\(\int \frac {1}{\sqrt {d+e x} (a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}} \, dx\) [2077]

Optimal result

Integrand size = 39, antiderivative size = 329 \[ \int \frac {1}{\sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {1}{2 \left (c d^2-a e^2\right ) \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {7 c d \sqrt {d+e x}}{6 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {35 c d e}{12 \left (c d^2-a e^2\right )^3 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {35 c^2 d^2 e \sqrt {d+e x}}{4 \left (c d^2-a e^2\right )^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {35 c^2 d^2 e^{3/2} \arctan \left (\frac {\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d^2-a e^2} \sqrt {d+e x}}\right )}{4 \left (c d^2-a e^2\right )^{9/2}} \]

[Out]

35/4*c^2*d^2*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)^(9/2)+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)/(e*x+d)^(1/2)-7/6*c*d*(e*x+d
)^(1/2)/(-a*e^2+c*d^2)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)-35/12*c*d*e/(-a*e^2+c*d^2)^3/(e*x+d)^(1/2)/(a
*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+35/4*c^2*d^2*e*(e*x+d)^(1/2)/(-a*e^2+c*d^2)^4/(a*d*e+(a*e^2+c*d^2)*x+c*d
*e*x^2)^(1/2)

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 329, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {686, 680, 674, 211} \[ \int \frac {1}{\sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {35 c^2 d^2 e^{3/2} \arctan \left (\frac {\sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d^2-a e^2}}\right )}{4 \left (c d^2-a e^2\right )^{9/2}}+\frac {35 c^2 d^2 e \sqrt {d+e x}}{4 \left (c d^2-a e^2\right )^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {35 c d e}{12 \sqrt {d+e x} \left (c d^2-a e^2\right )^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {7 c d \sqrt {d+e x}}{6 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac {1}{2 \sqrt {d+e x} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]

[In]

Int[1/(Sqrt[d + e*x]*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)),x]

[Out]

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*Sqrt[d + e*x])/(6*(
c*d^2 - a*e^2)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) - (35*c*d*e)/(12*(c*d^2 - a*e^2)^3*Sqrt[d + e*
x]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (35*c^2*d^2*e*Sqrt[d + e*x])/(4*(c*d^2 - a*e^2)^4*Sqrt[a*d*e
+ (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (35*c^2*d^2*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])])/(4*(c*d^2 - a*e^2)^(9/2))

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}{2 \left (c d^2-a e^2\right ) \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 d) \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}{4 \left (c d^2-a e^2\right )} \\ & = \frac {1}{2 \left (c d^2-a e^2\right ) \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 d \sqrt {d+e x}}{6 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {(35 c d 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}{12 \left (c d^2-a e^2\right )^2} \\ & = \frac {1}{2 \left (c d^2-a e^2\right ) \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 d \sqrt {d+e x}}{6 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {35 c d e}{12 \left (c d^2-a e^2\right )^3 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {\left (35 c^2 d^2 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}{8 \left (c d^2-a e^2\right )^3} \\ & = \frac {1}{2 \left (c d^2-a e^2\right ) \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 d \sqrt {d+e x}}{6 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {35 c d e}{12 \left (c d^2-a e^2\right )^3 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {35 c^2 d^2 e \sqrt {d+e x}}{4 \left (c d^2-a e^2\right )^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {\left (35 c^2 d^2 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}{8 \left (c d^2-a e^2\right )^4} \\ & = \frac {1}{2 \left (c d^2-a e^2\right ) \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 d \sqrt {d+e x}}{6 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {35 c d e}{12 \left (c d^2-a e^2\right )^3 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {35 c^2 d^2 e \sqrt {d+e x}}{4 \left (c d^2-a e^2\right )^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {\left (35 c^2 d^2 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 )}{4 \left (c d^2-a e^2\right )^4} \\ & = \frac {1}{2 \left (c d^2-a e^2\right ) \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 d \sqrt {d+e x}}{6 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {35 c d e}{12 \left (c d^2-a e^2\right )^3 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {35 c^2 d^2 e \sqrt {d+e x}}{4 \left (c d^2-a e^2\right )^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {35 c^2 d^2 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 )}{4 \left (c d^2-a e^2\right )^{9/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.65 (sec) , antiderivative size = 240, normalized size of antiderivative = 0.73 \[ \int \frac {1}{\sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {c^2 d^2 (d+e x)^{5/2} \left (-\frac {(a e+c d x) \left (6 a^3 e^6-3 a^2 c d e^4 (13 d+7 e x)-2 a c^2 d^2 e^2 \left (40 d^2+119 d e x+70 e^2 x^2\right )+c^3 d^3 \left (8 d^3-56 d^2 e x-175 d e^2 x^2-105 e^3 x^3\right )\right )}{c^2 d^2 \left (c d^2-a e^2\right )^4 (d+e x)^2}+\frac {105 e^{3/2} (a e+c d x)^{5/2} \arctan \left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{9/2}}\right )}{12 ((a e+c d x) (d+e x))^{5/2}} \]

