Integrand size = 28, antiderivative size = 398 \[ \int \frac {x^2}{\sqrt {a+b (d+e x)^3+c (d+e x)^6}} \, dx=\frac {d^2 (d+e x) \sqrt {1+\frac {2 c (d+e x)^3}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c (d+e x)^3}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},\frac {1}{2},\frac {4}{3},-\frac {2 c (d+e x)^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c (d+e x)^3}{b+\sqrt {b^2-4 a c}}\right )}{e^3 \sqrt {a+b (d+e x)^3+c (d+e x)^6}}-\frac {d (d+e x)^2 \sqrt {1+\frac {2 c (d+e x)^3}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c (d+e x)^3}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{2},\frac {1}{2},\frac {5}{3},-\frac {2 c (d+e x)^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c (d+e x)^3}{b+\sqrt {b^2-4 a c}}\right )}{e^3 \sqrt {a+b (d+e x)^3+c (d+e x)^6}}+\frac {\text {arctanh}\left (\frac {b+2 c (d+e x)^3}{2 \sqrt {c} \sqrt {a+b (d+e x)^3+c (d+e x)^6}}\right )}{3 \sqrt {c} e^3} \]
1/3*arctanh(1/2*(b+2*c*(e*x+d)^3)/c^(1/2)/(a+b*(e*x+d)^3+c*(e*x+d)^6)^(1/2 ))/e^3/c^(1/2)+d^2*(e*x+d)*AppellF1(1/3,1/2,1/2,4/3,-2*c*(e*x+d)^3/(b-(-4* a*c+b^2)^(1/2)),-2*c*(e*x+d)^3/(b+(-4*a*c+b^2)^(1/2)))*(1+2*c*(e*x+d)^3/(b -(-4*a*c+b^2)^(1/2)))^(1/2)*(1+2*c*(e*x+d)^3/(b+(-4*a*c+b^2)^(1/2)))^(1/2) /e^3/(a+b*(e*x+d)^3+c*(e*x+d)^6)^(1/2)-d*(e*x+d)^2*AppellF1(2/3,1/2,1/2,5/ 3,-2*c*(e*x+d)^3/(b-(-4*a*c+b^2)^(1/2)),-2*c*(e*x+d)^3/(b+(-4*a*c+b^2)^(1/ 2)))*(1+2*c*(e*x+d)^3/(b-(-4*a*c+b^2)^(1/2)))^(1/2)*(1+2*c*(e*x+d)^3/(b+(- 4*a*c+b^2)^(1/2)))^(1/2)/e^3/(a+b*(e*x+d)^3+c*(e*x+d)^6)^(1/2)
\[ \int \frac {x^2}{\sqrt {a+b (d+e x)^3+c (d+e x)^6}} \, dx=\int \frac {x^2}{\sqrt {a+b (d+e x)^3+c (d+e x)^6}} \, dx \]
Time = 0.54 (sec) , antiderivative size = 393, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {1726, 2322, 2009}
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.
| \(\displaystyle \int \frac {x^2}{\sqrt {a+b (d+e x)^3+c (d+e x)^6}} \, dx\) |
| \(\Big \downarrow \) 1726 |
| \(\displaystyle \frac {\int \frac {e^2 x^2}{\sqrt {c (d+e x)^6+b (d+e x)^3+a}}d(d+e x)}{e^3}\) |
| \(\Big \downarrow \) 2322 |
| \(\displaystyle \frac {\int \left (\frac {d^2}{\sqrt {c (d+e x)^6+b (d+e x)^3+a}}-\frac {2 (d+e x) d}{\sqrt {c (d+e x)^6+b (d+e x)^3+a}}+\frac {(d+e x)^2}{\sqrt {c (d+e x)^6+b (d+e x)^3+a}}\right )d(d+e x)}{e^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {d^2 (d+e x) \sqrt {\frac {2 c (d+e x)^3}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c (d+e x)^3}{\sqrt {b^2-4 a c}+b}+1} \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},\frac {1}{2},\frac {4}{3},-\frac {2 c (d+e x)^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c (d+e x)^3}{b+\sqrt {b^2-4 a c}}\right )}{\sqrt {a+b (d+e x)^3+c (d+e x)^6}}-\frac {d (d+e x)^2 \sqrt {\frac {2 c (d+e x)^3}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c (d+e x)^3}{\sqrt {b^2-4 a c}+b}+1} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{2},\frac {1}{2},\frac {5}{3},-\frac {2 c (d+e x)^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c (d+e x)^3}{b+\sqrt {b^2-4 a c}}\right )}{\sqrt {a+b (d+e x)^3+c (d+e x)^6}}+\frac {\text {arctanh}\left (\frac {b+2 c (d+e x)^3}{2 \sqrt {c} \sqrt {a+b (d+e x)^3+c (d+e x)^6}}\right )}{3 \sqrt {c}}}{e^3}\) |
