Integrand size = 38, antiderivative size = 238 \[ \int \frac {x^2 (d+e x)}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {x^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c d}-\frac {\left (\left (3 c d^2-5 a e^2\right ) \left (c d^2+3 a e^2\right )-2 c d e \left (c d^2-5 a e^2\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 c^3 d^3 e^2}+\frac {\left (c d^2-a e^2\right ) \left (c^2 d^4+2 a c d^2 e^2+5 a^2 e^4\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} (d+e x)}{\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{8 c^{7/2} d^{7/2} e^{5/2}} \] Output:
1/3*x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c/d-1/24*((-5*a*e^2+3*c*d^ 2)*(3*a*e^2+c*d^2)-2*c*d*e*(-5*a*e^2+c*d^2)*x)*(a*d*e+(a*e^2+c*d^2)*x+c*d* e*x^2)^(1/2)/c^3/d^3/e^2+1/8*(-a*e^2+c*d^2)*(5*a^2*e^4+2*a*c*d^2*e^2+c^2*d ^4)*arctanh(c^(1/2)*d^(1/2)*(e*x+d)/e^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x ^2)^(1/2))/c^(7/2)/d^(7/2)/e^(5/2)
Time = 10.58 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.29 \[ \int \frac {x^2 (d+e x)}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=-\frac {(d+e x) \left (\sqrt {c} \sqrt {d} \sqrt {e} \sqrt {\frac {c d (d+e x)}{c d^2-a e^2}} \left (-15 a^3 e^5+a^2 c d e^3 (4 d-5 e x)+c^3 d^3 x \left (3 d^2-2 d e x-8 e^2 x^2\right )+a c^2 d^2 e \left (3 d^2+2 d e x+2 e^2 x^2\right )\right )-3 \sqrt {c d} \sqrt {c d^2-a e^2} \left (c^2 d^4+2 a c d^2 e^2+5 a^2 e^4\right ) \sqrt {a e+c d x} \text {arcsinh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a e+c d x}}{\sqrt {c d} \sqrt {c d^2-a e^2}}\right )\right )}{24 c^{7/2} d^{7/2} e^{5/2} \sqrt {\frac {c d (d+e x)}{c d^2-a e^2}} \sqrt {(a e+c d x) (d+e x)}} \] Input:
Integrate[(x^2*(d + e*x))/Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2],x]
Output:
-1/24*((d + e*x)*(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[(c*d*(d + e*x))/(c*d^2 - a* e^2)]*(-15*a^3*e^5 + a^2*c*d*e^3*(4*d - 5*e*x) + c^3*d^3*x*(3*d^2 - 2*d*e* x - 8*e^2*x^2) + a*c^2*d^2*e*(3*d^2 + 2*d*e*x + 2*e^2*x^2)) - 3*Sqrt[c*d]* Sqrt[c*d^2 - a*e^2]*(c^2*d^4 + 2*a*c*d^2*e^2 + 5*a^2*e^4)*Sqrt[a*e + c*d*x ]*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d^ 2 - a*e^2])]))/(c^(7/2)*d^(7/2)*e^(5/2)*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^ 2)]*Sqrt[(a*e + c*d*x)*(d + e*x)])
Time = 0.71 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.11, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.132, Rules used = {1236, 27, 1225, 1092, 219}
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.
