Integrand size = 40, antiderivative size = 355 \[ \int \frac {x^3}{(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {2 c d x^4}{a e \left (c d^2-a e^2\right ) (d+e x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {2 \left (7 c d^2+a e^2\right ) x^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{7 a e \left (c d^2-a e^2\right )^2 (d+e x)^4}+\frac {12 \left (7 c d^2+a e^2\right ) x^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 \left (c d^2-a e^2\right )^3 (d+e x)^3}+\frac {16 a d \left (7 c d^2+a e^2\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 \left (c d^2-a e^2\right )^4 (d+e x)^2}-\frac {16 a \left (c d^2-3 a e^2\right ) \left (7 c d^2+a e^2\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 \left (c d^2-a e^2\right )^5 (d+e x)} \] Output:
2*c*d*x^4/a/e/(-a*e^2+c*d^2)/(e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^( 1/2)-2/7*(a*e^2+7*c*d^2)*x^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/a/e/( -a*e^2+c*d^2)^2/(e*x+d)^4+12/35*(a*e^2+7*c*d^2)*x^2*(a*d*e+(a*e^2+c*d^2)*x +c*d*e*x^2)^(1/2)/(-a*e^2+c*d^2)^3/(e*x+d)^3+16/35*a*d*(a*e^2+7*c*d^2)*(a* d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(-a*e^2+c*d^2)^4/(e*x+d)^2-16/35*a*(- 3*a*e^2+c*d^2)*(a*e^2+7*c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(-a *e^2+c*d^2)^5/(e*x+d)
Time = 0.28 (sec) , antiderivative size = 227, normalized size of antiderivative = 0.64 \[ \int \frac {x^3}{(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {2 \left (c^4 d^7 x^3 (7 d+2 e x)-2 a c^3 d^5 e x^2 \left (7 d^2+24 d e x+7 e^2 x^2\right )+a^4 e^5 \left (16 d^3+56 d^2 e x+70 d e^2 x^2+35 e^3 x^3\right )+2 a^2 c^2 d^3 e^2 x \left (28 d^3+97 d^2 e x+119 d e^2 x^2+35 e^3 x^3\right )+2 a^3 c d e^3 \left (56 d^4+200 d^3 e x+259 d^2 e^2 x^2+140 d e^3 x^3+35 e^4 x^4\right )\right )}{35 \left (c d^2-a e^2\right )^5 (d+e x)^3 \sqrt {(a e+c d x) (d+e x)}} \] Input:
Integrate[x^3/((d + e*x)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)), x]
Output:
(2*(c^4*d^7*x^3*(7*d + 2*e*x) - 2*a*c^3*d^5*e*x^2*(7*d^2 + 24*d*e*x + 7*e^ 2*x^2) + a^4*e^5*(16*d^3 + 56*d^2*e*x + 70*d*e^2*x^2 + 35*e^3*x^3) + 2*a^2 *c^2*d^3*e^2*x*(28*d^3 + 97*d^2*e*x + 119*d*e^2*x^2 + 35*e^3*x^3) + 2*a^3* c*d*e^3*(56*d^4 + 200*d^3*e*x + 259*d^2*e^2*x^2 + 140*d*e^3*x^3 + 35*e^4*x ^4)))/(35*(c*d^2 - a*e^2)^5*(d + e*x)^3*Sqrt[(a*e + c*d*x)*(d + e*x)])
Time = 1.70 (sec) , antiderivative size = 444, normalized size of antiderivative = 1.25, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1267, 27, 2169, 27, 1220, 1129, 1129, 1088}
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.
