Integrand size = 40, antiderivative size = 126 \[ \int \frac {x^2}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {2 x^2}{3 \left (c d^2-a e^2\right ) (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {8 a e \left (2 a d e+\left (c d^2+a e^2\right ) x\right )}{3 \left (c d^2-a e^2\right )^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \]
2/3*x^2/(-a*e^2+c*d^2)/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-8/3 *a*e*(2*a*d*e+(a*e^2+c*d^2)*x)/(-a*e^2+c*d^2)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d *e*x^2)^(1/2)
Time = 0.14 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.75 \[ \int \frac {x^2}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {2 (a e+c d x)^3 \left (d^2-\frac {6 a d e (d+e x)}{a e+c d x}-\frac {3 a^2 e^2 (d+e x)^2}{(a e+c d x)^2}\right )}{3 \left (c d^2-a e^2\right )^3 ((a e+c d x) (d+e x))^{3/2}} \]
(2*(a*e + c*d*x)^3*(d^2 - (6*a*d*e*(d + e*x))/(a*e + c*d*x) - (3*a^2*e^2*( d + e*x)^2)/(a*e + c*d*x)^2))/(3*(c*d^2 - a*e^2)^3*((a*e + c*d*x)*(d + e*x ))^(3/2))
Time = 0.30 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {1243, 1158}
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.
| \(\displaystyle \int \frac {x^2}{(d+e x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1243 |
| \(\displaystyle \frac {2 x^2}{3 (d+e x) \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {4 a e \int \frac {x}{\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}dx}{3 \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 1158 |
| \(\displaystyle \frac {2 x^2}{3 (d+e x) \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {8 a e \left (x \left (a e^2+c d^2\right )+2 a d e\right )}{3 \left (c d^2-a e^2\right )^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\) |
(2*x^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*a*e*(2*a*d*e + (c*d^2 + a*e^2)*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])
3.5.80.3.1 Defintions of rubi rules used
Int[((d_.) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(3/2), x_Symbo l] :> Simp[-2*((b*d - 2*a*e + (2*c*d - b*e)*x)/((b^2 - 4*a*c)*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] :> Simp[(-(2*c*d - b*e))*(f + g*x)^n*((a + b* x + c*x^2)^(p + 1)/(e*p*(b^2 - 4*a*c)*(d + e*x))), x] + Simp[n*((a*g*(2*c*d - b*e) - c*f*(b*d - 2*a*e))/(d*e*p*(b^2 - 4*a*c))) Int[(f + g*x)^(n - 1) *(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[c*d^ 2 - b*d*e + a*e^2, 0] && IGtQ[n, 1] && LtQ[p, -1] && EqQ[n + 2*p + 1, 0]
Time = 0.66 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.15
| method | result | size |
| gosper | \(\frac {2 \left (c d x +a e \right ) \left (3 a^{2} e^{4} x^{2}+6 a c \,d^{2} e^{2} x^{2}-c^{2} d^{4} x^{2}+12 a^{2} d \,e^{3} x +4 a c \,d^{3} e x +8 a^{2} d^{2} e^{2}\right )}{3 \left (a^{3} e^{6}-3 a^{2} c \,d^{2} e^{4}+3 a \,c^{2} d^{4} e^{2}-c^{3} d^{6}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}}}\) | \(145\) |
| trager | \(\frac {2 \left (3 a^{2} e^{4} x^{2}+6 a c \,d^{2} e^{2} x^{2}-c^{2} d^{4} x^{2}+12 a^{2} d \,e^{3} x +4 a c \,d^{3} e x +8 a^{2} d^{2} e^{2}\right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{3 \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \left (e x +d \right )^{2} \left (e^{2} a -c \,d^{2}\right ) \left (c d x +a e \right )}\) | \(153\) |
| default | \(\frac {-\frac {1}{c d e \sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}}-\frac {\left (e^{2} a +c \,d^{2}\right ) \left (2 c d e x +e^{2} a +c \,d^{2}\right )}{c d e \left (4 a c \,d^{2} e^{2}-\left (e^{2} a +c \,d^{2}\right )^{2}\right ) \sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}}}{e}-\frac {2 d \left (2 c d e x +e^{2} a +c \,d^{2}\right )}{e^{2} \left (4 a c \,d^{2} e^{2}-\left (e^{2} a +c \,d^{2}\right )^{2}\right ) \sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}}+\frac {d^{2} \left (-\frac {2}{3 \left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right ) \sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}+\frac {8 c d e \left (2 c d e \left (x +\frac {d}{e}\right )+e^{2} a -c \,d^{2}\right )}{3 \left (e^{2} a -c \,d^{2}\right )^{3} \sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}\right )}{e^{3}}\) | \(365\) |
2/3*(c*d*x+a*e)*(3*a^2*e^4*x^2+6*a*c*d^2*e^2*x^2-c^2*d^4*x^2+12*a^2*d*e^3* x+4*a*c*d^3*e*x+8*a^2*d^2*e^2)/(a^3*e^6-3*a^2*c*d^2*e^4+3*a*c^2*d^4*e^2-c^ 3*d^6)/(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. 308 vs. \(2 (118) = 236\).
