\(\int \frac {x (a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{d+e x} \, dx\) [459]

Optimal result

Integrand size = 38, antiderivative size = 381 \[ \int \frac {x \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{d+e x} \, dx=-\frac {\left (c d^2-a e^2\right )^3 \left (7 c d^2+5 a e^2\right ) \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{512 c^3 d^3 e^4}+\frac {\left (c d^2-a e^2\right ) \left (7 c d^2+5 a e^2\right ) \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{192 c^2 d^2 e^3}-\frac {1}{60} \left (\frac {5 a}{c d}+\frac {7 d}{e^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}+\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{6 c d e (d+e x)}+\frac {\left (c d^2-a e^2\right )^5 \left (7 c d^2+5 a e^2\right ) \text {arctanh}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{1024 c^{7/2} d^{7/2} e^{9/2}} \]

[Out]

1/192*(-a*e^2+c*d^2)*(5*a*e^2+7*c*d^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^3-1/60*(5*a/c/d+7*d/e^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)+1/6*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2
)/c/d/e/(e*x+d)+1/1024*(-a*e^2+c*d^2)^5*(5*a*e^2+7*c*d^2)*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^(9/2)-1/512*(-a*e^2+c*d^2)^3*(5*a*e^2+7*c*d
^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)^(1/2)/c^3/d^3/e^4

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 381, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.132, Rules used = {808, 678, 626, 635, 212} \[ \int \frac {x \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{d+e x} \, dx=\frac {\left (5 a e^2+7 c d^2\right ) \left (c d^2-a e^2\right )^5 \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{1024 c^{7/2} d^{7/2} e^{9/2}}-\frac {\left (5 a e^2+7 c d^2\right ) \left (c d^2-a e^2\right )^3 \left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{512 c^3 d^3 e^4}+\frac {\left (5 a e^2+7 c d^2\right ) \left (c d^2-a e^2\right ) \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{192 c^2 d^2 e^3}+\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{6 c d e (d+e x)}-\frac {1}{60} \left (\frac {5 a}{c d}+\frac {7 d}{e^2}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2} \]

[In]

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

[Out]

-1/512*((c*d^2 - a*e^2)^3*(7*c*d^2 + 5*a*e^2)*(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])/(c^3*d^3*e^4) + ((c*d^2 - a*e^2)*(7*c*d^2 + 5*a*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)^(3/2))/(192*c^2*d^2*e^3) - (((5*a)/(c*d) + (7*d)/e^2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x
^2)^(5/2))/60 + (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2)/(6*c*d*e*(d + e*x)) + ((c*d^2 - a*e^2)^5*(7*c*d^
2 + 5*a*e^2)*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^(9/2))

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]

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]

Rule 808

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[g*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 2))), x] + Dist[(m*(g*(c*d - b*e) + c*e*f) + e*(p + 1)
*(2*c*f - b*g))/(c*e*(m + 2*p + 2)), Int[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g
, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[m + 2*p + 2, 0] && (NeQ[m, 2] || Eq
Q[d, 0])

