Integrand size = 21, antiderivative size = 642 \[ \int x^2 \sqrt {a+b \left (c x^3\right )^{3/2}} \, dx=\frac {4}{21} x^3 \sqrt {a+b \left (c x^3\right )^{3/2}}+\frac {4 a \sqrt {a+b \left (c x^3\right )^{3/2}}}{7 b^{2/3} c \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^3}\right )}-\frac {2 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{4/3} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^3}\right ) \sqrt {\frac {a^{2/3}+b^{2/3} c x^3-\sqrt [3]{a} \sqrt [3]{b} \sqrt {c x^3}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^3}\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^3}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^3}}\right )|-7-4 \sqrt {3}\right )}{7 b^{2/3} c \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^3}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^3}\right )^2}} \sqrt {a+b \left (c x^3\right )^{3/2}}}+\frac {4 \sqrt {2} a^{4/3} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^3}\right ) \sqrt {\frac {a^{2/3}+b^{2/3} c x^3-\sqrt [3]{a} \sqrt [3]{b} \sqrt {c x^3}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^3}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^3}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^3}}\right ),-7-4 \sqrt {3}\right )}{7 \sqrt [4]{3} b^{2/3} c \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^3}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {c x^3}\right )^2}} \sqrt {a+b \left (c x^3\right )^{3/2}}} \]
4/21*x^3*(a+b*(c*x^3)^(3/2))^(1/2)+4/7*a*(a+b*(c*x^3)^(3/2))^(1/2)/b^(2/3) /c/(a^(1/3)*(1+3^(1/2))+b^(1/3)*(c*x^3)^(1/2))+4/21*a^(4/3)*EllipticF((a^( 1/3)*(1-3^(1/2))+b^(1/3)*(c*x^3)^(1/2))/(a^(1/3)*(1+3^(1/2))+b^(1/3)*(c*x^ 3)^(1/2)),I*3^(1/2)+2*I)*2^(1/2)*(a^(1/3)+b^(1/3)*(c*x^3)^(1/2))*((a^(2/3) +b^(2/3)*c*x^3-a^(1/3)*b^(1/3)*(c*x^3)^(1/2))/(a^(1/3)*(1+3^(1/2))+b^(1/3) *(c*x^3)^(1/2))^2)^(1/2)*3^(3/4)/b^(2/3)/c/(a+b*(c*x^3)^(3/2))^(1/2)/(a^(1 /3)*(a^(1/3)+b^(1/3)*(c*x^3)^(1/2))/(a^(1/3)*(1+3^(1/2))+b^(1/3)*(c*x^3)^( 1/2))^2)^(1/2)-2/7*3^(1/4)*a^(4/3)*EllipticE((a^(1/3)*(1-3^(1/2))+b^(1/3)* (c*x^3)^(1/2))/(a^(1/3)*(1+3^(1/2))+b^(1/3)*(c*x^3)^(1/2)),I*3^(1/2)+2*I)* (a^(1/3)+b^(1/3)*(c*x^3)^(1/2))*(1/2*6^(1/2)-1/2*2^(1/2))*((a^(2/3)+b^(2/3 )*c*x^3-a^(1/3)*b^(1/3)*(c*x^3)^(1/2))/(a^(1/3)*(1+3^(1/2))+b^(1/3)*(c*x^3 )^(1/2))^2)^(1/2)/b^(2/3)/c/(a+b*(c*x^3)^(3/2))^(1/2)/(a^(1/3)*(a^(1/3)+b^ (1/3)*(c*x^3)^(1/2))/(a^(1/3)*(1+3^(1/2))+b^(1/3)*(c*x^3)^(1/2))^2)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.08 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.11 \[ \int x^2 \sqrt {a+b \left (c x^3\right )^{3/2}} \, dx=\frac {x^3 \sqrt {a+b \left (c x^3\right )^{3/2}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {2}{3},\frac {5}{3},-\frac {b \left (c x^3\right )^{3/2}}{a}\right )}{3 \sqrt {1+\frac {b \left (c x^3\right )^{3/2}}{a}}} \]
(x^3*Sqrt[a + b*(c*x^3)^(3/2)]*Hypergeometric2F1[-1/2, 2/3, 5/3, -((b*(c*x ^3)^(3/2))/a)])/(3*Sqrt[1 + (b*(c*x^3)^(3/2))/a])
Time = 0.63 (sec) , antiderivative size = 727, normalized size of antiderivative = 1.13, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {893, 864, 807, 811, 832, 759, 2416}
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.