| \(\displaystyle \int \frac {x^3}{(d+e x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1267 |
| \(\displaystyle -\frac {\int \frac {3 \left (3 c d^2+a e^2\right ) x^2 e^3+6 d \left (c d^2+a e^2\right ) x e^2+d^2 \left (c d^2+3 a e^2\right ) e}{2 (d+e x)^3 \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}dx}{2 c d e^4}-\frac {1}{2 c d e^3 (d+e x) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {3 \left (3 c d^2+a e^2\right ) x^2 e^3+6 d \left (c d^2+a e^2\right ) x e^2+d^2 \left (c d^2+3 a e^2\right ) e}{(d+e x)^3 \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}dx}{4 c d e^4}-\frac {1}{2 c d e^3 (d+e x) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\) |
| \(\Big \downarrow \) 2169 |
| \(\displaystyle -\frac {-\frac {\int \frac {3 e^4 \left (d \left (c^2 d^4+10 a c e^2 d^2+5 a^2 e^4\right )+e \left (9 c^2 d^4+10 a c e^2 d^2+5 a^2 e^4\right ) x\right )}{2 (d+e x)^3 \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}dx}{3 c d e^3}-\frac {e \left (\frac {a e^2}{c d}+3 d\right )}{(d+e x)^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}}{4 c d e^4}-\frac {1}{2 c d e^3 (d+e x) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {e \int \frac {d \left (c^2 d^4+10 a c e^2 d^2+5 a^2 e^4\right )+e \left (9 c^2 d^4+10 a c e^2 d^2+5 a^2 e^4\right ) x}{(d+e x)^3 \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}dx}{2 c d}-\frac {e \left (\frac {a e^2}{c d}+3 d\right )}{(d+e x)^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}}{4 c d e^4}-\frac {1}{2 c d e^3 (d+e x) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\) |
| \(\Big \downarrow \) 1220 |
| \(\displaystyle -\frac {-\frac {e \left (-\frac {\left (35 a^3 e^6+35 a^2 c d^2 e^4-7 a c^2 d^4 e^2+c^3 d^6\right ) \int \frac {1}{(d+e x)^2 \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}dx}{7 \left (c d^2-a e^2\right )}-\frac {16 c^2 d^5}{7 (d+e x)^3 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 c d}-\frac {e \left (\frac {a e^2}{c d}+3 d\right )}{(d+e x)^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}}{4 c d e^4}-\frac {1}{2 c d e^3 (d+e x) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\) |
| \(\Big \downarrow \) 1129 |
| \(\displaystyle -\frac {-\frac {e \left (-\frac {\left (35 a^3 e^6+35 a^2 c d^2 e^4-7 a c^2 d^4 e^2+c^3 d^6\right ) \left (\frac {6 c d \int \frac {1}{(d+e x) \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}dx}{5 \left (c d^2-a e^2\right )}+\frac {2}{5 (d+e x)^2 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{7 \left (c d^2-a e^2\right )}-\frac {16 c^2 d^5}{7 (d+e x)^3 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 c d}-\frac {e \left (\frac {a e^2}{c d}+3 d\right )}{(d+e x)^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}}{4 c d e^4}-\frac {1}{2 c d e^3 (d+e x) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\) |
