Integrand size = 38, antiderivative size = 339 \[ \int \frac {x^3 (d+e x)}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {\left (c d^2-7 a e^2\right ) x^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 c^2 d^2 e}+\frac {x^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 c d}+\frac {\left (15 c^3 d^6+17 a c^2 d^4 e^2+25 a^2 c d^2 e^4-105 a^3 e^6-2 c d e \left (5 c^2 d^4+6 a c d^2 e^2-35 a^2 e^4\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{192 c^4 d^4 e^3}-\frac {\left (c d^2-a e^2\right ) \left (5 c^3 d^6+9 a c^2 d^4 e^2+15 a^2 c d^2 e^4+35 a^3 e^6\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} (d+e x)}{\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{64 c^{9/2} d^{9/2} e^{7/2}} \] Output:
1/24*(-7*a*e^2+c*d^2)*x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^2/d^2/ e+1/4*x^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c/d+1/192*(15*c^3*d^6+17 *a*c^2*d^4*e^2+25*a^2*c*d^2*e^4-105*a^3*e^6-2*c*d*e*(-35*a^2*e^4+6*a*c*d^2 *e^2+5*c^2*d^4)*x)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^4/d^4/e^3-1/6 4*(-a*e^2+c*d^2)*(35*a^3*e^6+15*a^2*c*d^2*e^4+9*a*c^2*d^4*e^2+5*c^3*d^6)*a rctanh(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^(9/2)/d^(9/2)/e^(7/2)
Time = 11.18 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.11 \[ \int \frac {x^3 (d+e x)}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {(d+e x) \left (\sqrt {c} \sqrt {d} \sqrt {e} \sqrt {\frac {c d (d+e x)}{c d^2-a e^2}} \left (-105 a^4 e^7+5 a^3 c d e^5 (5 d-7 e x)+a^2 c^2 d^2 e^3 \left (17 d^2+13 d e x+14 e^2 x^2\right )+a c^3 d^3 e \left (15 d^3+7 d^2 e x-4 d e^2 x^2-8 e^3 x^3\right )+c^4 d^4 x \left (15 d^3-10 d^2 e x+8 d e^2 x^2+48 e^3 x^3\right )\right )-3 \sqrt {c d} \sqrt {c d^2-a e^2} \left (5 c^3 d^6+9 a c^2 d^4 e^2+15 a^2 c d^2 e^4+35 a^3 e^6\right ) \sqrt {a e+c d x} \text {arcsinh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a e+c d x}}{\sqrt {c d} \sqrt {c d^2-a e^2}}\right )\right )}{192 c^{9/2} d^{9/2} e^{7/2} \sqrt {\frac {c d (d+e x)}{c d^2-a e^2}} \sqrt {(a e+c d x) (d+e x)}} \] Input:
Integrate[(x^3*(d + e*x))/Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2],x]
Output:
((d + e*x)*(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)]* (-105*a^4*e^7 + 5*a^3*c*d*e^5*(5*d - 7*e*x) + a^2*c^2*d^2*e^3*(17*d^2 + 13 *d*e*x + 14*e^2*x^2) + a*c^3*d^3*e*(15*d^3 + 7*d^2*e*x - 4*d*e^2*x^2 - 8*e ^3*x^3) + c^4*d^4*x*(15*d^3 - 10*d^2*e*x + 8*d*e^2*x^2 + 48*e^3*x^3)) - 3* Sqrt[c*d]*Sqrt[c*d^2 - a*e^2]*(5*c^3*d^6 + 9*a*c^2*d^4*e^2 + 15*a^2*c*d^2* e^4 + 35*a^3*e^6)*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])]))/(192*c^(9/2)*d^(9/2)*e^(7 /2)*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)]*Sqrt[(a*e + c*d*x)*(d + e*x)])
Time = 1.05 (sec) , antiderivative size = 374, normalized size of antiderivative = 1.10, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.184, Rules used = {1236, 27, 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.
