Integrand size = 40, antiderivative size = 410 \[ \int \frac {x^4}{(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {2 a^4 e^4}{c^4 d^4 \left (c d^2-a e^2\right ) (d+e x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {2 \left (c^4 d^8+5 a^4 e^8\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{5 c^4 d^4 e^3 \left (c d^2-a e^2\right )^2 (d+e x)^3}+\frac {2 \left (11 c^4 d^8-20 a c^3 d^6 e^2-15 a^4 e^8\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{15 c^3 d^3 e^3 \left (c d^2-a e^2\right )^3 (d+e x)^2}-\frac {2 \left (23 c^4 d^8-80 a c^3 d^6 e^2+90 a^2 c^2 d^4 e^4+15 a^4 e^8\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{15 c^2 d^2 e^3 \left (c d^2-a e^2\right )^4 (d+e x)}+\frac {2 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} (d+e x)}{\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{c^{3/2} d^{3/2} e^{7/2}} \] Output:
-2*a^4*e^4/c^4/d^4/(-a*e^2+c*d^2)/(e*x+d)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x ^2)^(1/2)-2/5*(5*a^4*e^8+c^4*d^8)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/ c^4/d^4/e^3/(-a*e^2+c*d^2)^2/(e*x+d)^3+2/15*(-15*a^4*e^8-20*a*c^3*d^6*e^2+ 11*c^4*d^8)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^3/d^3/e^3/(-a*e^2+c* d^2)^3/(e*x+d)^2-2/15*(15*a^4*e^8+90*a^2*c^2*d^4*e^4-80*a*c^3*d^6*e^2+23*c ^4*d^8)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^2/d^2/e^3/(-a*e^2+c*d^2) ^4/(e*x+d)+2*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^(3/2)/d^(3/2)/e^(7/2)
Time = 0.72 (sec) , antiderivative size = 300, normalized size of antiderivative = 0.73 \[ \int \frac {x^4}{(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {2 \left (-\frac {\sqrt {c} \sqrt {d} \sqrt {e} (a e+c d x) \left (15 a^4 e^7 (d+e x)^3+c^4 d^8 x \left (15 d^2+35 d e x+23 e^2 x^2\right )+a^3 c d^3 e^5 \left (73 d^2+160 d e x+90 e^2 x^2\right )+a c^3 d^6 e \left (15 d^3-20 d^2 e x-106 d e^2 x^2-80 e^3 x^3\right )+a^2 c^2 d^4 e^3 \left (-55 d^3-56 d^2 e x+80 d e^2 x^2+90 e^3 x^3\right )\right )}{\left (c d^2-a e^2\right )^4 (d+e x)}+15 (a e+c d x)^{3/2} (d+e x)^{3/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \sqrt {a e+c d x}}\right )\right )}{15 c^{3/2} d^{3/2} e^{7/2} ((a e+c d x) (d+e x))^{3/2}} \] Input:
Integrate[x^4/((d + e*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)), x]
Output:
(2*(-((Sqrt[c]*Sqrt[d]*Sqrt[e]*(a*e + c*d*x)*(15*a^4*e^7*(d + e*x)^3 + c^4 *d^8*x*(15*d^2 + 35*d*e*x + 23*e^2*x^2) + a^3*c*d^3*e^5*(73*d^2 + 160*d*e* x + 90*e^2*x^2) + a*c^3*d^6*e*(15*d^3 - 20*d^2*e*x - 106*d*e^2*x^2 - 80*e^ 3*x^3) + a^2*c^2*d^4*e^3*(-55*d^3 - 56*d^2*e*x + 80*d*e^2*x^2 + 90*e^3*x^3 )))/((c*d^2 - a*e^2)^4*(d + e*x))) + 15*(a*e + c*d*x)^(3/2)*(d + e*x)^(3/2 )*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[a*e + c*d*x])]))/( 15*c^(3/2)*d^(3/2)*e^(7/2)*((a*e + c*d*x)*(d + e*x))^(3/2))
Time = 0.93 (sec) , antiderivative size = 404, normalized size of antiderivative = 0.99, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1268, 109, 27, 167, 27, 162, 66, 221}
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.