| \(\Big \downarrow \) 1129 |
| \(\displaystyle -\frac {-\frac {e \left (-\frac {\left (35 a^3 e^6+35 a^2 c d^2 e^4-7 a c^2 d^4 e^2+c^3 d^6\right ) \left (\frac {6 c d \left (\frac {4 c d \int \frac {1}{\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}dx}{3 \left (c d^2-a e^2\right )}+\frac {2}{3 (d+e x) \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{5 \left (c d^2-a e^2\right )}+\frac {2}{5 (d+e x)^2 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{7 \left (c d^2-a e^2\right )}-\frac {16 c^2 d^5}{7 (d+e x)^3 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 c d}-\frac {e \left (\frac {a e^2}{c d}+3 d\right )}{(d+e x)^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}}{4 c d e^4}-\frac {1}{2 c d e^3 (d+e x) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\) |
| \(\Big \downarrow \) 1088 |
| \(\displaystyle -\frac {-\frac {e \left (-\frac {\left (35 a^3 e^6+35 a^2 c d^2 e^4-7 a c^2 d^4 e^2+c^3 d^6\right ) \left (\frac {6 c d \left (\frac {2}{3 (d+e x) \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {8 c d \left (a e^2+c d^2+2 c d e x\right )}{3 \left (c d^2-a e^2\right )^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{5 \left (c d^2-a e^2\right )}+\frac {2}{5 (d+e x)^2 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{7 \left (c d^2-a e^2\right )}-\frac {16 c^2 d^5}{7 (d+e x)^3 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 c d}-\frac {e \left (\frac {a e^2}{c d}+3 d\right )}{(d+e x)^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}}{4 c d e^4}-\frac {1}{2 c d e^3 (d+e x) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\) |
Input:
Int[x^3/((d + e*x)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)),x]
Output:
-1/2*1/(c*d*e^3*(d + e*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) - ( -((e*(3*d + (a*e^2)/(c*d)))/((d + e*x)^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])) - (e*((-16*c^2*d^5)/(7*(c*d^2 - a*e^2)*(d + e*x)^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) - ((c^3*d^6 - 7*a*c^2*d^4*e^2 + 35*a^2* c*d^2*e^4 + 35*a^3*e^6)*(2/(5*(c*d^2 - a*e^2)*(d + e*x)^2*Sqrt[a*d*e + (c* d^2 + a*e^2)*x + c*d*e*x^2]) + (6*c*d*(2/(3*(c*d^2 - a*e^2)*(d + e*x)*Sqrt [a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) - (8*c*d*(c*d^2 + a*e^2 + 2*c*d*e *x))/(3*(c*d^2 - a*e^2)^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])))/( 5*(c*d^2 - a*e^2))))/(7*(c*d^2 - a*e^2))))/(2*c*d))/(4*c*d*e^4)
