\(\int \frac {1}{(d+e x)^{3/2} (a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}} \, dx\) [2078]

Optimal result

Integrand size = 39, antiderivative size = 395 \[ \int \frac {1}{(d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {1}{3 \left (c d^2-a e^2\right ) (d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {3 c d}{4 \left (c d^2-a e^2\right )^2 \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {7 c^2 d^2 \sqrt {d+e x}}{4 \left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {35 c^2 d^2 e}{8 \left (c d^2-a e^2\right )^4 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {105 c^3 d^3 e \sqrt {d+e x}}{8 \left (c d^2-a e^2\right )^5 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {105 c^3 d^3 e^{3/2} \arctan \left (\frac {\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d^2-a e^2} \sqrt {d+e x}}\right )}{8 \left (c d^2-a e^2\right )^{11/2}} \]

[Out]

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)+105/8*c^3*d^3*e^(3/2)*arctan(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)+3/4*c*d/(-a*
e^2+c*d^2)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^(1/2)-7/4*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^2*d^2*e/(-a*e^2+c*d^2)^4/(e*x+d)^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c
*d*e*x^2)^(1/2)+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)

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 395, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {686, 680, 674, 211} \[ \int \frac {1}{(d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {105 c^3 d^3 e^{3/2} \arctan \left (\frac {\sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d^2-a e^2}}\right )}{8 \left (c d^2-a e^2\right )^{11/2}}+\frac {105 c^3 d^3 e \sqrt {d+e x}}{8 \left (c d^2-a e^2\right )^5 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {35 c^2 d^2 e}{8 \sqrt {d+e x} \left (c d^2-a e^2\right )^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {7 c^2 d^2 \sqrt {d+e x}}{4 \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac {3 c d}{4 \sqrt {d+e x} \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac {1}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]

[In]

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]

[Out]

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)/(4*(c*d^2 - a*e^
2)^2*Sqrt[d + e*x]*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) - (7*c^2*d^2*Sqrt[d + e*x])/(4*(c*d^2 - a*e^
2)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) - (35*c^2*d^2*e)/(8*(c*d^2 - a*e^2)^4*Sqrt[d + e*x]*Sqrt[a
*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (105*c^3*d^3*e*Sqrt[d + e*x])/(8*(c*d^2 - a*e^2)^5*Sqrt[a*d*e + (c*d^
2 + a*e^2)*x + c*d*e*x^2]) + (105*c^3*d^3*e^(3/2)*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])])/(8*(c*d^2 - a*e^2)^(11/2))

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 674

Int[1/(Sqrt[(d_.) + (e_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[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] && NeQ[b^
2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0]

Rule 680

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> 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] - Dist[(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] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^
2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && LtQ[0, m, 1] && IntegerQ[2*p]

Rule 686

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-e)*(d + e*x)^m*((a
+ b*x + c*x^2)^(p + 1)/((m + p + 1)*(2*c*d - b*e))), x] + Dist[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] && NeQ[b^2 - 4*a*c, 0] && E
qQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, 0] && NeQ[m + p + 1, 0] && IntegerQ[2*p]

