Integrand size = 27, antiderivative size = 318 \[ \int \frac {x^3 \left (a+b x^3+c x^6\right )}{\sqrt {d+e x^3}} \, dx=\frac {2 \left (112 c d^2-136 b d e+187 a e^2\right ) x \sqrt {d+e x^3}}{935 e^3}-\frac {2 (14 c d-17 b e) x^4 \sqrt {d+e x^3}}{187 e^2}+\frac {2 c x^7 \sqrt {d+e x^3}}{17 e}-\frac {4 \sqrt {2+\sqrt {3}} d \left (112 c d^2-136 b d e+187 a e^2\right ) \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \sqrt {\frac {d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x}\right ),-7-4 \sqrt {3}\right )}{935 \sqrt [4]{3} e^{10/3} \sqrt {\frac {\sqrt [3]{d} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \sqrt {d+e x^3}} \] Output:
2/935*(187*a*e^2-136*b*d*e+112*c*d^2)*x*(e*x^3+d)^(1/2)/e^3-2/187*(-17*b*e +14*c*d)*x^4*(e*x^3+d)^(1/2)/e^2+2/17*c*x^7*(e*x^3+d)^(1/2)/e-4/2805*(1/2* 6^(1/2)+1/2*2^(1/2))*d*(187*a*e^2-136*b*d*e+112*c*d^2)*(d^(1/3)+e^(1/3)*x) *((d^(2/3)-d^(1/3)*e^(1/3)*x+e^(2/3)*x^2)/((1+3^(1/2))*d^(1/3)+e^(1/3)*x)^ 2)^(1/2)*EllipticF(((1-3^(1/2))*d^(1/3)+e^(1/3)*x)/((1+3^(1/2))*d^(1/3)+e^ (1/3)*x),I*3^(1/2)+2*I)*3^(3/4)/e^(10/3)/(d^(1/3)*(d^(1/3)+e^(1/3)*x)/((1+ 3^(1/2))*d^(1/3)+e^(1/3)*x)^2)^(1/2)/(e*x^3+d)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.15 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.39 \[ \int \frac {x^3 \left (a+b x^3+c x^6\right )}{\sqrt {d+e x^3}} \, dx=\frac {2 x \left (\left (d+e x^3\right ) \left (17 e \left (-8 b d+11 a e+5 b e x^3\right )+c \left (112 d^2-70 d e x^3+55 e^2 x^6\right )\right )+d \left (-112 c d^2+17 e (8 b d-11 a e)\right ) \sqrt {1+\frac {e x^3}{d}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},-\frac {e x^3}{d}\right )\right )}{935 e^3 \sqrt {d+e x^3}} \] Input:
Integrate[(x^3*(a + b*x^3 + c*x^6))/Sqrt[d + e*x^3],x]
Output:
(2*x*((d + e*x^3)*(17*e*(-8*b*d + 11*a*e + 5*b*e*x^3) + c*(112*d^2 - 70*d* e*x^3 + 55*e^2*x^6)) + d*(-112*c*d^2 + 17*e*(8*b*d - 11*a*e))*Sqrt[1 + (e* x^3)/d]*Hypergeometric2F1[1/3, 1/2, 4/3, -((e*x^3)/d)]))/(935*e^3*Sqrt[d + e*x^3])
Time = 0.40 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {1810, 27, 959, 843, 759}
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.
