\(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{(d+e x)^{11/2}} \, dx\) [2053]

Optimal result

Integrand size = 39, antiderivative size = 236 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx=\frac {15 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 e^3 \sqrt {d+e x}}-\frac {5 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{4 e^2 (d+e x)^{5/2}}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{2 e (d+e x)^{9/2}}-\frac {15 c^2 d^2 \sqrt {c d^2-a e^2} \arctan \left (\frac {\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d^2-a e^2} \sqrt {d+e x}}\right )}{4 e^{7/2}} \]

[Out]

-5/4*c*d*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/e^2/(e*x+d)^(5/2)-1/2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)
/e/(e*x+d)^(9/2)-15/4*c^2*d^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)^(1/2)/e^(7/2)+15/4*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)^(1/2)

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 = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {676, 678, 674, 211} \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx=-\frac {15 c^2 d^2 \sqrt {c d^2-a e^2} \arctan \left (\frac {\sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d^2-a e^2}}\right )}{4 e^{7/2}}+\frac {15 c^2 d^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 e^3 \sqrt {d+e x}}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{2 e (d+e x)^{9/2}}-\frac {5 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 e^2 (d+e x)^{5/2}} \]

[In]

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^(11/2),x]

[Out]

(15*c^2*d^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(4*e^3*Sqrt[d + e*x]) - (5*c*d*(a*d*e + (c*d^2 + a*e^
2)*x + c*d*e*x^2)^(3/2))/(4*e^2*(d + e*x)^(5/2)) - (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(2*e*(d + e*x
)^(9/2)) - (15*c^2*d^2*Sqrt[c*d^2 - a*e^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*e^(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 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 678

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 + 2*p + 1))), x] - Dist[p*((2*c*d - b*e)/(e^2*(m + 2*p + 1))), 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] && GtQ[p, 0] && (LeQ[-2, m, 0] || EqQ[m + p + 1, 0]) && NeQ[m + 2*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}}{2 e (d+e x)^{9/2}}+\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)^{7/2}} \, dx}{4 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}}{4 e^2 (d+e x)^{5/2}}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{2 e (d+e x)^{9/2}}+\frac {\left (15 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)^{3/2}} \, dx}{8 e^2} \\ & = \frac {15 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 e^3 \sqrt {d+e x}}-\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}}{4 e^2 (d+e x)^{5/2}}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{2 e (d+e x)^{9/2}}-\frac {\left (15 c^2 d^2 \left (c d^2-a e^2\right )\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 e^3} \\ & = \frac {15 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 e^3 \sqrt {d+e x}}-\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}}{4 e^2 (d+e x)^{5/2}}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{2 e (d+e x)^{9/2}}-\frac {\left (15 c^2 d^2 \left (c d^2-a e^2\right )\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 e^2} \\ & = \frac {15 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 e^3 \sqrt {d+e x}}-\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}}{4 e^2 (d+e x)^{5/2}}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{2 e (d+e x)^{9/2}}-\frac {15 c^2 d^2 \sqrt {c d^2-a e^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 e^{7/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.42 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.78 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx=\frac {\sqrt {(a e+c d x) (d+e x)} \left (\sqrt {e} \sqrt {a e+c d x} \left (-2 a^2 e^4-a c d e^2 (5 d+9 e x)+c^2 d^2 \left (15 d^2+25 d e x+8 e^2 x^2\right )\right )-15 c^2 d^2 \sqrt {c d^2-a e^2} (d+e x)^2 \arctan \left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c d^2-a e^2}}\right )\right )}{4 e^{7/2} \sqrt {a e+c d x} (d+e x)^{5/2}} \]

[In]

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^(11/2),x]

[Out]

(Sqrt[(a*e + c*d*x)*(d + e*x)]*(Sqrt[e]*Sqrt[a*e + c*d*x]*(-2*a^2*e^4 - a*c*d*e^2*(5*d + 9*e*x) + c^2*d^2*(15*
d^2 + 25*d*e*x + 8*e^2*x^2)) - 15*c^2*d^2*Sqrt[c*d^2 - a*e^2]*(d + e*x)^2*ArcTan[(Sqrt[e]*Sqrt[a*e + c*d*x])/S
qrt[c*d^2 - a*e^2]]))/(4*e^(7/2)*Sqrt[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. \(516\) vs. \(2(204)=408\).

Time = 3.14 (sec) , antiderivative size = 517, normalized size of antiderivative = 2.19

method	result	size
default	\(-\frac {\sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (15 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) a \,c^{2} d^{2} e^{4} x^{2}-15 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c^{3} d^{4} e^{2} x^{2}+30 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) a \,c^{2} d^{3} e^{3} x -30 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c^{3} d^{5} e x +15 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) a \,c^{2} d^{4} e^{2}-15 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c^{3} d^{6}-8 c^{2} d^{2} e^{2} x^{2} \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}+9 a c d \,e^{3} x \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}-25 c^{2} d^{3} e x \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}+2 \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a^{2} e^{4}+5 \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a c \,d^{2} e^{2}-15 \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, c^{2} d^{4}\right )}{4 \left (e x +d \right )^{\frac {5}{2}} \sqrt {c d x +a e}\, e^{3} \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\)	\(517\)

