Integrand size = 40, antiderivative size = 336 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^4 (d+e x)} \, dx=-\frac {\left (5 c^2 d^4+12 a c d^2 e^2-a^2 e^4-2 c d e \left (7 c d^2+a e^2\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 d x}-\frac {\left (4 a d e+3 \left (3 c d^2+a e^2\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{12 d x^3}+c^{3/2} d^{3/2} \sqrt {e} \left (3 c d^2+5 a e^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} (d+e x)}{\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )-\frac {\left (5 c^3 d^6+45 a c^2 d^4 e^2+15 a^2 c d^2 e^4-a^3 e^6\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {e} (d+e x)}{\sqrt {d} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{8 \sqrt {a} d^{3/2} \sqrt {e}} \] Output:
-1/8*(5*c^2*d^4+12*a*c*d^2*e^2-a^2*e^4-2*c*d*e*(a*e^2+7*c*d^2)*x)*(a*d*e+( a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/d/x-1/12*(4*a*d*e+3*(a*e^2+3*c*d^2)*x)*(a* d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/d/x^3+c^(3/2)*d^(3/2)*e^(1/2)*(5*a*e^ 2+3*c*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))-1/8*(-a^3*e^6+15*a^2*c*d^2*e^4+45*a*c^2*d^4*e^2+5*c^3*d^ 6)*arctanh(a^(1/2)*e^(1/2)*(e*x+d)/d^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^ 2)^(1/2))/a^(1/2)/d^(3/2)/e^(1/2)
Time = 1.59 (sec) , antiderivative size = 316, normalized size of antiderivative = 0.94 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^4 (d+e x)} \, dx=-\frac {\sqrt {a e+c d x} \sqrt {d+e x} \left (\sqrt {a} \sqrt {d} \sqrt {e} \sqrt {a e+c d x} \sqrt {d+e x} \left (3 c^2 d^3 x^2 (11 d-8 e x)+2 a c d^2 e x (13 d+34 e x)+a^2 e^2 \left (8 d^2+14 d e x+3 e^2 x^2\right )\right )+3 \left (5 c^3 d^6+45 a c^2 d^4 e^2+15 a^2 c d^2 e^4-a^3 e^6\right ) x^3 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a e+c d x}}{\sqrt {a} \sqrt {e} \sqrt {d+e x}}\right )-24 \sqrt {a} c^{3/2} d^3 e \left (3 c d^2+5 a e^2\right ) x^3 \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \sqrt {d+e x}}\right )\right )}{24 \sqrt {a} d^{3/2} \sqrt {e} x^3 \sqrt {(a e+c d x) (d+e x)}} \] Input:
Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(x^4*(d + e*x)),x]
Output:
-1/24*(Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*(Sqrt[a]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*(3*c^2*d^3*x^2*(11*d - 8*e*x) + 2*a*c*d^2*e*x*(13*d + 34*e*x) + a^2*e^2*(8*d^2 + 14*d*e*x + 3*e^2*x^2)) + 3*(5*c^3*d^6 + 45*a* c^2*d^4*e^2 + 15*a^2*c*d^2*e^4 - a^3*e^6)*x^3*ArcTanh[(Sqrt[d]*Sqrt[a*e + c*d*x])/(Sqrt[a]*Sqrt[e]*Sqrt[d + e*x])] - 24*Sqrt[a]*c^(3/2)*d^3*e*(3*c*d ^2 + 5*a*e^2)*x^3*ArcTanh[(Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c]*Sqrt[d]*Sqr t[d + e*x])]))/(Sqrt[a]*d^(3/2)*Sqrt[e]*x^3*Sqrt[(a*e + c*d*x)*(d + e*x)])
Time = 1.25 (sec) , antiderivative size = 375, normalized size of antiderivative = 1.12, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1215, 1229, 27, 1230, 25, 1269, 1092, 219, 1154, 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.