| \(\displaystyle \int \frac {x^3 (d+e x)}{\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \, dx\) |
| \(\Big \downarrow \) 1236 |
| \(\displaystyle \frac {\int -\frac {e x^2 \left (6 a d e-\left (c d^2-7 a e^2\right ) x\right )}{2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{4 c d e}+\frac {x^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c d}-\frac {\int \frac {x^2 \left (6 a d e-\left (c d^2-7 a e^2\right ) x\right )}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{8 c d}\) |
| \(\Big \downarrow \) 1236 |
| \(\displaystyle \frac {x^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c d}-\frac {\frac {\int \frac {x \left (4 a d e \left (c d^2-7 a e^2\right )+\left (5 c^2 d^4+6 a c e^2 d^2-35 a^2 e^4\right ) x\right )}{2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{3 c d e}-\frac {1}{3} x^2 \left (\frac {d}{e}-\frac {7 a e}{c d}\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 c d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c d}-\frac {\frac {\int \frac {x \left (4 a d e \left (c d^2-7 a e^2\right )+\left (5 c^2 d^4+6 a c e^2 d^2-35 a^2 e^4\right ) x\right )}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{6 c d e}-\frac {1}{3} x^2 \left (\frac {d}{e}-\frac {7 a e}{c d}\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 c d}\) |
| \(\Big \downarrow \) 1225 |
| \(\displaystyle \frac {x^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c d}-\frac {\frac {\frac {3 \left (c d^2-a e^2\right ) \left (35 a^3 e^6+15 a^2 c d^2 e^4+9 a c^2 d^4 e^2+5 c^3 d^6\right ) \int \frac {1}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{8 c^2 d^2 e^2}-\frac {\left (-105 a^3 e^6-2 c d e x \left (-35 a^2 e^4+6 a c d^2 e^2+5 c^2 d^4\right )+25 a^2 c d^2 e^4+17 a c^2 d^4 e^2+15 c^3 d^6\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c^2 d^2 e^2}}{6 c d e}-\frac {1}{3} x^2 \left (\frac {d}{e}-\frac {7 a e}{c d}\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 c d}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {x^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c d}-\frac {\frac {\frac {3 \left (c d^2-a e^2\right ) \left (35 a^3 e^6+15 a^2 c d^2 e^4+9 a c^2 d^4 e^2+5 c^3 d^6\right ) \int \frac {1}{4 c d e-\frac {\left (c d^2+2 c e x d+a e^2\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\frac {c d^2+2 c e x d+a e^2}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}}{4 c^2 d^2 e^2}-\frac {\left (-105 a^3 e^6-2 c d e x \left (-35 a^2 e^4+6 a c d^2 e^2+5 c^2 d^4\right )+25 a^2 c d^2 e^4+17 a c^2 d^4 e^2+15 c^3 d^6\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c^2 d^2 e^2}}{6 c d e}-\frac {1}{3} x^2 \left (\frac {d}{e}-\frac {7 a e}{c d}\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 c d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {x^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c d}-\frac {\frac {\frac {3 \left (c d^2-a e^2\right ) \left (35 a^3 e^6+15 a^2 c d^2 e^4+9 a c^2 d^4 e^2+5 c^3 d^6\right ) \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 c^{5/2} d^{5/2} e^{5/2}}-\frac {\left (-105 a^3 e^6-2 c d e x \left (-35 a^2 e^4+6 a c d^2 e^2+5 c^2 d^4\right )+25 a^2 c d^2 e^4+17 a c^2 d^4 e^2+15 c^3 d^6\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c^2 d^2 e^2}}{6 c d e}-\frac {1}{3} x^2 \left (\frac {d}{e}-\frac {7 a e}{c d}\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 c d}\) |
Input:
Int[(x^3*(d + e*x))/Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2],x]
Output:
(x^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(4*c*d) - (-1/3*((d/e - (7*a*e)/(c*d))*x^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (-1/4*(( 15*c^3*d^6 + 17*a*c^2*d^4*e^2 + 25*a^2*c*d^2*e^4 - 105*a^3*e^6 - 2*c*d*e*( 5*c^2*d^4 + 6*a*c*d^2*e^2 - 35*a^2*e^4)*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(c^2*d^2*e^2) + (3*(c*d^2 - a*e^2)*(5*c^3*d^6 + 9*a*c^2*d^4* e^2 + 15*a^2*c*d^2*e^4 + 35*a^3*e^6)*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*e))/(8*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. \(1199\) vs. \(2(311)=622\).
