\(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{(d+e x)^{13/2}} \, dx\) [2054]

Optimal result

Integrand size = 39, antiderivative size = 236 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{13/2}} \, dx=-\frac {5 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 e^3 (d+e x)^{3/2}}-\frac {5 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{12 e^2 (d+e x)^{7/2}}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{3 e (d+e x)^{11/2}}+\frac {5 c^3 d^3 \arctan \left (\frac {\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d^2-a e^2} \sqrt {d+e x}}\right )}{8 e^{7/2} \sqrt {c d^2-a e^2}} \]

[Out]

-5/12*c*d*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/e^2/(e*x+d)^(7/2)-1/3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2
)/e/(e*x+d)^(11/2)+5/8*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^(7/2)/(-a*e^2+c*d^2)^(1/2)-5/8*c^2*d^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/e^3/(e*x+d)^(3/2)

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {676, 674, 211} \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{13/2}} \, dx=\frac {5 c^3 d^3 \arctan \left (\frac {\sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d^2-a e^2}}\right )}{8 e^{7/2} \sqrt {c d^2-a e^2}}-\frac {5 c^2 d^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 e^3 (d+e x)^{3/2}}-\frac {5 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{12 e^2 (d+e x)^{7/2}}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{3 e (d+e x)^{11/2}} \]

[In]

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^(13/2),x]

[Out]

(-5*c^2*d^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(8*e^3*(d + e*x)^(3/2)) - (5*c*d*(a*d*e + (c*d^2 + a*
e^2)*x + c*d*e*x^2)^(3/2))/(12*e^2*(d + e*x)^(7/2)) - (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(3*e*(d +
e*x)^(11/2)) + (5*c^3*d^3*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]*Sq
rt[d + e*x])])/(8*e^(7/2)*Sqrt[c*d^2 - a*e^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]

Rubi steps \begin{align*} \text {integral}& = -\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{3 e (d+e x)^{11/2}}+\frac {(5 c d) \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{9/2}} \, dx}{6 e} \\ & = -\frac {5 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{12 e^2 (d+e x)^{7/2}}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{3 e (d+e x)^{11/2}}+\frac {\left (5 c^2 d^2\right ) \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^{5/2}} \, dx}{8 e^2} \\ & = -\frac {5 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 e^3 (d+e x)^{3/2}}-\frac {5 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{12 e^2 (d+e x)^{7/2}}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{3 e (d+e x)^{11/2}}+\frac {\left (5 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^3} \\ & = -\frac {5 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 e^3 (d+e x)^{3/2}}-\frac {5 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{12 e^2 (d+e x)^{7/2}}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{3 e (d+e x)^{11/2}}+\frac {\left (5 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 e^2} \\ & = -\frac {5 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 e^3 (d+e x)^{3/2}}-\frac {5 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{12 e^2 (d+e x)^{7/2}}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{3 e (d+e x)^{11/2}}+\frac {5 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^{7/2} \sqrt {c d^2-a e^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.62 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.73 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{13/2}} \, dx=\frac {\sqrt {(a e+c d x) (d+e x)} \left (-\sqrt {e} \left (8 a^2 e^4+2 a c d e^2 (5 d+13 e x)+c^2 d^2 \left (15 d^2+40 d e x+33 e^2 x^2\right )\right )+\frac {15 c^3 d^3 (d+e x)^3 \arctan \left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c d^2-a e^2}}\right )}{\sqrt {c d^2-a e^2} \sqrt {a e+c d x}}\right )}{24 e^{7/2} (d+e x)^{7/2}} \]

[In]

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^(13/2),x]

[Out]

(Sqrt[(a*e + c*d*x)*(d + e*x)]*(-(Sqrt[e]*(8*a^2*e^4 + 2*a*c*d*e^2*(5*d + 13*e*x) + c^2*d^2*(15*d^2 + 40*d*e*x
+ 33*e^2*x^2))) + (15*c^3*d^3*(d + e*x)^3*ArcTan[(Sqrt[e]*Sqrt[a*e + c*d*x])/Sqrt[c*d^2 - a*e^2]])/(Sqrt[c*d^
2 - a*e^2]*Sqrt[a*e + c*d*x])))/(24*e^(7/2)*(d + e*x)^(7/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(432\) vs. \(2(204)=408\).

Time = 2.49 (sec) , antiderivative size = 433, normalized size of antiderivative = 1.83

method	result	size
default	\(-\frac {\sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (15 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c^{3} d^{3} e^{3} x^{3}+45 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c^{3} d^{4} e^{2} x^{2}+45 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c^{3} d^{5} e x +15 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c^{3} d^{6}+33 c^{2} d^{2} e^{2} x^{2} \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}+26 a c d \,e^{3} x \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}+40 c^{2} d^{3} e x \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}+8 \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a^{2} e^{4}+10 \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a c \,d^{2} e^{2}+15 \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, c^{2} d^{4}\right )}{24 \left (e x +d \right )^{\frac {7}{2}} \sqrt {c d x +a e}\, e^{3} \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\)	\(433\)

[In]

int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(13/2),x,method=_RETURNVERBOSE)

[Out]