| \(\displaystyle \int \frac {x^2 (d+e x)}{\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \, dx\) |
| \(\Big \downarrow \) 1236 |
| \(\displaystyle \frac {\int -\frac {e x \left (4 a d e-\left (c d^2-5 a e^2\right ) x\right )}{2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{3 c d e}+\frac {x^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 c d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 c d}-\frac {\int \frac {x \left (4 a d e-\left (c d^2-5 a e^2\right ) x\right )}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{6 c d}\) |
| \(\Big \downarrow \) 1225 |
| \(\displaystyle \frac {x^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 c d}-\frac {\frac {\left (\left (3 c d^2-5 a e^2\right ) \left (3 a e^2+c d^2\right )-2 c d e x \left (c d^2-5 a e^2\right )\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c^2 d^2 e^2}-\frac {3 \left (c d^2-a e^2\right ) \left (5 a^2 e^4+2 a c d^2 e^2+c^2 d^4\right ) \int \frac {1}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{8 c^2 d^2 e^2}}{6 c d}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {x^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 c d}-\frac {\frac {\left (\left (3 c d^2-5 a e^2\right ) \left (3 a e^2+c d^2\right )-2 c d e x \left (c d^2-5 a e^2\right )\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c^2 d^2 e^2}-\frac {3 \left (c d^2-a e^2\right ) \left (5 a^2 e^4+2 a c d^2 e^2+c^2 d^4\right ) \int \frac {1}{4 c d e-\frac {\left (c d^2+2 c e x d+a e^2\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\frac {c d^2+2 c e x d+a e^2}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}}{4 c^2 d^2 e^2}}{6 c d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {x^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 c d}-\frac {\frac {\left (\left (3 c d^2-5 a e^2\right ) \left (3 a e^2+c d^2\right )-2 c d e x \left (c d^2-5 a e^2\right )\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c^2 d^2 e^2}-\frac {3 \left (c d^2-a e^2\right ) \left (5 a^2 e^4+2 a c d^2 e^2+c^2 d^4\right ) \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 c^{5/2} d^{5/2} e^{5/2}}}{6 c d}\) |
Input:
Int[(x^2*(d + e*x))/Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2],x]
Output:
(x^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(3*c*d) - ((((3*c*d^2 - 5*a*e^2)*(c*d^2 + 3*a*e^2) - 2*c*d*e*(c*d^2 - 5*a*e^2)*x)*Sqrt[a*d*e + (c* d^2 + a*e^2)*x + c*d*e*x^2])/(4*c^2*d^2*e^2) - (3*(c*d^2 - a*e^2)*(c^2*d^4 + 2*a*c*d^2*e^2 + 5*a^2*e^4)*ArcTanh[(c*d^2 + a*e^2 + 2*c*d*e*x)/(2*Sqrt[ c]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])])/(8*c^(5/ 2)*d^(5/2)*e^(5/2)))/(6*c*d)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[I nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a , b, c}, x]
Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*( x_)^2)^(p_), x_Symbol] :> Simp[(-(b*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 2*c*e*g*(p + 1)*x))*((a + b*x + c*x^2)^(p + 1)/(2*c^2*(p + 1)*(2*p + 3))), x] + Simp[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p + 3))/(2*c^2*(2*p + 3)) Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c , d, e, f, g, p}, x] && !LeQ[p, -1]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 2))), x] + Simp[1/(c*(m + 2*p + 2)) Int[(d + e*x)^(m - 1 )*(a + b*x + c*x^2)^p*Simp[m*(c*d*f - a*e*g) + d*(2*c*f - b*g)*(p + 1) + (m *(c*e*f + c*d*g - b*e*g) + e*(p + 1)*(2*c*f - b*g))*x, x], x], x] /; FreeQ[ {a, b, c, d, e, f, g, p}, x] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (Intege rQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p]) && !(IGtQ[m, 0] && EqQ[f, 0])
Leaf count of result is larger than twice the leaf count of optimal. \(691\) vs. \(2(214)=428\).