Time = 2.63 (sec) , antiderivative size = 1200, normalized size of antiderivative = 3.54
Input:
int(x^3*(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*(1/3*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)))+e*(1/4*x^3/d/e/c*(a*d*e+(a*e^2+c*d^2 )*x+c*d*x^2*e)^(1/2)-7/8*(a*e^2+c*d^2)/d/e/c*(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)))-3/4*a/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*...
Time = 0.24 (sec) , antiderivative size = 676, normalized size of antiderivative = 1.99 \[ \int \frac {x^3 (d+e x)}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\left [-\frac {3 \, {\left (5 \, c^{4} d^{8} + 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} + 20 \, a^{3} c d^{2} e^{6} - 35 \, a^{4} e^{8}\right )} \sqrt {c d e} \log \left (8 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4} + 4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {c d e} + 8 \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right ) - 4 \, {\left (48 \, c^{4} d^{4} e^{4} x^{3} + 15 \, c^{4} d^{7} e + 17 \, a c^{3} d^{5} e^{3} + 25 \, a^{2} c^{2} d^{3} e^{5} - 105 \, a^{3} c d e^{7} + 8 \, {\left (c^{4} d^{5} e^{3} - 7 \, a c^{3} d^{3} e^{5}\right )} x^{2} - 2 \, {\left (5 \, c^{4} d^{6} e^{2} + 6 \, a c^{3} d^{4} e^{4} - 35 \, a^{2} c^{2} d^{2} e^{6}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{768 \, c^{5} d^{5} e^{4}}, \frac {3 \, {\left (5 \, c^{4} d^{8} + 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} + 20 \, a^{3} c d^{2} e^{6} - 35 \, a^{4} e^{8}\right )} \sqrt {-c d e} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {-c d e}}{2 \, {\left (c^{2} d^{2} e^{2} x^{2} + a c d^{2} e^{2} + {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right )}}\right ) + 2 \, {\left (48 \, c^{4} d^{4} e^{4} x^{3} + 15 \, c^{4} d^{7} e + 17 \, a c^{3} d^{5} e^{3} + 25 \, a^{2} c^{2} d^{3} e^{5} - 105 \, a^{3} c d e^{7} + 8 \, {\left (c^{4} d^{5} e^{3} - 7 \, a c^{3} d^{3} e^{5}\right )} x^{2} - 2 \, {\left (5 \, c^{4} d^{6} e^{2} + 6 \, a c^{3} d^{4} e^{4} - 35 \, a^{2} c^{2} d^{2} e^{6}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{384 \, c^{5} d^{5} e^{4}}\right ] \] Input:
integrate(x^3*(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorithm ="fricas")
Output:
[-1/768*(3*(5*c^4*d^8 + 4*a*c^3*d^6*e^2 + 6*a^2*c^2*d^4*e^4 + 20*a^3*c*d^2 *e^6 - 35*a^4*e^8)*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*(48*c^4*d^ 4*e^4*x^3 + 15*c^4*d^7*e + 17*a*c^3*d^5*e^3 + 25*a^2*c^2*d^3*e^5 - 105*a^3 *c*d*e^7 + 8*(c^4*d^5*e^3 - 7*a*c^3*d^3*e^5)*x^2 - 2*(5*c^4*d^6*e^2 + 6*a* c^3*d^4*e^4 - 35*a^2*c^2*d^2*e^6)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e ^2)*x))/(c^5*d^5*e^4), 1/384*(3*(5*c^4*d^8 + 4*a*c^3*d^6*e^2 + 6*a^2*c^2*d ^4*e^4 + 20*a^3*c*d^2*e^6 - 35*a^4*e^8)*sqrt(-c*d*e)*arctan(1/2*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) /(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*(48*c^4* d^4*e^4*x^3 + 15*c^4*d^7*e + 17*a*c^3*d^5*e^3 + 25*a^2*c^2*d^3*e^5 - 105*a ^3*c*d*e^7 + 8*(c^4*d^5*e^3 - 7*a*c^3*d^3*e^5)*x^2 - 2*(5*c^4*d^6*e^2 + 6* a*c^3*d^4*e^4 - 35*a^2*c^2*d^2*e^6)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a *e^2)*x))/(c^5*d^5*e^4)]
Leaf count of result is larger than twice the leaf count of optimal. 906 vs. \(2 (343) = 686\).
