Integrand size = 40, antiderivative size = 246 \[ \int \frac {(d+e x)^2}{x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {2 c^2 d^2 (d+e x)}{a^3 e^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 a^2 e^2 x^2}+\frac {\left (7 c d^2-a e^2\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 a^3 d e^3 x}-\frac {\left (15 c^2 d^4-6 a c d^2 e^2-a^2 e^4\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {e} (d+e x)}{\sqrt {d} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{4 a^{7/2} d^{3/2} e^{7/2}} \] Output:
2*c^2*d^2*(e*x+d)/a^3/e^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-1/2*(a*d *e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/a^2/e^2/x^2+1/4*(-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^3/d/e^3/x-1/4*(-a^2*e^4-6*a*c*d^2*e^2 +15*c^2*d^4)*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^(7/2)/d^(3/2)/e^(7/2)
Time = 0.52 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.78 \[ \int \frac {(d+e x)^2}{x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {-\sqrt {a} \sqrt {d} \sqrt {e} (d+e x) \left (-15 c^2 d^3 x^2+a c d e x (-5 d+e x)+a^2 e^2 (2 d+e x)\right )-\left (15 c^2 d^4-6 a c d^2 e^2-a^2 e^4\right ) x^2 \sqrt {a e+c d x} \sqrt {d+e x} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {e} \sqrt {d+e x}}{\sqrt {d} \sqrt {a e+c d x}}\right )}{4 a^{7/2} d^{3/2} e^{7/2} x^2 \sqrt {(a e+c d x) (d+e x)}} \] Input:
Integrate[(d + e*x)^2/(x^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)), x]
Output:
(-(Sqrt[a]*Sqrt[d]*Sqrt[e]*(d + e*x)*(-15*c^2*d^3*x^2 + a*c*d*e*x*(-5*d + e*x) + a^2*e^2*(2*d + e*x))) - (15*c^2*d^4 - 6*a*c*d^2*e^2 - a^2*e^4)*x^2* Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*ArcTanh[(Sqrt[a]*Sqrt[e]*Sqrt[d + e*x])/(S qrt[d]*Sqrt[a*e + c*d*x])])/(4*a^(7/2)*d^(3/2)*e^(7/2)*x^2*Sqrt[(a*e + c*d *x)*(d + e*x)])
Time = 1.38 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.17, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.175, Rules used = {1212, 25, 2181, 27, 1228, 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.
\(\displaystyle \int \frac {(d+e x)^2}{x^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 1212 |
\(\displaystyle \frac {2 c^2 d^2 (d+e x)}{a^3 e^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-c^2 d^2 e^4 \int -\frac {\frac {\left (c d^2-a e^2\right ) x^2}{a^3 c d e^7}-\frac {\left (c d^2-a e^2\right ) x}{a^2 c^2 d^2 e^6}+\frac {1}{a c^2 d e^5}}{x^3 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle c^2 d^2 e^4 \int \frac {\frac {\left (c d^2-a e^2\right ) x^2}{a^3 c d e^7}-\frac {\left (c d^2-a e^2\right ) x}{a^2 c^2 d^2 e^6}+\frac {1}{a c^2 d e^5}}{x^3 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx+\frac {2 c^2 d^2 (d+e x)}{a^3 e^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\) |
\(\Big \downarrow \) 2181 |
\(\displaystyle c^2 d^2 e^4 \left (-\frac {\int \frac {a e \left (7 c d^2-a e^2\right )-2 c d \left (2 c d^2-3 a e^2\right ) x}{2 a^2 c^2 d e^6 x^2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 a d e}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 a^2 c^2 d^2 e^6 x^2}\right )+\frac {2 c^2 d^2 (d+e x)}{a^3 e^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle c^2 d^2 e^4 \left (-\frac {\int \frac {a e \left (7 c d^2-a e^2\right )-2 c d \left (2 c d^2-3 a e^2\right ) x}{x^2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{4 a^3 c^2 d^2 e^7}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 a^2 c^2 d^2 e^6 x^2}\right )+\frac {2 c^2 d^2 (d+e x)}{a^3 e^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\) |
