\(\int (a d e+(c d^2+a e^2) x+c d e x^2)^{5/2} \, dx\) [1935]

Optimal result

Integrand size = 29, antiderivative size = 305 \[ \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx=\frac {5 \left (c d^2-a e^2\right )^4 \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{512 c^3 d^3 e^3}-\frac {5 \left (c d^2-a e^2\right )^2 \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{192 c^2 d^2 e^2}+\frac {\left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{12 c d e}-\frac {5 \left (c d^2-a e^2\right )^6 \text {arctanh}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{1024 c^{7/2} d^{7/2} e^{7/2}} \]

[Out]

-5/192*(-a*e^2+c*d^2)^2*(2*c*d*e*x+a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/c^2/d^2/e^2+1/12*(2*c*
d*e*x+a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/c/d/e-5/1024*(-a*e^2+c*d^2)^6*arctanh(1/2*(2*c*d*e*
x+a*e^2+c*d^2)/c^(1/2)/d^(1/2)/e^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/c^(7/2)/d^(7/2)/e^(7/2)+5/512*
(-a*e^2+c*d^2)^4*(2*c*d*e*x+a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^3/d^3/e^3

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {626, 635, 212} \[ \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx=-\frac {5 \left (c d^2-a e^2\right )^6 \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{1024 c^{7/2} d^{7/2} e^{7/2}}+\frac {5 \left (c d^2-a e^2\right )^4 \left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{512 c^3 d^3 e^3}-\frac {5 \left (c d^2-a e^2\right )^2 \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{192 c^2 d^2 e^2}+\frac {\left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{12 c d e} \]

[In]

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

[Out]

(5*(c*d^2 - a*e^2)^4*(c*d^2 + a*e^2 + 2*c*d*e*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(512*c^3*d^3*e^3
) - (5*(c*d^2 - a*e^2)^2*(c*d^2 + a*e^2 + 2*c*d*e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(192*c^2*d
^2*e^2) + ((c*d^2 + a*e^2 + 2*c*d*e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(12*c*d*e) - (5*(c*d^2 -
a*e^2)^6*ArcTanh[(c*d^2 + a*e^2 + 2*c*d*e*x)/(2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*
e*x^2])])/(1024*c^(7/2)*d^(7/2)*e^(7/2))

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 626

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x)*((a + b*x + c*x^2)^p/(2*c*(2*p + 1
))), x] - Dist[p*((b^2 - 4*a*c)/(2*c*(2*p + 1))), Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]

Rule 635

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{12 c d e}-\frac {\left (5 \left (c d^2-a e^2\right )^2\right ) \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx}{24 c d e} \\ & = -\frac {5 \left (c d^2-a e^2\right )^2 \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{192 c^2 d^2 e^2}+\frac {\left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{12 c d e}+\frac {\left (5 \left (c d^2-a e^2\right )^4\right ) \int \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{128 c^2 d^2 e^2} \\ & = \frac {5 \left (c d^2-a e^2\right )^4 \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{512 c^3 d^3 e^3}-\frac {5 \left (c d^2-a e^2\right )^2 \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{192 c^2 d^2 e^2}+\frac {\left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{12 c d e}-\frac {\left (5 \left (c d^2-a e^2\right )^6\right ) \int \frac {1}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{1024 c^3 d^3 e^3} \\ & = \frac {5 \left (c d^2-a e^2\right )^4 \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{512 c^3 d^3 e^3}-\frac {5 \left (c d^2-a e^2\right )^2 \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{192 c^2 d^2 e^2}+\frac {\left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{12 c d e}-\frac {\left (5 \left (c d^2-a e^2\right )^6\right ) \text {Subst}\left (\int \frac {1}{4 c d e-x^2} \, dx,x,\frac {c d^2+a e^2+2 c d e x}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{512 c^3 d^3 e^3} \\ & = \frac {5 \left (c d^2-a e^2\right )^4 \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{512 c^3 d^3 e^3}-\frac {5 \left (c d^2-a e^2\right )^2 \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{192 c^2 d^2 e^2}+\frac {\left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{12 c d e}-\frac {5 \left (c d^2-a e^2\right )^6 \tanh ^{-1}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{1024 c^{7/2} d^{7/2} e^{7/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.86 (sec) , antiderivative size = 361, normalized size of antiderivative = 1.18 \[ \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx=\frac {((a e+c d x) (d+e x))^{5/2} \left (\frac {\sqrt {c} \sqrt {d} \sqrt {e} \left (15 a^5 e^{10}-5 a^4 c d e^8 (17 d+2 e x)+2 a^3 c^2 d^2 e^6 \left (99 d^2+28 d e x+4 e^2 x^2\right )+6 a^2 c^3 d^3 e^4 \left (33 d^3+198 d^2 e x+212 d e^2 x^2+72 e^3 x^3\right )+a c^4 d^4 e^2 \left (-85 d^4+56 d^3 e x+1272 d^2 e^2 x^2+1696 d e^3 x^3+640 e^4 x^4\right )+c^5 d^5 \left (15 d^5-10 d^4 e x+8 d^3 e^2 x^2+432 d^2 e^3 x^3+640 d e^4 x^4+256 e^5 x^5\right )\right )}{(a e+c d x)^2 (d+e x)^2}-\frac {15 \left (c d^2-a e^2\right )^6 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \sqrt {a e+c d x}}\right )}{(a e+c d x)^{5/2} (d+e x)^{5/2}}\right )}{1536 c^{7/2} d^{7/2} e^{7/2}} \]

[In]

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

[Out]