Time = 1.17 (sec) , antiderivative size = 906, normalized size of antiderivative = 2.67 \[ \int \frac {x^3 (d+e x)}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx =\text {Too large to display} \] Input:
integrate(x**3*(e*x+d)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2),x)
Output:
Piecewise(((-a*(-3*a*e/(4*c) - (d - (7*a*e**2/2 + 7*c*d**2/2)/(4*c*d))*(5* a*e**2/2 + 5*c*d**2/2)/(3*c*d*e))/(2*c) - (a*e**2 + c*d**2)*(-2*a*(d - (7* a*e**2/2 + 7*c*d**2/2)/(4*c*d))/(3*c) - (3*a*e**2/2 + 3*c*d**2/2)*(-3*a*e/ (4*c) - (d - (7*a*e**2/2 + 7*c*d**2/2)/(4*c*d))*(5*a*e**2/2 + 5*c*d**2/2)/ (3*c*d*e))/(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**3/(4*c*d) + x**2*(d - (7*a*e**2/2 + 7*c*d**2/2)/(4*c*d))/( 3*c*d*e) + x*(-3*a*e/(4*c) - (d - (7*a*e**2/2 + 7*c*d**2/2)/(4*c*d))*(5*a* e**2/2 + 5*c*d**2/2)/(3*c*d*e))/(2*c*d*e) + (-2*a*(d - (7*a*e**2/2 + 7*c*d **2/2)/(4*c*d))/(3*c) - (3*a*e**2/2 + 3*c*d**2/2)*(-3*a*e/(4*c) - (d - (7* a*e**2/2 + 7*c*d**2/2)/(4*c*d))*(5*a*e**2/2 + 5*c*d**2/2)/(3*c*d*e))/(2*c* d*e))/(c*d*e)), Ne(c*d*e, 0)), (2*(-a**3*c*d**6*e**3*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))**(9/2)/(9 *(a*e**2 + c*d**2)) + (a*d*e + x*(a*e**2 + c*d**2))**(7/2)*(-3*a*d*e**2 + c*d**3)/(7*(a*e**2 + c*d**2)) + (a*d*e + x*(a*e**2 + c*d**2))**(5/2)*(3*a* *2*d**2*e**3 - 3*a*c*d**4*e)/(5*(a*e**2 + c*d**2)) + (a*d*e + x*(a*e**2 + c*d**2))**(3/2)*(-a**3*d**3*e**4 + 3*a**2*c*d**5*e**2)/(3*(a*e**2 + c*d...
Exception generated. \[ \int \frac {x^3 (d+e x)}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(x^3*(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 = 306, normalized size of antiderivative = 0.90 \[ \int \frac {x^3 (d+e x)}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {1}{192} \, \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e} {\left (2 \, {\left (4 \, x {\left (\frac {6 \, x}{c d} + \frac {c^{3} d^{4} e^{2} - 7 \, a c^{2} d^{2} e^{4}}{c^{4} d^{4} e^{3}}\right )} - \frac {5 \, c^{3} d^{5} e + 6 \, a c^{2} d^{3} e^{3} - 35 \, a^{2} c d e^{5}}{c^{4} d^{4} e^{3}}\right )} x + \frac {15 \, c^{3} d^{6} + 17 \, a c^{2} d^{4} e^{2} + 25 \, a^{2} c d^{2} e^{4} - 105 \, a^{3} e^{6}}{c^{4} d^{4} e^{3}}\right )} + \frac {{\left (5 \, c^{4} d^{8} + 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} + 20 \, a^{3} c d^{2} e^{6} - 35 \, a^{4} e^{8}\right )} \log \left ({\left | -c d^{2} - a e^{2} - 2 \, \sqrt {c d e} {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )} \right |}\right )}{128 \, \sqrt {c d e} c^{4} d^{4} e^{3}} \] Input:
integrate(x^3*(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorithm ="giac")
Output:
1/192*sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e)*(2*(4*x*(6*x/(c*d) + (c^ 3*d^4*e^2 - 7*a*c^2*d^2*e^4)/(c^4*d^4*e^3)) - (5*c^3*d^5*e + 6*a*c^2*d^3*e ^3 - 35*a^2*c*d*e^5)/(c^4*d^4*e^3))*x + (15*c^3*d^6 + 17*a*c^2*d^4*e^2 + 2 5*a^2*c*d^2*e^4 - 105*a^3*e^6)/(c^4*d^4*e^3)) + 1/128*(5*c^4*d^8 + 4*a*c^3 *d^6*e^2 + 6*a^2*c^2*d^4*e^4 + 20*a^3*c*d^2*e^6 - 35*a^4*e^8)*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^4*d^4*e^3)