Rubi steps \begin{align*} \text {integral}& = \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) \int \frac {1}{\sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx}{2 \left (c d^2-a e^2\right )} \\ & = \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 {\left (21 c^2 d^2\right ) \int \frac {\sqrt {d+e x}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx}{8 \left (c d^2-a e^2\right )^2} \\ & = \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 {7 c^2 d^2 \sqrt {d+e x}}{4 \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 {\left (35 c^2 d^2 e\right ) \int \frac {1}{\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}} \, dx}{8 \left (c d^2-a e^2\right )^3} \\ & = \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 {7 c^2 d^2 \sqrt {d+e x}}{4 \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^2 d^2 e}{8 \left (c d^2-a e^2\right )^4 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {\left (105 c^3 d^3 e\right ) \int \frac {\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}} \, dx}{16 \left (c d^2-a e^2\right )^4} \\ & = \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 {7 c^2 d^2 \sqrt {d+e x}}{4 \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^2 d^2 e}{8 \left (c d^2-a e^2\right )^4 \sqrt {d+e x} \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 \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 {\left (105 c^3 d^3 e^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{16 \left (c d^2-a e^2\right )^5} \\ & = \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 {7 c^2 d^2 \sqrt {d+e x}}{4 \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^2 d^2 e}{8 \left (c d^2-a e^2\right )^4 \sqrt {d+e x} \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 \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 {\left (105 c^3 d^3 e^3\right ) \text {Subst}\left (\int \frac {1}{2 c d^2 e-e \left (c d^2+a e^2\right )+e^2 x^2} \, dx,x,\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}}\right )}{8 \left (c d^2-a e^2\right )^5} \\ & = \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 {7 c^2 d^2 \sqrt {d+e x}}{4 \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^2 d^2 e}{8 \left (c d^2-a e^2\right )^4 \sqrt {d+e x} \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 \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} \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d^2-a e^2} \sqrt {d+e x}}\right )}{8 \left (c d^2-a e^2\right )^{11/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.99 (sec) , antiderivative size = 294, normalized size of antiderivative = 0.74 \[ \int \frac {1}{(d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {c^3 d^3 (d+e x)^{5/2} \left (\frac {(a e+c d x) \left (8 a^4 e^8-2 a^3 c d e^6 (25 d+9 e x)+3 a^2 c^2 d^2 e^4 \left (55 d^2+60 d e x+21 e^2 x^2\right )+2 a c^3 d^3 e^2 \left (104 d^3+477 d^2 e x+567 d e^2 x^2+210 e^3 x^3\right )+c^4 d^4 \left (-16 d^4+144 d^3 e x+693 d^2 e^2 x^2+840 d e^3 x^3+315 e^4 x^4\right )\right )}{c^3 d^3 \left (c d^2-a e^2\right )^5 (d+e x)^3}+\frac {315 e^{3/2} (a e+c d x)^{5/2} \arctan \left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{11/2}}\right )}{24 ((a e+c d x) (d+e x))^{5/2}} \]

[In]

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]

[Out]

(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)*ArcTan[(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))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(919\) vs. \(2(351)=702\).

Time = 2.93 (sec) , antiderivative size = 920, normalized size of antiderivative = 2.33

method	result	size
default	\(\frac {\sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (315 \sqrt {c d x +a e}\, \operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c^{4} d^{4} e^{5} x^{4}+315 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) a \,c^{3} d^{3} e^{6} x^{3} \sqrt {c d x +a e}+945 \sqrt {c d x +a e}\, \operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c^{4} d^{5} e^{4} x^{3}+945 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) a \,c^{3} d^{4} e^{5} x^{2} \sqrt {c d x +a e}+945 \sqrt {c d x +a e}\, \operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c^{4} d^{6} e^{3} x^{2}-315 \sqrt {\left (e^{2} a -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 (e^{2} a -c \,d^{2}\right ) e}}\right ) a \,c^{3} d^{5} e^{4} x \sqrt {c d x +a e}+315 \sqrt {c d x +a e}\, \operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c^{4} d^{7} e^{2} x -420 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a \,c^{3} d^{3} e^{5} x^{3}-840 \sqrt {\left (e^{2} a -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 (e^{2} a -c \,d^{2}\right ) e}}\right ) a \,c^{3} d^{6} e^{3} \sqrt {c d x +a e}-63 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a^{2} c^{2} d^{2} e^{6} x^{2}-1134 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a \,c^{3} d^{4} e^{4} x^{2}-693 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, c^{4} d^{6} e^{2} x^{2}+18 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a^{3} c d \,e^{7} x -180 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a^{2} c^{2} d^{3} e^{5} x -954 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a \,c^{3} d^{5} e^{3} x -144 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, c^{4} d^{7} e x -8 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a^{4} e^{8}+50 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a^{3} c \,d^{2} e^{6}-165 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a^{2} c^{2} d^{4} e^{4}-208 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a \,c^{3} d^{6} e^{2}+16 \sqrt {\left (e^{2} a -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 (e^{2} a -c \,d^{2}\right )^{5} \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\)	\(920\)

[In]

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,method=_RETURNVERBOSE)

[Out]

1/24*((c*d*x+a*e)*(e*x+d))^(1/2)*(315*(c*d*x+a*e)^(1/2)*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+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)+94
5*(c*d*x+a*e)^(1/2)*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+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*(c*d*x+a*e)^(1/2)*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^2-315*((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*(c*d*x+a*e)^(1/2)*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-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*arctanh(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)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1183 vs. \(2 (351) = 702\).