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_) + (c_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[-2*((b + 2*c*x)/((b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2])), x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(-e)*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/((m + p + 1)*(2* c*d - b*e))), x] + Simp[c*(Simplify[m + 2*p + 2]/((m + p + 1)*(2*c*d - b*e) )) Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d , e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && ILtQ[Simplify[m + 2*p + 2], 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d*g - e*f)*(d + e*x)^m*((a + b*x + c*x ^2)^(p + 1)/((2*c*d - b*e)*(m + p + 1))), x] + Simp[(m*(g*(c*d - b*e) + c*e *f) + e*(p + 1)*(2*c*f - b*g))/(e*(2*c*d - b*e)*(m + p + 1)) Int[(d + e*x )^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && ((LtQ[m, -1] && !IGtQ[m + p + 1, 0 ]) || (LtQ[m, 0] && LtQ[p, -1]) || EqQ[m + 2*p + 2, 0]) && NeQ[m + p + 1, 0 ]
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_ ) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[g^n*(d + e*x)^(m + n - 1)*((a + b *x + c*x^2)^(p + 1)/(c*e^(n - 1)*(m + n + 2*p + 1))), x] + Simp[1/(c*e^n*(m + n + 2*p + 1)) Int[(d + e*x)^m*(a + b*x + c*x^2)^p*ExpandToSum[c*e^n*(m + n + 2*p + 1)*(f + g*x)^n - c*g^n*(m + n + 2*p + 1)*(d + e*x)^n - g^n*(d + e*x)^(n - 2)*(b*d*e*(p + 1) + a*e^2*(m + n - 1) - c*d^2*(m + n + 2*p + 1) - e*(2*c*d - b*e)*(m + n + p)*x), x], x], x] /; FreeQ[{a, b, c, d, e, f, g , m, p}, x] && IGtQ[n, 1] && IntegerQ[m] && NeQ[m + n + 2*p + 1, 0]
Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p _), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, S imp[f*(d + e*x)^(m + q - 1)*((a + b*x + c*x^2)^(p + 1)/(c*e^(q - 1)*(m + q + 2*p + 1))), x] + Simp[1/(c*e^q*(m + q + 2*p + 1)) Int[(d + e*x)^m*(a + b*x + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*f*(m + q + 2*p + 1)*(d + e*x)^q + e*f*(m + p + q)*(d + e*x)^(q - 2)*(b*d - 2*a*e + (2*c*d - b*e)*x), x], x], x] /; NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, d, e, m, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2 , 0]
Time = 2.65 (sec) , antiderivative size = 361, normalized size of antiderivative = 1.02
| method | result | size |
| gosper | \(-\frac {2 \left (c d x +a e \right ) \left (70 a^{3} c d \,e^{7} x^{4}+70 a^{2} c^{2} d^{3} e^{5} x^{4}-14 a \,c^{3} d^{5} e^{3} x^{4}+2 c^{4} d^{7} e \,x^{4}+35 a^{4} e^{8} x^{3}+280 a^{3} c \,d^{2} e^{6} x^{3}+238 a^{2} c^{2} d^{4} e^{4} x^{3}-48 a \,c^{3} d^{6} e^{2} x^{3}+7 c^{4} d^{8} x^{3}+70 a^{4} d \,e^{7} x^{2}+518 a^{3} c \,d^{3} e^{5} x^{2}+194 a^{2} c^{2} d^{5} e^{3} x^{2}-14 a \,c^{3} d^{7} e \,x^{2}+56 a^{4} d^{2} e^{6} x +400 a^{3} c \,d^{4} e^{4} x +56 a^{2} c^{2} d^{6} e^{2} x +16 a^{4} d^{3} e^{5}+112 a^{3} c \,d^{5} e^{3}\right )}{35 \left (e x +d \right )^{2} \left (a^{5} e^{10}-5 a^{4} c \,d^{2} e^{8}+10 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}-c^{5} d^{10}\right ) \left (c d \,x^{2} e +a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}}}\) | \(361\) |