\(\displaystyle \int \frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{x^4 (d+e x)} \, dx\) |
\(\Big \downarrow \) 1215 |
\(\displaystyle \int \frac {(a e+c d x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{x^4}dx\) |
\(\Big \downarrow \) 1229 |
\(\displaystyle -\frac {\int -\frac {a e \left (5 c^2 d^4+12 a c e^2 d^2+2 c e \left (7 c d^2+a e^2\right ) x d-a^2 e^4\right ) \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{2 x^2}dx}{4 a d e}-\frac {\left (3 x \left (a e^2+3 c d^2\right )+4 a d e\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{12 d x^3}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {\left (5 c^2 d^4+12 a c e^2 d^2+2 c e \left (7 c d^2+a e^2\right ) x d-a^2 e^4\right ) \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{x^2}dx}{8 d}-\frac {\left (3 x \left (a e^2+3 c d^2\right )+4 a d e\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{12 d x^3}\) |
\(\Big \downarrow \) 1230 |
\(\displaystyle \frac {-\frac {1}{2} \int -\frac {5 c^3 d^6+45 a c^2 e^2 d^4+8 c^2 e \left (3 c d^2+5 a e^2\right ) x d^3+15 a^2 c e^4 d^2-a^3 e^6}{x \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (-a^2 e^4-2 c d e x \left (a e^2+7 c d^2\right )+12 a c d^2 e^2+5 c^2 d^4\right )}{x}}{8 d}-\frac {\left (3 x \left (a e^2+3 c d^2\right )+4 a d e\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{12 d x^3}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\frac {1}{2} \int \frac {5 c^3 d^6+45 a c^2 e^2 d^4+8 c^2 e \left (3 c d^2+5 a e^2\right ) x d^3+15 a^2 c e^4 d^2-a^3 e^6}{x \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx-\frac {\left (-a^2 e^4-2 c d e x \left (a e^2+7 c d^2\right )+12 a c d^2 e^2+5 c^2 d^4\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{x}}{8 d}-\frac {\left (3 x \left (a e^2+3 c d^2\right )+4 a d e\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{12 d x^3}\) |
\(\Big \downarrow \) 1269 |
\(\displaystyle \frac {\frac {1}{2} \left (\left (-a^3 e^6+15 a^2 c d^2 e^4+45 a c^2 d^4 e^2+5 c^3 d^6\right ) \int \frac {1}{x \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx+8 c^2 d^3 e \left (5 a e^2+3 c d^2\right ) \int \frac {1}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx\right )-\frac {\left (-a^2 e^4-2 c d e x \left (a e^2+7 c d^2\right )+12 a c d^2 e^2+5 c^2 d^4\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{x}}{8 d}-\frac {\left (3 x \left (a e^2+3 c d^2\right )+4 a d e\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{12 d x^3}\) |
\(\Big \downarrow \) 1092 |
\(\displaystyle \frac {\frac {1}{2} \left (\left (-a^3 e^6+15 a^2 c d^2 e^4+45 a c^2 d^4 e^2+5 c^3 d^6\right ) \int \frac {1}{x \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx+16 c^2 d^3 e \left (5 a e^2+3 c d^2\right ) \int \frac {1}{4 c d e-\frac {\left (c d^2+2 c e x d+a e^2\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\frac {c d^2+2 c e x d+a e^2}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}\right )-\frac {\left (-a^2 e^4-2 c d e x \left (a e^2+7 c d^2\right )+12 a c d^2 e^2+5 c^2 d^4\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{x}}{8 d}-\frac {\left (3 x \left (a e^2+3 c d^2\right )+4 a d e\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{12 d x^3}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {\frac {1}{2} \left (\left (-a^3 e^6+15 a^2 c d^2 e^4+45 a c^2 d^4 e^2+5 c^3 d^6\right ) \int \frac {1}{x \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx+8 c^{3/2} d^{5/2} \sqrt {e} \left (5 a e^2+3 c d^2\right ) \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )\right )-\frac {\left (-a^2 e^4-2 c d e x \left (a e^2+7 c d^2\right )+12 a c d^2 e^2+5 c^2 d^4\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{x}}{8 d}-\frac {\left (3 x \left (a e^2+3 c d^2\right )+4 a d e\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{12 d x^3}\) |
\(\Big \downarrow \) 1154 |