(((a*e + c*d*x)*(d + e*x))^(5/2)*((Sqrt[c]*Sqrt[d]*Sqrt[e]*(15*a^5*e^10 - 5*a^4*c*d*e^8*(17*d + 2*e*x) + 2*a^3
*c^2*d^2*e^6*(99*d^2 + 28*d*e*x + 4*e^2*x^2) + 6*a^2*c^3*d^3*e^4*(33*d^3 + 198*d^2*e*x + 212*d*e^2*x^2 + 72*e^
3*x^3) + a*c^4*d^4*e^2*(-85*d^4 + 56*d^3*e*x + 1272*d^2*e^2*x^2 + 1696*d*e^3*x^3 + 640*e^4*x^4) + c^5*d^5*(15*
d^5 - 10*d^4*e*x + 8*d^3*e^2*x^2 + 432*d^2*e^3*x^3 + 640*d*e^4*x^4 + 256*e^5*x^5)))/((a*e + c*d*x)^2*(d + e*x)
^2) - (15*(c*d^2 - a*e^2)^6*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[a*e + c*d*x])])/((a*e + c*d*
x)^(5/2)*(d + e*x)^(5/2))))/(1536*c^(7/2)*d^(7/2)*e^(7/2))

Maple [A] (verified)

Time = 2.59 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.11

method	result	size
default	\(\frac {\left (2 x c d e +e^{2} a +c \,d^{2}\right ) {\left (a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}\right )}^{\frac {5}{2}}}{12 c d e}+\frac {5 \left (4 a c \,d^{2} e^{2}-\left (e^{2} a +c \,d^{2}\right )^{2}\right ) \left (\frac {\left (2 x c d e +e^{2} a +c \,d^{2}\right ) {\left (a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}\right )}^{\frac {3}{2}}}{8 c d e}+\frac {3 \left (4 a c \,d^{2} e^{2}-\left (e^{2} a +c \,d^{2}\right )^{2}\right ) \left (\frac {\left (2 x c d e +e^{2} a +c \,d^{2}\right ) \sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}}{4 c d e}+\frac {\left (4 a c \,d^{2} e^{2}-\left (e^{2} a +c \,d^{2}\right )^{2}\right ) \ln \left (\frac {\frac {1}{2} e^{2} a +\frac {1}{2} c \,d^{2}+x c d e}{\sqrt {c d e}}+\sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}\right )}{8 c d e \sqrt {c d e}}\right )}{16 c d e}\right )}{24 c d e}\)	\(340\)

[In]

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

[Out]

1/12*(2*c*d*e*x+a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/c/d/e+5/24*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2
)/c/d/e*(1/8*(2*c*d*e*x+a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/c/d/e+3/16*(4*a*c*d^2*e^2-(a*e^2+
c*d^2)^2)/c/d/e*(1/4*(2*c*d*e*x+a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c/d/e+1/8*(4*a*c*d^2*e^2-
(a*e^2+c*d^2)^2)/c/d/e*ln((1/2*e^2*a+1/2*c*d^2+x*c*d*e)/(c*d*e)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))
/(c*d*e)^(1/2)))

Fricas [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 1034, normalized size of antiderivative = 3.39 \[ \int \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^{6} d^{12} - 6 \, a c^{5} d^{10} e^{2} + 15 \, a^{2} c^{4} d^{8} e^{4} - 20 \, a^{3} c^{3} d^{6} e^{6} + 15 \, a^{4} c^{2} d^{4} e^{8} - 6 \, a^{5} c d^{2} e^{10} + a^{6} e^{12}\right )} \sqrt {c d e} \log \left (8 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4} - 4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {c d e} + 8 \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right ) + 4 \, {\left (256 \, c^{6} d^{6} e^{6} x^{5} + 15 \, c^{6} d^{11} e - 85 \, a c^{5} d^{9} e^{3} + 198 \, a^{2} c^{4} d^{7} e^{5} + 198 \, a^{3} c^{3} d^{5} e^{7} - 85 \, a^{4} c^{2} d^{3} e^{9} + 15 \, a^{5} c d e^{11} + 640 \, {\left (c^{6} d^{7} e^{5} + a c^{5} d^{5} e^{7}\right )} x^{4} + 16 \, {\left (27 \, c^{6} d^{8} e^{4} + 106 \, a c^{5} d^{6} e^{6} + 27 \, a^{2} c^{4} d^{4} e^{8}\right )} x^{3} + 8 \, {\left (c^{6} d^{9} e^{3} + 159 \, a c^{5} d^{7} e^{5} + 159 \, a^{2} c^{4} d^{5} e^{7} + a^{3} c^{3} d^{3} e^{9}\right )} x^{2} - 2 \, {\left (5 \, c^{6} d^{10} e^{2} - 28 \, a c^{5} d^{8} e^{4} - 594 \, a^{2} c^{4} d^{6} e^{6} - 28 \, a^{3} c^{3} d^{4} e^{8} + 5 \, a^{4} c^{2} d^{2} e^{10}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{6144 \, c^{4} d^{4} e^{4}}, \frac {15 \, {\left (c^{6} d^{12} - 6 \, a c^{5} d^{10} e^{2} + 15 \, a^{2} c^{4} d^{8} e^{4} - 20 \, a^{3} c^{3} d^{6} e^{6} + 15 \, a^{4} c^{2} d^{4} e^{8} - 6 \, a^{5} c d^{2} e^{10} + a^{6} e^{12}\right )} \sqrt {-c d e} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {-c d e}}{2 \, {\left (c^{2} d^{2} e^{2} x^{2} + a c d^{2} e^{2} + {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right )}}\right ) + 2 \, {\left (256 \, c^{6} d^{6} e^{6} x^{5} + 15 \, c^{6} d^{11} e - 85 \, a c^{5} d^{9} e^{3} + 198 \, a^{2} c^{4} d^{7} e^{5} + 198 \, a^{3} c^{3} d^{5} e^{7} - 85 \, a^{4} c^{2} d^{3} e^{9} + 15 \, a^{5} c d e^{11} + 640 \, {\left (c^{6} d^{7} e^{5} + a c^{5} d^{5} e^{7}\right )} x^{4} + 16 \, {\left (27 \, c^{6} d^{8} e^{4} + 106 \, a c^{5} d^{6} e^{6} + 27 \, a^{2} c^{4} d^{4} e^{8}\right )} x^{3} + 8 \, {\left (c^{6} d^{9} e^{3} + 159 \, a c^{5} d^{7} e^{5} + 159 \, a^{2} c^{4} d^{5} e^{7} + a^{3} c^{3} d^{3} e^{9}\right )} x^{2} - 2 \, {\left (5 \, c^{6} d^{10} e^{2} - 28 \, a c^{5} d^{8} e^{4} - 594 \, a^{2} c^{4} d^{6} e^{6} - 28 \, a^{3} c^{3} d^{4} e^{8} + 5 \, a^{4} c^{2} d^{2} e^{10}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{3072 \, c^{4} d^{4} e^{4}}\right ] \]

