\(\int \frac {\sqrt {d+e x}}{(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}} \, dx\) [2076]

Optimal result

Integrand size = 39, antiderivative size = 257 \[ \int \frac {\sqrt {d+e x}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=-\frac {2 \sqrt {d+e x}}{3 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {5 e}{3 \left (c d^2-a e^2\right )^2 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {5 c d e \sqrt {d+e x}}{\left (c d^2-a e^2\right )^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {5 c d e^{3/2} \arctan \left (\frac {\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d^2-a e^2} \sqrt {d+e x}}\right )}{\left (c d^2-a e^2\right )^{7/2}} \]

[Out]

5*c*d*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)^(7/2)-2/3*(e*x+d)^(1/2)/(-a*e^2+c*d^2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)-5/3*e/(-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)^(1/2)+5*c*d*e*(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)^(1/2)

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {680, 686, 674, 211} \[ \int \frac {\sqrt {d+e x}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {5 c d e^{3/2} \arctan \left (\frac {\sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{7/2}}+\frac {5 c d e \sqrt {d+e x}}{\left (c d^2-a e^2\right )^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {5 e}{3 \sqrt {d+e x} \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {2 \sqrt {d+e x}}{3 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]

[In]

Int[Sqrt[d + e*x]/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]

[Out]

(-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)/(3*(c*d^2 - a*e^2
)^2*Sqrt[d + e*x]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (5*c*d*e*Sqrt[d + e*x])/((c*d^2 - a*e^2)^3*Sq
rt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (5*c*d*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])])/(c*d^2 - a*e^2)^(7/2)

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 {2 \sqrt {d+e x}}{3 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {(5 e) \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}{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 (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {5 e}{3 \left (c d^2-a e^2\right )^2 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {(5 c d e) \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}{2 \left (c d^2-a e^2\right )^2} \\ & = -\frac {2 \sqrt {d+e x}}{3 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {5 e}{3 \left (c d^2-a e^2\right )^2 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {5 c d e \sqrt {d+e x}}{\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}}+\frac {\left (5 c d 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}{2 \left (c d^2-a e^2\right )^3} \\ & = -\frac {2 \sqrt {d+e x}}{3 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {5 e}{3 \left (c d^2-a e^2\right )^2 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {5 c d e \sqrt {d+e x}}{\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}}+\frac {\left (5 c d 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 )}{\left (c d^2-a e^2\right )^3} \\ & = -\frac {2 \sqrt {d+e x}}{3 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {5 e}{3 \left (c d^2-a e^2\right )^2 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {5 c d e \sqrt {d+e x}}{\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}}+\frac {5 c d 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 )}{\left (c d^2-a e^2\right )^{7/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.69 \[ \int \frac {\sqrt {d+e x}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {\sqrt {d+e x} \left (\sqrt {c d^2-a e^2} \left (3 a^2 e^4+2 a c d e^2 (7 d+10 e x)+c^2 d^2 \left (-2 d^2+10 d e x+15 e^2 x^2\right )\right )+15 c d e^{3/2} (a e+c d x)^{3/2} (d+e x) \arctan \left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c d^2-a e^2}}\right )\right )}{3 \left (c d^2-a e^2\right )^{7/2} ((a e+c d x) (d+e x))^{3/2}} \]

[In]

Integrate[Sqrt[d + e*x]/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]

[Out]

(Sqrt[d + e*x]*(Sqrt[c*d^2 - a*e^2]*(3*a^2*e^4 + 2*a*c*d*e^2*(7*d + 10*e*x) + c^2*d^2*(-2*d^2 + 10*d*e*x + 15*
e^2*x^2)) + 15*c*d*e^(3/2)*(a*e + c*d*x)^(3/2)*(d + e*x)*ArcTan[(Sqrt[e]*Sqrt[a*e + c*d*x])/Sqrt[c*d^2 - a*e^2
]]))/(3*(c*d^2 - a*e^2)^(7/2)*((a*e + c*d*x)*(d + e*x))^(3/2))

Maple [A] (verified)