\(\displaystyle \frac {\frac {1}{2} \left (8 c^{3/2} d^{5/2} \sqrt {e} \left (5 a e^2+3 c d^2\right ) \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )-2 \left (-a^3 e^6+15 a^2 c d^2 e^4+45 a c^2 d^4 e^2+5 c^3 d^6\right ) \int \frac {1}{4 a d e-\frac {\left (2 a d e+\left (c d^2+a e^2\right ) x\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\frac {2 a d e+\left (c d^2+a e^2\right ) x}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}\right )-\frac {\left (-a^2 e^4-2 c d e x \left (a e^2+7 c d^2\right )+12 a c d^2 e^2+5 c^2 d^4\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{x}}{8 d}-\frac {\left (3 x \left (a e^2+3 c d^2\right )+4 a d e\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{12 d x^3}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {\frac {1}{2} \left (8 c^{3/2} d^{5/2} \sqrt {e} \left (5 a e^2+3 c d^2\right ) \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )-\frac {\left (-a^3 e^6+15 a^2 c d^2 e^4+45 a c^2 d^4 e^2+5 c^3 d^6\right ) \text {arctanh}\left (\frac {x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{\sqrt {a} \sqrt {d} \sqrt {e}}\right )-\frac {\left (-a^2 e^4-2 c d e x \left (a e^2+7 c d^2\right )+12 a c d^2 e^2+5 c^2 d^4\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{x}}{8 d}-\frac {\left (3 x \left (a e^2+3 c d^2\right )+4 a d e\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{12 d x^3}\) |
Input:
Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(x^4*(d + e*x)),x]
Output:
-1/12*((4*a*d*e + 3*(3*c*d^2 + a*e^2)*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d* e*x^2)^(3/2))/(d*x^3) + (-(((5*c^2*d^4 + 12*a*c*d^2*e^2 - a^2*e^4 - 2*c*d* e*(7*c*d^2 + a*e^2)*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/x) + ( 8*c^(3/2)*d^(5/2)*Sqrt[e]*(3*c*d^2 + 5*a*e^2)*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])] - ((5*c^3*d^6 + 45*a*c^2*d^4*e^2 + 15*a^2*c*d^2*e^4 - a^3*e^6)*ArcT anh[(2*a*d*e + (c*d^2 + a*e^2)*x)/(2*Sqrt[a]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])])/(Sqrt[a]*Sqrt[d]*Sqrt[e]))/2)/(8*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[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym bol] :> Simp[-2 Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c , d, e}, x]
Int[(((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_))/( (d_) + (e_.)*(x_)), x_Symbol] :> Int[(a/d + c*(x/e))*(f + g*x)^n*(a + b*x + c*x^2)^(p - 1), x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[p, 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*((a + b*x + c*x^2 )^p/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)))*((d*g - e*f*(m + 2))*(c* d^2 - b*d*e + a*e^2) - d*p*(2*c*d - b*e)*(e*f - d*g) - e*(g*(m + 1)*(c*d^2 - b*d*e + a*e^2) + p*(2*c*d - b*e)*(e*f - d*g))*x), x] - Simp[p/(e^2*(m + 1 )*(m + 2)*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2 )^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) + b^2*e*(d*g*(p + 1) - e*f*(m + p + 2)) + b*(a*e^2*g*(m + 1) - c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2))) - c *(2*c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2)) - e*(2*a*e*g*(m + 1) - b*(d*g*( m - 2*p) + e*f*(m + 2*p + 2))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g }, x] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] && !ILtQ[m + 2*p + 3, 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*((a + b*x + c*x^2)^p/(e^2*(m + 1)*(m + 2*p + 2))), x] + Simp[p/(e^2*(m + 1)*(m + 2*p + 2)) Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p - 1)*Simp[g*(b*d + 2*a*e + 2*a*e*m + 2*b*d*p) - f*b*e*(m + 2*p + 2) + (g*(2*c*d + b*e + b*e*m + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (LtQ[m, - 1] || EqQ[p, 1] || (IntegerQ[p] && !RationalQ[m])) && NeQ[m, -1] && !ILtQ [m + 2*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Simp[(e*f - d*g)/e Int[(d + e*x)^m*(a + b*x + c*x^2)^ p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && !IGtQ[m, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(6849\) vs. \(2(298)=596\).
