\(\int \frac {x^3 \sqrt {a d e+(c d^2+a e^2) x+c d e x^2}}{d+e x} \, dx\) [437]

Optimal result

Integrand size = 40, antiderivative size = 345 \[ \int \frac {x^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{d+e x} \, dx=\frac {1}{24} \left (\frac {a}{c d}-\frac {7 d}{e^2}\right ) x^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}+\frac {x^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 e}-\frac {\left (105 c^3 d^6-25 a c^2 d^4 e^2-17 a^2 c d^2 e^4-15 a^3 e^6-2 c d e \left (35 c^2 d^4-6 a c d^2 e^2-5 a^2 e^4\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{192 c^3 d^3 e^4}+\frac {\left (c d^2-a e^2\right ) \left (35 c^3 d^6+15 a c^2 d^4 e^2+9 a^2 c d^2 e^4+5 a^3 e^6\right ) \text {arctanh}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{128 c^{7/2} d^{7/2} e^{9/2}} \]

[Out]

1/128*(-a*e^2+c*d^2)*(5*a^3*e^6+9*a^2*c*d^2*e^4+15*a*c^2*d^4*e^2+35*c^3*d^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^(9/2)+1/24*(a/c/d-7*d/e^
2)*x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+1/4*x^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/e-1/192*(105*c^
3*d^6-25*a*c^2*d^4*e^2-17*a^2*c*d^2*e^4-15*a^3*e^6-2*c*d*e*(-5*a^2*e^4-6*a*c*d^2*e^2+35*c^2*d^4)*x)*(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 = 0.27 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {863, 846, 793, 635, 212} \[ \int \frac {x^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{d+e x} \, dx=\frac {\left (c d^2-a e^2\right ) \left (5 a^3 e^6+9 a^2 c d^2 e^4+15 a c^2 d^4 e^2+35 c^3 d^6\right ) \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{128 c^{7/2} d^{7/2} e^{9/2}}-\frac {\left (-15 a^3 e^6-2 c d e x \left (-5 a^2 e^4-6 a c d^2 e^2+35 c^2 d^4\right )-17 a^2 c d^2 e^4-25 a c^2 d^4 e^2+105 c^3 d^6\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{192 c^3 d^3 e^4}+\frac {1}{24} x^2 \left (\frac {a}{c d}-\frac {7 d}{e^2}\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}+\frac {x^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 e} \]

[In]

Int[(x^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(d + e*x),x]

[Out]

((a/(c*d) - (7*d)/e^2)*x^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/24 + (x^3*Sqrt[a*d*e + (c*d^2 + a*e^2)
*x + c*d*e*x^2])/(4*e) - ((105*c^3*d^6 - 25*a*c^2*d^4*e^2 - 17*a^2*c*d^2*e^4 - 15*a^3*e^6 - 2*c*d*e*(35*c^2*d^
4 - 6*a*c*d^2*e^2 - 5*a^2*e^4)*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(192*c^3*d^3*e^4) + ((c*d^2 - a
*e^2)*(35*c^3*d^6 + 15*a*c^2*d^4*e^2 + 9*a^2*c*d^2*e^4 + 5*a^3*e^6)*ArcTanh[(c*d^2 + a*e^2 + 2*c*d*e*x)/(2*Sqr
t[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])])/(128*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 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 793

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(b
*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 2*c*e*g*(p + 1)*x))*((a + b*x + c*x^2)^(p + 1)/(2*c^2*(p + 1)*(2*p +
3))), x] + Dist[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p + 3))/(2*c^2*(2*p + 3)), Int[(
a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[p, -1]

Rule 846

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

Rule 863

Int[((x_)^(n_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_))/((d_) + (e_.)*(x_)), x_Symbol] :> Int[x^n*(a/d + c*(
x/e))*(a + b*x + c*x^2)^(p - 1), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b
*d*e + a*e^2, 0] &&  !IntegerQ[p] && ( !IntegerQ[n] ||  !IntegerQ[2*p] || IGtQ[n, 2] || (GtQ[p, 0] && NeQ[n, 2
]))

