Integrand size = 19, antiderivative size = 296 \[ \int (c x)^{11/2} \sqrt {a+b x^3} \, dx=-\frac {21 a^2 c^5 \sqrt {c x} \sqrt {a+b x^3}}{320 b^2}+\frac {3 a c^2 (c x)^{7/2} \sqrt {a+b x^3}}{80 b}+\frac {(c x)^{13/2} \sqrt {a+b x^3}}{8 c}+\frac {7\ 3^{3/4} a^{8/3} c^5 \sqrt {c x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{a}+\left (1-\sqrt {3}\right ) \sqrt [3]{b} x}{\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{640 b^2 \sqrt {\frac {\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \] Output:
-21/320*a^2*c^5*(c*x)^(1/2)*(b*x^3+a)^(1/2)/b^2+3/80*a*c^2*(c*x)^(7/2)*(b* x^3+a)^(1/2)/b+1/8*(c*x)^(13/2)*(b*x^3+a)^(1/2)/c+7/640*3^(3/4)*a^(8/3)*c^ 5*(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))/b^2/(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 = 10.05 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.35 \[ \int (c x)^{11/2} \sqrt {a+b x^3} \, dx=\frac {c^5 \sqrt {c x} \sqrt {a+b x^3} \left (\sqrt {1+\frac {b x^3}{a}} \left (-7 a^2+3 a b x^3+10 b^2 x^6\right )+7 a^2 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{6},\frac {7}{6},-\frac {b x^3}{a}\right )\right )}{80 b^2 \sqrt {1+\frac {b x^3}{a}}} \] Input:
Integrate[(c*x)^(11/2)*Sqrt[a + b*x^3],x]
Output:
(c^5*Sqrt[c*x]*Sqrt[a + b*x^3]*(Sqrt[1 + (b*x^3)/a]*(-7*a^2 + 3*a*b*x^3 + 10*b^2*x^6) + 7*a^2*Hypergeometric2F1[-1/2, 1/6, 7/6, -((b*x^3)/a)]))/(80* b^2*Sqrt[1 + (b*x^3)/a])
Time = 0.67 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.14, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {811, 843, 843, 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.
| \(\displaystyle \int (c x)^{11/2} \sqrt {a+b x^3} \, dx\) |
| \(\Big \downarrow \) 811 |
| \(\displaystyle \frac {3}{16} a \int \frac {(c x)^{11/2}}{\sqrt {b x^3+a}}dx+\frac {(c x)^{13/2} \sqrt {a+b x^3}}{8 c}\) |
| \(\Big \downarrow \) 843 |
| \(\displaystyle \frac {3}{16} a \left (\frac {c^2 (c x)^{7/2} \sqrt {a+b x^3}}{5 b}-\frac {7 a c^3 \int \frac {(c x)^{5/2}}{\sqrt {b x^3+a}}dx}{10 b}\right )+\frac {(c x)^{13/2} \sqrt {a+b x^3}}{8 c}\) |
| \(\Big \downarrow \) 843 |
| \(\displaystyle \frac {3}{16} a \left (\frac {c^2 (c x)^{7/2} \sqrt {a+b x^3}}{5 b}-\frac {7 a c^3 \left (\frac {c^2 \sqrt {c x} \sqrt {a+b x^3}}{2 b}-\frac {a c^3 \int \frac {1}{\sqrt {c x} \sqrt {b x^3+a}}dx}{4 b}\right )}{10 b}\right )+\frac {(c x)^{13/2} \sqrt {a+b x^3}}{8 c}\) |
| \(\Big \downarrow \) 851 |
| \(\displaystyle \frac {3}{16} a \left (\frac {c^2 (c x)^{7/2} \sqrt {a+b x^3}}{5 b}-\frac {7 a c^3 \left (\frac {c^2 \sqrt {c x} \sqrt {a+b x^3}}{2 b}-\frac {a c^2 \int \frac {1}{\sqrt {b x^3+a}}d\sqrt {c x}}{2 b}\right )}{10 b}\right )+\frac {(c x)^{13/2} \sqrt {a+b x^3}}{8 c}\) |
| \(\Big \downarrow \) 766 |
| \(\displaystyle \frac {3}{16} a \left (\frac {c^2 (c x)^{7/2} \sqrt {a+b x^3}}{5 b}-\frac {7 a c^3 \left (\frac {c^2 \sqrt {c x} \sqrt {a+b x^3}}{2 b}-\frac {a^{2/3} c \sqrt {c x} \left (\sqrt [3]{a} c+\sqrt [3]{b} c x\right ) \sqrt {\frac {a^{2/3} c^2-\sqrt [3]{a} \sqrt [3]{b} c^2 x+b^{2/3} c^2 x^2}{\left (\sqrt [3]{a} c+\left (1+\sqrt {3}\right ) \sqrt [3]{b} c x\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} x c+\sqrt [3]{a} c}{\left (1+\sqrt {3}\right ) \sqrt [3]{b} x c+\sqrt [3]{a} c}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{4 \sqrt [4]{3} b \sqrt {a+b x^3} \sqrt {\frac {\sqrt [3]{b} c x \left (\sqrt [3]{a} c+\sqrt [3]{b} c x\right )}{\left (\sqrt [3]{a} c+\left (1+\sqrt {3}\right ) \sqrt [3]{b} c x\right )^2}}}\right )}{10 b}\right )+\frac {(c x)^{13/2} \sqrt {a+b x^3}}{8 c}\) |
Input:
Int[(c*x)^(11/2)*Sqrt[a + b*x^3],x]
Output:
((c*x)^(13/2)*Sqrt[a + b*x^3])/(8*c) + (3*a*((c^2*(c*x)^(7/2)*Sqrt[a + b*x ^3])/(5*b) - (7*a*c^3*((c^2*Sqrt[c*x]*Sqrt[a + b*x^3])/(2*b) - (a^(2/3)*c* 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]*Elliptic F[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*3^(1/4)*b*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])))/(10*b)))/16