[In]

Integrate[1/(Sqrt[d + e*x]*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)),x]

[Out]

(c^2*d^2*(d + e*x)^(5/2)*(-(((a*e + c*d*x)*(6*a^3*e^6 - 3*a^2*c*d*e^4*(13*d + 7*e*x) - 2*a*c^2*d^2*e^2*(40*d^2
+ 119*d*e*x + 70*e^2*x^2) + c^3*d^3*(8*d^3 - 56*d^2*e*x - 175*d*e^2*x^2 - 105*e^3*x^3)))/(c^2*d^2*(c*d^2 - a*
e^2)^4*(d + e*x)^2)) + (105*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)^(9/2)))/(12*((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. \(657\) vs. \(2(291)=582\).

Time = 2.79 (sec) , antiderivative size = 658, normalized size of antiderivative = 2.00

method	result	size
default	\(-\frac {\sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (105 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) \sqrt {c d x +a e}\, c^{3} d^{3} e^{4} x^{3}+105 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) a \,c^{2} d^{2} e^{5} x^{2} \sqrt {c d x +a e}+210 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) \sqrt {c d x +a e}\, c^{3} d^{4} e^{3} x^{2}+210 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) a \,c^{2} d^{3} e^{4} x \sqrt {c d x +a e}+105 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) \sqrt {c d x +a e}\, c^{3} d^{5} e^{2} x -105 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, c^{3} d^{3} e^{3} x^{3}+105 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) a \,c^{2} d^{4} e^{3} \sqrt {c d x +a e}-140 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a \,c^{2} d^{2} e^{4} x^{2}-175 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, c^{3} d^{4} e^{2} x^{2}-21 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a^{2} c d \,e^{5} x -238 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a \,c^{2} d^{3} e^{3} x -56 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, c^{3} d^{5} e x +6 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a^{3} e^{6}-39 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a^{2} c \,d^{2} e^{4}-80 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a \,c^{2} d^{4} e^{2}+8 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, c^{3} d^{6}\right )}{12 \left (e x +d \right )^{\frac {5}{2}} \left (c d x +a e \right )^{2} \left (e^{2} a -c \,d^{2}\right )^{4} \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\)	\(658\)

[In]

int(1/(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]

-1/12*((c*d*x+a*e)*(e*x+d))^(1/2)*(105*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^3*d^3*e^4*x^3+105*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*a*c^2*d^2*e^5*x^2*(c*d*x+a*e)^(1/2)+2
10*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^3*d^4*e^3*x^2+210*arctanh(e*(c*d*x
+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*a*c^2*d^3*e^4*x*(c*d*x+a*e)^(1/2)+105*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^3*d^5*e^2*x-105*((a*e^2-c*d^2)*e)^(1/2)*c^3*d^3*e^3*x^3+105*arctanh(e*(c
*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*a*c^2*d^4*e^3*(c*d*x+a*e)^(1/2)-140*((a*e^2-c*d^2)*e)^(1/2)*a*c^2*d^2
*e^4*x^2-175*((a*e^2-c*d^2)*e)^(1/2)*c^3*d^4*e^2*x^2-21*((a*e^2-c*d^2)*e)^(1/2)*a^2*c*d*e^5*x-238*((a*e^2-c*d^
2)*e)^(1/2)*a*c^2*d^3*e^3*x-56*((a*e^2-c*d^2)*e)^(1/2)*c^3*d^5*e*x+6*((a*e^2-c*d^2)*e)^(1/2)*a^3*e^6-39*((a*e^
2-c*d^2)*e)^(1/2)*a^2*c*d^2*e^4-80*((a*e^2-c*d^2)*e)^(1/2)*a*c^2*d^4*e^2+8*((a*e^2-c*d^2)*e)^(1/2)*c^3*d^6)/(e
*x+d)^(5/2)/(c*d*x+a*e)^2/(a*e^2-c*d^2)^4/((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. 878 vs. \(2 (291) = 582\).

