Integrand size = 19, antiderivative size = 265 \[ \int \frac {\sqrt {a+b x^3}}{(c x)^{13/2}} \, dx=-\frac {2 \sqrt {a+b x^3}}{11 c (c x)^{11/2}}-\frac {6 b \sqrt {a+b x^3}}{55 a c^4 (c x)^{5/2}}-\frac {2\ 3^{3/4} b^2 \sqrt {c x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{a}+\left (1-\sqrt {3}\right ) \sqrt [3]{b} x}{\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{55 a^{4/3} c^7 \sqrt {\frac {\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \] Output:
-2/11*(b*x^3+a)^(1/2)/c/(c*x)^(11/2)-6/55*b*(b*x^3+a)^(1/2)/a/c^4/(c*x)^(5 /2)-2/55*3^(3/4)*b^2*(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)*InverseJacobi AM(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))/a^(4/3)/c^7/(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 = 56, normalized size of antiderivative = 0.21 \[ \int \frac {\sqrt {a+b x^3}}{(c x)^{13/2}} \, dx=-\frac {2 x \sqrt {a+b x^3} \operatorname {Hypergeometric2F1}\left (-\frac {11}{6},-\frac {1}{2},-\frac {5}{6},-\frac {b x^3}{a}\right )}{11 (c x)^{13/2} \sqrt {1+\frac {b x^3}{a}}} \] Input:
Integrate[Sqrt[a + b*x^3]/(c*x)^(13/2),x]
Output:
(-2*x*Sqrt[a + b*x^3]*Hypergeometric2F1[-11/6, -1/2, -5/6, -((b*x^3)/a)])/ (11*(c*x)^(13/2)*Sqrt[1 + (b*x^3)/a])
Time = 0.57 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.13, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {809, 847, 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.
| \(\displaystyle \int \frac {\sqrt {a+b x^3}}{(c x)^{13/2}} \, dx\) |
| \(\Big \downarrow \) 809 |
| \(\displaystyle \frac {3 b \int \frac {1}{(c x)^{7/2} \sqrt {b x^3+a}}dx}{11 c^3}-\frac {2 \sqrt {a+b x^3}}{11 c (c x)^{11/2}}\) |
| \(\Big \downarrow \) 847 |
| \(\displaystyle \frac {3 b \left (-\frac {2 b \int \frac {1}{\sqrt {c x} \sqrt {b x^3+a}}dx}{5 a c^3}-\frac {2 \sqrt {a+b x^3}}{5 a c (c x)^{5/2}}\right )}{11 c^3}-\frac {2 \sqrt {a+b x^3}}{11 c (c x)^{11/2}}\) |
| \(\Big \downarrow \) 851 |
| \(\displaystyle \frac {3 b \left (-\frac {4 b \int \frac {1}{\sqrt {b x^3+a}}d\sqrt {c x}}{5 a c^4}-\frac {2 \sqrt {a+b x^3}}{5 a c (c x)^{5/2}}\right )}{11 c^3}-\frac {2 \sqrt {a+b x^3}}{11 c (c x)^{11/2}}\) |
| \(\Big \downarrow \) 766 |
| \(\displaystyle \frac {3 b \left (-\frac {2 b \sqrt {c x} \left (\sqrt [3]{a} c+\sqrt [3]{b} c x\right ) \sqrt {\frac {a^{2/3} c^2-\sqrt [3]{a} \sqrt [3]{b} c^2 x+b^{2/3} c^2 x^2}{\left (\sqrt [3]{a} c+\left (1+\sqrt {3}\right ) \sqrt [3]{b} c x\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} x c+\sqrt [3]{a} c}{\left (1+\sqrt {3}\right ) \sqrt [3]{b} x c+\sqrt [3]{a} c}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{5 \sqrt [4]{3} a^{4/3} c^5 \sqrt {a+b x^3} \sqrt {\frac {\sqrt [3]{b} c x \left (\sqrt [3]{a} c+\sqrt [3]{b} c x\right )}{\left (\sqrt [3]{a} c+\left (1+\sqrt {3}\right ) \sqrt [3]{b} c x\right )^2}}}-\frac {2 \sqrt {a+b x^3}}{5 a c (c x)^{5/2}}\right )}{11 c^3}-\frac {2 \sqrt {a+b x^3}}{11 c (c x)^{11/2}}\) |
Input:
Int[Sqrt[a + b*x^3]/(c*x)^(13/2),x]
Output:
(-2*Sqrt[a + b*x^3])/(11*c*(c*x)^(11/2)) + (3*b*((-2*Sqrt[a + b*x^3])/(5*a *c*(c*x)^(5/2)) - (2*b*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])/(5*3^(1/4)*a^(4/ 3)*c^5*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])))/(11*c^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 + 1))), x] - Simp[b*n*(p/(c^n*(m + 1))) I nt[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && IGtQ [n, 0] && GtQ[p, 0] && LtQ[m, -1] && !ILtQ[(m + n*p + n + 1)/n, 0] && IntB inomialQ[a, b, c, n, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x )^(m + 1)*((a + b*x^n)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))) Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a , b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && 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 = 1.15 (sec) , antiderivative size = 742, normalized size of antiderivative = 2.80