| \(\displaystyle \int \frac {x^3 \left (a+b x^3+c x^6\right )}{\sqrt {d+e x^3}} \, dx\) |
| \(\Big \downarrow \) 1810 |
| \(\displaystyle \frac {2 \int \frac {x^3 \left (17 a e-(14 c d-17 b e) x^3\right )}{2 \sqrt {e x^3+d}}dx}{17 e}+\frac {2 c x^7 \sqrt {d+e x^3}}{17 e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {x^3 \left (17 a e-(14 c d-17 b e) x^3\right )}{\sqrt {e x^3+d}}dx}{17 e}+\frac {2 c x^7 \sqrt {d+e x^3}}{17 e}\) |
| \(\Big \downarrow \) 959 |
| \(\displaystyle \frac {\frac {\left (112 c d^2-17 e (8 b d-11 a e)\right ) \int \frac {x^3}{\sqrt {e x^3+d}}dx}{11 e}-\frac {2 x^4 \sqrt {d+e x^3} (14 c d-17 b e)}{11 e}}{17 e}+\frac {2 c x^7 \sqrt {d+e x^3}}{17 e}\) |
| \(\Big \downarrow \) 843 |
| \(\displaystyle \frac {\frac {\left (112 c d^2-17 e (8 b d-11 a e)\right ) \left (\frac {2 x \sqrt {d+e x^3}}{5 e}-\frac {2 d \int \frac {1}{\sqrt {e x^3+d}}dx}{5 e}\right )}{11 e}-\frac {2 x^4 \sqrt {d+e x^3} (14 c d-17 b e)}{11 e}}{17 e}+\frac {2 c x^7 \sqrt {d+e x^3}}{17 e}\) |
| \(\Big \downarrow \) 759 |
| \(\displaystyle \frac {\frac {\left (112 c d^2-17 e (8 b d-11 a e)\right ) \left (\frac {2 x \sqrt {d+e x^3}}{5 e}-\frac {4 \sqrt {2+\sqrt {3}} d \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \sqrt {\frac {d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{e} x+\left (1-\sqrt {3}\right ) \sqrt [3]{d}}{\sqrt [3]{e} x+\left (1+\sqrt {3}\right ) \sqrt [3]{d}}\right ),-7-4 \sqrt {3}\right )}{5 \sqrt [4]{3} e^{4/3} \sqrt {\frac {\sqrt [3]{d} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \sqrt {d+e x^3}}\right )}{11 e}-\frac {2 x^4 \sqrt {d+e x^3} (14 c d-17 b e)}{11 e}}{17 e}+\frac {2 c x^7 \sqrt {d+e x^3}}{17 e}\) |
Input:
Int[(x^3*(a + b*x^3 + c*x^6))/Sqrt[d + e*x^3],x]
Output:
(2*c*x^7*Sqrt[d + e*x^3])/(17*e) + ((-2*(14*c*d - 17*b*e)*x^4*Sqrt[d + e*x ^3])/(11*e) + ((112*c*d^2 - 17*e*(8*b*d - 11*a*e))*((2*x*Sqrt[d + e*x^3])/ (5*e) - (4*Sqrt[2 + Sqrt[3]]*d*(d^(1/3) + e^(1/3)*x)*Sqrt[(d^(2/3) - d^(1/ 3)*e^(1/3)*x + e^(2/3)*x^2)/((1 + Sqrt[3])*d^(1/3) + e^(1/3)*x)^2]*Ellipti cF[ArcSin[((1 - Sqrt[3])*d^(1/3) + e^(1/3)*x)/((1 + Sqrt[3])*d^(1/3) + e^( 1/3)*x)], -7 - 4*Sqrt[3]])/(5*3^(1/4)*e^(4/3)*Sqrt[(d^(1/3)*(d^(1/3) + e^( 1/3)*x))/((1 + Sqrt[3])*d^(1/3) + e^(1/3)*x)^2]*Sqrt[d + e*x^3])))/(11*e)) /(17*e)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[((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[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n _)), x_Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(b*e*(m + n*(p + 1) + 1))), x] - Simp[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(b*(m + n*(p + 1) + 1)) Int[(e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0] && NeQ[m + n*(p + 1) + 1, 0]
Int[((f_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.)*( (d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> Simp[c^p*(f*x)^(m + 2*n*p - n + 1)*((d + e*x^n)^(q + 1)/(e*f^(2*n*p - n + 1)*(m + 2*n*p + n*q + 1))), x] + Simp[1/(e*(m + 2*n*p + n*q + 1)) Int[(f*x)^m*(d + e*x^n)^q*ExpandToSum[e *(m + 2*n*p + n*q + 1)*((a + b*x^n + c*x^(2*n))^p - c^p*x^(2*n*p)) - d*c^p* (m + 2*n*p - n + 1)*x^(2*n*p - n), x], x], x] /; FreeQ[{a, b, c, d, e, f, m , q}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && IGtQ[p, 0] && GtQ[2*n*p, n - 1] && !IntegerQ[q] && NeQ[m + 2*n*p + n*q + 1, 0]
Time = 1.09 (sec) , antiderivative size = 362, normalized size of antiderivative = 1.14
| method | result | size |