Time = 2.78 (sec) , antiderivative size = 423, normalized size of antiderivative = 1.65

method	result	size
default	\(\frac {\sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (15 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) \sqrt {c d x +a e}\, c^{2} d^{2} e^{3} x^{2}+15 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) a c d \,e^{4} x \sqrt {c d x +a e}+15 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) \sqrt {c d x +a e}\, c^{2} d^{3} e^{2} x +15 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) a c \,d^{2} e^{3} \sqrt {c d x +a e}-15 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, c^{2} d^{2} e^{2} x^{2}-20 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a c d \,e^{3} x -10 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, c^{2} d^{3} e x -3 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a^{2} e^{4}-14 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a c \,d^{2} e^{2}+2 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, c^{2} d^{4}\right )}{3 \left (e x +d \right )^{\frac {3}{2}} \left (c d x +a e \right )^{2} \left (e^{2} a -c \,d^{2}\right )^{3} \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\)	\(423\)

[In]

int((e*x+d)^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/3*((c*d*x+a*e)*(e*x+d))^(1/2)*(15*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*(c*d*x+a*e)^(1/2)*c^2
*d^2*e^3*x^2+15*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*a*c*d*e^4*x*(c*d*x+a*e)^(1/2)+15*arctanh(
e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*(c*d*x+a*e)^(1/2)*c^2*d^3*e^2*x+15*arctanh(e*(c*d*x+a*e)^(1/2)/((
a*e^2-c*d^2)*e)^(1/2))*a*c*d^2*e^3*(c*d*x+a*e)^(1/2)-15*((a*e^2-c*d^2)*e)^(1/2)*c^2*d^2*e^2*x^2-20*((a*e^2-c*d
^2)*e)^(1/2)*a*c*d*e^3*x-10*((a*e^2-c*d^2)*e)^(1/2)*c^2*d^3*e*x-3*((a*e^2-c*d^2)*e)^(1/2)*a^2*e^4-14*((a*e^2-c
*d^2)*e)^(1/2)*a*c*d^2*e^2+2*((a*e^2-c*d^2)*e)^(1/2)*c^2*d^4)/(e*x+d)^(3/2)/(c*d*x+a*e)^2/(a*e^2-c*d^2)^3/((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. 607 vs. \(2 (229) = 458\).