Time = 1.68 (sec) , antiderivative size = 308, normalized size of antiderivative = 2.44 \[ \int \frac {x^2}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {2 \, {\left (8 \, a^{2} d^{2} e^{2} - {\left (c^{2} d^{4} - 6 \, a c d^{2} e^{2} - 3 \, a^{2} e^{4}\right )} x^{2} + 4 \, {\left (a c d^{3} e + 3 \, a^{2} d e^{3}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{3 \, {\left (a c^{3} d^{8} e - 3 \, a^{2} c^{2} d^{6} e^{3} + 3 \, a^{3} c d^{4} e^{5} - a^{4} d^{2} e^{7} + {\left (c^{4} d^{7} e^{2} - 3 \, a c^{3} d^{5} e^{4} + 3 \, a^{2} c^{2} d^{3} e^{6} - a^{3} c d e^{8}\right )} x^{3} + {\left (2 \, c^{4} d^{8} e - 5 \, a c^{3} d^{6} e^{3} + 3 \, a^{2} c^{2} d^{4} e^{5} + a^{3} c d^{2} e^{7} - a^{4} e^{9}\right )} x^{2} + {\left (c^{4} d^{9} - a c^{3} d^{7} e^{2} - 3 \, a^{2} c^{2} d^{5} e^{4} + 5 \, a^{3} c d^{3} e^{6} - 2 \, a^{4} d e^{8}\right )} x\right )}} \]
-2/3*(8*a^2*d^2*e^2 - (c^2*d^4 - 6*a*c*d^2*e^2 - 3*a^2*e^4)*x^2 + 4*(a*c*d ^3*e + 3*a^2*d*e^3)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(a*c^3* d^8*e - 3*a^2*c^2*d^6*e^3 + 3*a^3*c*d^4*e^5 - a^4*d^2*e^7 + (c^4*d^7*e^2 - 3*a*c^3*d^5*e^4 + 3*a^2*c^2*d^3*e^6 - a^3*c*d*e^8)*x^3 + (2*c^4*d^8*e - 5 *a*c^3*d^6*e^3 + 3*a^2*c^2*d^4*e^5 + a^3*c*d^2*e^7 - a^4*e^9)*x^2 + (c^4*d ^9 - a*c^3*d^7*e^2 - 3*a^2*c^2*d^5*e^4 + 5*a^3*c*d^3*e^6 - 2*a^4*d*e^8)*x)
\[ \int \frac {x^2}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\int \frac {x^{2}}{\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {3}{2}} \left (d + e x\right )}\, dx \]
Exception generated. \[ \int \frac {x^2}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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
\[ \int \frac {x^2}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\int { \frac {x^{2}}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {3}{2}} {\left (e x + d\right )}} \,d x } \]