[In]

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

[Out]

[1/6144*(15*(c^6*d^12 - 6*a*c^5*d^10*e^2 + 15*a^2*c^4*d^8*e^4 - 20*a^3*c^3*d^6*e^6 + 15*a^4*c^2*d^4*e^8 - 6*a^
5*c*d^2*e^10 + a^6*e^12)*sqrt(c*d*e)*log(8*c^2*d^2*e^2*x^2 + c^2*d^4 + 6*a*c*d^2*e^2 + a^2*e^4 - 4*sqrt(c*d*e*
x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(2*c*d*e*x + c*d^2 + a*e^2)*sqrt(c*d*e) + 8*(c^2*d^3*e + a*c*d*e^3)*x) + 4*(2
56*c^6*d^6*e^6*x^5 + 15*c^6*d^11*e - 85*a*c^5*d^9*e^3 + 198*a^2*c^4*d^7*e^5 + 198*a^3*c^3*d^5*e^7 - 85*a^4*c^2
*d^3*e^9 + 15*a^5*c*d*e^11 + 640*(c^6*d^7*e^5 + a*c^5*d^5*e^7)*x^4 + 16*(27*c^6*d^8*e^4 + 106*a*c^5*d^6*e^6 +
27*a^2*c^4*d^4*e^8)*x^3 + 8*(c^6*d^9*e^3 + 159*a*c^5*d^7*e^5 + 159*a^2*c^4*d^5*e^7 + a^3*c^3*d^3*e^9)*x^2 - 2*
(5*c^6*d^10*e^2 - 28*a*c^5*d^8*e^4 - 594*a^2*c^4*d^6*e^6 - 28*a^3*c^3*d^4*e^8 + 5*a^4*c^2*d^2*e^10)*x)*sqrt(c*
d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^4*d^4*e^4), 1/3072*(15*(c^6*d^12 - 6*a*c^5*d^10*e^2 + 15*a^2*c^4*d^8*
e^4 - 20*a^3*c^3*d^6*e^6 + 15*a^4*c^2*d^4*e^8 - 6*a^5*c*d^2*e^10 + a^6*e^12)*sqrt(-c*d*e)*arctan(1/2*sqrt(c*d*
e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(2*c*d*e*x + c*d^2 + a*e^2)*sqrt(-c*d*e)/(c^2*d^2*e^2*x^2 + a*c*d^2*e^2 + (
c^2*d^3*e + a*c*d*e^3)*x)) + 2*(256*c^6*d^6*e^6*x^5 + 15*c^6*d^11*e - 85*a*c^5*d^9*e^3 + 198*a^2*c^4*d^7*e^5 +
198*a^3*c^3*d^5*e^7 - 85*a^4*c^2*d^3*e^9 + 15*a^5*c*d*e^11 + 640*(c^6*d^7*e^5 + a*c^5*d^5*e^7)*x^4 + 16*(27*c
^6*d^8*e^4 + 106*a*c^5*d^6*e^6 + 27*a^2*c^4*d^4*e^8)*x^3 + 8*(c^6*d^9*e^3 + 159*a*c^5*d^7*e^5 + 159*a^2*c^4*d^
5*e^7 + a^3*c^3*d^3*e^9)*x^2 - 2*(5*c^6*d^10*e^2 - 28*a*c^5*d^8*e^4 - 594*a^2*c^4*d^6*e^6 - 28*a^3*c^3*d^4*e^8
+ 5*a^4*c^2*d^2*e^10)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^4*d^4*e^4)]

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 654 vs. \(2 (298) = 596\).

Time = 7.96 (sec) , antiderivative size = 6698, normalized size of antiderivative = 21.96 \[ \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