| \(\displaystyle \int \frac {x^4}{(d+e x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1268 |
| \(\displaystyle \frac {\sqrt {d+e x} \sqrt {a e+c d x} \int \frac {x^4}{(a e+c d x)^{3/2} (d+e x)^{7/2}}dx}{\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {\sqrt {d+e x} \sqrt {a e+c d x} \left (\frac {2 a e x^3}{c d (d+e x)^{5/2} \left (c d^2-a e^2\right ) \sqrt {a e+c d x}}-\frac {2 \int \frac {x^2 \left (6 a d e-\left (c d^2-a e^2\right ) x\right )}{2 \sqrt {a e+c d x} (d+e x)^{7/2}}dx}{c d \left (c d^2-a e^2\right )}\right )}{\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {d+e x} \sqrt {a e+c d x} \left (\frac {2 a e x^3}{c d (d+e x)^{5/2} \left (c d^2-a e^2\right ) \sqrt {a e+c d x}}-\frac {\int \frac {x^2 \left (6 a d e-\left (c d^2-a e^2\right ) x\right )}{\sqrt {a e+c d x} (d+e x)^{7/2}}dx}{c d \left (c d^2-a e^2\right )}\right )}{\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle \frac {\sqrt {d+e x} \sqrt {a e+c d x} \left (\frac {2 a e x^3}{c d (d+e x)^{5/2} \left (c d^2-a e^2\right ) \sqrt {a e+c d x}}-\frac {\frac {2 d x^2 \left (5 a e^2+c d^2\right ) \sqrt {a e+c d x}}{5 e (d+e x)^{5/2} \left (c d^2-a e^2\right )}-\frac {2 \int \frac {x \left (5 x \left (c d^2-a e^2\right )^2+4 a d e \left (c d^2+5 a e^2\right )\right )}{2 \sqrt {a e+c d x} (d+e x)^{5/2}}dx}{5 e \left (c d^2-a e^2\right )}}{c d \left (c d^2-a e^2\right )}\right )}{\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {d+e x} \sqrt {a e+c d x} \left (\frac {2 a e x^3}{c d (d+e x)^{5/2} \left (c d^2-a e^2\right ) \sqrt {a e+c d x}}-\frac {\frac {2 d x^2 \left (5 a e^2+c d^2\right ) \sqrt {a e+c d x}}{5 e (d+e x)^{5/2} \left (c d^2-a e^2\right )}-\frac {\int \frac {x \left (5 x \left (c d^2-a e^2\right )^2+4 a d e \left (c d^2+5 a e^2\right )\right )}{\sqrt {a e+c d x} (d+e x)^{5/2}}dx}{5 e \left (c d^2-a e^2\right )}}{c d \left (c d^2-a e^2\right )}\right )}{\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\) |
| \(\Big \downarrow \) 162 |
| \(\displaystyle \frac {\sqrt {d+e x} \sqrt {a e+c d x} \left (\frac {2 a e x^3}{c d (d+e x)^{5/2} \left (c d^2-a e^2\right ) \sqrt {a e+c d x}}-\frac {\frac {2 d x^2 \left (5 a e^2+c d^2\right ) \sqrt {a e+c d x}}{5 e (d+e x)^{5/2} \left (c d^2-a e^2\right )}-\frac {\frac {5 \left (c d^2-a e^2\right )^2 \int \frac {1}{\sqrt {a e+c d x} \sqrt {d+e x}}dx}{e^2}-\frac {2 d \sqrt {a e+c d x} \left (2 e x \left (15 a^3 e^6+36 a^2 c d^2 e^4-37 a c^2 d^4 e^2+10 c^3 d^6\right )+d \left (15 a^3 e^6+73 a^2 c d^2 e^4-55 a c^2 d^4 e^2+15 c^3 d^6\right )\right )}{3 e^2 (d+e x)^{3/2} \left (c d^2-a e^2\right )^2}}{5 e \left (c d^2-a e^2\right )}}{c d \left (c d^2-a e^2\right )}\right )}{\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\) |
| \(\Big \downarrow \) 66 |
| \(\displaystyle \frac {\sqrt {d+e x} \sqrt {a e+c d x} \left (\frac {2 a e x^3}{c d (d+e x)^{5/2} \left (c d^2-a e^2\right ) \sqrt {a e+c d x}}-\frac {\frac {2 d x^2 \left (5 a e^2+c d^2\right ) \sqrt {a e+c d x}}{5 e (d+e x)^{5/2} \left (c d^2-a e^2\right )}-\frac {\frac {10 \left (c d^2-a e^2\right )^2 \int \frac {1}{c d-\frac {e (a e+c d x)}{d+e x}}d\frac {\sqrt {a e+c d x}}{\sqrt {d+e x}}}{e^2}-\frac {2 d \sqrt {a e+c d x} \left (2 e x \left (15 a^3 e^6+36 a^2 c d^2 e^4-37 a c^2 d^4 e^2+10 c^3 d^6\right )+d \left (15 a^3 e^6+73 a^2 c d^2 e^4-55 a c^2 d^4 e^2+15 c^3 d^6\right )\right )}{3 e^2 (d+e x)^{3/2} \left (c d^2-a e^2\right )^2}}{5 e \left (c d^2-a e^2\right )}}{c d \left (c