Time = 4.54 (sec) , antiderivative size = 6850, normalized size of antiderivative = 20.39
Input:
int((a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(5/2)/x^4/(e*x+d),x,method=_RETURNVE RBOSE)
Output:
result too large to display
Time = 4.90 (sec) , antiderivative size = 1741, normalized size of antiderivative = 5.18 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^4 (d+e x)} \, dx=\text {Too large to display} \] Input:
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/x^4/(e*x+d),x, algorithm ="fricas")
Output:
[1/96*(24*(3*a*c^2*d^5*e + 5*a^2*c*d^3*e^3)*sqrt(c*d*e)*x^3*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) - 3*(5*c^3*d^6 + 45*a*c^2*d^4*e^2 + 15*a^2*c*d^2*e^4 - a^3* e^6)*sqrt(a*d*e)*x^3*log((8*a^2*d^2*e^2 + (c^2*d^4 + 6*a*c*d^2*e^2 + a^2*e ^4)*x^2 + 4*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(2*a*d*e + (c*d^2 + a*e^2)*x)*sqrt(a*d*e) + 8*(a*c*d^3*e + a^2*d*e^3)*x)/x^2) + 4*(24*a*c^2* d^4*e^2*x^3 - 8*a^3*d^3*e^3 - (33*a*c^2*d^5*e + 68*a^2*c*d^3*e^3 + 3*a^3*d *e^5)*x^2 - 2*(13*a^2*c*d^4*e^2 + 7*a^3*d^2*e^4)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(a*d^2*e*x^3), -1/96*(48*(3*a*c^2*d^5*e + 5*a^2*c*d ^3*e^3)*sqrt(-c*d*e)*x^3*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)) + 3*(5*c^3*d^6 + 45*a*c^2*d^4*e^2 + 15*a ^2*c*d^2*e^4 - a^3*e^6)*sqrt(a*d*e)*x^3*log((8*a^2*d^2*e^2 + (c^2*d^4 + 6* a*c*d^2*e^2 + a^2*e^4)*x^2 + 4*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x) *(2*a*d*e + (c*d^2 + a*e^2)*x)*sqrt(a*d*e) + 8*(a*c*d^3*e + a^2*d*e^3)*x)/ x^2) - 4*(24*a*c^2*d^4*e^2*x^3 - 8*a^3*d^3*e^3 - (33*a*c^2*d^5*e + 68*a^2* c*d^3*e^3 + 3*a^3*d*e^5)*x^2 - 2*(13*a^2*c*d^4*e^2 + 7*a^3*d^2*e^4)*x)*sqr t(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(a*d^2*e*x^3), 1/48*(3*(5*c^3*d^ 6 + 45*a*c^2*d^4*e^2 + 15*a^2*c*d^2*e^4 - a^3*e^6)*sqrt(-a*d*e)*x^3*arc...
Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^4 (d+e x)} \, dx=\text {Timed out} \] Input:
integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2)/x**4/(e*x+d),x)
Output:
Timed out
\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^4 (d+e x)} \, dx=\int { \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {5}{2}}}{{\left (e x + d\right )} x^{4}} \,d x } \] Input:
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/x^4/(e*x+d),x, algorithm ="maxima")
Output:
integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)/((e*x + d)*x^4), x )
Leaf count of result is larger than twice the leaf count of optimal. 1195 vs. \(2 (298) = 596\).