Time = 2.47 (sec) , antiderivative size = 692, normalized size of antiderivative = 2.91
| method | result | size |
| default | \(d \left (\frac {x \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{2 d e c}-\frac {3 \left (a \,e^{2}+c \,d^{2}\right ) \left (\frac {\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{d e c}-\frac {\left (a \,e^{2}+c \,d^{2}\right ) \ln \left (\frac {\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}+c d x e}{\sqrt {d e c}}+\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}\right )}{2 d e c \sqrt {d e c}}\right )}{4 d e c}-\frac {a \ln \left (\frac {\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}+c d x e}{\sqrt {d e c}}+\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}\right )}{2 c \sqrt {d e c}}\right )+e \left (\frac {x^{2} \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{3 d e c}-\frac {5 \left (a \,e^{2}+c \,d^{2}\right ) \left (\frac {x \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{2 d e c}-\frac {3 \left (a \,e^{2}+c \,d^{2}\right ) \left (\frac {\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{d e c}-\frac {\left (a \,e^{2}+c \,d^{2}\right ) \ln \left (\frac {\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}+c d x e}{\sqrt {d e c}}+\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}\right )}{2 d e c \sqrt {d e c}}\right )}{4 d e c}-\frac {a \ln \left (\frac {\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}+c d x e}{\sqrt {d e c}}+\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}\right )}{2 c \sqrt {d e c}}\right )}{6 d e c}-\frac {2 a \left (\frac {\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{d e c}-\frac {\left (a \,e^{2}+c \,d^{2}\right ) \ln \left (\frac {\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}+c d x e}{\sqrt {d e c}}+\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}\right )}{2 d e c \sqrt {d e c}}\right )}{3 c}\right )\) | \(692\) |
Input:
int(x^2*(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2),x,method=_RETURNVE RBOSE)
Output:
d*(1/2*x/d/e/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)-3/4*(a*e^2+c*d^2)/d /e/c*(1/d/e/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)-1/2*(a*e^2+c*d^2)/d/ e/c*ln((1/2*a*e^2+1/2*c*d^2+c*d*x*e)/(d*e*c)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+ c*d*x^2*e)^(1/2))/(d*e*c)^(1/2))-1/2*a/c*ln((1/2*a*e^2+1/2*c*d^2+c*d*x*e)/ (d*e*c)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2))/(d*e*c)^(1/2))+e*(1 /3*x^2/d/e/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)-5/6*(a*e^2+c*d^2)/d/e /c*(1/2*x/d/e/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)-3/4*(a*e^2+c*d^2)/ d/e/c*(1/d/e/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)-1/2*(a*e^2+c*d^2)/d /e/c*ln((1/2*a*e^2+1/2*c*d^2+c*d*x*e)/(d*e*c)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x +c*d*x^2*e)^(1/2))/(d*e*c)^(1/2))-1/2*a/c*ln((1/2*a*e^2+1/2*c*d^2+c*d*x*e) /(d*e*c)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2))/(d*e*c)^(1/2))-2/3 *a/c*(1/d/e/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)-1/2*(a*e^2+c*d^2)/d/ e/c*ln((1/2*a*e^2+1/2*c*d^2+c*d*x*e)/(d*e*c)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+ c*d*x^2*e)^(1/2))/(d*e*c)^(1/2)))