[In]

int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(11/2),x,method=_RETURNVERBOSE)

[Out]

-1/4*((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))*a*c^2*d^2*e^4*x^2-15
*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+30*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c
*d^2)*e)^(1/2))*a*c^2*d^3*e^3*x-30*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))*a*c^2*d^4*e^2-15*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(
1/2))*c^3*d^6-8*c^2*d^2*e^2*x^2*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)+9*a*c*d*e^3*x*(c*d*x+a*e)^(1/2)*((a*
e^2-c*d^2)*e)^(1/2)-25*c^2*d^3*e*x*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)+2*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2
)*e)^(1/2)*a^2*e^4+5*(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)^(5/2)/(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 = 0.69 (sec) , antiderivative size = 562, normalized size of antiderivative = 2.38 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx=\left [\frac {15 \, {\left (c^{2} d^{2} e^{3} x^{3} + 3 \, c^{2} d^{3} e^{2} x^{2} + 3 \, c^{2} d^{4} e x + c^{2} d^{5}\right )} \sqrt {-\frac {c d^{2} - a e^{2}}{e}} \log \left (-\frac {c d e^{2} x^{2} + 2 \, a e^{3} x - c d^{3} + 2 \, a d e^{2} - 2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d} e \sqrt {-\frac {c d^{2} - a e^{2}}{e}}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) + 2 \, {\left (8 \, c^{2} d^{2} e^{2} x^{2} + 15 \, c^{2} d^{4} - 5 \, a c d^{2} e^{2} - 2 \, a^{2} e^{4} + {\left (25 \, c^{2} d^{3} e - 9 \, a c d e^{3}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{8 \, {\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}}, \frac {15 \, {\left (c^{2} d^{2} e^{3} x^{3} + 3 \, c^{2} d^{3} e^{2} x^{2} + 3 \, c^{2} d^{4} e x + c^{2} d^{5}\right )} \sqrt {\frac {c d^{2} - a e^{2}}{e}} \arctan \left (\frac {\sqrt {e x + d} \sqrt {\frac {c d^{2} - a e^{2}}{e}}}{\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}\right ) + {\left (8 \, c^{2} d^{2} e^{2} x^{2} + 15 \, c^{2} d^{4} - 5 \, a c d^{2} e^{2} - 2 \, a^{2} e^{4} + {\left (25 \, c^{2} d^{3} e - 9 \, a c d e^{3}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{4 \, {\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}}\right ] \]

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(11/2),x, algorithm="fricas")

[Out]

[1/8*(15*(c^2*d^2*e^3*x^3 + 3*c^2*d^3*e^2*x^2 + 3*c^2*d^4*e*x + c^2*d^5)*sqrt(-(c*d^2 - a*e^2)/e)*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(e*x + d)*e*sqrt(-(c
*d^2 - a*e^2)/e))/(e^2*x^2 + 2*d*e*x + d^2)) + 2*(8*c^2*d^2*e^2*x^2 + 15*c^2*d^4 - 5*a*c*d^2*e^2 - 2*a^2*e^4 +
(25*c^2*d^3*e - 9*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))/(e^6*x^3 + 3*d*e^5
*x^2 + 3*d^2*e^4*x + d^3*e^3), 1/4*(15*(c^2*d^2*e^3*x^3 + 3*c^2*d^3*e^2*x^2 + 3*c^2*d^4*e*x + c^2*d^5)*sqrt((c
*d^2 - a*e^2)/e)*arctan(sqrt(e*x + d)*sqrt((c*d^2 - a*e^2)/e)/sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)) + (
8*c^2*d^2*e^2*x^2 + 15*c^2*d^4 - 5*a*c*d^2*e^2 - 2*a^2*e^4 + (25*c^2*d^3*e - 9*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))/(e^6*x^3 + 3*d*e^5*x^2 + 3*d^2*e^4*x + d^3*e^3)]

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx=\text {Timed out} \]

[In]

integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2)/(e*x+d)**(11/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx=\int { \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{\frac {11}{2}}} \,d x } \]

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(11/2),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)^(11/2), x)

Giac [A] (verification not implemented)

none

Time = 0.44 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.42 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx=\frac {8 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} c^{3} d^{3} {\left | e \right |} - \frac {15 \, {\left (c^{4} d^{5} e {\left | e \right |} - a c^{3} d^{3} e^{3} {\left | e \right |}\right )} \arctan \left (\frac {\sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}}}{\sqrt {c d^{2} e - a e^{3}}}\right )}{\sqrt {c d^{2} e - a e^{3}}} + \frac {7 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} c^{5} d^{7} e^{2} {\left | e \right |} - 14 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} a c^{4} d^{5} e^{4} {\left | e \right |} + 7 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} a^{2} c^{3} d^{3} e^{6} {\left | e \right |} + 9 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c^{4} d^{5} e {\left | e \right |} - 9 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a c^{3} d^{3} e^{3} {\left | e \right |}}{{\left (e x + d\right )}^{2} c^{2} d^{2} e^{2}}}{4 \, c d e^{5}} \]

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(11/2),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx=\int \frac {{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2}}{{\left (d+e\,x\right )}^{11/2}} \,d x \]

[In]

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)/(d + e*x)^(11/2),x)

[Out]

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)/(d + e*x)^(11/2), x)