d^2-a e^2\right )}\right )}{\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\sqrt {d+e x} \sqrt {a e+c d x} \left (\frac {2 a e x^3}{c d (d+e x)^{5/2} \left (c d^2-a e^2\right ) \sqrt {a e+c d x}}-\frac {\frac {2 d x^2 \left (5 a e^2+c d^2\right ) \sqrt {a e+c d x}}{5 e (d+e x)^{5/2} \left (c d^2-a e^2\right )}-\frac {\frac {10 \left (c d^2-a e^2\right )^2 \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \sqrt {d+e x}}\right )}{\sqrt {c} \sqrt {d} e^{5/2}}-\frac {2 d \sqrt {a e+c d x} \left (2 e x \left (15 a^3 e^6+36 a^2 c d^2 e^4-37 a c^2 d^4 e^2+10 c^3 d^6\right )+d \left (15 a^3 e^6+73 a^2 c d^2 e^4-55 a c^2 d^4 e^2+15 c^3 d^6\right )\right )}{3 e^2 (d+e x)^{3/2} \left (c d^2-a e^2\right )^2}}{5 e \left (c d^2-a e^2\right )}}{c d \left (c d^2-a e^2\right )}\right )}{\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\) |
Input:
Int[x^4/((d + e*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)),x]
Output:
(Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*((2*a*e*x^3)/(c*d*(c*d^2 - a*e^2)*Sqrt[a* e + c*d*x]*(d + e*x)^(5/2)) - ((2*d*(c*d^2 + 5*a*e^2)*x^2*Sqrt[a*e + c*d*x ])/(5*e*(c*d^2 - a*e^2)*(d + e*x)^(5/2)) - ((-2*d*Sqrt[a*e + c*d*x]*(d*(15 *c^3*d^6 - 55*a*c^2*d^4*e^2 + 73*a^2*c*d^2*e^4 + 15*a^3*e^6) + 2*e*(10*c^3 *d^6 - 37*a*c^2*d^4*e^2 + 36*a^2*c*d^2*e^4 + 15*a^3*e^6)*x))/(3*e^2*(c*d^2 - a*e^2)^2*(d + e*x)^(3/2)) + (10*(c*d^2 - a*e^2)^2*ArcTanh[(Sqrt[e]*Sqrt [a*e + c*d*x])/(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])])/(Sqrt[c]*Sqrt[d]*e^(5/2)) )/(5*e*(c*d^2 - a*e^2)))/(c*d*(c*d^2 - a*e^2))))/Sqrt[a*d*e + (c*d^2 + a*e ^2)*x + c*d*e*x^2]
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_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 2 Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre eQ[{a, b, c, d}, x] && !GtQ[c - a*(d/b), 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(b*c - a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f *x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_) )*((g_.) + (h_.)*(x_)), x_] :> Simp[((b^3*c*e*g*(m + 2) - a^3*d*f*h*(n + 2) - a^2*b*(c*f*h*m - d*(f*g + e*h)*(m + n + 3)) - a*b^2*(c*(f*g + e*h) + d*e *g*(2*m + n + 4)) + b*(a^2*d*f*h*(m - n) - a*b*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(n + 1)) + b^2*(c*(f*g + e*h)*(m + 1) - d*e*g*(m + n + 2)))*x)/(b^2*(b *c - a*d)^2*(m + 1)*(m + 2)))*(a + b*x)^(m + 1)*(c + d*x)^(n + 1), x] + Sim p[(f*(h/b^2) - (d*(m + n + 3)*(a^2*d*f*h*(m - n) - a*b*(2*c*f*h*(m + 1) - d *(f*g + e*h)*(n + 1)) + b^2*(c*(f*g + e*h)*(m + 1) - d*e*g*(m + n + 2))))/( b^2*(b*c - a*d)^2*(m + 1)*(m + 2))) Int[(a + b*x)^(m + 2)*(c + d*x)^n, x] , x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && (LtQ[m, -2] || (EqQ[m + n + 3, 0] && !LtQ[n, -2]))
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] - Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b* c*(f*g - e*h)*(m + 1) + (b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h )*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegersQ[2*m, 2*n, 2*p]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((d_) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x_ ) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x + c*x^2)^FracPart[p]/((d + e*x)^FracPart[p]*(a/d + (c*x)/e)^FracPart[p]) Int[(d + e*x)^(m + p)*(f + g*x)^n*(a/d + (c/e)*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x ] && EqQ[c*d^2 - b*d*e + a*e^2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(867\) vs. \(2(382)=764\).