a**2*d**2*e**2*Piecewise(((x/2 + (a*e**2/4 + c*d**2/4)/(c*d*e))*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))
+ (a*d*e/2 - (a*e**2/4 + c*d**2/4)*(a*e**2 + c*d**2)/(2*c*d*e))*Piecewise((log(a*e**2 + c*d**2 + 2*c*d*e*x +
2*sqrt(c*d*e)*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2)))/sqrt(c*d*e), Ne(a*d*e - (a*e**2 + c*d**2)**2/(4*
c*d*e), 0)), ((x - (-a*e**2 - c*d**2)/(2*c*d*e))*log(x - (-a*e**2 - c*d**2)/(2*c*d*e))/sqrt(c*d*e*(x - (-a*e**
2 - c*d**2)/(2*c*d*e))**2), True)), Ne(c*d*e, 0)), (2*(a*d*e + x*(a*e**2 + c*d**2))**(3/2)/(3*(a*e**2 + c*d**2
)), Ne(a*e**2 + c*d**2, 0)), (x*sqrt(a*d*e), True)) + 2*a**2*d*e**3*Piecewise(((-a*(a*e**2/6 + c*d**2/6)/(2*c)
- (a*e**2 + c*d**2)*(a*d*e/3 - (a*e**2/6 + c*d**2/6)*(3*a*e**2/2 + 3*c*d**2/2)/(2*c*d*e))/(2*c*d*e))*Piecewis
e((log(a*e**2 + c*d**2 + 2*c*d*e*x + 2*sqrt(c*d*e)*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2)))/sqrt(c*d*e)
, Ne(a*d*e - (a*e**2 + c*d**2)**2/(4*c*d*e), 0)), ((x - (-a*e**2 - c*d**2)/(2*c*d*e))*log(x - (-a*e**2 - c*d**
2)/(2*c*d*e))/sqrt(c*d*e*(x - (-a*e**2 - c*d**2)/(2*c*d*e))**2), True)) + (x**2/3 + x*(a*e**2/6 + c*d**2/6)/(2
*c*d*e) + (a*d*e/3 - (a*e**2/6 + c*d**2/6)*(3*a*e**2/2 + 3*c*d**2/2)/(2*c*d*e))/(c*d*e))*sqrt(a*d*e + c*d*e*x*
*2 + x*(a*e**2 + c*d**2)), Ne(c*d*e, 0)), (2*(-a*d*e*(a*d*e + x*(a*e**2 + c*d**2))**(3/2)/3 + (a*d*e + x*(a*e*
*2 + c*d**2))**(5/2)/5)/(a*e**2 + c*d**2)**2, Ne(a*e**2 + c*d**2, 0)), (x**2*sqrt(a*d*e)/2, True)) + a**2*e**4
*Piecewise(((-a*(a*d*e/4 - (a*e**2/8 + c*d**2/8)*(5*a*e**2/2 + 5*c*d**2/2)/(3*c*d*e))/(2*c) - (a*e**2 + c*d**2
)*(-2*a*(a*e**2/8 + c*d**2/8)/(3*c) - (3*a*e**2/2 + 3*c*d**2/2)*(a*d*e/4 - (a*e**2/8 + c*d**2/8)*(5*a*e**2/2 +
5*c*d**2/2)/(3*c*d*e))/(2*c*d*e))/(2*c*d*e))*Piecewise((log(a*e**2 + c*d**2 + 2*c*d*e*x + 2*sqrt(c*d*e)*sqrt(
a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2)))/sqrt(c*d*e), Ne(a*d*e - (a*e**2 + c*d**2)**2/(4*c*d*e), 0)), ((x -
(-a*e**2 - c*d**2)/(2*c*d*e))*log(x - (-a*e**2 - c*d**2)/(2*c*d*e))/sqrt(c*d*e*(x - (-a*e**2 - c*d**2)/(2*c*d*
e))**2), True)) + sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))*(x**3/4 + x**2*(a*e**2/8 + c*d**2/8)/(3*c*d*e
) + x*(a*d*e/4 - (a*e**2/8 + c*d**2/8)*(5*a*e**2/2 + 5*c*d**2/2)/(3*c*d*e))/(2*c*d*e) + (-2*a*(a*e**2/8 + c*d*
*2/8)/(3*c) - (3*a*e**2/2 + 3*c*d**2/2)*(a*d*e/4 - (a*e**2/8 + c*d**2/8)*(5*a*e**2/2 + 5*c*d**2/2)/(3*c*d*e))/
(2*c*d*e))/(c*d*e)), Ne(c*d*e, 0)), (2*(a**2*d**2*e**2*(a*d*e + x*(a*e**2 + c*d**2))**(3/2)/3 - 2*a*d*e*(a*d*e
+ x*(a*e**2 + c*d**2))**(5/2)/5 + (a*d*e + x*(a*e**2 + c*d**2))**(7/2)/7)/(a*e**2 + c*d**2)**3, Ne(a*e**2 + c
*d**2, 0)), (x**3*sqrt(a*d*e)/3, True)) + 2*a*c*d**3*e*Piecewise(((-a*(a*e**2/6 + c*d**2/6)/(2*c) - (a*e**2 +
c*d**2)*(a*d*e/3 - (a*e**2/6 + c*d**2/6)*(3*a*e**2/2 + 3*c*d**2/2)/(2*c*d*e))/(2*c*d*e))*Piecewise((log(a*e**2
+ c*d**2 + 2*c*d*e*x + 2*sqrt(c*d*e)*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2)))/sqrt(c*d*e), Ne(a*d*e -