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 )^{7/2}}{6 c d e (d+e x)}+\frac {1}{12} \left (-\frac {7 d}{e}-\frac {5 a e}{c d}\right ) \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} \, dx \\ & = -\frac {1}{60} \left (\frac {5 a}{c d}+\frac {7 d}{e^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}+\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{6 c d e (d+e x)}+\frac {\left (\left (\frac {7 d}{e}+\frac {5 a e}{c d}\right ) \left (c d^2-a e^2\right )\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 e} \\ & = \frac {\left (c d^2-a e^2\right ) \left (7 c d^2+5 a e^2\right ) \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^3}-\frac {1}{60} \left (\frac {5 a}{c d}+\frac {7 d}{e^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}+\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{6 c d e (d+e x)}-\frac {\left (\left (c d^2-a e^2\right )^3 \left (7 c d^2+5 a e^2\right )\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^3} \\ & = -\frac {\left (c d^2-a e^2\right )^3 \left (7 c d^2+5 a e^2\right ) \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^4}+\frac {\left (c d^2-a e^2\right ) \left (7 c d^2+5 a e^2\right ) \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^3}-\frac {1}{60} \left (\frac {5 a}{c d}+\frac {7 d}{e^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}+\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{6 c d e (d+e x)}+\frac {\left (\left (c d^2-a e^2\right )^5 \left (7 c d^2+5 a e^2\right )\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^4} \\ & = -\frac {\left (c d^2-a e^2\right )^3 \left (7 c d^2+5 a e^2\right ) \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^4}+\frac {\left (c d^2-a e^2\right ) \left (7 c d^2+5 a e^2\right ) \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^3}-\frac {1}{60} \left (\frac {5 a}{c d}+\frac {7 d}{e^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}+\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{6 c d e (d+e x)}+\frac {\left (\left (c d^2-a e^2\right )^5 \left (7 c d^2+5 a e^2\right )\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^4} \\ & = -\frac {\left (c d^2-a e^2\right )^3 \left (7 c d^2+5 a e^2\right ) \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^4}+\frac {\left (c d^2-a e^2\right ) \left (7 c d^2+5 a e^2\right ) \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^3}-\frac {1}{60} \left (\frac {5 a}{c d}+\frac {7 d}{e^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}+\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{6 c d e (d+e x)}+\frac {\left (c d^2-a e^2\right )^5 \left (7 c d^2+5 a e^2\right ) \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^{9/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.89 (sec) , antiderivative size = 388, normalized size of antiderivative = 1.02 \[ \int \frac {x \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{d+e x} \, dx=\frac {\left (c d^2-a e^2\right )^5 ((a e+c d x) (d+e x))^{3/2} \left (\frac {\sqrt {c} \sqrt {d} \sqrt {e} \left (75 a^5 e^{10}-5 a^4 c d e^8 (49 d+10 e x)+10 a^3 c^2 d^2 e^6 \left (15 d^2+16 d e x+4 e^2 x^2\right )-6 a^2 c^3 d^3 e^4 \left (91 d^3-58 d^2 e x-564 d e^2 x^2-360 e^3 x^3\right )+a c^4 d^4 e^2 \left (415 d^4-272 d^3 e x+216 d^2 e^2 x^2+4448 d e^3 x^3+3200 e^4 x^4\right )+c^5 d^5 \left (-105 d^5+70 d^4 e x-56 d^3 e^2 x^2+48 d^2 e^3 x^3+1664 d e^4 x^4+1280 e^5 x^5\right )\right )}{\left (c d^2-a e^2\right )^5 (a e+c d x) (d+e x)}+\frac {15 \left (7 c d^2+5 a e^2\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \sqrt {d+e x}}\right )}{(a e+c d x)^{3/2} (d+e x)^{3/2}}\right )}{7680 c^{7/2} d^{7/2} e^{9/2}} \]

[In]

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

[Out]

((c*d^2 - a*e^2)^5*((a*e + c*d*x)*(d + e*x))^(3/2)*((Sqrt[c]*Sqrt[d]*Sqrt[e]*(75*a^5*e^10 - 5*a^4*c*d*e^8*(49*
d + 10*e*x) + 10*a^3*c^2*d^2*e^6*(15*d^2 + 16*d*e*x + 4*e^2*x^2) - 6*a^2*c^3*d^3*e^4*(91*d^3 - 58*d^2*e*x - 56
4*d*e^2*x^2 - 360*e^3*x^3) + a*c^4*d^4*e^2*(415*d^4 - 272*d^3*e*x + 216*d^2*e^2*x^2 + 4448*d*e^3*x^3 + 3200*e^
4*x^4) + c^5*d^5*(-105*d^5 + 70*d^4*e*x - 56*d^3*e^2*x^2 + 48*d^2*e^3*x^3 + 1664*d*e^4*x^4 + 1280*e^5*x^5)))/(
(c*d^2 - a*e^2)^5*(a*e + c*d*x)*(d + e*x)) + (15*(7*c*d^2 + 5*a*e^2)*ArcTanh[(Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt
[c]*Sqrt[d]*Sqrt[d + e*x])])/((a*e + c*d*x)^(3/2)*(d + e*x)^(3/2))))/(7680*c^(7/2)*d^(7/2)*e^(9/2))