Time = 0.40 (sec) , antiderivative size = 1236, normalized size of antiderivative = 4.81 \[ \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=\left [-\frac {15 \, {\left (c^{3} d^{3} e^{3} x^{4} + a^{2} c d^{3} e^{3} + 2 \, {\left (c^{3} d^{4} e^{2} + a c^{2} d^{2} e^{4}\right )} x^{3} + {\left (c^{3} d^{5} e + 4 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x^{2} + 2 \, {\left (a c^{2} d^{4} e^{2} + a^{2} c d^{2} e^{4}\right )} x\right )} \sqrt {-\frac {e}{c d^{2} - a e^{2}}} \log \left (-\frac {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 + {\left (c d^{2} + a e^{2}\right )} x} {\left (c d^{2} - a e^{2}\right )} \sqrt {e x + d} \sqrt {-\frac {e}{c d^{2} - a e^{2}}}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) - 2 \, {\left (15 \, c^{2} d^{2} e^{2} x^{2} - 2 \, c^{2} d^{4} + 14 \, a c d^{2} e^{2} + 3 \, a^{2} e^{4} + 10 \, {\left (c^{2} d^{3} e + 2 \, a c d e^{3}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{6 \, {\left (a^{2} c^{3} d^{8} e^{2} - 3 \, a^{3} c^{2} d^{6} e^{4} + 3 \, a^{4} c d^{4} e^{6} - a^{5} d^{2} e^{8} + {\left (c^{5} d^{8} e^{2} - 3 \, a c^{4} d^{6} e^{4} + 3 \, a^{2} c^{3} d^{4} e^{6} - a^{3} c^{2} d^{2} e^{8}\right )} x^{4} + 2 \, {\left (c^{5} d^{9} e - 2 \, a c^{4} d^{7} e^{3} + 2 \, a^{3} c^{2} d^{3} e^{7} - a^{4} c d e^{9}\right )} x^{3} + {\left (c^{5} d^{10} + a c^{4} d^{8} e^{2} - 8 \, a^{2} c^{3} d^{6} e^{4} + 8 \, a^{3} c^{2} d^{4} e^{6} - a^{4} c d^{2} e^{8} - a^{5} e^{10}\right )} x^{2} + 2 \, {\left (a c^{4} d^{9} e - 2 \, a^{2} c^{3} d^{7} e^{3} + 2 \, a^{4} c d^{3} e^{7} - a^{5} d e^{9}\right )} x\right )}}, \frac {15 \, {\left (c^{3} d^{3} e^{3} x^{4} + a^{2} c d^{3} e^{3} + 2 \, {\left (c^{3} d^{4} e^{2} + a c^{2} d^{2} e^{4}\right )} x^{3} + {\left (c^{3} d^{5} e + 4 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x^{2} + 2 \, {\left (a c^{2} d^{4} e^{2} + a^{2} c d^{2} e^{4}\right )} x\right )} \sqrt {\frac {e}{c d^{2} - a e^{2}}} \arctan \left (-\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (c d^{2} - a e^{2}\right )} \sqrt {e x + d} \sqrt {\frac {e}{c d^{2} - a e^{2}}}}{c d e^{2} x^{2} + a d e^{2} + {\left (c d^{2} e + a e^{3}\right )} x}\right ) + {\left (15 \, c^{2} d^{2} e^{2} x^{2} - 2 \, c^{2} d^{4} + 14 \, a c d^{2} e^{2} + 3 \, a^{2} e^{4} + 10 \, {\left (c^{2} d^{3} e + 2 \, a c d e^{3}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{3 \, {\left (a^{2} c^{3} d^{8} e^{2} - 3 \, a^{3} c^{2} d^{6} e^{4} + 3 \, a^{4} c d^{4} e^{6} - a^{5} d^{2} e^{8} + {\left (c^{5} d^{8} e^{2} - 3 \, a c^{4} d^{6} e^{4} + 3 \, a^{2} c^{3} d^{4} e^{6} - a^{3} c^{2} d^{2} e^{8}\right )} x^{4} + 2 \, {\left (c^{5} d^{9} e - 2 \, a c^{4} d^{7} e^{3} + 2 \, a^{3} c^{2} d^{3} e^{7} - a^{4} c d e^{9}\right )} x^{3} + {\left (c^{5} d^{10} + a c^{4} d^{8} e^{2} - 8 \, a^{2} c^{3} d^{6} e^{4} + 8 \, a^{3} c^{2} d^{4} e^{6} - a^{4} c d^{2} e^{8} - a^{5} e^{10}\right )} x^{2} + 2 \, {\left (a c^{4} d^{9} e - 2 \, a^{2} c^{3} d^{7} e^{3} + 2 \, a^{4} c d^{3} e^{7} - a^{5} d e^{9}\right )} x\right )}}\right ] \]

[In]

integrate((e*x+d)^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="fricas")

[Out]