(a*e**2 + c*d**2)**2/(4*c*d*e), 0)), ((x - (-a*e**2 - c*d**2)/(2*c*d*e))*log(x - (-a*e**2 - c*d**2)/(2*c*d*e))
/sqrt(c*d*e*(x - (-a*e**2 - c*d**2)/(2*c*d*e))**2), True)) + (x**2/3 + x*(a*e**2/6 + c*d**2/6)/(2*c*d*e) + (a*
d*e/3 - (a*e**2/6 + c*d**2/6)*(3*a*e**2/2 + 3*c*d**2/2)/(2*c*d*e))/(c*d*e))*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**
2 + c*d**2)), Ne(c*d*e, 0)), (2*(-a*d*e*(a*d*e + x*(a*e**2 + c*d**2))**(3/2)/3 + (a*d*e + x*(a*e**2 + c*d**2))
**(5/2)/5)/(a*e**2 + c*d**2)**2, Ne(a*e**2 + c*d**2, 0)), (x**2*sqrt(a*d*e)/2, True)) + 4*a*c*d**2*e**2*Piecew
ise(((-a*(a*d*e/4 - (a*e**2/8 + c*d**2/8)*(5*a*e**2/2 + 5*c*d**2/2)/(3*c*d*e))/(2*c) - (a*e**2 + c*d**2)*(-2*a
*(a*e**2/8 + c*d**2/8)/(3*c) - (3*a*e**2/2 + 3*c*d**2/2)*(a*d*e/4 - (a*e**2/8 + c*d**2/8)*(5*a*e**2/2 + 5*c*d*
*2/2)/(3*c*d*e))/(2*c*d*e))/(2*c*d*e))*Piecewise((log(a*e**2 + c*d**2 + 2*c*d*e*x + 2*sqrt(c*d*e)*sqrt(a*d*e +
c*d*e*x**2 + x*(a*e**2 + c*d**2)))/sqrt(c*d*e), Ne(a*d*e - (a*e**2 + c*d**2)**2/(4*c*d*e), 0)), ((x - (-a*e**
2 - c*d**2)/(2*c*d*e))*log(x - (-a*e**2 - c*d**2)/(2*c*d*e))/sqrt(c*d*e*(x - (-a*e**2 - c*d**2)/(2*c*d*e))**2)
, True)) + sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))*(x**3/4 + x**2*(a*e**2/8 + c*d**2/8)/(3*c*d*e) + x*(
a*d*e/4 - (a*e**2/8 + c*d**2/8)*(5*a*e**2/2 + 5*c*d**2/2)/(3*c*d*e))/(2*c*d*e) + (-2*a*(a*e**2/8 + c*d**2/8)/(
3*c) - (3*a*e**2/2 + 3*c*d**2/2)*(a*d*e/4 - (a*e**2/8 + c*d**2/8)*(5*a*e**2/2 + 5*c*d**2/2)/(3*c*d*e))/(2*c*d*
e))/(c*d*e)), Ne(c*d*e, 0)), (2*(a**2*d**2*e**2*(a*d*e + x*(a*e**2 + c*d**2))**(3/2)/3 - 2*a*d*e*(a*d*e + x*(a
*e**2 + c*d**2))**(5/2)/5 + (a*d*e + x*(a*e**2 + c*d**2))**(7/2)/7)/(a*e**2 + c*d**2)**3, Ne(a*e**2 + c*d**2,
0)), (x**3*sqrt(a*d*e)/3, True)) + 2*a*c*d*e**3*Piecewise(((-a*(-3*a*(a*e**2/10 + c*d**2/10)/(4*c) - (5*a*e**2
/2 + 5*c*d**2/2)*(a*d*e/5 - (a*e**2/10 + c*d**2/10)*(7*a*e**2/2 + 7*c*d**2/2)/(4*c*d*e))/(3*c*d*e))/(2*c) - (a
*e**2 + c*d**2)*(-2*a*(a*d*e/5 - (a*e**2/10 + c*d**2/10)*(7*a*e**2/2 + 7*c*d**2/2)/(4*c*d*e))/(3*c) - (3*a*e**
2/2 + 3*c*d**2/2)*(-3*a*(a*e**2/10 + c*d**2/10)/(4*c) - (5*a*e**2/2 + 5*c*d**2/2)*(a*d*e/5 - (a*e**2/10 + c*d*
*2/10)*(7*a*e**2/2 + 7*c*d**2/2)/(4*c*d*e))/(3*c*d*e))/(2*c*d*e))/(2*c*d*e))*Piecewise((log(a*e**2 + c*d**2 +
2*c*d*e*x + 2*sqrt(c*d*e)*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2)))/sqrt(c*d*e), Ne(a*d*e - (a*e**2 + c*
d**2)**2/(4*c*d*e), 0)), ((x - (-a*e**2 - c*d**2)/(2*c*d*e))*log(x - (-a*e**2 - c*d**2)/(2*c*d*e))/sqrt(c*d*e*
(x - (-a*e**2 - c*d**2)/(2*c*d*e))**2), True)) + sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))*(x**4/5 + x**3
*(a*e**2/10 + c*d**2/10)/(4*c*d*e) + x**2*(a*d*e/5 - (a*e**2/10 + c*d**2/10)*(7*a*e**2/2 + 7*c*d**2/2)/(4*c*d*
e))/(3*c*d*e) + x*(-3*a*(a*e**2/10 + c*d**2/10)/(4*c) - (5*a*e**2/2 + 5*c*d**2/2)*(a*d*e/5 - (a*e**2/10 + c*d*
*2/10)*(7*a*e**2/2 + 7*c*d**2/2)/(4*c*d*e))/(3*c*d*e))/(2*c*d*e) + (-2*a*(a*d*e/5 - (a*e**2/10 + c*d**2/10)*(7
*a*e**2/2 + 7*c*d**2/2)/(4*c*d*e))/(3*c) - (3*a*e**2/2 + 3*c*d**2/2)*(-3*a*(a*e**2/10 + c*d**2/10)/(4*c) - (5*