Time = 1.10 (sec) , antiderivative size = 2388, normalized size of antiderivative = 6.05 \[ \int \frac {1}{(d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\text {Too large to display} \]

[In]

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, algorithm="fricas")

[Out]

[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)*
sqrt(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^1
0 + 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 + 4*a^5*c^2*d^5*e^11 + 2*a^6*c*d^3*e^13 - a^7*d*e^15)*x^3 + (c^
7*d^16 + 3*a*c^6*d^14*e^2 - 24*a^2*c^5*d^12*e^4 + 40*a^3*c^4*d^10*e^6 - 15*a^4*c^3*d^8*e^8 - 21*a^5*c^2*d^6*e^
10 + 22*a^6*c*d^4*e^12 - 6*a^7*d^2*e^14)*x^2 + 2*(a*c^6*d^15*e - 3*a^2*c^5*d^13*e^3 + 10*a^4*c^3*d^9*e^7 - 15*
a^5*c^2*d^7*e^9 + 9*a^6*c*d^5*e^11 - 2*a^7*d^3*e^13)*x), 1/24*(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))*arctan(-sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(c*d^2
- a*e^2)*sqrt(e*x + d)*sqrt(e/(c*d^2 - a*e^2))/(c*d*e^2*x^2 + a*d*e^2 + (c*d^2*e + a*e^3)*x)) + (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 + 4*a^5*c^2*d^5*e^11 + 2*a^6*c*d^3*e^1
3 - a^7*d*e^15)*x^3 + (c^7*d^16 + 3*a*c^6*d^14*e^2 - 24*a^2*c^5*d^12*e^4 + 40*a^3*c^4*d^10*e^6 - 15*a^4*c^3*d^
8*e^8 - 21*a^5*c^2*d^6*e^10 + 22*a^6*c*d^4*e^12 - 6*a^7*d^2*e^14)*x^2 + 2*(a*c^6*d^15*e - 3*a^2*c^5*d^13*e^3 +
10*a^4*c^3*d^9*e^7 - 15*a^5*c^2*d^7*e^9 + 9*a^6*c*d^5*e^11 - 2*a^7*d^3*e^13)*x)]

Sympy [F]

\[ \int \frac {1}{(d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\int \frac {1}{\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {5}{2}} \left (d + e x\right )^{\frac {3}{2}}}\, dx \]

[In]

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)

[Out]

Integral(1/(((d + e*x)*(a*e + c*d*x))**(5/2)*(d + e*x)**(3/2)), x)

Maxima [F]

\[ \int \frac {1}{(d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\int { \frac {1}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {5}{2}} {\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]

[In]

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, algorithm="maxima")

[Out]

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)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 742 vs. \(2 (351) = 702\).

Time = 0.57 (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=\frac {1}{24} \, {\left (\frac {315 \, c^{3} d^{3} e \arctan \left (\frac {\sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}}}{\sqrt {c d^{2} e - a e^{3}}}\right )}{{\left (c^{5} d^{10} {\left | e \right |} - 5 \, a c^{4} d^{8} e^{2} {\left | e \right |} + 10 \, a^{2} c^{3} d^{6} e^{4} {\left | e \right |} - 10 \, a^{3} c^{2} d^{4} e^{6} {\left | e \right |} + 5 \, a^{4} c d^{2} e^{8} {\left | e \right |} - a^{5} e^{10} {\left | e \right |}\right )} \sqrt {c d^{2} e - a e^{3}}} - \frac {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 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )} c^{6} d^{9} e^{4} + 432 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )} a c^{5} d^{7} e^{6} - 432 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )} a^{2} c^{4} d^{5} e^{8} + 144 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )} a^{3} c^{3} d^{3} e^{10} - 693 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{2} c^{5} d^{7} e^{3} + 1386 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{2} a c^{4} d^{5} e^{5} - 693 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{2} a^{2} c^{3} d^{3} e^{7} - 840 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{3} c^{4} d^{5} e^{2} + 840 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{3} a c^{3} d^{3} e^{4} - 315 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{4} c^{3} d^{3} e}{{\left (c^{5} d^{10} {\left | e \right |} - 5 \, a c^{4} d^{8} e^{2} {\left | e \right |} + 10 \, a^{2} c^{3} d^{6} e^{4} {\left | e \right |} - 10 \, a^{3} c^{2} d^{4} e^{6} {\left | e \right |} + 5 \, a^{4} c d^{2} e^{8} {\left | e \right |} - a^{5} e^{10} {\left | e \right |}\right )} {\left (\sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} c d^{2} e - \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} a e^{3} + {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}}\right )}^{3}}\right )} e^{2} \]

[In]

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, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\int \frac {1}{{\left (d+e\,x\right )}^{3/2}\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2}} \,d x \]

[In]

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)

[Out]

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)