\(\Big \downarrow \) 1228 |
\(\displaystyle c^2 d^2 e^4 \left (-\frac {-\frac {\left (-a^2 e^4-6 a c d^2 e^2+15 c^2 d^4\right ) \int \frac {1}{x \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 d}-\frac {\left (7 c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d x}}{4 a^3 c^2 d^2 e^7}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 a^2 c^2 d^2 e^6 x^2}\right )+\frac {2 c^2 d^2 (d+e x)}{a^3 e^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\) |
\(\Big \downarrow \) 1154 |
\(\displaystyle c^2 d^2 e^4 \left (-\frac {\frac {\left (-a^2 e^4-6 a c d^2 e^2+15 c^2 d^4\right ) \int \frac {1}{4 a d e-\frac {\left (2 a d e+\left (c d^2+a e^2\right ) x\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\frac {2 a d e+\left (c d^2+a e^2\right ) x}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}}{d}-\frac {\left (7 c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d x}}{4 a^3 c^2 d^2 e^7}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 a^2 c^2 d^2 e^6 x^2}\right )+\frac {2 c^2 d^2 (d+e x)}{a^3 e^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {2 c^2 d^2 (d+e x)}{a^3 e^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+c^2 d^2 e^4 \left (-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 a^2 c^2 d^2 e^6 x^2}-\frac {\frac {\left (-a^2 e^4-6 a c d^2 e^2+15 c^2 d^4\right ) \text {arctanh}\left (\frac {x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 \sqrt {a} d^{3/2} \sqrt {e}}-\frac {\left (7 c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d x}}{4 a^3 c^2 d^2 e^7}\right )\) |
Input:
Int[(d + e*x)^2/(x^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)),x]
Output:
(2*c^2*d^2*(d + e*x))/(a^3*e^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2] ) + c^2*d^2*e^4*(-1/2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(a^2*c^2 *d^2*e^6*x^2) - (-(((7*c*d^2 - a*e^2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d *e*x^2])/(d*x)) + ((15*c^2*d^4 - 6*a*c*d^2*e^2 - a^2*e^4)*ArcTanh[(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])])/(2*Sqrt[a]*d^(3/2)*Sqrt[e]))/(4*a^3*c^2*d^2*e^7))
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/(((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[((x_)^(n_.)*((d_.) + (e_.)*(x_))^(m_.))/((a_) + (b_.)*(x_) + (c_.)*(x_) ^2)^(3/2), x_Symbol] :> Simp[-2*(2*c*d - b*e)^(m - 2)*(c*d - b*e)^n*((d + e *x)/(c^(m + n - 1)*e^(n - 1)*Sqrt[a + b*x + c*x^2])), x] - Simp[e^2/c^(m + n - 1) Int[ExpandToSum[(c^(m + n - 1)*(d + e*x)^(m - 1) - ((c*d - b*e)^n* (2*c*d - b*e)^(m - 1))/(e^n*x^n))/(c*d - b*e - c*e*x), x]/(Sqrt[a + b*x + c *x^2]/x^n), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c*d^2 - b*d*e + a*e^ 2, 0] && IGtQ[m, 0] && ILtQ[n, 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2))), x] - Simp[(b*(e *f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)) 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[Simplify[m + 2*p + 3], 0]
Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_ ), x_Symbol] :> With[{Qx = PolynomialQuotient[Pq, d + e*x, x], R = Polynomi alRemainder[Pq, d + e*x, x]}, Simp[(e*R*(d + e*x)^(m + 1)*(a + b*x + c*x^2) ^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Simp[1/((m + 1)*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*ExpandToSum[(m + 1)*(c*d^2 - b*d*e + a*e^2)*Qx + c*d*R*(m + 1) - b*e*R*(m + p + 2) - c*e*R *(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, e, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(1199\) vs. \(2(220)=440\).
