Integrand size = 39, antiderivative size = 394 \[ \int \frac {1}{(d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {1}{3 \left (c d^2-a e^2\right ) (d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {3 c d}{4 \left (c d^2-a e^2\right )^2 \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {21 c^2 d^2 \sqrt {d+e x}}{8 \left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {35 c^3 d^3 (d+e x)^{3/2}}{8 \left (c d^2-a e^2\right )^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {105 c^3 d^3 e \sqrt {d+e x}}{8 \left (c d^2-a e^2\right )^5 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {105 c^3 d^3 e^{3/2} \arctan \left (\frac {\sqrt {c d^2-a e^2} \sqrt {d+e x}}{\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{8 \left (c d^2-a e^2\right )^{11/2}} \] Output:
1/3/(-a*e^2+c*d^2)/(e*x+d)^(3/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+3 /4*c*d/(-a*e^2+c*d^2)^2/(e*x+d)^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3 /2)+21/8*c^2*d^2*(e*x+d)^(1/2)/(-a*e^2+c*d^2)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d *e*x^2)^(3/2)-35/8*c^3*d^3*(e*x+d)^(3/2)/(-a*e^2+c*d^2)^4/(a*d*e+(a*e^2+c* d^2)*x+c*d*e*x^2)^(3/2)+105/8*c^3*d^3*e*(e*x+d)^(1/2)/(-a*e^2+c*d^2)^5/(a* d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-105/8*c^3*d^3*e^(3/2)*arctan(1/e^(1/2 )/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*(-a*e^2+c*d^2)^(1/2)*(e*x+d)^(1/ 2))/(-a*e^2+c*d^2)^(11/2)
Time = 1.33 (sec) , antiderivative size = 294, normalized size of antiderivative = 0.75 \[ \int \frac {1}{(d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {c^3 d^3 (d+e x)^{5/2} \left (\frac {(a e+c d x) \left (8 a^4 e^8-2 a^3 c d e^6 (25 d+9 e x)+3 a^2 c^2 d^2 e^4 \left (55 d^2+60 d e x+21 e^2 x^2\right )+2 a c^3 d^3 e^2 \left (104 d^3+477 d^2 e x+567 d e^2 x^2+210 e^3 x^3\right )+c^4 d^4 \left (-16 d^4+144 d^3 e x+693 d^2 e^2 x^2+840 d e^3 x^3+315 e^4 x^4\right )\right )}{c^3 d^3 \left (c d^2-a e^2\right )^5 (d+e x)^3}+\frac {315 e^{3/2} (a e+c d x)^{5/2} \arctan \left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{11/2}}\right )}{24 ((a e+c d x) (d+e x))^{5/2}} \] Input:
Integrate[1/((d + e*x)^(3/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2) ),x]
Output:
(c^3*d^3*(d + e*x)^(5/2)*(((a*e + c*d*x)*(8*a^4*e^8 - 2*a^3*c*d*e^6*(25*d + 9*e*x) + 3*a^2*c^2*d^2*e^4*(55*d^2 + 60*d*e*x + 21*e^2*x^2) + 2*a*c^3*d^ 3*e^2*(104*d^3 + 477*d^2*e*x + 567*d*e^2*x^2 + 210*e^3*x^3) + c^4*d^4*(-16 *d^4 + 144*d^3*e*x + 693*d^2*e^2*x^2 + 840*d*e^3*x^3 + 315*e^4*x^4)))/(c^3 *d^3*(c*d^2 - a*e^2)^5*(d + e*x)^3) + (315*e^(3/2)*(a*e + c*d*x)^(5/2)*Arc Tan[(Sqrt[e]*Sqrt[a*e + c*d*x])/Sqrt[c*d^2 - a*e^2]])/(c*d^2 - a*e^2)^(11/ 2)))/(24*((a*e + c*d*x)*(d + e*x))^(5/2))
Time = 1.07 (sec) , antiderivative size = 443, normalized size of antiderivative = 1.12, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {1135, 1135, 1132, 1135, 1132, 1136, 218}
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.