| risch | \(\frac {2 x \left (55 c \,e^{2} x^{6}+85 b \,e^{2} x^{3}-70 c d e \,x^{3}+187 a \,e^{2}-136 b d e +112 c \,d^{2}\right ) \sqrt {e \,x^{3}+d}}{935 e^{3}}+\frac {4 i \left (187 a \,e^{2}-136 b d e +112 c \,d^{2}\right ) d \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (x +\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}-\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right ) \sqrt {3}\, e}{\left (-d \,e^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{e}}{-\frac {3 \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}+\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}}}\, \sqrt {-\frac {i \left (x +\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}+\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right ) \sqrt {3}\, e}{\left (-d \,e^{2}\right )^{\frac {1}{3}}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}-\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right ) \sqrt {3}\, e}{\left (-d \,e^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{e \left (-\frac {3 \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}+\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right )}}\right )}{2805 e^{4} \sqrt {e \,x^{3}+d}}\) | \(362\) |
| elliptic | \(\frac {2 c \,x^{7} \sqrt {e \,x^{3}+d}}{17 e}+\frac {2 \left (b -\frac {14 d c}{17 e}\right ) x^{4} \sqrt {e \,x^{3}+d}}{11 e}+\frac {2 \left (a -\frac {8 d \left (b -\frac {14 d c}{17 e}\right )}{11 e}\right ) x \sqrt {e \,x^{3}+d}}{5 e}+\frac {4 i d \left (a -\frac {8 d \left (b -\frac {14 d c}{17 e}\right )}{11 e}\right ) \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (x +\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}-\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right ) \sqrt {3}\, e}{\left (-d \,e^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{e}}{-\frac {3 \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}+\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}}}\, \sqrt {-\frac {i \left (x +\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}+\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right ) \sqrt {3}\, e}{\left (-d \,e^{2}\right )^{\frac {1}{3}}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}-\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right ) \sqrt {3}\, e}{\left (-d \,e^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{e \left (-\frac {3 \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}+\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right )}}\right )}{15 e^{2} \sqrt {e \,x^{3}+d}}\) | \(378\) |
| default | \(\text {Expression too large to display}\) | \(965\) |
Input:
int(x^3*(c*x^6+b*x^3+a)/(e*x^3+d)^(1/2),x,method=_RETURNVERBOSE)
Output:
2/935*x*(55*c*e^2*x^6+85*b*e^2*x^3-70*c*d*e*x^3+187*a*e^2-136*b*d*e+112*c* d^2)/e^3*(e*x^3+d)^(1/2)+4/2805*I*(187*a*e^2-136*b*d*e+112*c*d^2)*d/e^4*3^ (1/2)*(-d*e^2)^(1/3)*(I*(x+1/2/e*(-d*e^2)^(1/3)-1/2*I*3^(1/2)/e*(-d*e^2)^( 1/3))*3^(1/2)*e/(-d*e^2)^(1/3))^(1/2)*((x-1/e*(-d*e^2)^(1/3))/(-3/2/e*(-d* e^2)^(1/3)+1/2*I*3^(1/2)/e*(-d*e^2)^(1/3)))^(1/2)*(-I*(x+1/2/e*(-d*e^2)^(1 /3)+1/2*I*3^(1/2)/e*(-d*e^2)^(1/3))*3^(1/2)*e/(-d*e^2)^(1/3))^(1/2)/(e*x^3 +d)^(1/2)*EllipticF(1/3*3^(1/2)*(I*(x+1/2/e*(-d*e^2)^(1/3)-1/2*I*3^(1/2)/e *(-d*e^2)^(1/3))*3^(1/2)*e/(-d*e^2)^(1/3))^(1/2),(I*3^(1/2)/e*(-d*e^2)^(1/ 3)/(-3/2/e*(-d*e^2)^(1/3)+1/2*I*3^(1/2)/e*(-d*e^2)^(1/3)))^(1/2))
Time = 0.08 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.33 \[ \int \frac {x^3 \left (a+b x^3+c x^6\right )}{\sqrt {d+e x^3}} \, dx=-\frac {2 \, {\left (2 \, {\left (112 \, c d^{3} - 136 \, b d^{2} e + 187 \, a d e^{2}\right )} \sqrt {e} {\rm weierstrassPInverse}\left (0, -\frac {4 \, d}{e}, x\right ) - {\left (55 \, c e^{3} x^{7} - 5 \, {\left (14 \, c d e^{2} - 17 \, b e^{3}\right )} x^{4} + {\left (112 \, c d^{2} e - 136 \, b d e^{2} + 187 \, a e^{3}\right )} x\right )} \sqrt {e x^{3} + d}\right )}}{935 \, e^{4}} \] Input:
integrate(x^3*(c*x^6+b*x^3+a)/(e*x^3+d)^(1/2),x, algorithm="fricas")
Output:
-2/935*(2*(112*c*d^3 - 136*b*d^2*e + 187*a*d*e^2)*sqrt(e)*weierstrassPInve rse(0, -4*d/e, x) - (55*c*e^3*x^7 - 5*(14*c*d*e^2 - 17*b*e^3)*x^4 + (112*c *d^2*e - 136*b*d*e^2 + 187*a*e^3)*x)*sqrt(e*x^3 + d))/e^4
Time = 1.79 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.38 \[ \int \frac {x^3 \left (a+b x^3+c x^6\right )}{\sqrt {d+e x^3}} \, dx=\frac {a x^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {e x^{3} e^{i \pi }}{d}} \right )}}{3 \sqrt {d} \Gamma \left (\frac {7}{3}\right )} + \frac {b x^{7} \Gamma \left (\frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {7}{3} \\ \frac {10}{3} \end {matrix}\middle | {\frac {e x^{3} e^{i \pi }}{d}} \right )}}{3 \sqrt {d} \Gamma \left (\frac {10}{3}\right )} + \frac {c x^{10} \Gamma \left (\frac {10}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {10}{3} \\ \frac {13}{3} \end {matrix}\middle | {\frac {e x^{3} e^{i \pi }}{d}} \right )}}{3 \sqrt {d} \Gamma \left (\frac {13}{3}\right )} \] Input:
integrate(x**3*(c*x**6+b*x**3+a)/(e*x**3+d)**(1/2),x)
Output:
a*x**4*gamma(4/3)*hyper((1/2, 4/3), (7/3,), e*x**3*exp_polar(I*pi)/d)/(3*s qrt(d)*gamma(7/3)) + b*x**7*gamma(7/3)*hyper((1/2, 7/3), (10/3,), e*x**3*e xp_polar(I*pi)/d)/(3*sqrt(d)*gamma(10/3)) + c*x**10*gamma(10/3)*hyper((1/2 , 10/3), (13/3,), e*x**3*exp_polar(I*pi)/d)/(3*sqrt(d)*gamma(13/3))
\[ \int \frac {x^3 \left (a+b x^3+c x^6\right )}{\sqrt {d+e x^3}} \, dx=\int { \frac {{\left (c x^{6} + b x^{3} + a\right )} x^{3}}{\sqrt {e x^{3} + d}} \,d x } \] Input:
integrate(x^3*(c*x^6+b*x^3+a)/(e*x^3+d)^(1/2),x, algorithm="maxima")
Output:
integrate((c*x^6 + b*x^3 + a)*x^3/sqrt(e*x^3 + d), x)
\[ \int \frac {x^3 \left (a+b x^3+c x^6\right )}{\sqrt {d+e x^3}} \, dx=\int { \frac {{\left (c x^{6} + b x^{3} + a\right )} x^{3}}{\sqrt {e x^{3} + d}} \,d x } \] Input:
integrate(x^3*(c*x^6+b*x^3+a)/(e*x^3+d)^(1/2),x, algorithm="giac")
Output:
integrate((c*x^6 + b*x^3 + a)*x^3/sqrt(e*x^3 + d), x)
Timed out. \[ \int \frac {x^3 \left (a+b x^3+c x^6\right )}{\sqrt {d+e x^3}} \, dx=\int \frac {x^3\,\left (c\,x^6+b\,x^3+a\right )}{\sqrt {e\,x^3+d}} \,d x \] Input:
int((x^3*(a + b*x^3 + c*x^6))/(d + e*x^3)^(1/2),x)
Output:
int((x^3*(a + b*x^3 + c*x^6))/(d + e*x^3)^(1/2), x)
\[ \int \frac {x^3 \left (a+b x^3+c x^6\right )}{\sqrt {d+e x^3}} \, dx=\frac {\frac {2 \sqrt {e \,x^{3}+d}\, a \,e^{2} x}{5}-\frac {16 \sqrt {e \,x^{3}+d}\, b d e x}{55}+\frac {2 \sqrt {e \,x^{3}+d}\, b \,e^{2} x^{4}}{11}+\frac {224 \sqrt {e \,x^{3}+d}\, c \,d^{2} x}{935}-\frac {28 \sqrt {e \,x^{3}+d}\, c d e \,x^{4}}{187}+\frac {2 \sqrt {e \,x^{3}+d}\, c \,e^{2} x^{7}}{17}-\frac {2 \left (\int \frac {\sqrt {e \,x^{3}+d}}{e \,x^{3}+d}d x \right ) a d \,e^{2}}{5}+\frac {16 \left (\int \frac {\sqrt {e \,x^{3}+d}}{e \,x^{3}+d}d x \right ) b \,d^{2} e}{55}-\frac {224 \left (\int \frac {\sqrt {e \,x^{3}+d}}{e \,x^{3}+d}d x \right ) c \,d^{3}}{935}}{e^{3}} \] Input:
int(x^3*(c*x^6+b*x^3+a)/(e*x^3+d)^(1/2),x)
Output:
(2*(187*sqrt(d + e*x**3)*a*e**2*x - 136*sqrt(d + e*x**3)*b*d*e*x + 85*sqrt (d + e*x**3)*b*e**2*x**4 + 112*sqrt(d + e*x**3)*c*d**2*x - 70*sqrt(d + e*x **3)*c*d*e*x**4 + 55*sqrt(d + e*x**3)*c*e**2*x**7 - 187*int(sqrt(d + e*x** 3)/(d + e*x**3),x)*a*d*e**2 + 136*int(sqrt(d + e*x**3)/(d + e*x**3),x)*b*d **2*e - 112*int(sqrt(d + e*x**3)/(d + e*x**3),x)*c*d**3))/(935*e**3)