Time = 12.58 (sec) , antiderivative size = 1071, normalized size of antiderivative = 8.50 \[ \int \frac {x^2}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {4\,c\,d^3\,\sqrt {c\,d^2\,x+c\,d\,e\,x^2+a\,d\,e+a\,e^2\,x}}{3\,\left (a^3\,d\,e^7+x\,a^3\,e^8-3\,a^2\,c\,d^3\,e^5-3\,x\,a^2\,c\,d^2\,e^6+3\,a\,c^2\,d^5\,e^3+3\,x\,a\,c^2\,d^4\,e^4-c^3\,d^7\,e-x\,c^3\,d^6\,e^2\right )}-\frac {2\,d^2\,\sqrt {c\,d^2\,x+c\,d\,e\,x^2+a\,d\,e+a\,e^2\,x}}{3\,a^2\,d^2\,e^5+6\,a^2\,d\,e^6\,x+3\,a^2\,e^7\,x^2-6\,a\,c\,d^4\,e^3-12\,a\,c\,d^3\,e^4\,x-6\,a\,c\,d^2\,e^5\,x^2+3\,c^2\,d^6\,e+6\,c^2\,d^5\,e^2\,x+3\,c^2\,d^4\,e^3\,x^2}-\frac {4\,a\,d\,e^2\,\sqrt {c\,d^2\,x+c\,d\,e\,x^2+a\,d\,e+a\,e^2\,x}}{3\,\left (a^3\,d\,e^7+x\,a^3\,e^8-3\,a^2\,c\,d^3\,e^5-3\,x\,a^2\,c\,d^2\,e^6+3\,a\,c^2\,d^5\,e^3+3\,x\,a\,c^2\,d^4\,e^4-c^3\,d^7\,e-x\,c^3\,d^6\,e^2\right )}+\frac {2\,c^4\,d^7\,x}{\sqrt {c\,d^2\,x+c\,d\,e\,x^2+a\,d\,e+a\,e^2\,x}\,\left (a^4\,c\,d\,e^9-4\,a^3\,c^2\,d^3\,e^7+6\,a^2\,c^3\,d^5\,e^5-4\,a\,c^4\,d^7\,e^3+c^5\,d^9\,e\right )}+\frac {22\,a^3\,c\,d^2\,e^5}{3\,\sqrt {c\,d^2\,x+c\,d\,e\,x^2+a\,d\,e+a\,e^2\,x}\,\left (a^4\,c\,d\,e^9-4\,a^3\,c^2\,d^3\,e^7+6\,a^2\,c^3\,d^5\,e^5-4\,a\,c^4\,d^7\,e^3+c^5\,d^9\,e\right )}-\frac {28\,a^2\,c^2\,d^4\,e^3}{3\,\sqrt {c\,d^2\,x+c\,d\,e\,x^2+a\,d\,e+a\,e^2\,x}\,\left (a^4\,c\,d\,e^9-4\,a^3\,c^2\,d^3\,e^7+6\,a^2\,c^3\,d^5\,e^5-4\,a\,c^4\,d^7\,e^3+c^5\,d^9\,e\right )}+\frac {2\,a\,c^3\,d^6\,e}{\sqrt {c\,d^2\,x+c\,d\,e\,x^2+a\,d\,e+a\,e^2\,x}\,\left (a^4\,c\,d\,e^9-4\,a^3\,c^2\,d^3\,e^7+6\,a^2\,c^3\,d^5\,e^5-4\,a\,c^4\,d^7\,e^3+c^5\,d^9\,e\right )}+\frac {10\,a^2\,c^2\,d^3\,e^4\,x}{3\,\sqrt {c\,d^2\,x+c\,d\,e\,x^2+a\,d\,e+a\,e^2\,x}\,\left (a^4\,c\,d\,e^9-4\,a^3\,c^2\,d^3\,e^7+6\,a^2\,c^3\,d^5\,e^5-4\,a\,c^4\,d^7\,e^3+c^5\,d^9\,e\right )}+\frac {2\,a^3\,c\,d\,e^6\,x}{\sqrt {c\,d^2\,x+c\,d\,e\,x^2+a\,d\,e+a\,e^2\,x}\,\left (a^4\,c\,d\,e^9-4\,a^3\,c^2\,d^3\,e^7+6\,a^2\,c^3\,d^5\,e^5-4\,a\,c^4\,d^7\,e^3+c^5\,d^9\,e\right )}-\frac {22\,a\,c^3\,d^5\,e^2\,x}{3\,\sqrt {c\,d^2\,x+c\,d\,e\,x^2+a\,d\,e+a\,e^2\,x}\,\left (a^4\,c\,d\,e^9-4\,a^3\,c^2\,d^3\,e^7+6\,a^2\,c^3\,d^5\,e^5-4\,a\,c^4\,d^7\,e^3+c^5\,d^9\,e\right )} \]
(4*c*d^3*(a*d*e + a*e^2*x + c*d^2*x + c*d*e*x^2)^(1/2))/(3*(a^3*d*e^7 - c^ 3*d^7*e + a^3*e^8*x + 3*a*c^2*d^5*e^3 - 3*a^2*c*d^3*e^5 - c^3*d^6*e^2*x + 3*a*c^2*d^4*e^4*x - 3*a^2*c*d^2*e^6*x)) - (2*d^2*(a*d*e + a*e^2*x + c*d^2* x + c*d*e*x^2)^(1/2))/(3*c^2*d^6*e + 3*a^2*d^2*e^5 + 3*a^2*e^7*x^2 + 6*c^2 *d^5*e^2*x + 3*c^2*d^4*e^3*x^2 - 6*a*c*d^4*e^3 + 6*a^2*d*e^6*x - 12*a*c*d^ 3*e^4*x - 6*a*c*d^2*e^5*x^2) - (4*a*d*e^2*(a*d*e + a*e^2*x + c*d^2*x + c*d *e*x^2)^(1/2))/(3*(a^3*d*e^7 - c^3*d^7*e + a^3*e^8*x + 3*a*c^2*d^5*e^3 - 3 *a^2*c*d^3*e^5 - c^3*d^6*e^2*x + 3*a*c^2*d^4*e^4*x - 3*a^2*c*d^2*e^6*x)) + (2*c^4*d^7*x)/((a*d*e + a*e^2*x + c*d^2*x + c*d*e*x^2)^(1/2)*(c^5*d^9*e - 4*a*c^4*d^7*e^3 + 6*a^2*c^3*d^5*e^5 - 4*a^3*c^2*d^3*e^7 + a^4*c*d*e^9)) + (22*a^3*c*d^2*e^5)/(3*(a*d*e + a*e^2*x + c*d^2*x + c*d*e*x^2)^(1/2)*(c^5* d^9*e - 4*a*c^4*d^7*e^3 + 6*a^2*c^3*d^5*e^5 - 4*a^3*c^2*d^3*e^7 + a^4*c*d* e^9)) - (28*a^2*c^2*d^4*e^3)/(3*(a*d*e + a*e^2*x + c*d^2*x + c*d*e*x^2)^(1 /2)*(c^5*d^9*e - 4*a*c^4*d^7*e^3 + 6*a^2*c^3*d^5*e^5 - 4*a^3*c^2*d^3*e^7 + a^4*c*d*e^9)) + (2*a*c^3*d^6*e)/((a*d*e + a*e^2*x + c*d^2*x + c*d*e*x^2)^ (1/2)*(c^5*d^9*e - 4*a*c^4*d^7*e^3 + 6*a^2*c^3*d^5*e^5 - 4*a^3*c^2*d^3*e^7 + a^4*c*d*e^9)) + (10*a^2*c^2*d^3*e^4*x)/(3*(a*d*e + a*e^2*x + c*d^2*x + c*d*e*x^2)^(1/2)*(c^5*d^9*e - 4*a*c^4*d^7*e^3 + 6*a^2*c^3*d^5*e^5 - 4*a^3* c^2*d^3*e^7 + a^4*c*d*e^9)) + (2*a^3*c*d*e^6*x)/((a*d*e + a*e^2*x + c*d^2* x + c*d*e*x^2)^(1/2)*(c^5*d^9*e - 4*a*c^4*d^7*e^3 + 6*a^2*c^3*d^5*e^5 -...