Maple [A] (verified)

Time = 0.55 (sec) , antiderivative size = 673, normalized size of antiderivative = 1.77

method	result	size
default	\(\frac {\frac {\left (2 c d e x +e^{2} a +c \,d^{2}\right ) {\left (a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}\right )}^{\frac {5}{2}}}{12 c d e}+\frac {5 \left (4 a c \,d^{2} e^{2}-\left (e^{2} a +c \,d^{2}\right )^{2}\right ) \left (\frac {\left (2 c d e x +e^{2} a +c \,d^{2}\right ) {\left (a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}\right )}^{\frac {3}{2}}}{8 c d e}+\frac {3 \left (4 a c \,d^{2} e^{2}-\left (e^{2} a +c \,d^{2}\right )^{2}\right ) \left (\frac {\left (2 c d e x +e^{2} a +c \,d^{2}\right ) \sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}}{4 c d e}+\frac {\left (4 a c \,d^{2} e^{2}-\left (e^{2} a +c \,d^{2}\right )^{2}\right ) \ln \left (\frac {\frac {1}{2} e^{2} a +\frac {1}{2} c \,d^{2}+c d e x}{\sqrt {c d e}}+\sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}\right )}{8 c d e \sqrt {c d e}}\right )}{16 c d e}\right )}{24 c d e}}{e}-\frac {d \left (\frac {\left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{5}+\frac {\left (e^{2} a -c \,d^{2}\right ) \left (\frac {\left (2 c d e \left (x +\frac {d}{e}\right )+e^{2} a -c \,d^{2}\right ) \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{8 c d e}-\frac {3 \left (e^{2} a -c \,d^{2}\right )^{2} \left (\frac {\left (2 c d e \left (x +\frac {d}{e}\right )+e^{2} a -c \,d^{2}\right ) \sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}{4 c d e}-\frac {\left (e^{2} a -c \,d^{2}\right )^{2} \ln \left (\frac {\frac {e^{2} a}{2}-\frac {c \,d^{2}}{2}+c d e \left (x +\frac {d}{e}\right )}{\sqrt {c d e}}+\sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}\right )}{8 c d e \sqrt {c d e}}\right )}{16 c d e}\right )}{2}\right )}{e^{2}}\)	\(673\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 1046, normalized size of antiderivative = 2.75 \[ \int \frac {x \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{d+e x} \, dx=\left [-\frac {15 \, {\left (7 \, c^{6} d^{12} - 30 \, a c^{5} d^{10} e^{2} + 45 \, 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} + 18 \, a^{5} c d^{2} e^{10} - 5 \, 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 (1280 \, c^{6} d^{6} e^{6} x^{5} - 105 \, c^{6} d^{11} e + 415 \, a c^{5} d^{9} e^{3} - 546 \, a^{2} c^{4} d^{7} e^{5} + 150 \, a^{3} c^{3} d^{5} e^{7} - 245 \, a^{4} c^{2} d^{3} e^{9} + 75 \, a^{5} c d e^{11} + 128 \, {\left (13 \, c^{6} d^{7} e^{5} + 25 \, a c^{5} d^{5} e^{7}\right )} x^{4} + 16 \, {\left (3 \, c^{6} d^{8} e^{4} + 278 \, a c^{5} d^{6} e^{6} + 135 \, a^{2} c^{4} d^{4} e^{8}\right )} x^{3} - 8 \, {\left (7 \, c^{6} d^{9} e^{3} - 27 \, a c^{5} d^{7} e^{5} - 423 \, a^{2} c^{4} d^{5} e^{7} - 5 \, a^{3} c^{3} d^{3} e^{9}\right )} x^{2} + 2 \, {\left (35 \, c^{6} d^{10} e^{2} - 136 \, a c^{5} d^{8} e^{4} + 174 \, a^{2} c^{4} d^{6} e^{6} + 80 \, a^{3} c^{3} d^{4} e^{8} - 25 \, 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}}{30720 \, c^{4} d^{4} e^{5}}, -\frac {15 \, {\left (7 \, c^{6} d^{12} - 30 \, a c^{5} d^{10} e^{2} + 45 \, 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} + 18 \, a^{5} c d^{2} e^{10} - 5 \, 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 (1280 \, c^{6} d^{6} e^{6} x^{5} - 105 \, c^{6} d^{11} e + 415 \, a c^{5} d^{9} e^{3} - 546 \, a^{2} c^{4} d^{7} e^{5} + 150 \, a^{3} c^{3} d^{5} e^{7} - 245 \, a^{4} c^{2} d^{3} e^{9} + 75 \, a^{5} c d e^{11} + 128 \, {\left (13 \, c^{6} d^{7} e^{5} + 25 \, a c^{5} d^{5} e^{7}\right )} x^{4} + 16 \, {\left (3 \, c^{6} d^{8} e^{4} + 278 \, a c^{5} d^{6} e^{6} + 135 \, a^{2} c^{4} d^{4} e^{8}\right )} x^{3} - 8 \, {\left (7 \, c^{6} d^{9} e^{3} - 27 \, a c^{5} d^{7} e^{5} - 423 \, a^{2} c^{4} d^{5} e^{7} - 5 \, a^{3} c^{3} d^{3} e^{9}\right )} x^{2} + 2 \, {\left (35 \, c^{6} d^{10} e^{2} - 136 \, a c^{5} d^{8} e^{4} + 174 \, a^{2} c^{4} d^{6} e^{6} + 80 \, a^{3} c^{3} d^{4} e^{8} - 25 \, 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}}{15360 \, c^{4} d^{4} e^{5}}\right ] \]