Time = 2.25 (sec) , antiderivative size = 1200, normalized size of antiderivative = 4.88
Input:
int((e*x+d)^2/x^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/2),x,method=_RETURN VERBOSE)
Output:
d^2*(-1/2/a/d/e/x^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)-5/4*(a*e^2+c*d ^2)/a/d/e*(-1/a/d/e/x/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)-3/2*(a*e^2+c *d^2)/a/d/e*(1/a/d/e/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)-(a*e^2+c*d^2) /a/d/e*(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/a/d/e/(a*d*e)^(1/2)*ln((2*a*d*e+(a*e^2+c*d^ 2)*x+2*(a*d*e)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2))/x))-4*c/a*(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))-3/2*c/a*(1/a/d/e/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1 /2)-(a*e^2+c*d^2)/a/d/e*(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/a/d/e/(a*d*e)^(1/2)*ln((2* a*d*e+(a*e^2+c*d^2)*x+2*(a*d*e)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1 /2))/x)))+e^2*(1/a/d/e/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)-(a*e^2+c*d^ 2)/a/d/e*(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/a/d/e/(a*d*e)^(1/2)*ln((2*a*d*e+(a*e^2+c* d^2)*x+2*(a*d*e)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2))/x))+2*d*e* (-1/a/d/e/x/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)-3/2*(a*e^2+c*d^2)/a/d/ e*(1/a/d/e/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)-(a*e^2+c*d^2)/a/d/e*(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/a/d/e/(a*d*e)^(1/2)*ln((2*a*d*e+(a*e^2+c*d^2)*x+2*(a* d*e)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2))/x))-4*c/a*(2*c*d*e*...
Time = 1.68 (sec) , antiderivative size = 630, normalized size of antiderivative = 2.56 \[ \int \frac {(d+e x)^2}{x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\left [-\frac {{\left ({\left (15 \, c^{3} d^{5} - 6 \, a c^{2} d^{3} e^{2} - a^{2} c d e^{4}\right )} x^{3} + {\left (15 \, a c^{2} d^{4} e - 6 \, a^{2} c d^{2} e^{3} - a^{3} e^{5}\right )} x^{2}\right )} \sqrt {a d e} \log \left (\frac {8 \, a^{2} d^{2} e^{2} + {\left (c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} x^{2} + 4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, a d e + {\left (c d^{2} + a e^{2}\right )} x\right )} \sqrt {a d e} + 8 \, {\left (a c d^{3} e + a^{2} d e^{3}\right )} x}{x^{2}}\right ) + 4 \, {\left (2 \, a^{3} d^{2} e^{3} - {\left (15 \, a c^{2} d^{4} e - a^{2} c d^{2} e^{3}\right )} x^{2} - {\left (5 \, a^{2} c d^{3} e^{2} - a^{3} d e^{4}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{16 \, {\left (a^{4} c d^{3} e^{4} x^{3} + a^{5} d^{2} e^{5} x^{2}\right )}}, \frac {{\left ({\left (15 \, c^{3} d^{5} - 6 \, a c^{2} d^{3} e^{2} - a^{2} c d e^{4}\right )} x^{3} + {\left (15 \, a c^{2} d^{4} e - 6 \, a^{2} c d^{2} e^{3} - a^{3} e^{5}\right )} x^{2}\right )} \sqrt {-a d e} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, a d e + {\left (c d^{2} + a e^{2}\right )} x\right )} \sqrt {-a d e}}{2 \, {\left (a c d^{2} e^{2} x^{2} + a^{2} d^{2} e^{2} + {\left (a c d^{3} e + a^{2} d e^{3}\right )} x\right )}}\right ) - 2 \, {\left (2 \, a^{3} d^{2} e^{3} - {\left (15 \, a c^{2} d^{4} e - a^{2} c d^{2} e^{3}\right )} x^{2} - {\left (5 \, a^{2} c d^{3} e^{2} - a^{3} d e^{4}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{8 \, {\left (a^{4} c d^{3} e^{4} x^{3} + a^{5} d^{2} e^{5} x^{2}\right )}}\right ] \] Input:
integrate((e*x+d)^2/x^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorit hm="fricas")
Output:
[-1/16*(((15*c^3*d^5 - 6*a*c^2*d^3*e^2 - a^2*c*d*e^4)*x^3 + (15*a*c^2*d^4* e - 6*a^2*c*d^2*e^3 - a^3*e^5)*x^2)*sqrt(a*d*e)*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*(2*a^3*d^2*e^3 - (15*a*c^2*d^4*e - a^2*c*d^2*e^3)*x^2 - ( 5*a^2*c*d^3*e^2 - a^3*d*e^4)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x ))/(a^4*c*d^3*e^4*x^3 + a^5*d^2*e^5*x^2), 1/8*(((15*c^3*d^5 - 6*a*c^2*d^3* e^2 - a^2*c*d*e^4)*x^3 + (15*a*c^2*d^4*e - 6*a^2*c*d^2*e^3 - a^3*e^5)*x^2) *sqrt(-a*d*e)*arctan(1/2*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)/(a*c*d^2*e^2*x^2 + a^2*d^2*e^2 + (a* c*d^3*e + a^2*d*e^3)*x)) - 2*(2*a^3*d^2*e^3 - (15*a*c^2*d^4*e - a^2*c*d^2* e^3)*x^2 - (5*a^2*c*d^3*e^2 - a^3*d*e^4)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^ 2 + a*e^2)*x))/(a^4*c*d^3*e^4*x^3 + a^5*d^2*e^5*x^2)]
\[ \int \frac {(d+e x)^2}{x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\int \frac {\left (d + e x\right )^{2}}{x^{3} \left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((e*x+d)**2/x**3/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2),x)
Output:
Integral((d + e*x)**2/(x**3*((d + e*x)*(a*e + c*d*x))**(3/2)), x)
Exception generated. \[ \int \frac {(d+e x)^2}{x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((e*x+d)^2/x^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>0)', see `assume?` for more de tails)Is e
Exception generated. \[ \int \frac {(d+e x)^2}{x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((e*x+d)^2/x^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorit hm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro unding error%%%{%%%{1,[1,1,11]%%%},[2,7]%%%}+%%%{%%%{-4,[2,3,9]%%%},[2,6]% %%}+%%%{%
Timed out. \[ \int \frac {(d+e x)^2}{x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\int \frac {{\left (d+e\,x\right )}^2}{x^3\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/2}} \,d x \] Input:
int((d + e*x)^2/(x^3*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2)),x)
Output:
int((d + e*x)^2/(x^3*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2)), x)
Time = 0.55 (sec) , antiderivative size = 1140, normalized size of antiderivative = 4.63 \[ \int \frac {(d+e x)^2}{x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:
int((e*x+d)^2/x^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x)
Output:
( - 3*sqrt(e)*sqrt(d)*sqrt(a)*sqrt(a*e + c*d*x)*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**3*e**6*x**2 - 23*sqrt(e)*sqrt(d)*sqrt(a)*sqrt(a*e + c*d*x)*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**2*c*d**2*e**4*x**2 + 15*sqrt(e)*sqr t(d)*sqrt(a)*sqrt(a*e + c*d*x)*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*c**2 *d**4*e**2*x**2 + 75*sqrt(e)*sqrt(d)*sqrt(a)*sqrt(a*e + c*d*x)*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**3*d**6*x**2 - 3*sqrt(e)*sqrt(d)*sqrt(a)*sqrt( a*e + c*d*x)*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**3*e**6*x**2 - 23*sqrt (e)*sqrt(d)*sqrt(a)*sqrt(a*e + c*d*x)*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*d**2*e**4*x**2 + 15*sqrt(e)*sqrt(d)*sqrt(a)*sqrt(a*e + c*d*x)*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*c**2*d**4*e**2*x**2 + 75*sqrt(e)*sqrt(d )*sqrt(a)*sqrt(a*e + c*d*x)*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**3*d**6 *x**2 + 3*sqrt(e)*sqrt(d)*sqrt(a)*sqrt(a*e + c*d*x)*log(2*sqrt(e)*sqrt(...