Timed out. \[ \int \frac {x^3 (d+e x)}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int \frac {x^3\,\left (d+e\,x\right )}{\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}} \,d x \] Input:
int((x^3*(d + e*x))/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2),x)
Output:
int((x^3*(d + e*x))/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2), x)
Time = 0.28 (sec) , antiderivative size = 579, normalized size of antiderivative = 1.71 \[ \int \frac {x^3 (d+e x)}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {-105 \sqrt {e x +d}\, \sqrt {c d x +a e}\, a^{3} c d \,e^{7}+25 \sqrt {e x +d}\, \sqrt {c d x +a e}\, a^{2} c^{2} d^{3} e^{5}+70 \sqrt {e x +d}\, \sqrt {c d x +a e}\, a^{2} c^{2} d^{2} e^{6} x +17 \sqrt {e x +d}\, \sqrt {c d x +a e}\, a \,c^{3} d^{5} e^{3}-12 \sqrt {e x +d}\, \sqrt {c d x +a e}\, a \,c^{3} d^{4} e^{4} x -56 \sqrt {e x +d}\, \sqrt {c d x +a e}\, a \,c^{3} d^{3} e^{5} x^{2}+15 \sqrt {e x +d}\, \sqrt {c d x +a e}\, c^{4} d^{7} e -10 \sqrt {e x +d}\, \sqrt {c d x +a e}\, c^{4} d^{6} e^{2} x +8 \sqrt {e x +d}\, \sqrt {c d x +a e}\, c^{4} d^{5} e^{3} x^{2}+48 \sqrt {e x +d}\, \sqrt {c d x +a e}\, c^{4} d^{4} e^{4} x^{3}+105 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) a^{4} e^{8}-60 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) a^{3} c \,d^{2} e^{6}-18 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) a^{2} c^{2} d^{4} e^{4}-12 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) a \,c^{3} d^{6} e^{2}-15 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) c^{4} d^{8}}{192 c^{5} d^{5} e^{4}} \] Input:
int(x^3*(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x)
Output:
( - 105*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**3*c*d*e**7 + 25*sqrt(d + e*x)*s qrt(a*e + c*d*x)*a**2*c**2*d**3*e**5 + 70*sqrt(d + e*x)*sqrt(a*e + c*d*x)* a**2*c**2*d**2*e**6*x + 17*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*c**3*d**5*e** 3 - 12*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*c**3*d**4*e**4*x - 56*sqrt(d + e* x)*sqrt(a*e + c*d*x)*a*c**3*d**3*e**5*x**2 + 15*sqrt(d + e*x)*sqrt(a*e + c *d*x)*c**4*d**7*e - 10*sqrt(d + e*x)*sqrt(a*e + c*d*x)*c**4*d**6*e**2*x + 8*sqrt(d + e*x)*sqrt(a*e + c*d*x)*c**4*d**5*e**3*x**2 + 48*sqrt(d + e*x)*s qrt(a*e + c*d*x)*c**4*d**4*e**4*x**3 + 105*sqrt(e)*sqrt(d)*sqrt(c)*log((sq rt(e)*sqrt(a*e + c*d*x) + sqrt(d)*sqrt(c)*sqrt(d + e*x))/sqrt(a*e**2 - c*d **2))*a**4*e**8 - 60*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*c*d**2*e**6 - 18*sqrt(e)*sqrt(d)*sqrt(c)*log((sqrt(e)*sqrt(a*e + c*d*x) + sqrt(d)*sqr t(c)*sqrt(d + e*x))/sqrt(a*e**2 - c*d**2))*a**2*c**2*d**4*e**4 - 12*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**3*d**6*e**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))*c**4*d**8)/(192*c**5*d**5*e**4)