[In]

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

[Out]

[-1/30720*(15*(7*c^6*d^12 - 30*a*c^5*d^10*e^2 + 45*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 +
18*a^5*c*d^2*e^10 - 5*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*sqr
t(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*(1280*c^6*d^6*e^6*x^5 - 105*c^6*d^11*e + 415*a*c^5*d^9*e^3 - 546*a^2*c^4*d^7*e^5 + 150*a^3*c^3*d^5*e^7 -
245*a^4*c^2*d^3*e^9 + 75*a^5*c*d*e^11 + 128*(13*c^6*d^7*e^5 + 25*a*c^5*d^5*e^7)*x^4 + 16*(3*c^6*d^8*e^4 + 278
*a*c^5*d^6*e^6 + 135*a^2*c^4*d^4*e^8)*x^3 - 8*(7*c^6*d^9*e^3 - 27*a*c^5*d^7*e^5 - 423*a^2*c^4*d^5*e^7 - 5*a^3*
c^3*d^3*e^9)*x^2 + 2*(35*c^6*d^10*e^2 - 136*a*c^5*d^8*e^4 + 174*a^2*c^4*d^6*e^6 + 80*a^3*c^3*d^4*e^8 - 25*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^5), -1/15360*(15*(7*c^6*d^12 - 30*a*c
^5*d^10*e^2 + 45*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 + 18*a^5*c*d^2*e^10 - 5*a^6*e^12)*s
qrt(-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*(1280*c^6*d^6*e^6*x^5 - 105*c^6*d^11*e + 415*a*
c^5*d^9*e^3 - 546*a^2*c^4*d^7*e^5 + 150*a^3*c^3*d^5*e^7 - 245*a^4*c^2*d^3*e^9 + 75*a^5*c*d*e^11 + 128*(13*c^6*
d^7*e^5 + 25*a*c^5*d^5*e^7)*x^4 + 16*(3*c^6*d^8*e^4 + 278*a*c^5*d^6*e^6 + 135*a^2*c^4*d^4*e^8)*x^3 - 8*(7*c^6*
d^9*e^3 - 27*a*c^5*d^7*e^5 - 423*a^2*c^4*d^5*e^7 - 5*a^3*c^3*d^3*e^9)*x^2 + 2*(35*c^6*d^10*e^2 - 136*a*c^5*d^8
*e^4 + 174*a^2*c^4*d^6*e^6 + 80*a^3*c^3*d^4*e^8 - 25*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^5)]

