\(\int \frac {\sqrt {a d e+(c d^2+a e^2) x+c d e x^2}}{(d+e x)^{9/2}} \, dx\) [2035]

Optimal result

Integrand size = 39, antiderivative size = 264 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^{9/2}} \, dx=-\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 e (d+e x)^{7/2}}+\frac {c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 e \left (c d^2-a e^2\right ) (d+e x)^{5/2}}+\frac {c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 e \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}+\frac {c^3 d^3 \arctan \left (\frac {\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d^2-a e^2} \sqrt {d+e x}}\right )}{8 e^{3/2} \left (c d^2-a e^2\right )^{5/2}} \]

[Out]

1/8*c^3*d^3*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))/e^(3/2)
/(-a*e^2+c*d^2)^(5/2)-1/3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/e/(e*x+d)^(7/2)+1/12*c*d*(a*d*e+(a*e^2+c*d^2
)*x+c*d*e*x^2)^(1/2)/e/(-a*e^2+c*d^2)/(e*x+d)^(5/2)+1/8*c^2*d^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/e/(-a*
e^2+c*d^2)^2/(e*x+d)^(3/2)

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 264, 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 = {676, 686, 674, 211} \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^{9/2}} \, dx=\frac {c^3 d^3 \arctan \left (\frac {\sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d^2-a e^2}}\right )}{8 e^{3/2} \left (c d^2-a e^2\right )^{5/2}}+\frac {c^2 d^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 e (d+e x)^{3/2} \left (c d^2-a e^2\right )^2}+\frac {c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{12 e (d+e x)^{5/2} \left (c d^2-a e^2\right )}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 e (d+e x)^{7/2}} \]

[In]

Int[Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(d + e*x)^(9/2),x]

[Out]

-1/3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(e*(d + e*x)^(7/2)) + (c*d*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c
*d*e*x^2])/(12*e*(c*d^2 - a*e^2)*(d + e*x)^(5/2)) + (c^2*d^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(8*e
*(c*d^2 - a*e^2)^2*(d + e*x)^(3/2)) + (c^3*d^3*ArcTan[(Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(S
qrt[c*d^2 - a*e^2]*Sqrt[d + e*x])])/(8*e^(3/2)*(c*d^2 - a*e^2)^(5/2))

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 676

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*((
a + b*x + c*x^2)^p/(e*(m + p + 1))), x] - Dist[c*(p/(e^2*(m + p + 1))), Int[(d + e*x)^(m + 2)*(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] && GtQ[
p, 0] && (LtQ[m, -2] || EqQ[m + 2*p + 1, 0]) && NeQ[m + p + 1, 0] && 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 {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 e (d+e x)^{7/2}}+\frac {(c d) \int \frac {1}{(d+e x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{6 e} \\ & = -\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 e (d+e x)^{7/2}}+\frac {c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 e \left (c d^2-a e^2\right ) (d+e x)^{5/2}}+\frac {\left (c^2 d^2\right ) \int \frac {1}{(d+e x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{8 e \left (c d^2-a e^2\right )} \\ & = -\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 e (d+e x)^{7/2}}+\frac {c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 e \left (c d^2-a e^2\right ) (d+e x)^{5/2}}+\frac {c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 e \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}+\frac {\left (c^3 d^3\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 e \left (c d^2-a e^2\right )^2} \\ & = -\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 e (d+e x)^{7/2}}+\frac {c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 e \left (c d^2-a e^2\right ) (d+e x)^{5/2}}+\frac {c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 e \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}+\frac {\left (c^3 d^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 )^2} \\ & = -\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 e (d+e x)^{7/2}}+\frac {c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 e \left (c d^2-a e^2\right ) (d+e x)^{5/2}}+\frac {c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 e \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}+\frac {c^3 d^3 \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 e^{3/2} \left (c d^2-a e^2\right )^{5/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.54 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.76 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^{9/2}} \, dx=\frac {\sqrt {(a e+c d x) (d+e x)} \left (\sqrt {e} \sqrt {c d^2-a e^2} \sqrt {a e+c d x} \left (-8 a^2 e^4+2 a c d e^2 (7 d-e x)+c^2 d^2 \left (-3 d^2+8 d e x+3 e^2 x^2\right )\right )+3 c^3 d^3 (d+e x)^3 \arctan \left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c d^2-a e^2}}\right )\right )}{24 e^{3/2} \left (c d^2-a e^2\right )^{5/2} \sqrt {a e+c d x} (d+e x)^{7/2}} \]