a*e**2/2 + 5*c*d**2/2)*(a*d*e/5 - (a*e**2/10 + c*d**2/10)*(7*a*e**2/2 + 7*c*d**2/2)/(4*c*d*e))/(3*c*d*e))/(2*c
*d*e))/(c*d*e)), Ne(c*d*e, 0)), (2*(-a**3*d**3*e**3*(a*d*e + x*(a*e**2 + c*d**2))**(3/2)/3 + 3*a**2*d**2*e**2*
(a*d*e + x*(a*e**2 + c*d**2))**(5/2)/5 - 3*a*d*e*(a*d*e + x*(a*e**2 + c*d**2))**(7/2)/7 + (a*d*e + x*(a*e**2 +
c*d**2))**(9/2)/9)/(a*e**2 + c*d**2)**4, Ne(a*e**2 + c*d**2, 0)), (x**4*sqrt(a*d*e)/4, True)) + c**2*d**4*Pie
cewise(((-a*(a*d*e/4 - (a*e**2/8 + c*d**2/8)*(5*a*e**2/2 + 5*c*d**2/2)/(3*c*d*e))/(2*c) - (a*e**2 + c*d**2)*(-
2*a*(a*e**2/8 + c*d**2/8)/(3*c) - (3*a*e**2/2 + 3*c*d**2/2)*(a*d*e/4 - (a*e**2/8 + c*d**2/8)*(5*a*e**2/2 + 5*c
*d**2/2)/(3*c*d*e))/(2*c*d*e))/(2*c*d*e))*Piecewise((log(a*e**2 + c*d**2 + 2*c*d*e*x + 2*sqrt(c*d*e)*sqrt(a*d*
e + c*d*e*x**2 + x*(a*e**2 + c*d**2)))/sqrt(c*d*e), Ne(a*d*e - (a*e**2 + c*d**2)**2/(4*c*d*e), 0)), ((x - (-a*
e**2 - c*d**2)/(2*c*d*e))*log(x - (-a*e**2 - c*d**2)/(2*c*d*e))/sqrt(c*d*e*(x - (-a*e**2 - c*d**2)/(2*c*d*e))*
*2), True)) + sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))*(x**3/4 + x**2*(a*e**2/8 + c*d**2/8)/(3*c*d*e) +
x*(a*d*e/4 - (a*e**2/8 + c*d**2/8)*(5*a*e**2/2 + 5*c*d**2/2)/(3*c*d*e))/(2*c*d*e) + (-2*a*(a*e**2/8 + c*d**2/8
)/(3*c) - (3*a*e**2/2 + 3*c*d**2/2)*(a*d*e/4 - (a*e**2/8 + c*d**2/8)*(5*a*e**2/2 + 5*c*d**2/2)/(3*c*d*e))/(2*c
*d*e))/(c*d*e)), Ne(c*d*e, 0)), (2*(a**2*d**2*e**2*(a*d*e + x*(a*e**2 + c*d**2))**(3/2)/3 - 2*a*d*e*(a*d*e + x
*(a*e**2 + c*d**2))**(5/2)/5 + (a*d*e + x*(a*e**2 + c*d**2))**(7/2)/7)/(a*e**2 + c*d**2)**3, Ne(a*e**2 + c*d**
2, 0)), (x**3*sqrt(a*d*e)/3, True)) + 2*c**2*d**3*e*Piecewise(((-a*(-3*a*(a*e**2/10 + c*d**2/10)/(4*c) - (5*a*
e**2/2 + 5*c*d**2/2)*(a*d*e/5 - (a*e**2/10 + c*d**2/10)*(7*a*e**2/2 + 7*c*d**2/2)/(4*c*d*e))/(3*c*d*e))/(2*c)
- (a*e**2 + c*d**2)*(-2*a*(a*d*e/5 - (a*e**2/10 + c*d**2/10)*(7*a*e**2/2 + 7*c*d**2/2)/(4*c*d*e))/(3*c) - (3*a
*e**2/2 + 3*c*d**2/2)*(-3*a*(a*e**2/10 + c*d**2/10)/(4*c) - (5*a*e**2/2 + 5*c*d**2/2)*(a*d*e/5 - (a*e**2/10 +
c*d**2/10)*(7*a*e**2/2 + 7*c*d**2/2)/(4*c*d*e))/(3*c*d*e))/(2*c*d*e))/(2*c*d*e))*Piecewise((log(a*e**2 + c*d**
2 + 2*c*d*e*x + 2*sqrt(c*d*e)*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2)))/sqrt(c*d*e), Ne(a*d*e - (a*e**2
+ c*d**2)**2/(4*c*d*e), 0)), ((x - (-a*e**2 - c*d**2)/(2*c*d*e))*log(x - (-a*e**2 - c*d**2)/(2*c*d*e))/sqrt(c*
d*e*(x - (-a*e**2 - c*d**2)/(2*c*d*e))**2), True)) + sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))*(x**4/5 +
x**3*(a*e**2/10 + c*d**2/10)/(4*c*d*e) + x**2*(a*d*e/5 - (a*e**2/10 + c*d**2/10)*(7*a*e**2/2 + 7*c*d**2/2)/(4*
c*d*e))/(3*c*d*e) + x*(-3*a*(a*e**2/10 + c*d**2/10)/(4*c) - (5*a*e**2/2 + 5*c*d**2/2)*(a*d*e/5 - (a*e**2/10 +
c*d**2/10)*(7*a*e**2/2 + 7*c*d**2/2)/(4*c*d*e))/(3*c*d*e))/(2*c*d*e) + (-2*a*(a*d*e/5 - (a*e**2/10 + c*d**2/10
)*(7*a*e**2/2 + 7*c*d**2/2)/(4*c*d*e))/(3*c) - (3*a*e**2/2 + 3*c*d**2/2)*(-3*a*(a*e**2/10 + c*d**2/10)/(4*c) -
(5*a*e**2/2 + 5*c*d**2/2)*(a*d*e/5 - (a*e**2/10 + c*d**2/10)*(7*a*e**2/2 + 7*c*d**2/2)/(4*c*d*e))/(3*c*d*e))/
