Integrand size = 19, antiderivative size = 513 \[ \int (c x)^{3/2} \sqrt {a+b x^3} \, dx=\frac {(c x)^{5/2} \sqrt {a+b x^3}}{4 c}+\frac {3 \left (1+\sqrt {3}\right ) a c \sqrt {c x} \sqrt {a+b x^3}}{8 b^{2/3} \left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )}-\frac {3 \sqrt [4]{3} a^{4/3} c \sqrt {c x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} E\left (\arccos \left (\frac {\sqrt [3]{a}+\left (1-\sqrt {3}\right ) \sqrt [3]{b} x}{\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{8 b^{2/3} \sqrt {\frac {\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {3^{3/4} \left (1-\sqrt {3}\right ) a^{4/3} c \sqrt {c x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{a}+\left (1-\sqrt {3}\right ) \sqrt [3]{b} x}{\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{16 b^{2/3} \sqrt {\frac {\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \] Output:
1/4*(c*x)^(5/2)*(b*x^3+a)^(1/2)/c+3/8*(1+3^(1/2))*a*c*(c*x)^(1/2)*(b*x^3+a )^(1/2)/b^(2/3)/(a^(1/3)+(1+3^(1/2))*b^(1/3)*x)-3/8*3^(1/4)*a^(4/3)*c*(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)*EllipticE((1-(a^(1/3)+(1-3^(1/2))*b^(1 /3)*x)^2/(a^(1/3)+(1+3^(1/2))*b^(1/3)*x)^2)^(1/2),1/4*6^(1/2)+1/4*2^(1/2)) /b^(2/3)/(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)-1/16*3^(3/4)*(1-3^(1/2))*a^(4/3)*c*(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^(2/3)/(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 = 0.11 \[ \int (c x)^{3/2} \sqrt {a+b x^3} \, dx=\frac {2 x (c x)^{3/2} \sqrt {a+b x^3} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {5}{6},\frac {11}{6},-\frac {b x^3}{a}\right )}{5 \sqrt {1+\frac {b x^3}{a}}} \] Input:
Integrate[(c*x)^(3/2)*Sqrt[a + b*x^3],x]
Output:
(2*x*(c*x)^(3/2)*Sqrt[a + b*x^3]*Hypergeometric2F1[-1/2, 5/6, 11/6, -((b*x ^3)/a)])/(5*Sqrt[1 + (b*x^3)/a])
Time = 1.07 (sec) , antiderivative size = 579, normalized size of antiderivative = 1.13, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {811, 851, 837, 25, 766, 2420}
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.
\(\displaystyle \int (c x)^{3/2} \sqrt {a+b x^3} \, dx\) |
\(\Big \downarrow \) 811 |
\(\displaystyle \frac {3}{8} a \int \frac {(c x)^{3/2}}{\sqrt {b x^3+a}}dx+\frac {(c x)^{5/2} \sqrt {a+b x^3}}{4 c}\) |
\(\Big \downarrow \) 851 |
\(\displaystyle \frac {3 a \int \frac {c^2 x^2}{\sqrt {b x^3+a}}d\sqrt {c x}}{4 c}+\frac {(c x)^{5/2} \sqrt {a+b x^3}}{4 c}\) |
\(\Big \downarrow \) 837 |
\(\displaystyle \frac {3 a \left (-\frac {\left (1-\sqrt {3}\right ) a^{2/3} c^2 \int \frac {1}{\sqrt {b x^3+a}}d\sqrt {c x}}{2 b^{2/3}}-\frac {\int -\frac {2 b^{2/3} x^2 c^2+\left (1-\sqrt {3}\right ) a^{2/3} c^2}{\sqrt {b x^3+a}}d\sqrt {c x}}{2 b^{2/3}}\right )}{4 c}+\frac {(c x)^{5/2} \sqrt {a+b x^3}}{4 c}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {3 a \left (\frac {\int \frac {2 b^{2/3} x^2 c^2+\left (1-\sqrt {3}\right ) a^{2/3} c^2}{\sqrt {b x^3+a}}d\sqrt {c x}}{2 b^{2/3}}-\frac {\left (1-\sqrt {3}\right ) a^{2/3} c^2 \int \frac {1}{\sqrt {b x^3+a}}d\sqrt {c x}}{2 b^{2/3}}\right )}{4 c}+\frac {(c x)^{5/2} \sqrt {a+b x^3}}{4 c}\) |
\(\Big \downarrow \) 766 |