Time = 0.55 (sec) , antiderivative size = 1778, normalized size of antiderivative = 5.40 \[ \int \frac {1}{\sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\text {Too large to display} \]

[In]

integrate(1/(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]

[1/24*(105*(c^4*d^4*e^4*x^5 + a^2*c^2*d^5*e^3 + (3*c^4*d^5*e^3 + 2*a*c^3*d^3*e^5)*x^4 + (3*c^4*d^6*e^2 + 6*a*c
^3*d^4*e^4 + a^2*c^2*d^2*e^6)*x^3 + (c^4*d^7*e + 6*a*c^3*d^5*e^3 + 3*a^2*c^2*d^3*e^5)*x^2 + (2*a*c^3*d^6*e^2 +
3*a^2*c^2*d^4*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*(105*c^3*d^3*e^3*x^3 - 8*c^3*d^6 + 80*a*c^2*d^4*e^2 + 39*a^2*c*d^2*e^4 - 6*a^3*e^6 + 35*(5*c^3*d^
4*e^2 + 4*a*c^2*d^2*e^4)*x^2 + 7*(8*c^3*d^5*e + 34*a*c^2*d^3*e^3 + 3*a^2*c*d*e^5)*x)*sqrt(c*d*e*x^2 + a*d*e +
(c*d^2 + a*e^2)*x)*sqrt(e*x + d))/(a^2*c^4*d^11*e^2 - 4*a^3*c^3*d^9*e^4 + 6*a^4*c^2*d^7*e^6 - 4*a^5*c*d^5*e^8
+ a^6*d^3*e^10 + (c^6*d^10*e^3 - 4*a*c^5*d^8*e^5 + 6*a^2*c^4*d^6*e^7 - 4*a^3*c^3*d^4*e^9 + a^4*c^2*d^2*e^11)*x
^5 + (3*c^6*d^11*e^2 - 10*a*c^5*d^9*e^4 + 10*a^2*c^4*d^7*e^6 - 5*a^4*c^2*d^3*e^10 + 2*a^5*c*d*e^12)*x^4 + (3*c
^6*d^12*e - 6*a*c^5*d^10*e^3 - 5*a^2*c^4*d^8*e^5 + 20*a^3*c^3*d^6*e^7 - 15*a^4*c^2*d^4*e^9 + 2*a^5*c*d^2*e^11
+ a^6*e^13)*x^3 + (c^6*d^13 + 2*a*c^5*d^11*e^2 - 15*a^2*c^4*d^9*e^4 + 20*a^3*c^3*d^7*e^6 - 5*a^4*c^2*d^5*e^8 -
6*a^5*c*d^3*e^10 + 3*a^6*d*e^12)*x^2 + (2*a*c^5*d^12*e - 5*a^2*c^4*d^10*e^3 + 10*a^4*c^2*d^6*e^7 - 10*a^5*c*d
^4*e^9 + 3*a^6*d^2*e^11)*x), 1/12*(105*(c^4*d^4*e^4*x^5 + a^2*c^2*d^5*e^3 + (3*c^4*d^5*e^3 + 2*a*c^3*d^3*e^5)*
x^4 + (3*c^4*d^6*e^2 + 6*a*c^3*d^4*e^4 + a^2*c^2*d^2*e^6)*x^3 + (c^4*d^7*e + 6*a*c^3*d^5*e^3 + 3*a^2*c^2*d^3*e
^5)*x^2 + (2*a*c^3*d^6*e^2 + 3*a^2*c^2*d^4*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)) + (105*c^3*d^3*e^3*x^3 - 8*c^3*d^6 + 80*a*c^2*d^4*e^2 + 39*a^2*c*d^2*e^4 - 6*a^3*e^6 + 35*(5*c^3*d^4*e
^2 + 4*a*c^2*d^2*e^4)*x^2 + 7*(8*c^3*d^5*e + 34*a*c^2*d^3*e^3 + 3*a^2*c*d*e^5)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*
d^2 + a*e^2)*x)*sqrt(e*x + d))/(a^2*c^4*d^11*e^2 - 4*a^3*c^3*d^9*e^4 + 6*a^4*c^2*d^7*e^6 - 4*a^5*c*d^5*e^8 + a
^6*d^3*e^10 + (c^6*d^10*e^3 - 4*a*c^5*d^8*e^5 + 6*a^2*c^4*d^6*e^7 - 4*a^3*c^3*d^4*e^9 + a^4*c^2*d^2*e^11)*x^5
+ (3*c^6*d^11*e^2 - 10*a*c^5*d^9*e^4 + 10*a^2*c^4*d^7*e^6 - 5*a^4*c^2*d^3*e^10 + 2*a^5*c*d*e^12)*x^4 + (3*c^6*
d^12*e - 6*a*c^5*d^10*e^3 - 5*a^2*c^4*d^8*e^5 + 20*a^3*c^3*d^6*e^7 - 15*a^4*c^2*d^4*e^9 + 2*a^5*c*d^2*e^11 + a
^6*e^13)*x^3 + (c^6*d^13 + 2*a*c^5*d^11*e^2 - 15*a^2*c^4*d^9*e^4 + 20*a^3*c^3*d^7*e^6 - 5*a^4*c^2*d^5*e^8 - 6*
a^5*c*d^3*e^10 + 3*a^6*d*e^12)*x^2 + (2*a*c^5*d^12*e - 5*a^2*c^4*d^10*e^3 + 10*a^4*c^2*d^6*e^7 - 10*a^5*c*d^4*
e^9 + 3*a^6*d^2*e^11)*x)]