[-1/6*(15*(c^3*d^3*e^3*x^4 + a^2*c*d^3*e^3 + 2*(c^3*d^4*e^2 + a*c^2*d^2*e^4)*x^3 + (c^3*d^5*e + 4*a*c^2*d^3*e^
3 + a^2*c*d*e^5)*x^2 + 2*(a*c^2*d^4*e^2 + a^2*c*d^2*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*(15*c^2*d^2*e^2*x^2 - 2*c^2*d^4 + 14*a*c*d^2*e^2 + 3*a^2*e^4
+ 10*(c^2*d^3*e + 2*a*c*d*e^3)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d))/(a^2*c^3*d^8*e^2
- 3*a^3*c^2*d^6*e^4 + 3*a^4*c*d^4*e^6 - a^5*d^2*e^8 + (c^5*d^8*e^2 - 3*a*c^4*d^6*e^4 + 3*a^2*c^3*d^4*e^6 - a^
3*c^2*d^2*e^8)*x^4 + 2*(c^5*d^9*e - 2*a*c^4*d^7*e^3 + 2*a^3*c^2*d^3*e^7 - a^4*c*d*e^9)*x^3 + (c^5*d^10 + a*c^4
*d^8*e^2 - 8*a^2*c^3*d^6*e^4 + 8*a^3*c^2*d^4*e^6 - a^4*c*d^2*e^8 - a^5*e^10)*x^2 + 2*(a*c^4*d^9*e - 2*a^2*c^3*
d^7*e^3 + 2*a^4*c*d^3*e^7 - a^5*d*e^9)*x), 1/3*(15*(c^3*d^3*e^3*x^4 + a^2*c*d^3*e^3 + 2*(c^3*d^4*e^2 + a*c^2*d
^2*e^4)*x^3 + (c^3*d^5*e + 4*a*c^2*d^3*e^3 + a^2*c*d*e^5)*x^2 + 2*(a*c^2*d^4*e^2 + a^2*c*d^2*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)) + (15*c^2*d^2*e^2*x^2 - 2*c^2*d^4 + 14*a*c*d^2*e^2 +
3*a^2*e^4 + 10*(c^2*d^3*e + 2*a*c*d*e^3)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d))/(a^2*c
^3*d^8*e^2 - 3*a^3*c^2*d^6*e^4 + 3*a^4*c*d^4*e^6 - a^5*d^2*e^8 + (c^5*d^8*e^2 - 3*a*c^4*d^6*e^4 + 3*a^2*c^3*d^
4*e^6 - a^3*c^2*d^2*e^8)*x^4 + 2*(c^5*d^9*e - 2*a*c^4*d^7*e^3 + 2*a^3*c^2*d^3*e^7 - a^4*c*d*e^9)*x^3 + (c^5*d^
10 + a*c^4*d^8*e^2 - 8*a^2*c^3*d^6*e^4 + 8*a^3*c^2*d^4*e^6 - a^4*c*d^2*e^8 - a^5*e^10)*x^2 + 2*(a*c^4*d^9*e -
2*a^2*c^3*d^7*e^3 + 2*a^4*c*d^3*e^7 - a^5*d*e^9)*x)]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

integrate((e*x+d)^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="maxima")

[Out]

integrate(sqrt(e*x + d)/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2), x)

Giac [A] (verification not implemented)

none

Time = 0.43 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.25 \[ \int \frac {\sqrt {d+e x}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {1}{3} \, {\left (\frac {15 \, c d e \arctan \left (\frac {\sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}}}{\sqrt {c d^{2} e - a e^{3}}}\right )}{{\left (c^{3} d^{6} {\left | e \right |} - 3 \, a c^{2} d^{4} e^{2} {\left | e \right |} + 3 \, a^{2} c d^{2} e^{4} {\left | e \right |} - a^{3} e^{6} {\left | e \right |}\right )} \sqrt {c d^{2} e - a e^{3}}} - \frac {2 \, {\left (c^{2} d^{3} e^{2} - a c d e^{4} - 6 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )} c d e\right )}}{{\left (c^{3} d^{6} {\left | e \right |} - 3 \, a c^{2} d^{4} e^{2} {\left | e \right |} + 3 \, a^{2} c d^{2} e^{4} {\left | e \right |} - a^{3} e^{6} {\left | e \right |}\right )} {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}}} + \frac {3 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}}}{{\left (c^{3} d^{6} {\left | e \right |} - 3 \, a c^{2} d^{4} e^{2} {\left | e \right |} + 3 \, a^{2} c d^{2} e^{4} {\left | e \right |} - a^{3} e^{6} {\left | e \right |}\right )} {\left (e x + d\right )}}\right )} e^{2} \]

[In]

integrate((e*x+d)^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="giac")

[Out]

1/3*(15*c*d*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^3*d^6*abs(e) - 3*a*c^2
*d^4*e^2*abs(e) + 3*a^2*c*d^2*e^4*abs(e) - a^3*e^6*abs(e))*sqrt(c*d^2*e - a*e^3)) - 2*(c^2*d^3*e^2 - a*c*d*e^4
- 6*((e*x + d)*c*d*e - c*d^2*e + a*e^3)*c*d*e)/((c^3*d^6*abs(e) - 3*a*c^2*d^4*e^2*abs(e) + 3*a^2*c*d^2*e^4*ab
s(e) - a^3*e^6*abs(e))*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)) + 3*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)/
((c^3*d^6*abs(e) - 3*a*c^2*d^4*e^2*abs(e) + 3*a^2*c*d^2*e^4*abs(e) - a^3*e^6*abs(e))*(e*x + d)))*e^2

Mupad [F(-1)]

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

[In]

int((d + e*x)^(1/2)/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2),x)

[Out]

int((d + e*x)^(1/2)/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2), x)