Time = 2.84 (sec) , antiderivative size = 868, normalized size of antiderivative = 2.12
| method | result | size |
| default | \(\frac {-\frac {x}{d e c \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}-\frac {\left (a \,e^{2}+c \,d^{2}\right ) \left (-\frac {1}{d e c \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}-\frac {\left (a \,e^{2}+c \,d^{2}\right ) \left (2 c d x e +a \,e^{2}+c \,d^{2}\right )}{d e c \left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}\right )}{2 d e c}+\frac {\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 )}{d e c \sqrt {d e c}}}{e^{2}}+\frac {d^{4} \left (-\frac {2}{5 \left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{2} \sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}-\frac {6 d e c \left (-\frac {2}{3 \left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right ) \sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}+\frac {8 d e c \left (2 d e c \left (x +\frac {d}{e}\right )+a \,e^{2}-c \,d^{2}\right )}{3 \left (a \,e^{2}-c \,d^{2}\right )^{3} \sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}\right )}{5 \left (a \,e^{2}-c \,d^{2}\right )}\right )}{e^{6}}+\frac {6 d^{2} \left (2 c d x e +a \,e^{2}+c \,d^{2}\right )}{e^{4} \left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}-\frac {2 d \left (-\frac {1}{d e c \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}-\frac {\left (a \,e^{2}+c \,d^{2}\right ) \left (2 c d x e +a \,e^{2}+c \,d^{2}\right )}{d e c \left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}\right )}{e^{3}}-\frac {4 d^{3} \left (-\frac {2}{3 \left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right ) \sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}+\frac {8 d e c \left (2 d e c \left (x +\frac {d}{e}\right )+a \,e^{2}-c \,d^{2}\right )}{3 \left (a \,e^{2}-c \,d^{2}\right )^{3} \sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}\right )}{e^{5}}\) | \(868\) |
Input:
int(x^4/(e*x+d)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/2),x,method=_RETURN VERBOSE)
Output:
1/e^2*(-x/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*(-1/d/e/c/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)-(a*e^2+c*d^2)/d/e/ c*(2*c*d*e*x+a*e^2+c*d^2)/(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/(a*d*e+(a*e^2+c* d^2)*x+c*d*x^2*e)^(1/2))+1/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))+d^4/e^6*(-2/ 5/(a*e^2-c*d^2)/(x+d/e)^2/(d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)-6/ 5*d*e*c/(a*e^2-c*d^2)*(-2/3/(a*e^2-c*d^2)/(x+d/e)/(d*e*c*(x+d/e)^2+(a*e^2- c*d^2)*(x+d/e))^(1/2)+8/3*d*e*c/(a*e^2-c*d^2)^3*(2*d*e*c*(x+d/e)+a*e^2-c*d ^2)/(d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)))+6*d^2/e^4*(2*c*d*e*x+a *e^2+c*d^2)/(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2 *e)^(1/2)-2*d/e^3*(-1/d/e/c/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)-(a*e^2 +c*d^2)/d/e/c*(2*c*d*e*x+a*e^2+c*d^2)/(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/(a*d *e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2))-4/e^5*d^3*(-2/3/(a*e^2-c*d^2)/(x+d/e) /(d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)+8/3*d*e*c/(a*e^2-c*d^2)^3*( 2*d*e*c*(x+d/e)+a*e^2-c*d^2)/(d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2) )
Leaf count of result is larger than twice the leaf count of optimal. 1105 vs. \(2 (382) = 764\).