[In]

Integrate[Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(d + e*x)^(9/2),x]

[Out]

(Sqrt[(a*e + c*d*x)*(d + e*x)]*(Sqrt[e]*Sqrt[c*d^2 - a*e^2]*Sqrt[a*e + c*d*x]*(-8*a^2*e^4 + 2*a*c*d*e^2*(7*d -
e*x) + c^2*d^2*(-3*d^2 + 8*d*e*x + 3*e^2*x^2)) + 3*c^3*d^3*(d + e*x)^3*ArcTan[(Sqrt[e]*Sqrt[a*e + c*d*x])/Sqr
t[c*d^2 - a*e^2]]))/(24*e^(3/2)*(c*d^2 - a*e^2)^(5/2)*Sqrt[a*e + c*d*x]*(d + e*x)^(7/2))

Maple [A] (verified)

Time = 2.70 (sec) , antiderivative size = 447, normalized size of antiderivative = 1.69

method	result	size
default	\(-\frac {\sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (3 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c^{3} d^{3} e^{3} x^{3}+9 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c^{3} d^{4} e^{2} x^{2}+9 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c^{3} d^{5} e x +3 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c^{3} d^{6}-3 c^{2} d^{2} e^{2} x^{2} \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}+2 a c d \,e^{3} x \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}-8 c^{2} d^{3} e x \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}+8 \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a^{2} e^{4}-14 \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a c \,d^{2} e^{2}+3 \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, c^{2} d^{4}\right )}{24 \left (e x +d \right )^{\frac {7}{2}} \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, e \left (e^{2} a -c \,d^{2}\right )^{2} \sqrt {c d x +a e}}\)	\(447\)

[In]

int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)^(9/2),x,method=_RETURNVERBOSE)

[Out]

-1/24*((c*d*x+a*e)*(e*x+d))^(1/2)*(3*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*c^3*d^3*e^3*x^3+9*ar
ctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*c^3*d^4*e^2*x^2+9*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2
)*e)^(1/2))*c^3*d^5*e*x+3*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*c^3*d^6-3*c^2*d^2*e^2*x^2*(c*d*
x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)+2*a*c*d*e^3*x*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)-8*c^2*d^3*e*x*(c*
d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)+8*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)*a^2*e^4-14*(c*d*x+a*e)^(1/2
)*((a*e^2-c*d^2)*e)^(1/2)*a*c*d^2*e^2+3*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)*c^2*d^4)/(e*x+d)^(7/2)/((a*e
^2-c*d^2)*e)^(1/2)/e/(a*e^2-c*d^2)^2/(c*d*x+a*e)^(1/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 547 vs. \(2 (232) = 464\).