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] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n)^(p + 1)/(b*(m + n*p + 1))), x] - Simp[ a*c^n*((m - n + 1)/(b*(m + n*p + 1))) Int[(c*x)^(m - n)*(a + b*x^n)^p, x] , x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && 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 = 1.63 (sec) , antiderivative size = 751, normalized size of antiderivative = 2.54
| method | result | size |
| risch | \(-\frac {\left (-40 b^{2} x^{6}-12 a b \,x^{3}+21 a^{2}\right ) x \sqrt {b \,x^{3}+a}\, c^{6}}{320 b^{2} \sqrt {c x}}+\frac {21 a^{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 ) c^{6} \sqrt {c x \left (b \,x^{3}+a \right )}}{320 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 ) \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}}\) | \(751\) |
| elliptic | \(\frac {\sqrt {c x}\, \sqrt {c x \left (b \,x^{3}+a \right )}\, \left (\frac {c^{5} x^{6} \sqrt {b c \,x^{4}+a c x}}{8}+\frac {3 a \,c^{5} x^{3} \sqrt {b c \,x^{4}+a c x}}{80 b}-\frac {21 c^{5} a^{2} \sqrt {b c \,x^{4}+a c x}}{320 b^{2}}+\frac {21 c^{6} a^{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 )}{320 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 ) \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 )}{c x \sqrt {b \,x^{3}+a}}\) | \(784\) |
| default | \(\text {Expression too large to display}\) | \(2247\) |
Input:
int((c*x)^(11/2)*(b*x^3+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/320*(-40*b^2*x^6-12*a*b*x^3+21*a^2)*x*(b*x^3+a)^(1/2)/b^2*c^6/(c*x)^(1/ 2)+21/320*a^3/b*(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)
\[ \int (c x)^{11/2} \sqrt {a+b x^3} \, dx=\int { \sqrt {b x^{3} + a} \left (c x\right )^{\frac {11}{2}} \,d x } \] Input:
integrate((c*x)^(11/2)*(b*x^3+a)^(1/2),x, algorithm="fricas")
Output:
integral(sqrt(b*x^3 + a)*sqrt(c*x)*c^5*x^5, x)
Result contains complex when optimal does not.
Time = 134.38 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.16 \[ \int (c x)^{11/2} \sqrt {a+b x^3} \, dx=\frac {\sqrt {a} c^{\frac {11}{2}} x^{\frac {13}{2}} \Gamma \left (\frac {13}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {13}{6} \\ \frac {19}{6} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {19}{6}\right )} \] Input:
integrate((c*x)**(11/2)*(b*x**3+a)**(1/2),x)
Output:
sqrt(a)*c**(11/2)*x**(13/2)*gamma(13/6)*hyper((-1/2, 13/6), (19/6,), b*x** 3*exp_polar(I*pi)/a)/(3*gamma(19/6))
\[ \int (c x)^{11/2} \sqrt {a+b x^3} \, dx=\int { \sqrt {b x^{3} + a} \left (c x\right )^{\frac {11}{2}} \,d x } \] Input:
integrate((c*x)^(11/2)*(b*x^3+a)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(b*x^3 + a)*(c*x)^(11/2), x)
\[ \int (c x)^{11/2} \sqrt {a+b x^3} \, dx=\int { \sqrt {b x^{3} + a} \left (c x\right )^{\frac {11}{2}} \,d x } \] Input:
integrate((c*x)^(11/2)*(b*x^3+a)^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(b*x^3 + a)*(c*x)^(11/2), x)
Timed out. \[ \int (c x)^{11/2} \sqrt {a+b x^3} \, dx=\int {\left (c\,x\right )}^{11/2}\,\sqrt {b\,x^3+a} \,d x \] Input:
int((c*x)^(11/2)*(a + b*x^3)^(1/2),x)
Output:
int((c*x)^(11/2)*(a + b*x^3)^(1/2), x)
\[ \int (c x)^{11/2} \sqrt {a+b x^3} \, dx=\frac {\sqrt {c}\, c^{5} \left (-42 \sqrt {x}\, \sqrt {b \,x^{3}+a}\, a^{2}+24 \sqrt {x}\, \sqrt {b \,x^{3}+a}\, a b \,x^{3}+80 \sqrt {x}\, \sqrt {b \,x^{3}+a}\, b^{2} x^{6}+21 \left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{3}+a}}{b \,x^{4}+a x}d x \right ) a^{3}\right )}{640 b^{2}} \] Input:
int((c*x)^(11/2)*(b*x^3+a)^(1/2),x)
Output:
(sqrt(c)*c**5*( - 42*sqrt(x)*sqrt(a + b*x**3)*a**2 + 24*sqrt(x)*sqrt(a + b *x**3)*a*b*x**3 + 80*sqrt(x)*sqrt(a + b*x**3)*b**2*x**6 + 21*int((sqrt(x)* sqrt(a + b*x**3))/(a*x + b*x**4),x)*a**3))/(640*b**2)