Time = 4.80 (sec) , antiderivative size = 2224, normalized size of antiderivative = 5.42 \[ \int \frac {x^4}{(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\text {Too large to display} \] Input:
integrate(x^4/(e*x+d)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorit hm="fricas")
Output:
[1/30*(15*(a*c^4*d^11*e - 4*a^2*c^3*d^9*e^3 + 6*a^3*c^2*d^7*e^5 - 4*a^4*c* d^5*e^7 + a^5*d^3*e^9 + (c^5*d^9*e^3 - 4*a*c^4*d^7*e^5 + 6*a^2*c^3*d^5*e^7 - 4*a^3*c^2*d^3*e^9 + a^4*c*d*e^11)*x^4 + (3*c^5*d^10*e^2 - 11*a*c^4*d^8* e^4 + 14*a^2*c^3*d^6*e^6 - 6*a^3*c^2*d^4*e^8 - a^4*c*d^2*e^10 + a^5*e^12)* x^3 + 3*(c^5*d^11*e - 3*a*c^4*d^9*e^3 + 2*a^2*c^3*d^7*e^5 + 2*a^3*c^2*d^5* e^7 - 3*a^4*c*d^3*e^9 + a^5*d*e^11)*x^2 + (c^5*d^12 - a*c^4*d^10*e^2 - 6*a ^2*c^3*d^8*e^4 + 14*a^3*c^2*d^6*e^6 - 11*a^4*c*d^4*e^8 + 3*a^5*d^2*e^10)*x )*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*(15*a*c^4*d^10*e^2 - 55*a^2 *c^3*d^8*e^4 + 73*a^3*c^2*d^6*e^6 + 15*a^4*c*d^4*e^8 + (23*c^5*d^9*e^3 - 8 0*a*c^4*d^7*e^5 + 90*a^2*c^3*d^5*e^7 + 15*a^4*c*d*e^11)*x^3 + (35*c^5*d^10 *e^2 - 106*a*c^4*d^8*e^4 + 80*a^2*c^3*d^6*e^6 + 90*a^3*c^2*d^4*e^8 + 45*a^ 4*c*d^2*e^10)*x^2 + (15*c^5*d^11*e - 20*a*c^4*d^9*e^3 - 56*a^2*c^3*d^7*e^5 + 160*a^3*c^2*d^5*e^7 + 45*a^4*c*d^3*e^9)*x)*sqrt(c*d*e*x^2 + a*d*e + (c* d^2 + a*e^2)*x))/(a*c^6*d^13*e^5 - 4*a^2*c^5*d^11*e^7 + 6*a^3*c^4*d^9*e^9 - 4*a^4*c^3*d^7*e^11 + a^5*c^2*d^5*e^13 + (c^7*d^11*e^7 - 4*a*c^6*d^9*e^9 + 6*a^2*c^5*d^7*e^11 - 4*a^3*c^4*d^5*e^13 + a^4*c^3*d^3*e^15)*x^4 + (3*c^7 *d^12*e^6 - 11*a*c^6*d^10*e^8 + 14*a^2*c^5*d^8*e^10 - 6*a^3*c^4*d^6*e^12 - a^4*c^3*d^4*e^14 + a^5*c^2*d^2*e^16)*x^3 + 3*(c^7*d^13*e^5 - 3*a*c^6*d...
\[ \int \frac {x^4}{(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\int \frac {x^{4}}{\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {3}{2}} \left (d + e x\right )^{2}}\, dx \] Input:
integrate(x**4/(e*x+d)**2/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2),x)
Output:
Integral(x**4/(((d + e*x)*(a*e + c*d*x))**(3/2)*(d + e*x)**2), x)
Exception generated. \[ \int \frac {x^4}{(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(x^4/(e*x+d)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorit hm="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(e>0)', see `assume?` for more de tails)Is e
Leaf count of result is larger than twice the leaf count of optimal. 4572 vs. \(2 (382) = 764\).