\(\displaystyle \frac {3 a \left (\frac {\int \frac {2 b^{2/3} x^2 c^2+\left (1-\sqrt {3}\right ) a^{2/3} c^2}{\sqrt {b x^3+a}}d\sqrt {c x}}{2 b^{2/3}}-\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a} c \sqrt {c x} \left (\sqrt [3]{a} c+\sqrt [3]{b} c x\right ) \sqrt {\frac {a^{2/3} c^2-\sqrt [3]{a} \sqrt [3]{b} c^2 x+b^{2/3} c^2 x^2}{\left (\sqrt [3]{a} c+\left (1+\sqrt {3}\right ) \sqrt [3]{b} c x\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} x c+\sqrt [3]{a} c}{\left (1+\sqrt {3}\right ) \sqrt [3]{b} x c+\sqrt [3]{a} c}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{4 \sqrt [4]{3} b^{2/3} \sqrt {a+b x^3} \sqrt {\frac {\sqrt [3]{b} c x \left (\sqrt [3]{a} c+\sqrt [3]{b} c x\right )}{\left (\sqrt [3]{a} c+\left (1+\sqrt {3}\right ) \sqrt [3]{b} c x\right )^2}}}\right )}{4 c}+\frac {(c x)^{5/2} \sqrt {a+b x^3}}{4 c}\) |
\(\Big \downarrow \) 2420 |
\(\displaystyle \frac {3 a \left (\frac {\frac {\left (1+\sqrt {3}\right ) c^3 \sqrt {c x} \sqrt {a+b x^3}}{\sqrt [3]{a} c+\left (1+\sqrt {3}\right ) \sqrt [3]{b} c x}-\frac {\sqrt [4]{3} \sqrt [3]{a} c \sqrt {c x} \left (\sqrt [3]{a} c+\sqrt [3]{b} c x\right ) \sqrt {\frac {a^{2/3} c^2-\sqrt [3]{a} \sqrt [3]{b} c^2 x+b^{2/3} c^2 x^2}{\left (\sqrt [3]{a} c+\left (1+\sqrt {3}\right ) \sqrt [3]{b} c x\right )^2}} E\left (\arccos \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} x c+\sqrt [3]{a} c}{\left (1+\sqrt {3}\right ) \sqrt [3]{b} x c+\sqrt [3]{a} c}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{\sqrt {a+b x^3} \sqrt {\frac {\sqrt [3]{b} c x \left (\sqrt [3]{a} c+\sqrt [3]{b} c x\right )}{\left (\sqrt [3]{a} c+\left (1+\sqrt {3}\right ) \sqrt [3]{b} c x\right )^2}}}}{2 b^{2/3}}-\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a} c \sqrt {c x} \left (\sqrt [3]{a} c+\sqrt [3]{b} c x\right ) \sqrt {\frac {a^{2/3} c^2-\sqrt [3]{a} \sqrt [3]{b} c^2 x+b^{2/3} c^2 x^2}{\left (\sqrt [3]{a} c+\left (1+\sqrt {3}\right ) \sqrt [3]{b} c x\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} x c+\sqrt [3]{a} c}{\left (1+\sqrt {3}\right ) \sqrt [3]{b} x c+\sqrt [3]{a} c}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{4 \sqrt [4]{3} b^{2/3} \sqrt {a+b x^3} \sqrt {\frac {\sqrt [3]{b} c x \left (\sqrt [3]{a} c+\sqrt [3]{b} c x\right )}{\left (\sqrt [3]{a} c+\left (1+\sqrt {3}\right ) \sqrt [3]{b} c x\right )^2}}}\right )}{4 c}+\frac {(c x)^{5/2} \sqrt {a+b x^3}}{4 c}\) |
Input:
Int[(c*x)^(3/2)*Sqrt[a + b*x^3],x]
Output:
((c*x)^(5/2)*Sqrt[a + b*x^3])/(4*c) + (3*a*((((1 + Sqrt[3])*c^3*Sqrt[c*x]* Sqrt[a + b*x^3])/(a^(1/3)*c + (1 + Sqrt[3])*b^(1/3)*c*x) - (3^(1/4)*a^(1/3 )*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]*Elli pticE[ArcCos[(a^(1/3)*c + (1 - Sqrt[3])*b^(1/3)*c*x)/(a^(1/3)*c + (1 + Sqr t[3])*b^(1/3)*c*x)], (2 + Sqrt[3])/4])/(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]))/(2 *b^(2/3)) - ((1 - Sqrt[3])*a^(1/3)*c*Sqrt[c*x]*(a^(1/3)*c + b^(1/3)*c*x)*S qrt[(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 *3^(1/4)*b^(2/3)*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])))/(4*c)
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[(x_)^4/Sqrt[(a_) + (b_.)*(x_)^6], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(Sqrt[3] - 1)*(s^2/(2*r^2)) Int[1/Sqrt[ a + b*x^6], x], x] - Simp[1/(2*r^2) Int[((Sqrt[3] - 1)*s^2 - 2*r^2*x^4)/S qrt[a + b*x^6], x], x]] /; FreeQ[{a, b}, 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]
Int[((c_) + (d_.)*(x_)^4)/Sqrt[(a_) + (b_.)*(x_)^6], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(1 + Sqrt[3])*d*s^3*x*(Sqr t[a + b*x^6]/(2*a*r^2*(s + (1 + Sqrt[3])*r*x^2))), x] - Simp[3^(1/4)*d*s*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 *r^2*Sqrt[(r*x^2*(s + r*x^2))/(s + (1 + Sqrt[3])*r*x^2)^2]*Sqrt[a + b*x^6]) )*EllipticE[ArcCos[(s + (1 - Sqrt[3])*r*x^2)/(s + (1 + Sqrt[3])*r*x^2)], (2 + Sqrt[3])/4], x]] /; FreeQ[{a, b, c, d}, x] && EqQ[2*Rt[b/a, 3]^2*c - (1 - Sqrt[3])*d, 0]
Result contains complex when optimal does not.