Sympy [F]

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

[In]

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

Maxima [F]

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

[In]

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

Giac [A] (verification not implemented)

none

Time = 0.49 (sec) , antiderivative size = 469, normalized size of antiderivative = 1.43 \[ \int \frac {1}{\sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {1}{12} \, {\left (\frac {105 \, c^{2} d^{2} e \arctan \left (\frac {\sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}}}{\sqrt {c d^{2} e - a e^{3}}}\right )}{{\left (c^{4} d^{8} {\left | e \right |} - 4 \, a c^{3} d^{6} e^{2} {\left | e \right |} + 6 \, a^{2} c^{2} d^{4} e^{4} {\left | e \right |} - 4 \, a^{3} c d^{2} e^{6} {\left | e \right |} + a^{4} e^{8} {\left | e \right |}\right )} \sqrt {c d^{2} e - a e^{3}}} - \frac {8 \, {\left (c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4} - 9 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )} c^{2} d^{2} e\right )}}{{\left (c^{4} d^{8} {\left | e \right |} - 4 \, a c^{3} d^{6} e^{2} {\left | e \right |} + 6 \, a^{2} c^{2} d^{4} e^{4} {\left | e \right |} - 4 \, a^{3} c d^{2} e^{6} {\left | e \right |} + a^{4} e^{8} {\left | e \right |}\right )} {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}}} + \frac {3 \, {\left (13 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} c^{3} d^{4} e^{2} - 13 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} a c^{2} d^{2} e^{4} + 11 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c^{2} d^{2} e\right )}}{{\left (c^{4} d^{8} {\left | e \right |} - 4 \, a c^{3} d^{6} e^{2} {\left | e \right |} + 6 \, a^{2} c^{2} d^{4} e^{4} {\left | e \right |} - 4 \, a^{3} c d^{2} e^{6} {\left | e \right |} + a^{4} e^{8} {\left | e \right |}\right )} {\left (e x + d\right )}^{2} c^{2} d^{2} e^{2}}\right )} e^{2} \]

[In]

integrate(1/(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]

1/12*(105*c^2*d^2*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^4*d^8*abs(e) - 4
*a*c^3*d^6*e^2*abs(e) + 6*a^2*c^2*d^4*e^4*abs(e) - 4*a^3*c*d^2*e^6*abs(e) + a^4*e^8*abs(e))*sqrt(c*d^2*e - a*e
^3)) - 8*(c^3*d^4*e^2 - a*c^2*d^2*e^4 - 9*((e*x + d)*c*d*e - c*d^2*e + a*e^3)*c^2*d^2*e)/((c^4*d^8*abs(e) - 4*
a*c^3*d^6*e^2*abs(e) + 6*a^2*c^2*d^4*e^4*abs(e) - 4*a^3*c*d^2*e^6*abs(e) + a^4*e^8*abs(e))*((e*x + d)*c*d*e -
c*d^2*e + a*e^3)^(3/2)) + 3*(13*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*c^3*d^4*e^2 - 13*sqrt((e*x + d)*c*d*e
- c*d^2*e + a*e^3)*a*c^2*d^2*e^4 + 11*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*c^2*d^2*e)/((c^4*d^8*abs(e) -
4*a*c^3*d^6*e^2*abs(e) + 6*a^2*c^2*d^4*e^4*abs(e) - 4*a^3*c*d^2*e^6*abs(e) + a^4*e^8*abs(e))*(e*x + d)^2*c^2*d
^2*e^2))*e^2

Mupad [F(-1)]

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

[In]

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