Time = 0.34 (sec) , antiderivative size = 1115, normalized size of antiderivative = 4.22 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^{9/2}} \, dx=\left [-\frac {3 \, {\left (c^{3} d^{3} e^{4} x^{4} + 4 \, c^{3} d^{4} e^{3} x^{3} + 6 \, c^{3} d^{5} e^{2} x^{2} + 4 \, c^{3} d^{6} e x + c^{3} d^{7}\right )} \sqrt {-c d^{2} e + a e^{3}} \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} \sqrt {-c d^{2} e + a e^{3}} \sqrt {e x + d}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) + 2 \, {\left (3 \, c^{3} d^{6} e - 17 \, a c^{2} d^{4} e^{3} + 22 \, a^{2} c d^{2} e^{5} - 8 \, a^{3} e^{7} - 3 \, {\left (c^{3} d^{4} e^{3} - a c^{2} d^{2} e^{5}\right )} x^{2} - 2 \, {\left (4 \, c^{3} d^{5} e^{2} - 5 \, a c^{2} d^{3} e^{4} + a^{2} c d e^{6}\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}}{48 \, {\left (c^{3} d^{10} e^{2} - 3 \, a c^{2} d^{8} e^{4} + 3 \, a^{2} c d^{6} e^{6} - a^{3} d^{4} e^{8} + {\left (c^{3} d^{6} e^{6} - 3 \, a c^{2} d^{4} e^{8} + 3 \, a^{2} c d^{2} e^{10} - a^{3} e^{12}\right )} x^{4} + 4 \, {\left (c^{3} d^{7} e^{5} - 3 \, a c^{2} d^{5} e^{7} + 3 \, a^{2} c d^{3} e^{9} - a^{3} d e^{11}\right )} x^{3} + 6 \, {\left (c^{3} d^{8} e^{4} - 3 \, a c^{2} d^{6} e^{6} + 3 \, a^{2} c d^{4} e^{8} - a^{3} d^{2} e^{10}\right )} x^{2} + 4 \, {\left (c^{3} d^{9} e^{3} - 3 \, a c^{2} d^{7} e^{5} + 3 \, a^{2} c d^{5} e^{7} - a^{3} d^{3} e^{9}\right )} x\right )}}, -\frac {3 \, {\left (c^{3} d^{3} e^{4} x^{4} + 4 \, c^{3} d^{4} e^{3} x^{3} + 6 \, c^{3} d^{5} e^{2} x^{2} + 4 \, c^{3} d^{6} e x + c^{3} d^{7}\right )} \sqrt {c d^{2} e - a e^{3}} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {c d^{2} e - a e^{3}} \sqrt {e x + d}}{c d e^{2} x^{2} + a d e^{2} + {\left (c d^{2} e + a e^{3}\right )} x}\right ) + {\left (3 \, c^{3} d^{6} e - 17 \, a c^{2} d^{4} e^{3} + 22 \, a^{2} c d^{2} e^{5} - 8 \, a^{3} e^{7} - 3 \, {\left (c^{3} d^{4} e^{3} - a c^{2} d^{2} e^{5}\right )} x^{2} - 2 \, {\left (4 \, c^{3} d^{5} e^{2} - 5 \, a c^{2} d^{3} e^{4} + a^{2} c d e^{6}\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}}{24 \, {\left (c^{3} d^{10} e^{2} - 3 \, a c^{2} d^{8} e^{4} + 3 \, a^{2} c d^{6} e^{6} - a^{3} d^{4} e^{8} + {\left (c^{3} d^{6} e^{6} - 3 \, a c^{2} d^{4} e^{8} + 3 \, a^{2} c d^{2} e^{10} - a^{3} e^{12}\right )} x^{4} + 4 \, {\left (c^{3} d^{7} e^{5} - 3 \, a c^{2} d^{5} e^{7} + 3 \, a^{2} c d^{3} e^{9} - a^{3} d e^{11}\right )} x^{3} + 6 \, {\left (c^{3} d^{8} e^{4} - 3 \, a c^{2} d^{6} e^{6} + 3 \, a^{2} c d^{4} e^{8} - a^{3} d^{2} e^{10}\right )} x^{2} + 4 \, {\left (c^{3} d^{9} e^{3} - 3 \, a c^{2} d^{7} e^{5} + 3 \, a^{2} c d^{5} e^{7} - a^{3} d^{3} e^{9}\right )} x\right )}}\right ] \]

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)^(9/2),x, algorithm="fricas")

[Out]