| \(\displaystyle \int x^2 \sqrt {a+b \left (c x^3\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 893 |
| \(\displaystyle \int x^2 \sqrt {a+b c^{3/2} x^{9/2}}dx\) |
| \(\Big \downarrow \) 864 |
| \(\displaystyle 2 \int \frac {\left (c x^3\right )^{5/2} \sqrt {\frac {b \left (c x^3\right )^{9/2}}{c^3 x^9}+a}}{c^{5/2} x^5}d\frac {\sqrt {c x^3}}{\sqrt {c} x}\) |
| \(\Big \downarrow \) 807 |
| \(\displaystyle \frac {2}{3} \int \frac {\left (c x^3\right )^{3/2} \sqrt {\frac {b \left (c x^3\right )^{3/2}}{x^3}+a}}{c^{3/2} x^3}d\frac {\left (c x^3\right )^{3/2}}{c^{3/2} x^3}\) |
| \(\Big \downarrow \) 811 |
| \(\displaystyle \frac {2}{3} \left (\frac {3}{7} a \int \frac {\left (c x^3\right )^{3/2}}{c^{3/2} x^3 \sqrt {\frac {b \left (c x^3\right )^{3/2}}{x^3}+a}}d\frac {\left (c x^3\right )^{3/2}}{c^{3/2} x^3}+\frac {2}{7} x \sqrt {a+\frac {b \left (c x^3\right )^{3/2}}{x^3}}\right )\) |
| \(\Big \downarrow \) 832 |
| \(\displaystyle \frac {2}{3} \left (\frac {3}{7} a \left (\frac {\int \frac {\frac {\sqrt [3]{b} \sqrt {c x^3}}{x}+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt {\frac {b \left (c x^3\right )^{3/2}}{x^3}+a}}d\frac {\left (c x^3\right )^{3/2}}{c^{3/2} x^3}}{\sqrt [3]{b} \sqrt {c}}-\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a} \int \frac {1}{\sqrt {\frac {b \left (c x^3\right )^{3/2}}{x^3}+a}}d\frac {\left (c x^3\right )^{3/2}}{c^{3/2} x^3}}{\sqrt [3]{b} \sqrt {c}}\right )+\frac {2}{7} x \sqrt {a+\frac {b \left (c x^3\right )^{3/2}}{x^3}}\right )\) |
| \(\Big \downarrow \) 759 |
| \(\displaystyle \frac {2}{3} \left (\frac {3}{7} a \left (\frac {\int \frac {\frac {\sqrt [3]{b} \sqrt {c x^3}}{x}+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt {\frac {b \left (c x^3\right )^{3/2}}{x^3}+a}}d\frac {\left (c x^3\right )^{3/2}}{c^{3/2} x^3}}{\sqrt [3]{b} \sqrt {c}}-\frac {2 \left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b} \sqrt {c x^3}}{x}\right ) \sqrt {\frac {a^{2/3}-\frac {\sqrt [3]{a} \sqrt [3]{b} \sqrt {c x^3}}{x}+b^{2/3} c x}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} \sqrt {c x^3}}{x}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\frac {\sqrt [3]{b} \sqrt {c x^3}}{x}+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\frac {\sqrt [3]{b} \sqrt {c x^3}}{x}+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} b^{2/3} c \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b} \sqrt {c x^3}}{x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} \sqrt {c x^3}}{x}\right )^2}} \sqrt {a+\frac {b \left (c x^3\right )^{3/2}}{x^3}}}\right )+\frac {2}{7} x \sqrt {a+\frac {b \left (c x^3\right )^{3/2}}{x^3}}\right )\) |
| \(\Big \downarrow \) 2416 |
| \(\displaystyle \frac {2}{3} \left (\frac {3}{7} a \left (\frac {\frac {2 \sqrt {a+\frac {b \left (c x^3\right )^{3/2}}{x^3}}}{\sqrt [3]{b} \sqrt {c} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} \sqrt {c x^3}}{x}\right )}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b} \sqrt {c x^3}}{x}\right ) \sqrt {\frac {a^{2/3}-\frac {\sqrt [3]{a} \sqrt [3]{b} \sqrt {c x^3}}{x}+b^{2/3} c x}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} \sqrt {c x^3}}{x}\right )^2}} E\left (\arcsin \left (\frac {\frac {\sqrt [3]{b} \sqrt {c x^3}}{x}+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\frac {\sqrt [3]{b} \sqrt {c x^3}}{x}+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{\sqrt [3]{b} \sqrt {c} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b} \sqrt {c x^3}}{x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} \sqrt {c x^3}}{x}\right )^2}} \sqrt {a+\frac {b \left (c x^3\right )^{3/2}}{x^3}}}}{\sqrt [3]{b} \sqrt {c}}-\frac {2 \left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b} \sqrt {c x^3}}{x}\right ) \sqrt {\frac {a^{2/3}-\frac {\sqrt [3]{a} \sqrt [3]{b} \sqrt {c x^3}}{x}+b^{2/3} c x}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} \sqrt {c x^3}}{x}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\frac {\sqrt [3]{b} \sqrt {c x^3}}{x}+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\frac {\sqrt [3]{b} \sqrt {c x^3}}{x}+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} b^{2/3} c \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b} \sqrt {c x^3}}{x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} \sqrt {c x^3}}{x}\right )^2}} \sqrt {a+\frac {b \left (c x^3\right )^{3/2}}{x^3}}}\right )+\frac {2}{7} x \sqrt {a+\frac {b \left (c x^3\right )^{3/2}}{x^3}}\right )\) |
(2*((2*x*Sqrt[a + (b*(c*x^3)^(3/2))/x^3])/7 + (3*a*(((2*Sqrt[a + (b*(c*x^3 )^(3/2))/x^3])/(b^(1/3)*Sqrt[c]*((1 + Sqrt[3])*a^(1/3) + (b^(1/3)*Sqrt[c*x ^3])/x)) - (3^(1/4)*Sqrt[2 - Sqrt[3]]*a^(1/3)*(a^(1/3) + (b^(1/3)*Sqrt[c*x ^3])/x)*Sqrt[(a^(2/3) + b^(2/3)*c*x - (a^(1/3)*b^(1/3)*Sqrt[c*x^3])/x)/((1 + Sqrt[3])*a^(1/3) + (b^(1/3)*Sqrt[c*x^3])/x)^2]*EllipticE[ArcSin[((1 - S qrt[3])*a^(1/3) + (b^(1/3)*Sqrt[c*x^3])/x)/((1 + Sqrt[3])*a^(1/3) + (b^(1/ 3)*Sqrt[c*x^3])/x)], -7 - 4*Sqrt[3]])/(b^(1/3)*Sqrt[c]*Sqrt[(a^(1/3)*(a^(1 /3) + (b^(1/3)*Sqrt[c*x^3])/x))/((1 + Sqrt[3])*a^(1/3) + (b^(1/3)*Sqrt[c*x ^3])/x)^2]*Sqrt[a + (b*(c*x^3)^(3/2))/x^3]))/(b^(1/3)*Sqrt[c]) - (2*(1 - S qrt[3])*Sqrt[2 + Sqrt[3]]*a^(1/3)*(a^(1/3) + (b^(1/3)*Sqrt[c*x^3])/x)*Sqrt [(a^(2/3) + b^(2/3)*c*x - (a^(1/3)*b^(1/3)*Sqrt[c*x^3])/x)/((1 + Sqrt[3])* a^(1/3) + (b^(1/3)*Sqrt[c*x^3])/x)^2]*EllipticF[ArcSin[((1 - Sqrt[3])*a^(1 /3) + (b^(1/3)*Sqrt[c*x^3])/x)/((1 + Sqrt[3])*a^(1/3) + (b^(1/3)*Sqrt[c*x^ 3])/x)], -7 - 4*Sqrt[3]])/(3^(1/4)*b^(2/3)*c*Sqrt[(a^(1/3)*(a^(1/3) + (b^( 1/3)*Sqrt[c*x^3])/x))/((1 + Sqrt[3])*a^(1/3) + (b^(1/3)*Sqrt[c*x^3])/x)^2] *Sqrt[a + (b*(c*x^3)^(3/2))/x^3])))/7))/3
3.30.74.3.1 Defintions of rubi rules used
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] & & PosQ[a]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Simp[1/k Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]
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[(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] && PosQ[a]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denomi nator[n]}, Simp[k Subst[Int[x^(k*(m + 1) - 1)*(a + b*x^(k*n))^p, x], x, x ^(1/k)], x]] /; FreeQ[{a, b, m, p}, x] && FractionQ[n]
Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*((c_.)*(x_)^(q_))^(n_))^(p_.), x_Symbo l] :> With[{k = Denominator[n]}, Subst[Int[(d*x)^m*(a + b*c^n*x^(n*q))^p, x ], x^(1/k), (c*x^q)^(1/k)/(c^(1/k)*(x^(1/k))^(q - 1))]] /; FreeQ[{a, b, c, d, m, p, q}, x] && FractionQ[n]