| method | result | size |
| risch | \(-\frac {2 \sqrt {b \,x^{3}+a}\, \left (3 b \,x^{3}+5 a \right )}{55 x^{5} a \,c^{6} \sqrt {c x}}-\frac {12 b^{3} \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 )}}\, \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 )}}{55 a \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 )}\, c^{6} \sqrt {c x}\, \sqrt {b \,x^{3}+a}}\) | \(742\) |
| elliptic | \(\frac {\sqrt {c x \left (b \,x^{3}+a \right )}\, \left (-\frac {2 \sqrt {b c \,x^{4}+a c x}}{11 c^{7} x^{6}}-\frac {6 b \sqrt {b c \,x^{4}+a c x}}{55 a \,c^{7} x^{3}}-\frac {12 b^{3} \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 )}}\, \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 )}{55 a \,c^{6} \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}}\) | \(754\) |
| default | \(\text {Expression too large to display}\) | \(2022\) |
Input:
int((b*x^3+a)^(1/2)/(c*x)^(13/2),x,method=_RETURNVERBOSE)
Output:
-2/55*(b*x^3+a)^(1/2)*(3*b*x^3+5*a)/x^5/a/c^6/(c*x)^(1/2)-12/55/a*b^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))*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 ))/(-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^6*(c*x*(b*x^3+a))^(1/2)/(c*x)^(1/2)/(b*x^3+a) ^(1/2)
Time = 0.09 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.25 \[ \int \frac {\sqrt {a+b x^3}}{(c x)^{13/2}} \, dx=\frac {2 \, {\left (6 \, \sqrt {a c} b^{2} x^{6} {\rm weierstrassPInverse}\left (0, -\frac {4 \, b}{a}, \frac {1}{x}\right ) - {\left (3 \, a b x^{3} + 5 \, a^{2}\right )} \sqrt {b x^{3} + a} \sqrt {c x}\right )}}{55 \, a^{2} c^{7} x^{6}} \] Input:
integrate((b*x^3+a)^(1/2)/(c*x)^(13/2),x, algorithm="fricas")
Output:
2/55*(6*sqrt(a*c)*b^2*x^6*weierstrassPInverse(0, -4*b/a, 1/x) - (3*a*b*x^3 + 5*a^2)*sqrt(b*x^3 + a)*sqrt(c*x))/(a^2*c^7*x^6)
Result contains complex when optimal does not.
Time = 158.39 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.20 \[ \int \frac {\sqrt {a+b x^3}}{(c x)^{13/2}} \, dx=\frac {\sqrt {a} \Gamma \left (- \frac {11}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {11}{6}, - \frac {1}{2} \\ - \frac {5}{6} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 c^{\frac {13}{2}} x^{\frac {11}{2}} \Gamma \left (- \frac {5}{6}\right )} \] Input:
integrate((b*x**3+a)**(1/2)/(c*x)**(13/2),x)
Output:
sqrt(a)*gamma(-11/6)*hyper((-11/6, -1/2), (-5/6,), b*x**3*exp_polar(I*pi)/ a)/(3*c**(13/2)*x**(11/2)*gamma(-5/6))
\[ \int \frac {\sqrt {a+b x^3}}{(c x)^{13/2}} \, dx=\int { \frac {\sqrt {b x^{3} + a}}{\left (c x\right )^{\frac {13}{2}}} \,d x } \] Input:
integrate((b*x^3+a)^(1/2)/(c*x)^(13/2),x, algorithm="maxima")
Output:
integrate(sqrt(b*x^3 + a)/(c*x)^(13/2), x)
\[ \int \frac {\sqrt {a+b x^3}}{(c x)^{13/2}} \, dx=\int { \frac {\sqrt {b x^{3} + a}}{\left (c x\right )^{\frac {13}{2}}} \,d x } \] Input:
integrate((b*x^3+a)^(1/2)/(c*x)^(13/2),x, algorithm="giac")
Output:
integrate(sqrt(b*x^3 + a)/(c*x)^(13/2), x)
Timed out. \[ \int \frac {\sqrt {a+b x^3}}{(c x)^{13/2}} \, dx=\int \frac {\sqrt {b\,x^3+a}}{{\left (c\,x\right )}^{13/2}} \,d x \] Input:
int((a + b*x^3)^(1/2)/(c*x)^(13/2),x)
Output:
int((a + b*x^3)^(1/2)/(c*x)^(13/2), x)
\[ \int \frac {\sqrt {a+b x^3}}{(c x)^{13/2}} \, dx=\frac {\sqrt {c}\, \left (-2 \sqrt {b \,x^{3}+a}-3 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{3}+a}}{b \,x^{10}+a \,x^{7}}d x \right ) a \,x^{5}\right )}{8 \sqrt {x}\, c^{7} x^{5}} \] Input:
int((b*x^3+a)^(1/2)/(c*x)^(13/2),x)
Output:
(sqrt(c)*( - 2*sqrt(a + b*x**3) - 3*sqrt(x)*int((sqrt(x)*sqrt(a + b*x**3)) /(a*x**7 + b*x**10),x)*a*x**5))/(8*sqrt(x)*c**7*x**5)