Time = 0.27 (sec) , antiderivative size = 4572, normalized size of antiderivative = 11.15 \[ \int \frac {x^4}{(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\text {Too large to display} \] Input:
integrate(x^4/(e*x+d)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorit hm="giac")
Output:
-2/15*(15*a^4*e^5*abs(e)/((c^5*d^9*sgn(1/(e*x + d))*sgn(e) - 4*a*c^4*d^7*e ^2*sgn(1/(e*x + d))*sgn(e) + 6*a^2*c^3*d^5*e^4*sgn(1/(e*x + d))*sgn(e) - 4 *a^3*c^2*d^3*e^6*sgn(1/(e*x + d))*sgn(e) + a^4*c*d*e^8*sgn(1/(e*x + d))*sg n(e))*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))) - (15*sqrt(c*d*e) *c^4*d^8*abs(e)*arctan(sqrt(c*d*e)/sqrt(-c*d*e)) - 60*sqrt(c*d*e)*a*c^3*d^ 6*e^2*abs(e)*arctan(sqrt(c*d*e)/sqrt(-c*d*e)) + 90*sqrt(c*d*e)*a^2*c^2*d^4 *e^4*abs(e)*arctan(sqrt(c*d*e)/sqrt(-c*d*e)) - 60*sqrt(c*d*e)*a^3*c*d^2*e^ 6*abs(e)*arctan(sqrt(c*d*e)/sqrt(-c*d*e)) + 15*sqrt(c*d*e)*a^4*e^8*abs(e)* arctan(sqrt(c*d*e)/sqrt(-c*d*e)) + 23*sqrt(-c*d*e)*c^4*d^8*abs(e) - 80*sqr t(-c*d*e)*a*c^3*d^6*e^2*abs(e) + 90*sqrt(-c*d*e)*a^2*c^2*d^4*e^4*abs(e) + 15*sqrt(-c*d*e)*a^4*e^8*abs(e))*sgn(1/(e*x + d))*sgn(e)/(sqrt(c*d*e)*sqrt( -c*d*e)*c^5*d^9*e^3 - 4*sqrt(c*d*e)*sqrt(-c*d*e)*a*c^4*d^7*e^5 + 6*sqrt(c* d*e)*sqrt(-c*d*e)*a^2*c^3*d^5*e^7 - 4*sqrt(c*d*e)*sqrt(-c*d*e)*a^3*c^2*d^3 *e^9 + sqrt(c*d*e)*sqrt(-c*d*e)*a^4*c*d*e^11) + (15*sqrt(c*d*e - c*d^2*e/( e*x + d) + a*e^3/(e*x + d))*c^18*d^38*e^26*abs(e)*sgn(1/(e*x + d))^4*sgn(e )^4 - 300*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))*a*c^17*d^36*e^ 28*abs(e)*sgn(1/(e*x + d))^4*sgn(e)^4 + 2850*sqrt(c*d*e - c*d^2*e/(e*x + d ) + a*e^3/(e*x + d))*a^2*c^16*d^34*e^30*abs(e)*sgn(1/(e*x + d))^4*sgn(e)^4 - 17040*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))*a^3*c^15*d^32*e ^32*abs(e)*sgn(1/(e*x + d))^4*sgn(e)^4 + 71700*sqrt(c*d*e - c*d^2*e/(e*...
Timed out. \[ \int \frac {x^4}{(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\int \frac {x^4}{{\left (d+e\,x\right )}^2\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/2}} \,d x \] Input:
int(x^4/((d + e*x)^2*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2)),x)
Output:
int(x^4/((d + e*x)^2*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2)), x)
Time = 37.32 (sec) , antiderivative size = 2483, normalized size of antiderivative = 6.06 \[ \int \frac {x^4}{(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:
int(x^4/(e*x+d)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x)
Output:
(30*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*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**4*d**3*e**8 + 90*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*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**4*d**2*e**9 *x + 90*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*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**4*d*e**1 0*x**2 + 30*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*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**4*e* *11*x**3 - 120*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*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**5*e**6 - 360*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*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**4*e**7*x - 360*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*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**3*c*d**3*e**8*x**2 - 120*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x )*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**9*x**3 + 180*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a* e + c*d*x)*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**2*d**7*e**4 + 540*sqrt(e)*sqrt(d)*sqrt(c)* sqrt(a*e + c*d*x)*log((sqrt(e)*sqrt(a*e + c*d*x) + sqrt(d)*sqrt(c)*sqrt...