[-1/48*(3*(c^3*d^3*e^4*x^4 + 4*c^3*d^4*e^3*x^3 + 6*c^3*d^5*e^2*x^2 + 4*c^3*d^6*e*x + c^3*d^7)*sqrt(-c*d^2*e +
a*e^3)*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)*sqrt(
-c*d^2*e + a*e^3)*sqrt(e*x + d))/(e^2*x^2 + 2*d*e*x + d^2)) + 2*(3*c^3*d^6*e - 17*a*c^2*d^4*e^3 + 22*a^2*c*d^2
*e^5 - 8*a^3*e^7 - 3*(c^3*d^4*e^3 - a*c^2*d^2*e^5)*x^2 - 2*(4*c^3*d^5*e^2 - 5*a*c^2*d^3*e^4 + a^2*c*d*e^6)*x)*
sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d))/(c^3*d^10*e^2 - 3*a*c^2*d^8*e^4 + 3*a^2*c*d^6*e^6 -
a^3*d^4*e^8 + (c^3*d^6*e^6 - 3*a*c^2*d^4*e^8 + 3*a^2*c*d^2*e^10 - a^3*e^12)*x^4 + 4*(c^3*d^7*e^5 - 3*a*c^2*d^
5*e^7 + 3*a^2*c*d^3*e^9 - a^3*d*e^11)*x^3 + 6*(c^3*d^8*e^4 - 3*a*c^2*d^6*e^6 + 3*a^2*c*d^4*e^8 - a^3*d^2*e^10)
*x^2 + 4*(c^3*d^9*e^3 - 3*a*c^2*d^7*e^5 + 3*a^2*c*d^5*e^7 - a^3*d^3*e^9)*x), -1/24*(3*(c^3*d^3*e^4*x^4 + 4*c^3
*d^4*e^3*x^3 + 6*c^3*d^5*e^2*x^2 + 4*c^3*d^6*e*x + c^3*d^7)*sqrt(c*d^2*e - a*e^3)*arctan(sqrt(c*d*e*x^2 + a*d*
e + (c*d^2 + a*e^2)*x)*sqrt(c*d^2*e - a*e^3)*sqrt(e*x + d)/(c*d*e^2*x^2 + a*d*e^2 + (c*d^2*e + a*e^3)*x)) + (3
*c^3*d^6*e - 17*a*c^2*d^4*e^3 + 22*a^2*c*d^2*e^5 - 8*a^3*e^7 - 3*(c^3*d^4*e^3 - a*c^2*d^2*e^5)*x^2 - 2*(4*c^3*
d^5*e^2 - 5*a*c^2*d^3*e^4 + a^2*c*d*e^6)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d))/(c^3*d^
10*e^2 - 3*a*c^2*d^8*e^4 + 3*a^2*c*d^6*e^6 - a^3*d^4*e^8 + (c^3*d^6*e^6 - 3*a*c^2*d^4*e^8 + 3*a^2*c*d^2*e^10 -
a^3*e^12)*x^4 + 4*(c^3*d^7*e^5 - 3*a*c^2*d^5*e^7 + 3*a^2*c*d^3*e^9 - a^3*d*e^11)*x^3 + 6*(c^3*d^8*e^4 - 3*a*c
^2*d^6*e^6 + 3*a^2*c*d^4*e^8 - a^3*d^2*e^10)*x^2 + 4*(c^3*d^9*e^3 - 3*a*c^2*d^7*e^5 + 3*a^2*c*d^5*e^7 - a^3*d^
3*e^9)*x)]

Sympy [F(-1)]

Timed out. \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^{9/2}} \, dx=\text {Timed out} \]

[In]

integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2)/(e*x+d)**(9/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^{9/2}} \, dx=\int { \frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{{\left (e x + d\right )}^{\frac {9}{2}}} \,d x } \]

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)^(9/2),x, algorithm="maxima")

[Out]

integrate(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(e*x + d)^(9/2), x)

Giac [A] (verification not implemented)

none

Time = 0.55 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.38 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^{9/2}} \, dx=\frac {{\left (\frac {3 \, c^{4} d^{4} e \arctan \left (\frac {\sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}}}{\sqrt {c d^{2} e - a e^{3}}}\right )}{{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt {c d^{2} e - a e^{3}}} - \frac {3 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} c^{6} d^{8} e^{3} - 6 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} a c^{5} d^{6} e^{5} + 3 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} a^{2} c^{4} d^{4} e^{7} - 8 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c^{5} d^{6} e^{2} + 8 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a c^{4} d^{4} e^{4} - 3 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} c^{4} d^{4} e}{{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} {\left (e x + d\right )}^{3} c^{3} d^{3} e^{3}}\right )} {\left | e \right |}}{24 \, c d e^{3}} \]

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)^(9/2),x, algorithm="giac")

[Out]

1/24*(3*c^4*d^4*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^2*d^4 - 2*a*c*d^2*
e^2 + a^2*e^4)*sqrt(c*d^2*e - a*e^3)) - (3*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*c^6*d^8*e^3 - 6*sqrt((e*x +
d)*c*d*e - c*d^2*e + a*e^3)*a*c^5*d^6*e^5 + 3*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*a^2*c^4*d^4*e^7 - 8*((e
*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*c^5*d^6*e^2 + 8*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a*c^4*d^4*e^4
- 3*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*c^4*d^4*e)/((c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*(e*x + d)^3*c^3
*d^3*e^3))*abs(e)/(c*d*e^3)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^{9/2}} \, dx=\int \frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{{\left (d+e\,x\right )}^{9/2}} \,d x \]

[In]

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)/(d + e*x)^(9/2),x)

[Out]

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)/(d + e*x)^(9/2), x)