| trager | \(-\frac {2 \left (70 a^{3} c d \,e^{7} x^{4}+70 a^{2} c^{2} d^{3} e^{5} x^{4}-14 a \,c^{3} d^{5} e^{3} x^{4}+2 c^{4} d^{7} e \,x^{4}+35 a^{4} e^{8} x^{3}+280 a^{3} c \,d^{2} e^{6} x^{3}+238 a^{2} c^{2} d^{4} e^{4} x^{3}-48 a \,c^{3} d^{6} e^{2} x^{3}+7 c^{4} d^{8} x^{3}+70 a^{4} d \,e^{7} x^{2}+518 a^{3} c \,d^{3} e^{5} x^{2}+194 a^{2} c^{2} d^{5} e^{3} x^{2}-14 a \,c^{3} d^{7} e \,x^{2}+56 a^{4} d^{2} e^{6} x +400 a^{3} c \,d^{4} e^{4} x +56 a^{2} c^{2} d^{6} e^{2} x +16 a^{4} d^{3} e^{5}+112 a^{3} c \,d^{5} e^{3}\right ) \sqrt {c d \,x^{2} e +a \,e^{2} x +c \,d^{2} x +a d e}}{35 \left (c d x +a e \right ) \left (a \,e^{2}-c \,d^{2}\right ) \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right ) \left (e x +d \right )^{4}}\) | \(362\) |
| orering | \(-\frac {2 \left (70 a^{3} c d \,e^{7} x^{4}+70 a^{2} c^{2} d^{3} e^{5} x^{4}-14 a \,c^{3} d^{5} e^{3} x^{4}+2 c^{4} d^{7} e \,x^{4}+35 a^{4} e^{8} x^{3}+280 a^{3} c \,d^{2} e^{6} x^{3}+238 a^{2} c^{2} d^{4} e^{4} x^{3}-48 a \,c^{3} d^{6} e^{2} x^{3}+7 c^{4} d^{8} x^{3}+70 a^{4} d \,e^{7} x^{2}+518 a^{3} c \,d^{3} e^{5} x^{2}+194 a^{2} c^{2} d^{5} e^{3} x^{2}-14 a \,c^{3} d^{7} e \,x^{2}+56 a^{4} d^{2} e^{6} x +400 a^{3} c \,d^{4} e^{4} x +56 a^{2} c^{2} d^{6} e^{2} x +16 a^{4} d^{3} e^{5}+112 a^{3} c \,d^{5} e^{3}\right ) \left (c d x +a e \right )}{35 \left (a^{5} e^{10}-5 a^{4} c \,d^{2} e^{8}+10 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}-c^{5} d^{10}\right ) \left (e x +d \right )^{2} {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{\frac {3}{2}}}\) | \(362\) |
| default | \(\frac {4 c d x e +2 a \,e^{2}+2 c \,d^{2}}{e^{3} \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 {3 d \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^{4}}+\frac {3 d^{2} \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^{5}}-\frac {d^{3} \left (-\frac {2}{7 \left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\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 )}}-\frac {8 d e c \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 )}{7 \left (a \,e^{2}-c \,d^{2}\right )}\right )}{e^{6}}\) | \(767\) |
Input:
int(x^3/(e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/2),x,method=_RETURN VERBOSE)
Output:
-2/35*(c*d*x+a*e)*(70*a^3*c*d*e^7*x^4+70*a^2*c^2*d^3*e^5*x^4-14*a*c^3*d^5* e^3*x^4+2*c^4*d^7*e*x^4+35*a^4*e^8*x^3+280*a^3*c*d^2*e^6*x^3+238*a^2*c^2*d ^4*e^4*x^3-48*a*c^3*d^6*e^2*x^3+7*c^4*d^8*x^3+70*a^4*d*e^7*x^2+518*a^3*c*d ^3*e^5*x^2+194*a^2*c^2*d^5*e^3*x^2-14*a*c^3*d^7*e*x^2+56*a^4*d^2*e^6*x+400 *a^3*c*d^4*e^4*x+56*a^2*c^2*d^6*e^2*x+16*a^4*d^3*e^5+112*a^3*c*d^5*e^3)/(e *x+d)^2/(a^5*e^10-5*a^4*c*d^2*e^8+10*a^3*c^2*d^4*e^6-10*a^2*c^3*d^6*e^4+5* a*c^4*d^8*e^2-c^5*d^10)/(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(3/2)