| \(\displaystyle \int \frac {1}{(d+e x)^{3/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1135 |
| \(\displaystyle \frac {3 c d \int \frac {1}{\sqrt {d+e x} \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{5/2}}dx}{2 \left (c d^2-a e^2\right )}+\frac {1}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1135 |
| \(\displaystyle \frac {3 c d \left (\frac {7 c d \int \frac {\sqrt {d+e x}}{\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{5/2}}dx}{4 \left (c d^2-a e^2\right )}+\frac {1}{2 \sqrt {d+e x} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\right )}{2 \left (c d^2-a e^2\right )}+\frac {1}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1132 |
| \(\displaystyle \frac {3 c d \left (\frac {7 c d \left (-\frac {5 e \int \frac {1}{\sqrt {d+e x} \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}dx}{3 \left (c d^2-a e^2\right )}-\frac {2 \sqrt {d+e x}}{3 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\right )}{4 \left (c d^2-a e^2\right )}+\frac {1}{2 \sqrt {d+e x} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\right )}{2 \left (c d^2-a e^2\right )}+\frac {1}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1135 |
| \(\displaystyle \frac {3 c d \left (\frac {7 c d \left (-\frac {5 e \left (\frac {3 c d \int \frac {\sqrt {d+e x}}{\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}dx}{2 \left (c d^2-a e^2\right )}+\frac {1}{\sqrt {d+e x} \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{3 \left (c d^2-a e^2\right )}-\frac {2 \sqrt {d+e x}}{3 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\right )}{4 \left (c d^2-a e^2\right )}+\frac {1}{2 \sqrt {d+e x} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\right )}{2 \left (c d^2-a e^2\right )}+\frac {1}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1132 |
| \(\displaystyle \frac {3 c d \left (\frac {7 c d \left (-\frac {5 e \left (\frac {3 c d \left (-\frac {e \int \frac {1}{\sqrt {d+e x} \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{c d^2-a e^2}-\frac {2 \sqrt {d+e x}}{\left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 \left (c d^2-a e^2\right )}+\frac {1}{\sqrt {d+e x} \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{3 \left (c d^2-a e^2\right )}-\frac {2 \sqrt {d+e x}}{3 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\right )}{4 \left (c d^2-a e^2\right )}+\frac {1}{2 \sqrt {d+e x} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\right )}{2 \left (c d^2-a e^2\right )}+\frac {1}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1136 |
| \(\displaystyle \frac {3 c d \left (\frac {7 c d \left (-\frac {5 e \left (\frac {3 c d \left (-\frac {2 e^2 \int \frac {1}{\frac {\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right ) e^2}{d+e x}+\left (c d^2-a e^2\right ) e}d\frac {\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{\sqrt {d+e x}}}{c d^2-a e^2}-\frac {2 \sqrt {d+e x}}{\left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 \left (c d^2-a e^2\right )}+\frac {1}{\sqrt {d+e x} \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{3 \left (c d^2-a e^2\right )}-\frac {2 \sqrt {d+e x}}{3 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\right )}{4 \left (c d^2-a e^2\right )}+\frac {1}{2 \sqrt {d+e x} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\right )}{2 \left (c d^2-a e^2\right )}+\frac {1}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {3 c d \left (\frac {7 c d \left (-\frac {5 e \left (\frac {3 c d \left (-\frac {2 \sqrt {e} \arctan \left (\frac {\sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{3/2}}-\frac {2 \sqrt {d+e x}}{\left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 \left (c d^2-a e^2\right )}+\frac {1}{\sqrt {d+e x} \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{3 \left (c d^2-a e^2\right )}-\frac {2 \sqrt {d+e x}}{3 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\right )}{4 \left (c d^2-a e^2\right )}+\frac {1}{2 \sqrt {d+e x} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\right )}{2 \left (c d^2-a e^2\right )}+\frac {1}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\) |
Input:
Int[1/((d + e*x)^(3/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)),x]
Output:
1/(3*(c*d^2 - a*e^2)*(d + e*x)^(3/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^ 2)^(3/2)) + (3*c*d*(1/(2*(c*d^2 - a*e^2)*Sqrt[d + e*x]*(a*d*e + (c*d^2 + a *e^2)*x + c*d*e*x^2)^(3/2)) + (7*c*d*((-2*Sqrt[d + e*x])/(3*(c*d^2 - a*e^2 )*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) - (5*e*(1/((c*d^2 - a*e^2 )*Sqrt[d + e*x]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (3*c*d*((-2 *Sqrt[d + e*x])/((c*d^2 - a*e^2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^ 2]) - (2*Sqrt[e]*ArcTan[(Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^ 2])/(Sqrt[c*d^2 - a*e^2]*Sqrt[d + e*x])])/(c*d^2 - a*e^2)^(3/2)))/(2*(c*d^ 2 - a*e^2))))/(3*(c*d^2 - a*e^2))))/(4*(c*d^2 - a*e^2))))/(2*(c*d^2 - a*e^ 2))
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(2*c*d - b*e)*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/(e*(p + 1)*(b^2 - 4*a*c))), x] - Simp[(2*c*d - b*e)*((m + 2*p + 2)/((p + 1)*(b^2 - 4*a*c))) Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1), x], x] /; Free Q[{a, b, c, d, e}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && LtQ [0, m, 1] && IntegerQ[2*p]
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*((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, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, 0] && NeQ[m + p + 1, 0] && I ntegerQ[2*p]
Int[1/(Sqrt[(d_.) + (e_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x _Symbol] :> Simp[2*e Subst[Int[1/(2*c*d - b*e + e^2*x^2), x], x, Sqrt[a + b*x + c*x^2]/Sqrt[d + e*x]], x] /; FreeQ[{a, b, c, d, e}, 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. \(919\) vs. \(2(350)=700\).
Time = 1.08 (sec) , antiderivative size = 920, normalized size of antiderivative = 2.34
| method | result | size |
| default | \(\frac {\sqrt {\left (e x +d \right ) \left (c d x +a e \right )}\, \left (315 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right ) c^{4} d^{4} e^{5} x^{4} \sqrt {c d x +a e}+315 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right ) a \,c^{3} d^{3} e^{6} x^{3} \sqrt {c d x +a e}+945 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right ) c^{4} d^{5} e^{4} x^{3} \sqrt {c d x +a e}+945 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right ) a \,c^{3} d^{4} e^{5} x^{2} \sqrt {c d x +a e}+945 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right ) c^{4} d^{6} e^{3} x^{2} \sqrt {c d x +a e}-315 \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, c^{4} d^{4} e^{4} x^{4}+945 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right ) a \,c^{3} d^{5} e^{4} x \sqrt {c d x +a e}+315 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right ) c^{4} d^{7} e^{2} x \sqrt {c d x +a e}-420 \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, a \,c^{3} d^{3} e^{5} x^{3}-840 \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, c^{4} d^{5} e^{3} x^{3}+315 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right ) a \,c^{3} d^{6} e^{3} \sqrt {c d x +a e}-63 \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, a^{2} c^{2} d^{2} e^{6} x^{2}-1134 \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, a \,c^{3} d^{4} e^{4} x^{2}-693 \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, c^{4} d^{6} e^{2} x^{2}+18 \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, a^{3} c d \,e^{7} x -180 \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, a^{2} c^{2} d^{3} e^{5} x -954 \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, a \,c^{3} d^{5} e^{3} x -144 \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, c^{4} d^{7} e x -8 \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, a^{4} e^{8}+50 \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, a^{3} c \,d^{2} e^{6}-165 \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, a^{2} c^{2} d^{4} e^{4}-208 \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, a \,c^{3} d^{6} e^{2}+16 \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, c^{4} d^{8}\right )}{24 \left (e x +d \right )^{\frac {7}{2}} \left (c d x +a e \right )^{2} \left (a \,e^{2}-c \,d^{2}\right )^{5} \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\) | \(920\) |
Input:
int(1/(e*x+d)^(3/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(5/2),x,method=_RETU RNVERBOSE)
Output:
1/24*((e*x+d)*(c*d*x+a*e))^(1/2)*(315*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2- c*d^2)*e)^(1/2))*c^4*d^4*e^5*x^4*(c*d*x+a*e)^(1/2)+315*arctanh(e*(c*d*x+a* e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*a*c^3*d^3*e^6*x^3*(c*d*x+a*e)^(1/2)+945* arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*c^4*d^5*e^4*x^3*(c*d* x+a*e)^(1/2)+945*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*a*c^ 3*d^4*e^5*x^2*(c*d*x+a*e)^(1/2)+945*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c* d^2)*e)^(1/2))*c^4*d^6*e^3*x^2*(c*d*x+a*e)^(1/2)-315*((a*e^2-c*d^2)*e)^(1/ 2)*c^4*d^4*e^4*x^4+945*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2) )*a*c^3*d^5*e^4*x*(c*d*x+a*e)^(1/2)+315*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^ 2-c*d^2)*e)^(1/2))*c^4*d^7*e^2*x*(c*d*x+a*e)^(1/2)-420*((a*e^2-c*d^2)*e)^( 1/2)*a*c^3*d^3*e^5*x^3-840*((a*e^2-c*d^2)*e)^(1/2)*c^4*d^5*e^3*x^3+315*arc tanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*a*c^3*d^6*e^3*(c*d*x+a*e )^(1/2)-63*((a*e^2-c*d^2)*e)^(1/2)*a^2*c^2*d^2*e^6*x^2-1134*((a*e^2-c*d^2) *e)^(1/2)*a*c^3*d^4*e^4*x^2-693*((a*e^2-c*d^2)*e)^(1/2)*c^4*d^6*e^2*x^2+18 *((a*e^2-c*d^2)*e)^(1/2)*a^3*c*d*e^7*x-180*((a*e^2-c*d^2)*e)^(1/2)*a^2*c^2 *d^3*e^5*x-954*((a*e^2-c*d^2)*e)^(1/2)*a*c^3*d^5*e^3*x-144*((a*e^2-c*d^2)* e)^(1/2)*c^4*d^7*e*x-8*((a*e^2-c*d^2)*e)^(1/2)*a^4*e^8+50*((a*e^2-c*d^2)*e )^(1/2)*a^3*c*d^2*e^6-165*((a*e^2-c*d^2)*e)^(1/2)*a^2*c^2*d^4*e^4-208*((a* e^2-c*d^2)*e)^(1/2)*a*c^3*d^6*e^2+16*((a*e^2-c*d^2)*e)^(1/2)*c^4*d^8)/(e*x +d)^(7/2)/(c*d*x+a*e)^2/(a*e^2-c*d^2)^5/((a*e^2-c*d^2)*e)^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 1183 vs. \(2 (350) = 700\).
