Integrand size = 27, antiderivative size = 284 \[ \int \frac {a+b x^3+c x^6}{x^6 \sqrt {d+e x^3}} \, dx=-\frac {a \sqrt {d+e x^3}}{5 d x^5}-\frac {(10 b d-7 a e) \sqrt {d+e x^3}}{20 d^2 x^2}+\frac {\sqrt {2+\sqrt {3}} \left (40 c d^2-e (10 b d-7 a e)\right ) \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \sqrt {\frac {d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x}\right ),-7-4 \sqrt {3}\right )}{20 \sqrt [4]{3} d^2 \sqrt [3]{e} \sqrt {\frac {\sqrt [3]{d} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \sqrt {d+e x^3}} \] Output:
-1/5*a*(e*x^3+d)^(1/2)/d/x^5-1/20*(-7*a*e+10*b*d)*(e*x^3+d)^(1/2)/d^2/x^2+ 1/60*(1/2*6^(1/2)+1/2*2^(1/2))*(40*c*d^2-e*(-7*a*e+10*b*d))*(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)/d^2/e^(1/3)/(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 = 10.06 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.37 \[ \int \frac {a+b x^3+c x^6}{x^6 \sqrt {d+e x^3}} \, dx=\frac {\sqrt {1+\frac {e x^3}{d}} \left (-2 a \operatorname {Hypergeometric2F1}\left (-\frac {5}{3},\frac {1}{2},-\frac {2}{3},-\frac {e x^3}{d}\right )-5 b x^3 \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},\frac {1}{2},\frac {1}{3},-\frac {e x^3}{d}\right )+10 c x^6 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},-\frac {e x^3}{d}\right )\right )}{10 x^5 \sqrt {d+e x^3}} \] Input:
Integrate[(a + b*x^3 + c*x^6)/(x^6*Sqrt[d + e*x^3]),x]
Output:
(Sqrt[1 + (e*x^3)/d]*(-2*a*Hypergeometric2F1[-5/3, 1/2, -2/3, -((e*x^3)/d) ] - 5*b*x^3*Hypergeometric2F1[-2/3, 1/2, 1/3, -((e*x^3)/d)] + 10*c*x^6*Hyp ergeometric2F1[1/3, 1/2, 4/3, -((e*x^3)/d)]))/(10*x^5*Sqrt[d + e*x^3])
Time = 0.39 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.09, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {1810, 27, 955, 847, 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.
| \(\displaystyle \int \frac {a+b x^3+c x^6}{x^6 \sqrt {d+e x^3}} \, dx\) |
| \(\Big \downarrow \) 1810 |
| \(\displaystyle -\frac {2 \int -\frac {a e-(4 c d-b e) x^3}{2 x^6 \sqrt {e x^3+d}}dx}{e}-\frac {2 c \sqrt {d+e x^3}}{e x^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {a e-(4 c d-b e) x^3}{x^6 \sqrt {e x^3+d}}dx}{e}-\frac {2 c \sqrt {d+e x^3}}{e x^2}\) |
| \(\Big \downarrow \) 955 |
| \(\displaystyle \frac {-\frac {\left (40 c d^2-e (10 b d-7 a e)\right ) \int \frac {1}{x^3 \sqrt {e x^3+d}}dx}{10 d}-\frac {a e \sqrt {d+e x^3}}{5 d x^5}}{e}-\frac {2 c \sqrt {d+e x^3}}{e x^2}\) |
| \(\Big \downarrow \) 847 |
| \(\displaystyle \frac {-\frac {\left (40 c d^2-e (10 b d-7 a e)\right ) \left (-\frac {e \int \frac {1}{\sqrt {e x^3+d}}dx}{4 d}-\frac {\sqrt {d+e x^3}}{2 d x^2}\right )}{10 d}-\frac {a e \sqrt {d+e x^3}}{5 d x^5}}{e}-\frac {2 c \sqrt {d+e x^3}}{e x^2}\) |
| \(\Big \downarrow \) 759 |
| \(\displaystyle \frac {-\frac {\left (40 c d^2-e (10 b d-7 a e)\right ) \left (-\frac {\sqrt {2+\sqrt {3}} e^{2/3} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \sqrt {\frac {d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{e} x+\left (1-\sqrt {3}\right ) \sqrt [3]{d}}{\sqrt [3]{e} x+\left (1+\sqrt {3}\right ) \sqrt [3]{d}}\right ),-7-4 \sqrt {3}\right )}{2 \sqrt [4]{3} d \sqrt {\frac {\sqrt [3]{d} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \sqrt {d+e x^3}}-\frac {\sqrt {d+e x^3}}{2 d x^2}\right )}{10 d}-\frac {a e \sqrt {d+e x^3}}{5 d x^5}}{e}-\frac {2 c \sqrt {d+e x^3}}{e x^2}\) |
Input:
Int[(a + b*x^3 + c*x^6)/(x^6*Sqrt[d + e*x^3]),x]
Output:
(-2*c*Sqrt[d + e*x^3])/(e*x^2) + (-1/5*(a*e*Sqrt[d + e*x^3])/(d*x^5) - ((4 0*c*d^2 - e*(10*b*d - 7*a*e))*(-1/2*Sqrt[d + e*x^3]/(d*x^2) - (Sqrt[2 + Sq rt[3]]*e^(2/3)*(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]*EllipticF[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]])/(2*3^(1/4)*d*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])))/(10*d))/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*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[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n _)), x_Symbol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*e*(m + 1))), x] + Simp[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)) Int[(e *x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b* c - a*d, 0] && (IntegerQ[n] || GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) && !ILtQ[p, -1]
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 = 0.68 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.20
| method | result | size |
| risch | \(-\frac {\sqrt {e \,x^{3}+d}\, \left (-7 a e \,x^{3}+10 b d \,x^{3}+4 a d \right )}{20 d^{2} x^{5}}-\frac {i \left (7 a \,e^{2}-10 b d e +40 c \,d^{2}\right ) \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (x +\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}-\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right ) \sqrt {3}\, e}{\left (-d \,e^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{e}}{-\frac {3 \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}+\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}}}\, \sqrt {-\frac {i \left (x +\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}+\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right ) \sqrt {3}\, e}{\left (-d \,e^{2}\right )^{\frac {1}{3}}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}-\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right ) \sqrt {3}\, e}{\left (-d \,e^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{e \left (-\frac {3 \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}+\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right )}}\right )}{60 d^{2} e \sqrt {e \,x^{3}+d}}\) | \(341\) |
| elliptic | \(-\frac {a \sqrt {e \,x^{3}+d}}{5 d \,x^{5}}+\frac {\left (7 a e -10 b d \right ) \sqrt {e \,x^{3}+d}}{20 d^{2} x^{2}}-\frac {2 i \left (c +\frac {e \left (7 a e -10 b d \right )}{40 d^{2}}\right ) \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (x +\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}-\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right ) \sqrt {3}\, e}{\left (-d \,e^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{e}}{-\frac {3 \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}+\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}}}\, \sqrt {-\frac {i \left (x +\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}+\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right ) \sqrt {3}\, e}{\left (-d \,e^{2}\right )^{\frac {1}{3}}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}-\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right ) \sqrt {3}\, e}{\left (-d \,e^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{e \left (-\frac {3 \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}+\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right )}}\right )}{3 e \sqrt {e \,x^{3}+d}}\) | \(345\) |
| default | \(\text {Expression too large to display}\) | \(908\) |
Input:
int((c*x^6+b*x^3+a)/x^6/(e*x^3+d)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/20*(e*x^3+d)^(1/2)*(-7*a*e*x^3+10*b*d*x^3+4*a*d)/d^2/x^5-1/60*I*(7*a*e^ 2-10*b*d*e+40*c*d^2)/d^2*3^(1/2)/e*(-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 = 0.07 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.28 \[ \int \frac {a+b x^3+c x^6}{x^6 \sqrt {d+e x^3}} \, dx=\frac {{\left (40 \, c d^{2} - 10 \, b d e + 7 \, a e^{2}\right )} \sqrt {e} x^{5} {\rm weierstrassPInverse}\left (0, -\frac {4 \, d}{e}, x\right ) - {\left ({\left (10 \, b d e - 7 \, a e^{2}\right )} x^{3} + 4 \, a d e\right )} \sqrt {e x^{3} + d}}{20 \, d^{2} e x^{5}} \] Input:
integrate((c*x^6+b*x^3+a)/x^6/(e*x^3+d)^(1/2),x, algorithm="fricas")
Output:
1/20*((40*c*d^2 - 10*b*d*e + 7*a*e^2)*sqrt(e)*x^5*weierstrassPInverse(0, - 4*d/e, x) - ((10*b*d*e - 7*a*e^2)*x^3 + 4*a*d*e)*sqrt(e*x^3 + d))/(d^2*e*x ^5)
Time = 1.46 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.45 \[ \int \frac {a+b x^3+c x^6}{x^6 \sqrt {d+e x^3}} \, dx=\frac {a \Gamma \left (- \frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{3}, \frac {1}{2} \\ - \frac {2}{3} \end {matrix}\middle | {\frac {e x^{3} e^{i \pi }}{d}} \right )}}{3 \sqrt {d} x^{5} \Gamma \left (- \frac {2}{3}\right )} + \frac {b \Gamma \left (- \frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {1}{2} \\ \frac {1}{3} \end {matrix}\middle | {\frac {e x^{3} e^{i \pi }}{d}} \right )}}{3 \sqrt {d} x^{2} \Gamma \left (\frac {1}{3}\right )} + \frac {c x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{2} \\ \frac {4}{3} \end {matrix}\middle | {\frac {e x^{3} e^{i \pi }}{d}} \right )}}{3 \sqrt {d} \Gamma \left (\frac {4}{3}\right )} \] Input:
integrate((c*x**6+b*x**3+a)/x**6/(e*x**3+d)**(1/2),x)
Output:
a*gamma(-5/3)*hyper((-5/3, 1/2), (-2/3,), e*x**3*exp_polar(I*pi)/d)/(3*sqr t(d)*x**5*gamma(-2/3)) + b*gamma(-2/3)*hyper((-2/3, 1/2), (1/3,), e*x**3*e xp_polar(I*pi)/d)/(3*sqrt(d)*x**2*gamma(1/3)) + c*x*gamma(1/3)*hyper((1/3, 1/2), (4/3,), e*x**3*exp_polar(I*pi)/d)/(3*sqrt(d)*gamma(4/3))
\[ \int \frac {a+b x^3+c x^6}{x^6 \sqrt {d+e x^3}} \, dx=\int { \frac {c x^{6} + b x^{3} + a}{\sqrt {e x^{3} + d} x^{6}} \,d x } \] Input:
integrate((c*x^6+b*x^3+a)/x^6/(e*x^3+d)^(1/2),x, algorithm="maxima")
Output:
integrate((c*x^6 + b*x^3 + a)/(sqrt(e*x^3 + d)*x^6), x)
\[ \int \frac {a+b x^3+c x^6}{x^6 \sqrt {d+e x^3}} \, dx=\int { \frac {c x^{6} + b x^{3} + a}{\sqrt {e x^{3} + d} x^{6}} \,d x } \] Input:
integrate((c*x^6+b*x^3+a)/x^6/(e*x^3+d)^(1/2),x, algorithm="giac")
Output:
integrate((c*x^6 + b*x^3 + a)/(sqrt(e*x^3 + d)*x^6), x)
Timed out. \[ \int \frac {a+b x^3+c x^6}{x^6 \sqrt {d+e x^3}} \, dx=\int \frac {c\,x^6+b\,x^3+a}{x^6\,\sqrt {e\,x^3+d}} \,d x \] Input:
int((a + b*x^3 + c*x^6)/(x^6*(d + e*x^3)^(1/2)),x)
Output:
int((a + b*x^3 + c*x^6)/(x^6*(d + e*x^3)^(1/2)), x)
\[ \int \frac {a+b x^3+c x^6}{x^6 \sqrt {d+e x^3}} \, dx=\frac {-2 \sqrt {e \,x^{3}+d}\, b e +8 \sqrt {e \,x^{3}+d}\, c d -14 \sqrt {e \,x^{3}+d}\, c e \,x^{3}+7 \left (\int \frac {\sqrt {e \,x^{3}+d}}{e \,x^{9}+d \,x^{6}}d x \right ) a \,e^{2} x^{5}-10 \left (\int \frac {\sqrt {e \,x^{3}+d}}{e \,x^{9}+d \,x^{6}}d x \right ) b d e \,x^{5}+40 \left (\int \frac {\sqrt {e \,x^{3}+d}}{e \,x^{9}+d \,x^{6}}d x \right ) c \,d^{2} x^{5}}{7 e^{2} x^{5}} \] Input:
int((c*x^6+b*x^3+a)/x^6/(e*x^3+d)^(1/2),x)
Output:
( - 2*sqrt(d + e*x**3)*b*e + 8*sqrt(d + e*x**3)*c*d - 14*sqrt(d + e*x**3)* c*e*x**3 + 7*int(sqrt(d + e*x**3)/(d*x**6 + e*x**9),x)*a*e**2*x**5 - 10*in t(sqrt(d + e*x**3)/(d*x**6 + e*x**9),x)*b*d*e*x**5 + 40*int(sqrt(d + e*x** 3)/(d*x**6 + e*x**9),x)*c*d**2*x**5)/(7*e**2*x**5)