Leaf count of result is larger than twice the leaf count of optimal. 770 vs. \(2 (337) = 674\).
Time = 11.93 (sec) , antiderivative size = 770, normalized size of antiderivative = 2.17 \[ \int \frac {x^3}{(d+e x)^3 \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 (112 \, a^{3} c d^{5} e^{3} + 16 \, a^{4} d^{3} e^{5} + 2 \, {\left (c^{4} d^{7} e - 7 \, a c^{3} d^{5} e^{3} + 35 \, a^{2} c^{2} d^{3} e^{5} + 35 \, a^{3} c d e^{7}\right )} x^{4} + {\left (7 \, c^{4} d^{8} - 48 \, a c^{3} d^{6} e^{2} + 238 \, a^{2} c^{2} d^{4} e^{4} + 280 \, a^{3} c d^{2} e^{6} + 35 \, a^{4} e^{8}\right )} x^{3} - 2 \, {\left (7 \, a c^{3} d^{7} e - 97 \, a^{2} c^{2} d^{5} e^{3} - 259 \, a^{3} c d^{3} e^{5} - 35 \, a^{4} d e^{7}\right )} x^{2} + 8 \, {\left (7 \, a^{2} c^{2} d^{6} e^{2} + 50 \, a^{3} c d^{4} e^{4} + 7 \, a^{4} 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}}{35 \, {\left (a c^{5} d^{14} e - 5 \, a^{2} c^{4} d^{12} e^{3} + 10 \, a^{3} c^{3} d^{10} e^{5} - 10 \, a^{4} c^{2} d^{8} e^{7} + 5 \, a^{5} c d^{6} e^{9} - a^{6} d^{4} e^{11} + {\left (c^{6} d^{11} e^{4} - 5 \, a c^{5} d^{9} e^{6} + 10 \, a^{2} c^{4} d^{7} e^{8} - 10 \, a^{3} c^{3} d^{5} e^{10} + 5 \, a^{4} c^{2} d^{3} e^{12} - a^{5} c d e^{14}\right )} x^{5} + {\left (4 \, c^{6} d^{12} e^{3} - 19 \, a c^{5} d^{10} e^{5} + 35 \, a^{2} c^{4} d^{8} e^{7} - 30 \, a^{3} c^{3} d^{6} e^{9} + 10 \, a^{4} c^{2} d^{4} e^{11} + a^{5} c d^{2} e^{13} - a^{6} e^{15}\right )} x^{4} + 2 \, {\left (3 \, c^{6} d^{13} e^{2} - 13 \, a c^{5} d^{11} e^{4} + 20 \, a^{2} c^{4} d^{9} e^{6} - 10 \, a^{3} c^{3} d^{7} e^{8} - 5 \, a^{4} c^{2} d^{5} e^{10} + 7 \, a^{5} c d^{3} e^{12} - 2 \, a^{6} d e^{14}\right )} x^{3} + 2 \, {\left (2 \, c^{6} d^{14} e - 7 \, a c^{5} d^{12} e^{3} + 5 \, a^{2} c^{4} d^{10} e^{5} + 10 \, a^{3} c^{3} d^{8} e^{7} - 20 \, a^{4} c^{2} d^{6} e^{9} + 13 \, a^{5} c d^{4} e^{11} - 3 \, a^{6} d^{2} e^{13}\right )} x^{2} + {\left (c^{6} d^{15} - a c^{5} d^{13} e^{2} - 10 \, a^{2} c^{4} d^{11} e^{4} + 30 \, a^{3} c^{3} d^{9} e^{6} - 35 \, a^{4} c^{2} d^{7} e^{8} + 19 \, a^{5} c d^{5} e^{10} - 4 \, a^{6} d^{3} e^{12}\right )} x\right )}} \] Input:
integrate(x^3/(e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorit hm="fricas")
Output:
2/35*(112*a^3*c*d^5*e^3 + 16*a^4*d^3*e^5 + 2*(c^4*d^7*e - 7*a*c^3*d^5*e^3 + 35*a^2*c^2*d^3*e^5 + 35*a^3*c*d*e^7)*x^4 + (7*c^4*d^8 - 48*a*c^3*d^6*e^2 + 238*a^2*c^2*d^4*e^4 + 280*a^3*c*d^2*e^6 + 35*a^4*e^8)*x^3 - 2*(7*a*c^3* d^7*e - 97*a^2*c^2*d^5*e^3 - 259*a^3*c*d^3*e^5 - 35*a^4*d*e^7)*x^2 + 8*(7* a^2*c^2*d^6*e^2 + 50*a^3*c*d^4*e^4 + 7*a^4*d^2*e^6)*x)*sqrt(c*d*e*x^2 + a* d*e + (c*d^2 + a*e^2)*x)/(a*c^5*d^14*e - 5*a^2*c^4*d^12*e^3 + 10*a^3*c^3*d ^10*e^5 - 10*a^4*c^2*d^8*e^7 + 5*a^5*c*d^6*e^9 - a^6*d^4*e^11 + (c^6*d^11* e^4 - 5*a*c^5*d^9*e^6 + 10*a^2*c^4*d^7*e^8 - 10*a^3*c^3*d^5*e^10 + 5*a^4*c ^2*d^3*e^12 - a^5*c*d*e^14)*x^5 + (4*c^6*d^12*e^3 - 19*a*c^5*d^10*e^5 + 35 *a^2*c^4*d^8*e^7 - 30*a^3*c^3*d^6*e^9 + 10*a^4*c^2*d^4*e^11 + a^5*c*d^2*e^ 13 - a^6*e^15)*x^4 + 2*(3*c^6*d^13*e^2 - 13*a*c^5*d^11*e^4 + 20*a^2*c^4*d^ 9*e^6 - 10*a^3*c^3*d^7*e^8 - 5*a^4*c^2*d^5*e^10 + 7*a^5*c*d^3*e^12 - 2*a^6 *d*e^14)*x^3 + 2*(2*c^6*d^14*e - 7*a*c^5*d^12*e^3 + 5*a^2*c^4*d^10*e^5 + 1 0*a^3*c^3*d^8*e^7 - 20*a^4*c^2*d^6*e^9 + 13*a^5*c*d^4*e^11 - 3*a^6*d^2*e^1 3)*x^2 + (c^6*d^15 - a*c^5*d^13*e^2 - 10*a^2*c^4*d^11*e^4 + 30*a^3*c^3*d^9 *e^6 - 35*a^4*c^2*d^7*e^8 + 19*a^5*c*d^5*e^10 - 4*a^6*d^3*e^12)*x)
\[ \int \frac {x^3}{(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\int \frac {x^{3}}{\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {3}{2}} \left (d + e x\right )^{3}}\, dx \] Input:
integrate(x**3/(e*x+d)**3/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2),x)
Output:
Integral(x**3/(((d + e*x)*(a*e + c*d*x))**(3/2)*(d + e*x)**3), x)
Exception generated. \[ \int \frac {x^3}{(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(x^3/(e*x+d)^3/(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*(a*e^2-c*d^2)>0)', see `assume ?` for mor
\[ \int \frac {x^3}{(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\int { \frac {x^{3}}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{3}} \,d x } \] Input:
integrate(x^3/(e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorit hm="giac")
Output:
integrate(x^3/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)*(e*x + d)^3), x)
Time = 11.49 (sec) , antiderivative size = 11309, normalized size of antiderivative = 31.86 \[ \int \frac {x^3}{(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\text {Too large to display} \] Input:
int(x^3/((d + e*x)^3*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2)),x)
Output:
(2*d^3*(a*d*e + a*e^2*x + c*d^2*x + c*d*e*x^2)^(1/2))/(7*a^2*d^4*e^6 + 7*c ^2*d^8*e^2 + 7*a^2*e^10*x^4 + 28*a^2*d^3*e^7*x + 28*a^2*d*e^9*x^3 + 28*c^2 *d^7*e^3*x + 42*a^2*d^2*e^8*x^2 + 42*c^2*d^6*e^4*x^2 + 28*c^2*d^5*e^5*x^3 + 7*c^2*d^4*e^6*x^4 - 14*a*c*d^6*e^4 - 56*a*c*d^5*e^5*x - 84*a*c*d^4*e^6*x ^2 - 56*a*c*d^3*e^7*x^3 - 14*a*c*d^2*e^8*x^4) - (16*d^2*(a*d*e + a*e^2*x + c*d^2*x + c*d*e*x^2)^(1/2))/(7*(5*a^2*d^3*e^6 + 5*c^2*d^7*e^2 + 5*a^2*e^9 *x^3 + 15*a^2*d^2*e^7*x + 15*a^2*d*e^8*x^2 + 15*c^2*d^6*e^3*x + 15*c^2*d^5 *e^4*x^2 + 5*c^2*d^4*e^5*x^3 - 10*a*c*d^5*e^4 - 30*a*c*d^4*e^5*x - 30*a*c* d^3*e^6*x^2 - 10*a*c*d^2*e^7*x^3)) + (32*a^5*e^10*(a*d*e + a*e^2*x + c*d^2 *x + c*d*e*x^2)^(1/2))/(35*(a^7*d*e^16 + a^7*e^17*x - c^7*d^15*e^2 + 7*a*c ^6*d^13*e^4 - 7*a^6*c*d^3*e^14 - c^7*d^14*e^3*x - 21*a^2*c^5*d^11*e^6 + 35 *a^3*c^4*d^9*e^8 - 35*a^4*c^3*d^7*e^10 + 21*a^5*c^2*d^5*e^12 + 7*a*c^6*d^1 2*e^5*x - 7*a^6*c*d^2*e^15*x - 21*a^2*c^5*d^10*e^7*x + 35*a^3*c^4*d^8*e^9* x - 35*a^4*c^3*d^6*e^11*x + 21*a^5*c^2*d^4*e^13*x)) - (10*c^5*d^10*(a*d*e + a*e^2*x + c*d^2*x + c*d*e*x^2)^(1/2))/(3*(a^7*d*e^16 + a^7*e^17*x - c^7* d^15*e^2 + 7*a*c^6*d^13*e^4 - 7*a^6*c*d^3*e^14 - c^7*d^14*e^3*x - 21*a^2*c ^5*d^11*e^6 + 35*a^3*c^4*d^9*e^8 - 35*a^4*c^3*d^7*e^10 + 21*a^5*c^2*d^5*e^ 12 + 7*a*c^6*d^12*e^5*x - 7*a^6*c*d^2*e^15*x - 21*a^2*c^5*d^10*e^7*x + 35* a^3*c^4*d^8*e^9*x - 35*a^4*c^3*d^6*e^11*x + 21*a^5*c^2*d^4*e^13*x)) + (2*a ^2*e^4*(a*d*e + a*e^2*x + c*d^2*x + c*d*e*x^2)^(1/2))/(7*(a^4*d*e^10 + ...
Time = 141.20 (sec) , antiderivative size = 1359, normalized size of antiderivative = 3.83 \[ \int \frac {x^3}{(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:
int(x^3/(e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x)
Output:
(2*(70*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*a**3*d**4*e**6 + 280*sqrt (e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*a**3*d**3*e**7*x + 420*sqrt(e)*sqrt( d)*sqrt(c)*sqrt(a*e + c*d*x)*a**3*d**2*e**8*x**2 + 280*sqrt(e)*sqrt(d)*sqr t(c)*sqrt(a*e + c*d*x)*a**3*d*e**9*x**3 + 70*sqrt(e)*sqrt(d)*sqrt(c)*sqrt( a*e + c*d*x)*a**3*e**10*x**4 + 70*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x )*a**2*c*d**6*e**4 + 280*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*a**2*c* d**5*e**5*x + 420*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*a**2*c*d**4*e* *6*x**2 + 280*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*a**2*c*d**3*e**7*x **3 + 70*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*a**2*c*d**2*e**8*x**4 - 14*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*a*c**2*d**8*e**2 - 56*sqrt(e )*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*a*c**2*d**7*e**3*x - 84*sqrt(e)*sqrt(d )*sqrt(c)*sqrt(a*e + c*d*x)*a*c**2*d**6*e**4*x**2 - 56*sqrt(e)*sqrt(d)*sqr t(c)*sqrt(a*e + c*d*x)*a*c**2*d**5*e**5*x**3 - 14*sqrt(e)*sqrt(d)*sqrt(c)* sqrt(a*e + c*d*x)*a*c**2*d**4*e**6*x**4 + 2*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a *e + c*d*x)*c**3*d**10 + 8*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*c**3* d**9*e*x + 12*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*c**3*d**8*e**2*x** 2 + 8*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*c**3*d**7*e**3*x**3 + 2*sq rt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*c**3*d**6*e**4*x**4 - 16*sqrt(d + e*x)*a**4*d**3*e**8 - 56*sqrt(d + e*x)*a**4*d**2*e**9*x - 70*sqrt(d + e*x) *a**4*d*e**10*x**2 - 35*sqrt(d + e*x)*a**4*e**11*x**3 - 112*sqrt(d + e*...