(2*c*d*e))/(c*d*e)), Ne(c*d*e, 0)), (2*(-a**3*d**3*e**3*(a*d*e + x*(a*e**2 + c*d**2))**(3/2)/3 + 3*a**2*d**2*e
**2*(a*d*e + x*(a*e**2 + c*d**2))**(5/2)/5 - 3*a*d*e*(a*d*e + x*(a*e**2 + c*d**2))**(7/2)/7 + (a*d*e + x*(a*e*
*2 + c*d**2))**(9/2)/9)/(a*e**2 + c*d**2)**4, Ne(a*e**2 + c*d**2, 0)), (x**4*sqrt(a*d*e)/4, True)) + c**2*d**2
*e**2*Piecewise(((-a*(-3*a*(a*d*e/6 - (a*e**2/12 + c*d**2/12)*(9*a*e**2/2 + 9*c*d**2/2)/(5*c*d*e))/(4*c) - (5*
a*e**2/2 + 5*c*d**2/2)*(-4*a*(a*e**2/12 + c*d**2/12)/(5*c) - (7*a*e**2/2 + 7*c*d**2/2)*(a*d*e/6 - (a*e**2/12 +
c*d**2/12)*(9*a*e**2/2 + 9*c*d**2/2)/(5*c*d*e))/(4*c*d*e))/(3*c*d*e))/(2*c) - (a*e**2 + c*d**2)*(-2*a*(-4*a*(
a*e**2/12 + c*d**2/12)/(5*c) - (7*a*e**2/2 + 7*c*d**2/2)*(a*d*e/6 - (a*e**2/12 + c*d**2/12)*(9*a*e**2/2 + 9*c*
d**2/2)/(5*c*d*e))/(4*c*d*e))/(3*c) - (3*a*e**2/2 + 3*c*d**2/2)*(-3*a*(a*d*e/6 - (a*e**2/12 + c*d**2/12)*(9*a*
e**2/2 + 9*c*d**2/2)/(5*c*d*e))/(4*c) - (5*a*e**2/2 + 5*c*d**2/2)*(-4*a*(a*e**2/12 + c*d**2/12)/(5*c) - (7*a*e
**2/2 + 7*c*d**2/2)*(a*d*e/6 - (a*e**2/12 + c*d**2/12)*(9*a*e**2/2 + 9*c*d**2/2)/(5*c*d*e))/(4*c*d*e))/(3*c*d*
e))/(2*c*d*e))/(2*c*d*e))*Piecewise((log(a*e**2 + c*d**2 + 2*c*d*e*x + 2*sqrt(c*d*e)*sqrt(a*d*e + c*d*e*x**2 +
x*(a*e**2 + c*d**2)))/sqrt(c*d*e), Ne(a*d*e - (a*e**2 + c*d**2)**2/(4*c*d*e), 0)), ((x - (-a*e**2 - c*d**2)/(
2*c*d*e))*log(x - (-a*e**2 - c*d**2)/(2*c*d*e))/sqrt(c*d*e*(x - (-a*e**2 - c*d**2)/(2*c*d*e))**2), True)) + sq
rt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))*(x**5/6 + x**4*(a*e**2/12 + c*d**2/12)/(5*c*d*e) + x**3*(a*d*e/6
- (a*e**2/12 + c*d**2/12)*(9*a*e**2/2 + 9*c*d**2/2)/(5*c*d*e))/(4*c*d*e) + x**2*(-4*a*(a*e**2/12 + c*d**2/12)/
(5*c) - (7*a*e**2/2 + 7*c*d**2/2)*(a*d*e/6 - (a*e**2/12 + c*d**2/12)*(9*a*e**2/2 + 9*c*d**2/2)/(5*c*d*e))/(4*c
*d*e))/(3*c*d*e) + x*(-3*a*(a*d*e/6 - (a*e**2/12 + c*d**2/12)*(9*a*e**2/2 + 9*c*d**2/2)/(5*c*d*e))/(4*c) - (5*
a*e**2/2 + 5*c*d**2/2)*(-4*a*(a*e**2/12 + c*d**2/12)/(5*c) - (7*a*e**2/2 + 7*c*d**2/2)*(a*d*e/6 - (a*e**2/12 +
c*d**2/12)*(9*a*e**2/2 + 9*c*d**2/2)/(5*c*d*e))/(4*c*d*e))/(3*c*d*e))/(2*c*d*e) + (-2*a*(-4*a*(a*e**2/12 + c*
d**2/12)/(5*c) - (7*a*e**2/2 + 7*c*d**2/2)*(a*d*e/6 - (a*e**2/12 + c*d**2/12)*(9*a*e**2/2 + 9*c*d**2/2)/(5*c*d
*e))/(4*c*d*e))/(3*c) - (3*a*e**2/2 + 3*c*d**2/2)*(-3*a*(a*d*e/6 - (a*e**2/12 + c*d**2/12)*(9*a*e**2/2 + 9*c*d
**2/2)/(5*c*d*e))/(4*c) - (5*a*e**2/2 + 5*c*d**2/2)*(-4*a*(a*e**2/12 + c*d**2/12)/(5*c) - (7*a*e**2/2 + 7*c*d*
*2/2)*(a*d*e/6 - (a*e**2/12 + c*d**2/12)*(9*a*e**2/2 + 9*c*d**2/2)/(5*c*d*e))/(4*c*d*e))/(3*c*d*e))/(2*c*d*e))
/(c*d*e)), Ne(c*d*e, 0)), (2*(a**4*d**4*e**4*(a*d*e + x*(a*e**2 + c*d**2))**(3/2)/3 - 4*a**3*d**3*e**3*(a*d*e
+ x*(a*e**2 + c*d**2))**(5/2)/5 + 6*a**2*d**2*e**2*(a*d*e + x*(a*e**2 + c*d**2))**(7/2)/7 - 4*a*d*e*(a*d*e + x
*(a*e**2 + c*d**2))**(9/2)/9 + (a*d*e + x*(a*e**2 + c*d**2))**(11/2)/11)/(a*e**2 + c*d**2)**5, Ne(a*e**2 + c*d
**2, 0)), (x**5*sqrt(a*d*e)/5, True))