Time = 0.86 (sec) , antiderivative size = 1109, normalized size of antiderivative = 2.16
method | result | size |
risch | \(\text {Expression too large to display}\) | \(1109\) |
elliptic | \(\text {Expression too large to display}\) | \(1113\) |
default | \(\text {Expression too large to display}\) | \(2599\) |
Input:
int((c*x)^(3/2)*(b*x^3+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/4*x^3*(b*x^3+a)^(1/2)*c^2/(c*x)^(1/2)+3/8*a*(x*(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/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)*(((-1/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1 /2)/b*(-a*b^2)^(1/3))/b*(-a*b^2)^(1/3)+1/b^2*(-a*b^2)^(2/3))/(-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)*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))+(1/2/b* (-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*EllipticE(((-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...
\[ \int (c x)^{3/2} \sqrt {a+b x^3} \, dx=\int { \sqrt {b x^{3} + a} \left (c x\right )^{\frac {3}{2}} \,d x } \] Input:
integrate((c*x)^(3/2)*(b*x^3+a)^(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 = 1.33 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.09 \[ \int (c x)^{3/2} \sqrt {a+b x^3} \, dx=\frac {\sqrt {a} c^{\frac {3}{2}} x^{\frac {5}{2}} \Gamma \left (\frac {5}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{6} \\ \frac {11}{6} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {11}{6}\right )} \] Input:
integrate((c*x)**(3/2)*(b*x**3+a)**(1/2),x)
Output:
sqrt(a)*c**(3/2)*x**(5/2)*gamma(5/6)*hyper((-1/2, 5/6), (11/6,), b*x**3*ex p_polar(I*pi)/a)/(3*gamma(11/6))
\[ \int (c x)^{3/2} \sqrt {a+b x^3} \, dx=\int { \sqrt {b x^{3} + a} \left (c x\right )^{\frac {3}{2}} \,d x } \] Input:
integrate((c*x)^(3/2)*(b*x^3+a)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(b*x^3 + a)*(c*x)^(3/2), x)
\[ \int (c x)^{3/2} \sqrt {a+b x^3} \, dx=\int { \sqrt {b x^{3} + a} \left (c x\right )^{\frac {3}{2}} \,d x } \] Input:
integrate((c*x)^(3/2)*(b*x^3+a)^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(b*x^3 + a)*(c*x)^(3/2), x)
Timed out. \[ \int (c x)^{3/2} \sqrt {a+b x^3} \, dx=\int {\left (c\,x\right )}^{3/2}\,\sqrt {b\,x^3+a} \,d x \] Input:
int((c*x)^(3/2)*(a + b*x^3)^(1/2),x)
Output:
int((c*x)^(3/2)*(a + b*x^3)^(1/2), x)
\[ \int (c x)^{3/2} \sqrt {a+b x^3} \, dx=\frac {\sqrt {c}\, c \left (2 \sqrt {x}\, \sqrt {b \,x^{3}+a}\, x^{2}+3 \left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{3}+a}\, x}{b \,x^{3}+a}d x \right ) a \right )}{8} \] Input:
int((c*x)^(3/2)*(b*x^3+a)^(1/2),x)
Output:
(sqrt(c)*c*(2*sqrt(x)*sqrt(a + b*x**3)*x**2 + 3*int((sqrt(x)*sqrt(a + b*x* *3)*x)/(a + b*x**3),x)*a))/8