2/3} c^2 x^2}{\left (\sqrt [3]{a} c+\left (1+\sqrt {3}\right ) \sqrt [3]{b} c x\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} x c+\sqrt [3]{a} c}{\left (1+\sqrt {3}\right ) \sqrt [3]{b} x c+\sqrt [3]{a} c}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{4 c^2 \sqrt {a+b x^3} \sqrt {\frac {\sqrt [3]{b} c x \left (\sqrt [3]{a} c+\sqrt [3]{b} c x\right )}{\left (\sqrt [3]{a} c+\left (1+\sqrt {3}\right ) \sqrt [3]{b} c x\right )^2}}}+\frac {\sqrt {c x} \sqrt {a+b x^3}}{2 c}\) |
Input:
Int[Sqrt[a + b*x^3]/Sqrt[c*x],x]
Output:
(Sqrt[c*x]*Sqrt[a + b*x^3])/(2*c) + (3^(3/4)*a^(2/3)*Sqrt[c*x]*(a^(1/3)*c + b^(1/3)*c*x)*Sqrt[(a^(2/3)*c^2 - a^(1/3)*b^(1/3)*c^2*x + b^(2/3)*c^2*x^2 )/(a^(1/3)*c + (1 + Sqrt[3])*b^(1/3)*c*x)^2]*EllipticF[ArcCos[(a^(1/3)*c + (1 - Sqrt[3])*b^(1/3)*c*x)/(a^(1/3)*c + (1 + Sqrt[3])*b^(1/3)*c*x)], (2 + Sqrt[3])/4])/(4*c^2*Sqrt[(b^(1/3)*c*x*(a^(1/3)*c + b^(1/3)*c*x))/(a^(1/3) *c + (1 + Sqrt[3])*b^(1/3)*c*x)^2]*Sqrt[a + b*x^3])
Int[1/Sqrt[(a_) + (b_.)*(x_)^6], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[x*(s + r*x^2)*(Sqrt[(s^2 - r*s*x^2 + r^2*x^4)/ (s + (1 + Sqrt[3])*r*x^2)^2]/(2*3^(1/4)*s*Sqrt[a + b*x^6]*Sqrt[r*x^2*((s + r*x^2)/(s + (1 + Sqrt[3])*r*x^2)^2)]))*EllipticF[ArcCos[(s + (1 - Sqrt[3])* r*x^2)/(s + (1 + Sqrt[3])*r*x^2)], (2 + Sqrt[3])/4], x]] /; FreeQ[{a, b}, x ]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c* x)^(m + 1)*((a + b*x^n)^p/(c*(m + n*p + 1))), x] + Simp[a*n*(p/(m + n*p + 1 )) Int[(c*x)^m*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x] && I GtQ[n, 0] && GtQ[p, 0] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b, c, n, m , p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^ n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]
Result contains complex when optimal does not.
Time = 0.80 (sec) , antiderivative size = 717, normalized size of antiderivative = 3.08
| method | result | size |
| risch | \(\frac {x \sqrt {b \,x^{3}+a}}{2 \sqrt {c x}}+\frac {3 a \left (\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 {\frac {\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 ) x}{\left (-\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 ) \left (x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right )}}\, {\left (x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right )}^{2} \sqrt {\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}} \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 )}{b \left (-\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 ) \left (x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right )}}\, \sqrt {\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}} \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 )}{b \left (-\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 ) \left (x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right )}}\, b \operatorname {EllipticF}\left (\sqrt {\frac {\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 ) x}{\left (-\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 ) \left (x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right )}}, \sqrt {\frac {\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 ) \left (\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 )}{\left (\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 ) \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 ) \sqrt {c x \left (b \,x^{3}+a \right )}}{2 \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 ) \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {b c x \left (x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right ) \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 ) \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 {c x}\, \sqrt {b \,x^{3}+a}}\) | \(717\) |
| elliptic | \(\frac {\sqrt {c x \left (b \,x^{3}+a \right )}\, \left (\frac {\sqrt {b c \,x^{4}+a c x}}{2 c}+\frac {3 a \left (\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 {\frac {\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 ) x}{\left (-\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 ) \left (x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right )}}\, {\left (x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right )}^{2} \sqrt {\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}} \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 )}{b \left (-\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 ) \left (x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right )}}\, \sqrt {\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}} \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 )}{b \left (-\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 ) \left (x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right )}}\, b \operatorname {EllipticF}\left (\sqrt {\frac {\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 ) x}{\left (-\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 ) \left (x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right )}}, \sqrt {\frac {\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 ) \left (\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 )}{\left (\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 ) \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 )}{2 \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 ) \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {b c x \left (x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right ) \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 ) \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 )}}\right )}{\sqrt {c x}\, \sqrt {b \,x^{3}+a}}\) | \(719\) |
| default | \(\text {Expression too large to display}\) | \(1797\) |
Input:
int((b*x^3+a)^(1/2)/(c*x)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/2*x*(b*x^3+a)^(1/2)/(c*x)^(1/2)+3/2*a*(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) )*x/(-1/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))/(x-1/b*(-a*b^2) ^(1/3)))^(1/2)*(x-1/b*(-a*b^2)^(1/3))^2*(1/b*(-a*b^2)^(1/3)*(x+1/2/b*(-a*b ^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))/(-1/2/b*(-a*b^2)^(1/3)-1/2*I*3^( 1/2)/b*(-a*b^2)^(1/3))/(x-1/b*(-a*b^2)^(1/3)))^(1/2)*(1/b*(-a*b^2)^(1/3)*( x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))/(-1/2/b*(-a*b^2)^(1 /3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))/(x-1/b*(-a*b^2)^(1/3)))^(1/2)/(-3/2/b* (-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*b/(-a*b^2)^(1/3)/(b*c*x*(x- 1/b*(-a*b^2)^(1/3))*(x+1/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3) )*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2)*EllipticF (((-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*x/(-1/2/b*(-a*b^2 )^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))/(x-1/b*(-a*b^2)^(1/3)))^(1/2),((3/ 2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*(1/2/b*(-a*b^2)^(1/3)-1 /2*I*3^(1/2)/b*(-a*b^2)^(1/3))/(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))*( c*x*(b*x^3+a))^(1/2)/(c*x)^(1/2)/(b*x^3+a)^(1/2)
\[ \int \frac {\sqrt {a+b x^3}}{\sqrt {c x}} \, dx=\int { \frac {\sqrt {b x^{3} + a}}{\sqrt {c x}} \,d x } \] Input:
integrate((b*x^3+a)^(1/2)/(c*x)^(1/2),x, algorithm="fricas")
Output:
integral(sqrt(b*x^3 + a)*sqrt(c*x)/(c*x), x)
Result contains complex when optimal does not.
Time = 0.59 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.20 \[ \int \frac {\sqrt {a+b x^3}}{\sqrt {c x}} \, dx=\frac {\sqrt {a} \sqrt {x} \Gamma \left (\frac {1}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{6} \\ \frac {7}{6} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt {c} \Gamma \left (\frac {7}{6}\right )} \] Input:
integrate((b*x**3+a)**(1/2)/(c*x)**(1/2),x)
Output:
sqrt(a)*sqrt(x)*gamma(1/6)*hyper((-1/2, 1/6), (7/6,), b*x**3*exp_polar(I*p i)/a)/(3*sqrt(c)*gamma(7/6))
\[ \int \frac {\sqrt {a+b x^3}}{\sqrt {c x}} \, dx=\int { \frac {\sqrt {b x^{3} + a}}{\sqrt {c x}} \,d x } \] Input:
integrate((b*x^3+a)^(1/2)/(c*x)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(b*x^3 + a)/sqrt(c*x), x)
\[ \int \frac {\sqrt {a+b x^3}}{\sqrt {c x}} \, dx=\int { \frac {\sqrt {b x^{3} + a}}{\sqrt {c x}} \,d x } \] Input:
integrate((b*x^3+a)^(1/2)/(c*x)^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(b*x^3 + a)/sqrt(c*x), x)
Timed out. \[ \int \frac {\sqrt {a+b x^3}}{\sqrt {c x}} \, dx=\int \frac {\sqrt {b\,x^3+a}}{\sqrt {c\,x}} \,d x \] Input:
int((a + b*x^3)^(1/2)/(c*x)^(1/2),x)
Output:
int((a + b*x^3)^(1/2)/(c*x)^(1/2), x)
\[ \int \frac {\sqrt {a+b x^3}}{\sqrt {c x}} \, dx=\frac {\sqrt {c}\, \left (2 \sqrt {x}\, \sqrt {b \,x^{3}+a}+3 \left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{3}+a}}{b \,x^{4}+a x}d x \right ) a \right )}{4 c} \] Input:
int((b*x^3+a)^(1/2)/(c*x)^(1/2),x)
Output:
(sqrt(c)*(2*sqrt(x)*sqrt(a + b*x**3) + 3*int((sqrt(x)*sqrt(a + b*x**3))/(a *x + b*x**4),x)*a))/(4*c)