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F(-2)]

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

[In]

integrate(x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d),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 = 0.38 (sec) , antiderivative size = 526, normalized size of antiderivative = 1.38 \[ \int \frac {x \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{d+e x} \, dx=\frac {1}{7680} \, \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, c^{2} d^{2} e x + \frac {13 \, c^{7} d^{8} e^{5} + 25 \, a c^{6} d^{6} e^{7}}{c^{5} d^{5} e^{5}}\right )} x + \frac {3 \, c^{7} d^{9} e^{4} + 278 \, a c^{6} d^{7} e^{6} + 135 \, a^{2} c^{5} d^{5} e^{8}}{c^{5} d^{5} e^{5}}\right )} x - \frac {7 \, c^{7} d^{10} e^{3} - 27 \, a c^{6} d^{8} e^{5} - 423 \, a^{2} c^{5} d^{6} e^{7} - 5 \, a^{3} c^{4} d^{4} e^{9}}{c^{5} d^{5} e^{5}}\right )} x + \frac {35 \, c^{7} d^{11} e^{2} - 136 \, a c^{6} d^{9} e^{4} + 174 \, a^{2} c^{5} d^{7} e^{6} + 80 \, a^{3} c^{4} d^{5} e^{8} - 25 \, a^{4} c^{3} d^{3} e^{10}}{c^{5} d^{5} e^{5}}\right )} x - \frac {105 \, c^{7} d^{12} e - 415 \, a c^{6} d^{10} e^{3} + 546 \, a^{2} c^{5} d^{8} e^{5} - 150 \, a^{3} c^{4} d^{6} e^{7} + 245 \, a^{4} c^{3} d^{4} e^{9} - 75 \, a^{5} c^{2} d^{2} e^{11}}{c^{5} d^{5} e^{5}}\right )} - \frac {{\left (7 \, c^{6} d^{12} - 30 \, a c^{5} d^{10} e^{2} + 45 \, a^{2} c^{4} d^{8} e^{4} - 20 \, a^{3} c^{3} d^{6} e^{6} - 15 \, a^{4} c^{2} d^{4} e^{8} + 18 \, a^{5} c d^{2} e^{10} - 5 \, a^{6} e^{12}\right )} \log \left ({\left | -c d^{2} - a e^{2} - 2 \, \sqrt {c d e} {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )} \right |}\right )}{1024 \, \sqrt {c d e} c^{3} d^{3} e^{4}} \]

[In]

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

[Out]

1/7680*sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e)*(2*(4*(2*(8*(10*c^2*d^2*e*x + (13*c^7*d^8*e^5 + 25*a*c^6*d^
6*e^7)/(c^5*d^5*e^5))*x + (3*c^7*d^9*e^4 + 278*a*c^6*d^7*e^6 + 135*a^2*c^5*d^5*e^8)/(c^5*d^5*e^5))*x - (7*c^7*
d^10*e^3 - 27*a*c^6*d^8*e^5 - 423*a^2*c^5*d^6*e^7 - 5*a^3*c^4*d^4*e^9)/(c^5*d^5*e^5))*x + (35*c^7*d^11*e^2 - 1
36*a*c^6*d^9*e^4 + 174*a^2*c^5*d^7*e^6 + 80*a^3*c^4*d^5*e^8 - 25*a^4*c^3*d^3*e^10)/(c^5*d^5*e^5))*x - (105*c^7
*d^12*e - 415*a*c^6*d^10*e^3 + 546*a^2*c^5*d^8*e^5 - 150*a^3*c^4*d^6*e^7 + 245*a^4*c^3*d^4*e^9 - 75*a^5*c^2*d^
2*e^11)/(c^5*d^5*e^5)) - 1/1024*(7*c^6*d^12 - 30*a*c^5*d^10*e^2 + 45*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 + 18*a^5*c*d^2*e^10 - 5*a^6*e^12)*log(abs(-c*d^2 - a*e^2 - 2*sqrt(c*d*e)*(sqrt(c*d*e)*x - sqr
t(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^4)

Mupad [F(-1)]

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

[In]

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

[Out]

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