Time = 0.15 (sec) , antiderivative size = 530, normalized size of antiderivative = 2.23 \[ \int \frac {x^2 (d+e x)}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\left [-\frac {3 \, {\left (c^{3} d^{6} + a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - 5 \, a^{3} e^{6}\right )} \sqrt {c d e} \log \left (8 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4} - 4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {c d e} + 8 \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right ) - 4 \, {\left (8 \, c^{3} d^{3} e^{3} x^{2} - 3 \, c^{3} d^{5} e - 4 \, a c^{2} d^{3} e^{3} + 15 \, a^{2} c d e^{5} + 2 \, {\left (c^{3} d^{4} e^{2} - 5 \, a c^{2} d^{2} e^{4}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{96 \, c^{4} d^{4} e^{3}}, -\frac {3 \, {\left (c^{3} d^{6} + a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - 5 \, a^{3} e^{6}\right )} \sqrt {-c d e} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {-c d e}}{2 \, {\left (c^{2} d^{2} e^{2} x^{2} + a c d^{2} e^{2} + {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right )}}\right ) - 2 \, {\left (8 \, c^{3} d^{3} e^{3} x^{2} - 3 \, c^{3} d^{5} e - 4 \, a c^{2} d^{3} e^{3} + 15 \, a^{2} c d e^{5} + 2 \, {\left (c^{3} d^{4} e^{2} - 5 \, a c^{2} d^{2} e^{4}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{48 \, c^{4} d^{4} e^{3}}\right ] \] Input:
integrate(x^2*(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorithm ="fricas")
Output:
[-1/96*(3*(c^3*d^6 + a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 - 5*a^3*e^6)*sqrt(c*d *e)*log(8*c^2*d^2*e^2*x^2 + c^2*d^4 + 6*a*c*d^2*e^2 + a^2*e^4 - 4*sqrt(c*d *e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(2*c*d*e*x + c*d^2 + a*e^2)*sqrt(c*d*e ) + 8*(c^2*d^3*e + a*c*d*e^3)*x) - 4*(8*c^3*d^3*e^3*x^2 - 3*c^3*d^5*e - 4* a*c^2*d^3*e^3 + 15*a^2*c*d*e^5 + 2*(c^3*d^4*e^2 - 5*a*c^2*d^2*e^4)*x)*sqrt (c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^4*d^4*e^3), -1/48*(3*(c^3*d^6 + a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 - 5*a^3*e^6)*sqrt(-c*d*e)*arctan(1/2*sqr t(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(2*c*d*e*x + c*d^2 + a*e^2)*sqrt( -c*d*e)/(c^2*d^2*e^2*x^2 + a*c*d^2*e^2 + (c^2*d^3*e + a*c*d*e^3)*x)) - 2*( 8*c^3*d^3*e^3*x^2 - 3*c^3*d^5*e - 4*a*c^2*d^3*e^3 + 15*a^2*c*d*e^5 + 2*(c^ 3*d^4*e^2 - 5*a*c^2*d^2*e^4)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x ))/(c^4*d^4*e^3)]
Leaf count of result is larger than twice the leaf count of optimal. 631 vs. \(2 (230) = 460\).
Time = 1.11 (sec) , antiderivative size = 631, normalized size of antiderivative = 2.65 \[ \int \frac {x^2 (d+e x)}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\begin {cases} \left (- \frac {a \left (d - \frac {\frac {5 a e^{2}}{2} + \frac {5 c d^{2}}{2}}{3 c d}\right )}{2 c} - \frac {\left (a e^{2} + c d^{2}\right ) \left (- \frac {2 a e}{3 c} - \frac {\left (d - \frac {\frac {5 a e^{2}}{2} + \frac {5 c d^{2}}{2}}{3 c d}\right ) \left (\frac {3 a e^{2}}{2} + \frac {3 c d^{2}}{2}\right )}{2 c d e}\right )}{2 c d e}\right ) \left (\begin {cases} \frac {\log {\left (a e^{2} + c d^{2} + 2 c d e x + 2 \sqrt {c d e} \sqrt {a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )} \right )}}{\sqrt {c d e}} & \text {for}\: a d e - \frac {\left (a e^{2} + c d^{2}\right )^{2}}{4 c d e} \neq 0 \\\frac {\left (x - \frac {- a e^{2} - c d^{2}}{2 c d e}\right ) \log {\left (x - \frac {- a e^{2} - c d^{2}}{2 c d e} \right )}}{\sqrt {c d e \left (x - \frac {- a e^{2} - c d^{2}}{2 c d e}\right )^{2}}} & \text {otherwise} \end {cases}\right ) + \sqrt {a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )} \left (\frac {x^{2}}{3 c d} + \frac {x \left (d - \frac {\frac {5 a e^{2}}{2} + \frac {5 c d^{2}}{2}}{3 c d}\right )}{2 c d e} + \frac {- \frac {2 a e}{3 c} - \frac {\left (d - \frac {\frac {5 a e^{2}}{2} + \frac {5 c d^{2}}{2}}{3 c d}\right ) \left (\frac {3 a e^{2}}{2} + \frac {3 c d^{2}}{2}\right )}{2 c d e}}{c d e}\right ) & \text {for}\: c d e \neq 0 \\\frac {2 \left (\frac {a^{2} c d^{5} e^{2} \sqrt {a d e + x \left (a e^{2} + c d^{2}\right )}}{a e^{2} + c d^{2}} + \frac {e \left (a d e + x \left (a e^{2} + c d^{2}\right )\right )^{\frac {7}{2}}}{7 \left (a e^{2} + c d^{2}\right )} + \frac {\left (a d e + x \left (a e^{2} + c d^{2}\right )\right )^{\frac {5}{2}} \left (- 2 a d e^{2} + c d^{3}\right )}{5 \left (a e^{2} + c d^{2}\right )} + \frac {\left (a d e + x \left (a e^{2} + c d^{2}\right )\right )^{\frac {3}{2}} \left (a^{2} d^{2} e^{3} - 2 a c d^{4} e\right )}{3 \left (a e^{2} + c d^{2}\right )}\right )}{\left (a e^{2} + c d^{2}\right )^{3}} & \text {for}\: a e^{2} + c d^{2} \neq 0 \\\frac {\frac {d x^{3}}{3} + \frac {e x^{4}}{4}}{\sqrt {a d e}} & \text {otherwise} \end {cases} \] Input:
integrate(x**2*(e*x+d)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2),x)
Output:
Piecewise(((-a*(d - (5*a*e**2/2 + 5*c*d**2/2)/(3*c*d))/(2*c) - (a*e**2 + c *d**2)*(-2*a*e/(3*c) - (d - (5*a*e**2/2 + 5*c*d**2/2)/(3*c*d))*(3*a*e**2/2 + 3*c*d**2/2)/(2*c*d*e))/(2*c*d*e))*Piecewise((log(a*e**2 + c*d**2 + 2*c* d*e*x + 2*sqrt(c*d*e)*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2)))/sqrt (c*d*e), Ne(a*d*e - (a*e**2 + c*d**2)**2/(4*c*d*e), 0)), ((x - (-a*e**2 - c*d**2)/(2*c*d*e))*log(x - (-a*e**2 - c*d**2)/(2*c*d*e))/sqrt(c*d*e*(x - ( -a*e**2 - c*d**2)/(2*c*d*e))**2), True)) + sqrt(a*d*e + c*d*e*x**2 + x*(a* e**2 + c*d**2))*(x**2/(3*c*d) + x*(d - (5*a*e**2/2 + 5*c*d**2/2)/(3*c*d))/ (2*c*d*e) + (-2*a*e/(3*c) - (d - (5*a*e**2/2 + 5*c*d**2/2)/(3*c*d))*(3*a*e **2/2 + 3*c*d**2/2)/(2*c*d*e))/(c*d*e)), Ne(c*d*e, 0)), (2*(a**2*c*d**5*e* *2*sqrt(a*d*e + x*(a*e**2 + c*d**2))/(a*e**2 + c*d**2) + e*(a*d*e + x*(a*e **2 + c*d**2))**(7/2)/(7*(a*e**2 + c*d**2)) + (a*d*e + x*(a*e**2 + c*d**2) )**(5/2)*(-2*a*d*e**2 + c*d**3)/(5*(a*e**2 + c*d**2)) + (a*d*e + x*(a*e**2 + c*d**2))**(3/2)*(a**2*d**2*e**3 - 2*a*c*d**4*e)/(3*(a*e**2 + c*d**2)))/ (a*e**2 + c*d**2)**3, Ne(a*e**2 + c*d**2, 0)), ((d*x**3/3 + e*x**4/4)/sqrt (a*d*e), True))
Exception generated. \[ \int \frac {x^2 (d+e x)}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(x^2*(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorithm ="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(a*e^2-c*d^2>0)', see `assume?` f or more de