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 + Sqrt [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] && PosQ[a] && Eq Q[b*c^3 - 2*(5 - 3*Sqrt[3])*a*d^3, 0]
Time = 3.96 (sec) , antiderivative size = 495, normalized size of antiderivative = 0.77
| method | result | size |
| derivativedivides | \(\frac {\frac {4 c \,x^{3} \sqrt {a +b \left (c \,x^{3}\right )^{\frac {3}{2}}}}{7}-\frac {4 i a \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (\sqrt {c \,x^{3}}+\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 {\sqrt {c \,x^{3}}-\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 (\sqrt {c \,x^{3}}+\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 ) E\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (\sqrt {c \,x^{3}}+\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}} F\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (\sqrt {c \,x^{3}}+\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 )}{7 b \sqrt {a +b \left (c \,x^{3}\right )^{\frac {3}{2}}}}}{3 c}\) | \(495\) |
| default | \(\frac {\frac {4 c \,x^{3} \sqrt {a +b \left (c \,x^{3}\right )^{\frac {3}{2}}}}{7}-\frac {4 i a \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (\sqrt {c \,x^{3}}+\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 {\sqrt {c \,x^{3}}-\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 (\sqrt {c \,x^{3}}+\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 ) E\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (\sqrt {c \,x^{3}}+\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}} F\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (\sqrt {c \,x^{3}}+\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 )}{7 b \sqrt {a +b \left (c \,x^{3}\right )^{\frac {3}{2}}}}}{3 c}\) | \(495\) |
1/3/c*(4/7*c*x^3*(a+b*(c*x^3)^(3/2))^(1/2)-4/7*I*a*3^(1/2)/b*(-a*b^2)^(1/3 )*(I*((c*x^3)^(1/2)+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)*(((c*x^3)^(1/2)-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*((c*x^3)^(1/2)+ 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)/(a+b*(c*x^3)^(3/2))^(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*((c*x^3)^(1/2)+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*((c*x^3)^(1/2)+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))))
\[ \int x^2 \sqrt {a+b \left (c x^3\right )^{3/2}} \, dx=\int { \sqrt {\left (c x^{3}\right )^{\frac {3}{2}} b + a} x^{2} \,d x } \]
\[ \int x^2 \sqrt {a+b \left (c x^3\right )^{3/2}} \, dx=\int x^{2} \sqrt {a + b \left (c x^{3}\right )^{\frac {3}{2}}}\, dx \]
\[ \int x^2 \sqrt {a+b \left (c x^3\right )^{3/2}} \, dx=\int { \sqrt {\left (c x^{3}\right )^{\frac {3}{2}} b + a} x^{2} \,d x } \]
\[ \int x^2 \sqrt {a+b \left (c x^3\right )^{3/2}} \, dx=\int { \sqrt {\left (c x^{3}\right )^{\frac {3}{2}} b + a} x^{2} \,d x } \]
Time = 6.51 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.08 \[ \int x^2 \sqrt {a+b \left (c x^3\right )^{3/2}} \, dx=\frac {x^3\,\sqrt {a+b\,{\left (c\,x^3\right )}^{3/2}}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{2},\frac {2}{3};\ \frac {5}{3};\ -\frac {b\,{\left (c\,x^3\right )}^{3/2}}{a}\right )}{3\,\sqrt {\frac {b\,{\left (c\,x^3\right )}^{3/2}}{a}+1}} \]