((d^2*(d + e*x)*Sqrt[1 + (2*c*(d + e*x)^3)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*(d + e*x)^3)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[1/3, 1/2, 1/2, 4/3, (-2*c*(d + e*x)^3)/(b - Sqrt[b^2 - 4*a*c]), (-2*c*(d + e*x)^3)/(b + Sqrt[ b^2 - 4*a*c])])/Sqrt[a + b*(d + e*x)^3 + c*(d + e*x)^6] - (d*(d + e*x)^2*S qrt[1 + (2*c*(d + e*x)^3)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*(d + e*x) ^3)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[2/3, 1/2, 1/2, 5/3, (-2*c*(d + e*x)^ 3)/(b - Sqrt[b^2 - 4*a*c]), (-2*c*(d + e*x)^3)/(b + Sqrt[b^2 - 4*a*c])])/S qrt[a + b*(d + e*x)^3 + c*(d + e*x)^6] + ArcTanh[(b + 2*c*(d + e*x)^3)/(2* Sqrt[c]*Sqrt[a + b*(d + e*x)^3 + c*(d + e*x)^6])]/(3*Sqrt[c]))/e^3
3.7.62.3.1 Defintions of rubi rules used
Int[((a_.) + (c_.)*(v_)^(n2_.) + (b_.)*(v_)^(n_))^(p_.)*(x_)^(m_.), x_Symbo l] :> Simp[1/Coefficient[v, x, 1]^(m + 1) Subst[Int[SimplifyIntegrand[(x - Coefficient[v, x, 0])^m*(a + b*x^n + c*x^(2*n))^p, x], x], x, v], x] /; F reeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] && LinearQ[v, x] && IntegerQ[m] && NeQ[v, x]
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(p_), x_Symbol] :> Mo dule[{q = Expon[Pq, x], j, k}, Int[Sum[x^j*Sum[Coeff[Pq, x, j + k*n]*x^(k*n ), {k, 0, (q - j)/n + 1}]*(a + b*x^n + c*x^(2*n))^p, {j, 0, n - 1}], x]] /; FreeQ[{a, b, c, p}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && !PolyQ[Pq, x^n]
\[\int \frac {x^{2}}{\sqrt {a +b \left (e x +d \right )^{3}+c \left (e x +d \right )^{6}}}d x\]
\[ \int \frac {x^2}{\sqrt {a+b (d+e x)^3+c (d+e x)^6}} \, dx=\int { \frac {x^{2}}{\sqrt {{\left (e x + d\right )}^{6} c + {\left (e x + d\right )}^{3} b + a}} \,d x } \]
integral(x^2/sqrt(c*e^6*x^6 + 6*c*d*e^5*x^5 + 15*c*d^2*e^4*x^4 + c*d^6 + ( 20*c*d^3 + b)*e^3*x^3 + 3*(5*c*d^4 + b*d)*e^2*x^2 + b*d^3 + 3*(2*c*d^5 + b *d^2)*e*x + a), x)
\[ \int \frac {x^2}{\sqrt {a+b (d+e x)^3+c (d+e x)^6}} \, dx=\int \frac {x^{2}}{\sqrt {a + b d^{3} + 3 b d^{2} e x + 3 b d e^{2} x^{2} + b e^{3} x^{3} + c d^{6} + 6 c d^{5} e x + 15 c d^{4} e^{2} x^{2} + 20 c d^{3} e^{3} x^{3} + 15 c d^{2} e^{4} x^{4} + 6 c d e^{5} x^{5} + c e^{6} x^{6}}}\, dx \]
Integral(x**2/sqrt(a + b*d**3 + 3*b*d**2*e*x + 3*b*d*e**2*x**2 + b*e**3*x* *3 + c*d**6 + 6*c*d**5*e*x + 15*c*d**4*e**2*x**2 + 20*c*d**3*e**3*x**3 + 1 5*c*d**2*e**4*x**4 + 6*c*d*e**5*x**5 + c*e**6*x**6), x)
\[ \int \frac {x^2}{\sqrt {a+b (d+e x)^3+c (d+e x)^6}} \, dx=\int { \frac {x^{2}}{\sqrt {{\left (e x + d\right )}^{6} c + {\left (e x + d\right )}^{3} b + a}} \,d x } \]
\[ \int \frac {x^2}{\sqrt {a+b (d+e x)^3+c (d+e x)^6}} \, dx=\int { \frac {x^{2}}{\sqrt {{\left (e x + d\right )}^{6} c + {\left (e x + d\right )}^{3} b + a}} \,d x } \]
Timed out. \[ \int \frac {x^2}{\sqrt {a+b (d+e x)^3+c (d+e x)^6}} \, dx=\int \frac {x^2}{\sqrt {a+b\,{\left (d+e\,x\right )}^3+c\,{\left (d+e\,x\right )}^6}} \,d x \]