Maxima [F(-2)]

Exception generated. \[ \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx=\text {Exception raised: ValueError} \]

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more
details)Is e

Giac [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 524, normalized size of antiderivative = 1.72 \[ \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx=\frac {1}{1536} \, \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (2 \, c^{2} d^{2} e^{2} x + \frac {5 \, {\left (c^{7} d^{8} e^{6} + a c^{6} d^{6} e^{8}\right )}}{c^{5} d^{5} e^{5}}\right )} x + \frac {27 \, c^{7} d^{9} e^{5} + 106 \, a c^{6} d^{7} e^{7} + 27 \, a^{2} c^{5} d^{5} e^{9}}{c^{5} d^{5} e^{5}}\right )} x + \frac {c^{7} d^{10} e^{4} + 159 \, a c^{6} d^{8} e^{6} + 159 \, a^{2} c^{5} d^{6} e^{8} + a^{3} c^{4} d^{4} e^{10}}{c^{5} d^{5} e^{5}}\right )} x - \frac {5 \, c^{7} d^{11} e^{3} - 28 \, a c^{6} d^{9} e^{5} - 594 \, a^{2} c^{5} d^{7} e^{7} - 28 \, a^{3} c^{4} d^{5} e^{9} + 5 \, a^{4} c^{3} d^{3} e^{11}}{c^{5} d^{5} e^{5}}\right )} x + \frac {15 \, c^{7} d^{12} e^{2} - 85 \, a c^{6} d^{10} e^{4} + 198 \, a^{2} c^{5} d^{8} e^{6} + 198 \, a^{3} c^{4} d^{6} e^{8} - 85 \, a^{4} c^{3} d^{4} e^{10} + 15 \, a^{5} c^{2} d^{2} e^{12}}{c^{5} d^{5} e^{5}}\right )} + \frac {5 \, {\left (c^{6} d^{12} - 6 \, a c^{5} d^{10} e^{2} + 15 \, a^{2} c^{4} d^{8} e^{4} - 20 \, a^{3} c^{3} d^{6} e^{6} + 15 \, a^{4} c^{2} d^{4} e^{8} - 6 \, a^{5} c d^{2} e^{10} + a^{6} e^{12}\right )} \log \left ({\left | -c d^{2} - a e^{2} - 2 \, \sqrt {c d e} {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )} \right |}\right )}{1024 \, \sqrt {c d e} c^{3} d^{3} e^{3}} \]

[In]

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

[Out]

1/1536*sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e)*(2*(4*(2*(8*(2*c^2*d^2*e^2*x + 5*(c^7*d^8*e^6 + a*c^6*d^6*e
^8)/(c^5*d^5*e^5))*x + (27*c^7*d^9*e^5 + 106*a*c^6*d^7*e^7 + 27*a^2*c^5*d^5*e^9)/(c^5*d^5*e^5))*x + (c^7*d^10*
e^4 + 159*a*c^6*d^8*e^6 + 159*a^2*c^5*d^6*e^8 + a^3*c^4*d^4*e^10)/(c^5*d^5*e^5))*x - (5*c^7*d^11*e^3 - 28*a*c^
6*d^9*e^5 - 594*a^2*c^5*d^7*e^7 - 28*a^3*c^4*d^5*e^9 + 5*a^4*c^3*d^3*e^11)/(c^5*d^5*e^5))*x + (15*c^7*d^12*e^2
- 85*a*c^6*d^10*e^4 + 198*a^2*c^5*d^8*e^6 + 198*a^3*c^4*d^6*e^8 - 85*a^4*c^3*d^4*e^10 + 15*a^5*c^2*d^2*e^12)/
(c^5*d^5*e^5)) + 5/1024*(c^6*d^12 - 6*a*c^5*d^10*e^2 + 15*a^2*c^4*d^8*e^4 - 20*a^3*c^3*d^6*e^6 + 15*a^4*c^2*d^
4*e^8 - 6*a^5*c*d^2*e^10 + a^6*e^12)*log(abs(-c*d^2 - a*e^2 - 2*sqrt(c*d*e)*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 +
c*d^2*x + a*e^2*x + a*d*e))))/(sqrt(c*d*e)*c^3*d^3*e^3)

Mupad [B] (verification not implemented)

Time = 9.95 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.05 \[ \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx=\frac {\left (\frac {c\,d^2}{2}+c\,x\,d\,e+\frac {a\,e^2}{2}\right )\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2}}{6\,c\,d\,e}-\frac {\left (\frac {5\,{\left (c\,d^2+a\,e^2\right )}^2}{4}-5\,a\,c\,d^2\,e^2\right )\,\left (\frac {\left (\frac {c\,d^2}{2}+c\,x\,d\,e+\frac {a\,e^2}{2}\right )\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/2}}{4\,c\,d\,e}-\frac {\left (\frac {3\,{\left (c\,d^2+a\,e^2\right )}^2}{4}-3\,a\,c\,d^2\,e^2\right )\,\left (\left (\frac {x}{2}+\frac {c\,d^2+a\,e^2}{4\,c\,d\,e}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}-\frac {\ln \left (2\,\sqrt {\left (a\,e+c\,d\,x\right )\,\left (d+e\,x\right )}\,\sqrt {c\,d\,e}+a\,e^2+c\,d^2+2\,c\,d\,e\,x\right )\,\left (\frac {{\left (c\,d^2+a\,e^2\right )}^2}{4}-a\,c\,d^2\,e^2\right )}{2\,{\left (c\,d\,e\right )}^{3/2}}\right )}{4\,c\,d\,e}\right )}{6\,c\,d\,e} \]

[In]

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

[Out]

(((a*e^2)/2 + (c*d^2)/2 + c*d*e*x)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2))/(6*c*d*e) - (((5*(a*e^2 + c*
d^2)^2)/4 - 5*a*c*d^2*e^2)*((((a*e^2)/2 + (c*d^2)/2 + c*d*e*x)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2))/
(4*c*d*e) - (((3*(a*e^2 + c*d^2)^2)/4 - 3*a*c*d^2*e^2)*((x/2 + (a*e^2 + c*d^2)/(4*c*d*e))*(x*(a*e^2 + c*d^2) +
a*d*e + c*d*e*x^2)^(1/2) - (log(2*((a*e + c*d*x)*(d + e*x))^(1/2)*(c*d*e)^(1/2) + a*e^2 + c*d^2 + 2*c*d*e*x)*
((a*e^2 + c*d^2)^2/4 - a*c*d^2*e^2))/(2*(c*d*e)^(3/2))))/(4*c*d*e)))/(6*c*d*e)