-1/24*((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^3*d^3*e^3*x^3+45*
arctanh(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+45*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*
d^2)*e)^(1/2))*c^3*d^5*e*x+15*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*c^3*d^6+33*c^2*d^2*e^2*x^2*
(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)+26*a*c*d*e^3*x*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)+40*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+10*(c*d*x+a*
e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)*a*c*d^2*e^2+15*(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)/(c*d*x+a*e)^(1/2)/e^3/((a*e^2-c*d^2)*e)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 837, normalized size of antiderivative = 3.55 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{13/2}} \, dx=\left [-\frac {15 \, {\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 (15 \, c^{3} d^{6} e - 5 \, a c^{2} d^{4} e^{3} - 2 \, a^{2} c d^{2} e^{5} - 8 \, a^{3} e^{7} + 33 \, {\left (c^{3} d^{4} e^{3} - a c^{2} d^{2} e^{5}\right )} x^{2} + 2 \, {\left (20 \, c^{3} d^{5} e^{2} - 7 \, a c^{2} d^{3} e^{4} - 13 \, 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 d^{6} e^{4} - a d^{4} e^{6} + {\left (c d^{2} e^{8} - a e^{10}\right )} x^{4} + 4 \, {\left (c d^{3} e^{7} - a d e^{9}\right )} x^{3} + 6 \, {\left (c d^{4} e^{6} - a d^{2} e^{8}\right )} x^{2} + 4 \, {\left (c d^{5} e^{5} - a d^{3} e^{7}\right )} x\right )}}, -\frac {15 \, {\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 (15 \, c^{3} d^{6} e - 5 \, a c^{2} d^{4} e^{3} - 2 \, a^{2} c d^{2} e^{5} - 8 \, a^{3} e^{7} + 33 \, {\left (c^{3} d^{4} e^{3} - a c^{2} d^{2} e^{5}\right )} x^{2} + 2 \, {\left (20 \, c^{3} d^{5} e^{2} - 7 \, a c^{2} d^{3} e^{4} - 13 \, 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 d^{6} e^{4} - a d^{4} e^{6} + {\left (c d^{2} e^{8} - a e^{10}\right )} x^{4} + 4 \, {\left (c d^{3} e^{7} - a d e^{9}\right )} x^{3} + 6 \, {\left (c d^{4} e^{6} - a d^{2} e^{8}\right )} x^{2} + 4 \, {\left (c d^{5} e^{5} - a d^{3} e^{7}\right )} x\right )}}\right ] \]

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(13/2),x, algorithm="fricas")

[Out]

[-1/48*(15*(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*(15*c^3*d^6*e - 5*a*c^2*d^4*e^3 - 2*a^2*c*d^2
*e^5 - 8*a^3*e^7 + 33*(c^3*d^4*e^3 - a*c^2*d^2*e^5)*x^2 + 2*(20*c^3*d^5*e^2 - 7*a*c^2*d^3*e^4 - 13*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*d^6*e^4 - a*d^4*e^6 + (c*d^2*e^8 - a*e^10)*
x^4 + 4*(c*d^3*e^7 - a*d*e^9)*x^3 + 6*(c*d^4*e^6 - a*d^2*e^8)*x^2 + 4*(c*d^5*e^5 - a*d^3*e^7)*x), -1/24*(15*(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)) + (15*c^3*d^6*e - 5*a*c^2*d^4*e^3 - 2*a^2*c*d^2*e^5 - 8*a^3*e^7 + 33*(c^3*d^4*e^3 - a*c^2*d^
2*e^5)*x^2 + 2*(20*c^3*d^5*e^2 - 7*a*c^2*d^3*e^4 - 13*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*d^6*e^4 - a*d^4*e^6 + (c*d^2*e^8 - a*e^10)*x^4 + 4*(c*d^3*e^7 - a*d*e^9)*x^3 + 6*(c*d^4*
e^6 - a*d^2*e^8)*x^2 + 4*(c*d^5*e^5 - a*d^3*e^7)*x)]

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{13/2}} \, dx=\text {Timed out} \]

[In]

integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2)/(e*x+d)**(13/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{13/2}} \, dx=\int { \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{\frac {13}{2}}} \,d x } \]

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(13/2),x, algorithm="maxima")

[Out]

integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)/(e*x + d)^(13/2), x)

Giac [A] (verification not implemented)

none

Time = 0.49 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.37 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{13/2}} \, dx=\frac {\frac {15 \, c^{4} d^{4} e {\left | e \right |} \arctan \left (\frac {\sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}}}{\sqrt {c d^{2} e - a e^{3}}}\right )}{\sqrt {c d^{2} e - a e^{3}}} - \frac {15 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} c^{6} d^{8} e^{3} {\left | e \right |} - 30 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} a c^{5} d^{6} e^{5} {\left | e \right |} + 15 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} a^{2} c^{4} d^{4} e^{7} {\left | e \right |} + 40 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c^{5} d^{6} e^{2} {\left | e \right |} - 40 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a c^{4} d^{4} e^{4} {\left | e \right |} + 33 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} c^{4} d^{4} e {\left | e \right |}}{{\left (e x + d\right )}^{3} c^{3} d^{3} e^{3}}}{24 \, c d e^{5}} \]

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(13/2),x, algorithm="giac")

[Out]

1/24*(15*c^4*d^4*e*abs(e)*arctan(sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)/sqrt(c*d^2*e - a*e^3))/sqrt(c*d^2*e -
a*e^3) - (15*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*c^6*d^8*e^3*abs(e) - 30*sqrt((e*x + d)*c*d*e - c*d^2*e +
a*e^3)*a*c^5*d^6*e^5*abs(e) + 15*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*a^2*c^4*d^4*e^7*abs(e) + 40*((e*x +
d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*c^5*d^6*e^2*abs(e) - 40*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a*c^4*d^4*
e^4*abs(e) + 33*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*c^4*d^4*e*abs(e))/((e*x + d)^3*c^3*d^3*e^3))/(c*d*e^
5)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{13/2}} \, dx=\int \frac {{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2}}{{\left (d+e\,x\right )}^{13/2}} \,d x \]

[In]

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)/(d + e*x)^(13/2),x)

[Out]

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)/(d + e*x)^(13/2), x)