Time = 0.27 (sec) , antiderivative size = 1195, normalized size of antiderivative = 3.56 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^4 (d+e x)} \, dx =\text {Too large to display} \] Input:
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/x^4/(e*x+d),x, algorithm ="giac")
Output:
sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e)*c^2*d^2*e - 1/2*(3*c^3*d^4*e + 5*a*c^2*d^2*e^3)*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) + 1/8*(5*c^3*d^ 6 + 45*a*c^2*d^4*e^2 + 15*a^2*c*d^2*e^4 - a^3*e^6)*arctan(-(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))/sqrt(-a*d*e))/(sqrt(-a*d*e) *d) - 1/24*(15*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e ))*a^2*c^3*d^8*e^2 + 39*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2* x + a*d*e))*a^3*c^2*d^6*e^4 + 45*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))*a^4*c*d^4*e^6 - 3*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c *d^2*x + a*e^2*x + a*d*e))*a^5*d^2*e^8 + 48*sqrt(c*d*e)*a^3*c^2*d^7*e^3 + 112*sqrt(c*d*e)*a^4*c*d^5*e^5 - 40*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2 *x + a*e^2*x + a*d*e))^3*a*c^3*d^7*e - 72*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^3*a^2*c^2*d^5*e^3 - 24*(sqrt(c*d*e)*x - sqrt (c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^3*a^3*c*d^3*e^5 + 8*(sqrt(c*d*e)* x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^3*a^4*d*e^7 - 144*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))^2*a^2*c ^2*d^6*e^2 - 240*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))^2*a^3*c*d^4*e^4 + 33*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c* d^2*x + a*e^2*x + a*d*e))^5*c^3*d^6 + 153*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^5*a*c^2*d^4*e^2 + 99*(sqrt(c*d*e)*x - sqr...
Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^4 (d+e x)} \, dx=\int \frac {{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2}}{x^4\,\left (d+e\,x\right )} \,d x \] Input:
int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)/(x^4*(d + e*x)),x)
Output:
int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)/(x^4*(d + e*x)), x)
Time = 7.95 (sec) , antiderivative size = 1564, normalized size of antiderivative = 4.65 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^4 (d+e x)} \, dx =\text {Too large to display} \] Input:
int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/x^4/(e*x+d),x)
Output:
( - 16*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**4*d**3*e**5 - 28*sqrt(d + e*x)*s qrt(a*e + c*d*x)*a**4*d**2*e**6*x - 6*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**4 *d*e**7*x**2 - 16*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**3*c*d**5*e**3 - 80*sq rt(d + e*x)*sqrt(a*e + c*d*x)*a**3*c*d**4*e**4*x - 142*sqrt(d + e*x)*sqrt( a*e + c*d*x)*a**3*c*d**3*e**5*x**2 - 52*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a* *2*c**2*d**6*e**2*x - 202*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**2*c**2*d**5*e **3*x**2 + 48*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**2*c**2*d**4*e**4*x**3 - 6 6*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*c**3*d**7*e*x**2 + 48*sqrt(d + e*x)*sq rt(a*e + c*d*x)*a*c**3*d**6*e**2*x**3 - 3*sqrt(e)*sqrt(d)*sqrt(a)*log(sqrt (e)*sqrt(a*e + c*d*x) - sqrt(2*sqrt(c)*sqrt(a)*d*e + a*e**2 + c*d**2) + sq rt(d)*sqrt(c)*sqrt(d + e*x))*a**4*e**8*x**3 + 42*sqrt(e)*sqrt(d)*sqrt(a)*l og(sqrt(e)*sqrt(a*e + c*d*x) - sqrt(2*sqrt(c)*sqrt(a)*d*e + a*e**2 + c*d** 2) + sqrt(d)*sqrt(c)*sqrt(d + e*x))*a**3*c*d**2*e**6*x**3 + 180*sqrt(e)*sq rt(d)*sqrt(a)*log(sqrt(e)*sqrt(a*e + c*d*x) - sqrt(2*sqrt(c)*sqrt(a)*d*e + a*e**2 + c*d**2) + sqrt(d)*sqrt(c)*sqrt(d + e*x))*a**2*c**2*d**4*e**4*x** 3 + 150*sqrt(e)*sqrt(d)*sqrt(a)*log(sqrt(e)*sqrt(a*e + c*d*x) - sqrt(2*sqr t(c)*sqrt(a)*d*e + a*e**2 + c*d**2) + sqrt(d)*sqrt(c)*sqrt(d + e*x))*a*c** 3*d**6*e**2*x**3 + 15*sqrt(e)*sqrt(d)*sqrt(a)*log(sqrt(e)*sqrt(a*e + c*d*x ) - sqrt(2*sqrt(c)*sqrt(a)*d*e + a*e**2 + c*d**2) + sqrt(d)*sqrt(c)*sqrt(d + e*x))*c**4*d**8*x**3 - 3*sqrt(e)*sqrt(d)*sqrt(a)*log(sqrt(e)*sqrt(a*...