Time = 0.15 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.94 \[ \int \frac {x^2 (d+e x)}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {1}{24} \, \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e} {\left (2 \, x {\left (\frac {4 \, x}{c d} + \frac {c^{2} d^{3} e - 5 \, a c d e^{3}}{c^{3} d^{3} e^{2}}\right )} - \frac {3 \, c^{2} d^{4} + 4 \, a c d^{2} e^{2} - 15 \, a^{2} e^{4}}{c^{3} d^{3} e^{2}}\right )} - \frac {{\left (c^{3} d^{6} + a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - 5 \, a^{3} e^{6}\right )} \log \left ({\left | -c d^{2} - a e^{2} - 2 \, \sqrt {c d e} {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )} \right |}\right )}{16 \, \sqrt {c d e} c^{3} d^{3} e^{2}} \] Input:
integrate(x^2*(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorithm ="giac")
Output:
1/24*sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e)*(2*x*(4*x/(c*d) + (c^2*d^ 3*e - 5*a*c*d*e^3)/(c^3*d^3*e^2)) - (3*c^2*d^4 + 4*a*c*d^2*e^2 - 15*a^2*e^ 4)/(c^3*d^3*e^2)) - 1/16*(c^3*d^6 + a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 - 5*a^ 3*e^6)*log(abs(-c*d^2 - a*e^2 - 2*sqrt(c*d*e)*(sqrt(c*d*e)*x - sqrt(c*d*e* x^2 + c*d^2*x + a*e^2*x + a*d*e))))/(sqrt(c*d*e)*c^3*d^3*e^2)
Timed out. \[ \int \frac {x^2 (d+e x)}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int \frac {x^2\,\left (d+e\,x\right )}{\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}} \,d x \] Input:
int((x^2*(d + e*x))/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2),x)
Output:
int((x^2*(d + e*x))/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2), x)
Time = 0.26 (sec) , antiderivative size = 400, normalized size of antiderivative = 1.68 \[ \int \frac {x^2 (d+e x)}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {15 \sqrt {e x +d}\, \sqrt {c d x +a e}\, a^{2} c d \,e^{5}-4 \sqrt {e x +d}\, \sqrt {c d x +a e}\, a \,c^{2} d^{3} e^{3}-10 \sqrt {e x +d}\, \sqrt {c d x +a e}\, a \,c^{2} d^{2} e^{4} x -3 \sqrt {e x +d}\, \sqrt {c d x +a e}\, c^{3} d^{5} e +2 \sqrt {e x +d}\, \sqrt {c d x +a e}\, c^{3} d^{4} e^{2} x +8 \sqrt {e x +d}\, \sqrt {c d x +a e}\, c^{3} d^{3} e^{3} x^{2}-15 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) a^{3} e^{6}+9 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) a^{2} c \,d^{2} e^{4}+3 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) a \,c^{2} d^{4} e^{2}+3 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) c^{3} d^{6}}{24 c^{4} d^{4} e^{3}} \] Input:
int(x^2*(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x)
Output:
(15*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**2*c*d*e**5 - 4*sqrt(d + e*x)*sqrt(a *e + c*d*x)*a*c**2*d**3*e**3 - 10*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*c**2*d **2*e**4*x - 3*sqrt(d + e*x)*sqrt(a*e + c*d*x)*c**3*d**5*e + 2*sqrt(d + e* x)*sqrt(a*e + c*d*x)*c**3*d**4*e**2*x + 8*sqrt(d + e*x)*sqrt(a*e + c*d*x)* c**3*d**3*e**3*x**2 - 15*sqrt(e)*sqrt(d)*sqrt(c)*log((sqrt(e)*sqrt(a*e + c *d*x) + sqrt(d)*sqrt(c)*sqrt(d + e*x))/sqrt(a*e**2 - c*d**2))*a**3*e**6 + 9*sqrt(e)*sqrt(d)*sqrt(c)*log((sqrt(e)*sqrt(a*e + c*d*x) + sqrt(d)*sqrt(c) *sqrt(d + e*x))/sqrt(a*e**2 - c*d**2))*a**2*c*d**2*e**4 + 3*sqrt(e)*sqrt(d )*sqrt(c)*log((sqrt(e)*sqrt(a*e + c*d*x) + sqrt(d)*sqrt(c)*sqrt(d + e*x))/ sqrt(a*e**2 - c*d**2))*a*c**2*d**4*e**2 + 3*sqrt(e)*sqrt(d)*sqrt(c)*log((s qrt(e)*sqrt(a*e + c*d*x) + sqrt(d)*sqrt(c)*sqrt(d + e*x))/sqrt(a*e**2 - c* d**2))*c**3*d**6)/(24*c**4*d**4*e**3)