Time = 0.84 (sec) , antiderivative size = 2388, normalized size of antiderivative = 6.06 \[ \int \frac {1}{(d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:
integrate(1/(e*x+d)^(3/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algor ithm="fricas")
Output:
[-1/48*(315*(c^5*d^5*e^5*x^6 + a^2*c^3*d^7*e^3 + 2*(2*c^5*d^6*e^4 + a*c^4* d^4*e^6)*x^5 + (6*c^5*d^7*e^3 + 8*a*c^4*d^5*e^5 + a^2*c^3*d^3*e^7)*x^4 + 4 *(c^5*d^8*e^2 + 3*a*c^4*d^6*e^4 + a^2*c^3*d^4*e^6)*x^3 + (c^5*d^9*e + 8*a* c^4*d^7*e^3 + 6*a^2*c^3*d^5*e^5)*x^2 + 2*(a*c^4*d^8*e^2 + 2*a^2*c^3*d^6*e^ 4)*x)*sqrt(-e/(c*d^2 - a*e^2))*log(-(c*d*e^2*x^2 + 2*a*e^3*x - c*d^3 + 2*a *d*e^2 - 2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(c*d^2 - a*e^2)*sqr t(e*x + d)*sqrt(-e/(c*d^2 - a*e^2)))/(e^2*x^2 + 2*d*e*x + d^2)) - 2*(315*c ^4*d^4*e^4*x^4 - 16*c^4*d^8 + 208*a*c^3*d^6*e^2 + 165*a^2*c^2*d^4*e^4 - 50 *a^3*c*d^2*e^6 + 8*a^4*e^8 + 420*(2*c^4*d^5*e^3 + a*c^3*d^3*e^5)*x^3 + 63* (11*c^4*d^6*e^2 + 18*a*c^3*d^4*e^4 + a^2*c^2*d^2*e^6)*x^2 + 18*(8*c^4*d^7* e + 53*a*c^3*d^5*e^3 + 10*a^2*c^2*d^3*e^5 - a^3*c*d*e^7)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d))/(a^2*c^5*d^14*e^2 - 5*a^3*c^4 *d^12*e^4 + 10*a^4*c^3*d^10*e^6 - 10*a^5*c^2*d^8*e^8 + 5*a^6*c*d^6*e^10 - a^7*d^4*e^12 + (c^7*d^12*e^4 - 5*a*c^6*d^10*e^6 + 10*a^2*c^5*d^8*e^8 - 10* a^3*c^4*d^6*e^10 + 5*a^4*c^3*d^4*e^12 - a^5*c^2*d^2*e^14)*x^6 + 2*(2*c^7*d ^13*e^3 - 9*a*c^6*d^11*e^5 + 15*a^2*c^5*d^9*e^7 - 10*a^3*c^4*d^7*e^9 + 3*a ^5*c^2*d^3*e^13 - a^6*c*d*e^15)*x^5 + (6*c^7*d^14*e^2 - 22*a*c^6*d^12*e^4 + 21*a^2*c^5*d^10*e^6 + 15*a^3*c^4*d^8*e^8 - 40*a^4*c^3*d^6*e^10 + 24*a^5* c^2*d^4*e^12 - 3*a^6*c*d^2*e^14 - a^7*e^16)*x^4 + 4*(c^7*d^15*e - 2*a*c^6* d^13*e^3 - 4*a^2*c^5*d^11*e^5 + 15*a^3*c^4*d^9*e^7 - 15*a^4*c^3*d^7*e^9...
\[ \int \frac {1}{(d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\int \frac {1}{\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {5}{2}} \left (d + e x\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(1/(e*x+d)**(3/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2),x)
Output:
Integral(1/(((d + e*x)*(a*e + c*d*x))**(5/2)*(d + e*x)**(3/2)), x)
\[ \int \frac {1}{(d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\int { \frac {1}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {5}{2}} {\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(e*x+d)^(3/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algor ithm="maxima")
Output:
integrate(1/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)*(e*x + d)^(3/2) ), x)
Leaf count of result is larger than twice the leaf count of optimal. 742 vs. \(2 (350) = 700\).
Time = 0.19 (sec) , antiderivative size = 742, normalized size of antiderivative = 1.88 \[ \int \frac {1}{(d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:
integrate(1/(e*x+d)^(3/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algor ithm="giac")
Output:
1/24*(315*c^3*d^3*e*arctan(sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)/sqrt(c* d^2*e - a*e^3))/((c^5*d^10*abs(e) - 5*a*c^4*d^8*e^2*abs(e) + 10*a^2*c^3*d^ 6*e^4*abs(e) - 10*a^3*c^2*d^4*e^6*abs(e) + 5*a^4*c*d^2*e^8*abs(e) - a^5*e^ 10*abs(e))*sqrt(c*d^2*e - a*e^3)) - (16*c^7*d^11*e^5 - 64*a*c^6*d^9*e^7 + 96*a^2*c^5*d^7*e^9 - 64*a^3*c^4*d^5*e^11 + 16*a^4*c^3*d^3*e^13 - 144*((e*x + d)*c*d*e - c*d^2*e + a*e^3)*c^6*d^9*e^4 + 432*((e*x + d)*c*d*e - c*d^2* e + a*e^3)*a*c^5*d^7*e^6 - 432*((e*x + d)*c*d*e - c*d^2*e + a*e^3)*a^2*c^4 *d^5*e^8 + 144*((e*x + d)*c*d*e - c*d^2*e + a*e^3)*a^3*c^3*d^3*e^10 - 693* ((e*x + d)*c*d*e - c*d^2*e + a*e^3)^2*c^5*d^7*e^3 + 1386*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^2*a*c^4*d^5*e^5 - 693*((e*x + d)*c*d*e - c*d^2*e + a*e^ 3)^2*a^2*c^3*d^3*e^7 - 840*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^3*c^4*d^5*e ^2 + 840*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^3*a*c^3*d^3*e^4 - 315*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^4*c^3*d^3*e)/((c^5*d^10*abs(e) - 5*a*c^4*d^8* e^2*abs(e) + 10*a^2*c^3*d^6*e^4*abs(e) - 10*a^3*c^2*d^4*e^6*abs(e) + 5*a^4 *c*d^2*e^8*abs(e) - a^5*e^10*abs(e))*(sqrt((e*x + d)*c*d*e - c*d^2*e + a*e ^3)*c*d^2*e - sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*a*e^3 + ((e*x + d)*c *d*e - c*d^2*e + a*e^3)^(3/2))^3))*e^2
Timed out. \[ \int \frac {1}{(d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\int \frac {1}{{\left (d+e\,x\right )}^{3/2}\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2}} \,d x \] Input:
int(1/((d + e*x)^(3/2)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)),x)
Output:
int(1/((d + e*x)^(3/2)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)), x)
Time = 0.30 (sec) , antiderivative size = 1379, normalized size of antiderivative = 3.50 \[ \int \frac {1}{(d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:
int(1/(e*x+d)^(3/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x)
Output:
(315*sqrt(e)*sqrt(a*e + c*d*x)*sqrt( - a*e**2 + c*d**2)*atan((sqrt(a*e + c *d*x)*e)/(sqrt(e)*sqrt( - a*e**2 + c*d**2)))*a*c**3*d**6*e**2 + 945*sqrt(e )*sqrt(a*e + c*d*x)*sqrt( - a*e**2 + c*d**2)*atan((sqrt(a*e + c*d*x)*e)/(s qrt(e)*sqrt( - a*e**2 + c*d**2)))*a*c**3*d**5*e**3*x + 945*sqrt(e)*sqrt(a* e + c*d*x)*sqrt( - a*e**2 + c*d**2)*atan((sqrt(a*e + c*d*x)*e)/(sqrt(e)*sq rt( - a*e**2 + c*d**2)))*a*c**3*d**4*e**4*x**2 + 315*sqrt(e)*sqrt(a*e + c* d*x)*sqrt( - a*e**2 + c*d**2)*atan((sqrt(a*e + c*d*x)*e)/(sqrt(e)*sqrt( - a*e**2 + c*d**2)))*a*c**3*d**3*e**5*x**3 + 315*sqrt(e)*sqrt(a*e + c*d*x)*s qrt( - a*e**2 + c*d**2)*atan((sqrt(a*e + c*d*x)*e)/(sqrt(e)*sqrt( - a*e**2 + c*d**2)))*c**4*d**7*e*x + 945*sqrt(e)*sqrt(a*e + c*d*x)*sqrt( - a*e**2 + c*d**2)*atan((sqrt(a*e + c*d*x)*e)/(sqrt(e)*sqrt( - a*e**2 + c*d**2)))*c **4*d**6*e**2*x**2 + 945*sqrt(e)*sqrt(a*e + c*d*x)*sqrt( - a*e**2 + c*d**2 )*atan((sqrt(a*e + c*d*x)*e)/(sqrt(e)*sqrt( - a*e**2 + c*d**2)))*c**4*d**5 *e**3*x**3 + 315*sqrt(e)*sqrt(a*e + c*d*x)*sqrt( - a*e**2 + c*d**2)*atan(( sqrt(a*e + c*d*x)*e)/(sqrt(e)*sqrt( - a*e**2 + c*d**2)))*c**4*d**4*e**4*x* *4 - 8*a**5*e**10 + 58*a**4*c*d**2*e**8 + 18*a**4*c*d*e**9*x - 215*a**3*c* *2*d**4*e**6 - 198*a**3*c**2*d**3*e**7*x - 63*a**3*c**2*d**2*e**8*x**2 - 4 3*a**2*c**3*d**6*e**4 - 774*a**2*c**3*d**5*e**5*x - 1071*a**2*c**3*d**4*e* *6*x**2 - 420*a**2*c**3*d**3*e**7*x**3 + 224*a*c**4*d**8*e**2 + 810*a*c**4 *d**7*e**3*x + 441*a*c**4*d**6*e**4*x**2 - 420*a*c**4*d**5*e**5*x**3 - ...