Rubi steps \begin{align*} \text {integral}& = \int \frac {x^3 (a e+c d x)}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx \\ & = \frac {x^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 e}+\frac {\int \frac {x^2 \left (-3 a c d^2 e-\frac {1}{2} c d \left (7 c d^2-a e^2\right ) x\right )}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{4 c d e} \\ & = \frac {1}{24} \left (\frac {a}{c d}-\frac {7 d}{e^2}\right ) x^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}+\frac {x^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 e}+\frac {\int \frac {x \left (a c d^2 e \left (7 c d^2-a e^2\right )+\frac {1}{4} c d \left (35 c^2 d^4-6 a c d^2 e^2-5 a^2 e^4\right ) x\right )}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{12 c^2 d^2 e^2} \\ & = \frac {1}{24} \left (\frac {a}{c d}-\frac {7 d}{e^2}\right ) x^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}+\frac {x^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 e}-\frac {\left (105 c^3 d^6-25 a c^2 d^4 e^2-17 a^2 c d^2 e^4-15 a^3 e^6-2 c d e \left (35 c^2 d^4-6 a c d^2 e^2-5 a^2 e^4\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{192 c^3 d^3 e^4}+\frac {\left (\left (c d^2-a e^2\right ) \left (35 c^3 d^6+15 a c^2 d^4 e^2+9 a^2 c d^2 e^4+5 a^3 e^6\right )\right ) \int \frac {1}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{128 c^3 d^3 e^4} \\ & = \frac {1}{24} \left (\frac {a}{c d}-\frac {7 d}{e^2}\right ) x^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}+\frac {x^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 e}-\frac {\left (105 c^3 d^6-25 a c^2 d^4 e^2-17 a^2 c d^2 e^4-15 a^3 e^6-2 c d e \left (35 c^2 d^4-6 a c d^2 e^2-5 a^2 e^4\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{192 c^3 d^3 e^4}+\frac {\left (\left (c d^2-a e^2\right ) \left (35 c^3 d^6+15 a c^2 d^4 e^2+9 a^2 c d^2 e^4+5 a^3 e^6\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 )}{64 c^3 d^3 e^4} \\ & = \frac {1}{24} \left (\frac {a}{c d}-\frac {7 d}{e^2}\right ) x^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}+\frac {x^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 e}-\frac {\left (105 c^3 d^6-25 a c^2 d^4 e^2-17 a^2 c d^2 e^4-15 a^3 e^6-2 c d e \left (35 c^2 d^4-6 a c d^2 e^2-5 a^2 e^4\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{192 c^3 d^3 e^4}+\frac {\left (c d^2-a e^2\right ) \left (35 c^3 d^6+15 a c^2 d^4 e^2+9 a^2 c d^2 e^4+5 a^3 e^6\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 )}{128 c^{7/2} d^{7/2} e^{9/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 11.13 (sec) , antiderivative size = 304, normalized size of antiderivative = 0.88 \[ \int \frac {x^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{d+e x} \, dx=\frac {\sqrt {(a e+c d x) (d+e x)} \left (-\sqrt {c} \sqrt {d} \sqrt {e} \left (-15 a^3 e^6+a^2 c d e^4 (-17 d+10 e x)+a c^2 d^2 e^2 \left (-25 d^2+12 d e x-8 e^2 x^2\right )+c^3 d^3 \left (105 d^3-70 d^2 e x+56 d e^2 x^2-48 e^3 x^3\right )\right )+\frac {3 \sqrt {c d} \sqrt {c d^2-a e^2} \left (35 c^3 d^6+15 a c^2 d^4 e^2+9 a^2 c d^2 e^4+5 a^3 e^6\right ) \text {arcsinh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a e+c d x}}{\sqrt {c d} \sqrt {c d^2-a e^2}}\right )}{\sqrt {a e+c d x} \sqrt {\frac {c d (d+e x)}{c d^2-a e^2}}}\right )}{192 c^{7/2} d^{7/2} e^{9/2}} \]

[In]

Integrate[(x^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(d + e*x),x]

[Out]

(Sqrt[(a*e + c*d*x)*(d + e*x)]*(-(Sqrt[c]*Sqrt[d]*Sqrt[e]*(-15*a^3*e^6 + a^2*c*d*e^4*(-17*d + 10*e*x) + a*c^2*
d^2*e^2*(-25*d^2 + 12*d*e*x - 8*e^2*x^2) + c^3*d^3*(105*d^3 - 70*d^2*e*x + 56*d*e^2*x^2 - 48*e^3*x^3))) + (3*S
qrt[c*d]*Sqrt[c*d^2 - a*e^2]*(35*c^3*d^6 + 15*a*c^2*d^4*e^2 + 9*a^2*c*d^2*e^4 + 5*a^3*e^6)*ArcSinh[(Sqrt[c]*Sq
rt[d]*Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d^2 - a*e^2])])/(Sqrt[a*e + c*d*x]*Sqrt[(c*d*(d + e*x))/(c*
d^2 - a*e^2)])))/(192*c^(7/2)*d^(7/2)*e^(9/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(954\) vs. \(2(315)=630\).

Time = 0.68 (sec) , antiderivative size = 955, normalized size of antiderivative = 2.77

method	result	size
default	\(\frac {\frac {x {\left (a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}\right )}^{\frac {3}{2}}}{4 c d e}-\frac {5 \left (e^{2} a +c \,d^{2}\right ) \left (\frac {{\left (a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}\right )}^{\frac {3}{2}}}{3 c d e}-\frac {\left (e^{2} a +c \,d^{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 )}{2 c d e}\right )}{8 c d e}-\frac {a \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 )}{4 c}}{e}+\frac {d^{2} \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 )}{e^{3}}-\frac {d \left (\frac {{\left (a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}\right )}^{\frac {3}{2}}}{3 c d e}-\frac {\left (e^{2} a +c \,d^{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 )}{2 c d e}\right )}{e^{2}}-\frac {d^{3} \left (\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 )}+\frac {\left (e^{2} a -c \,d^{2}\right ) \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 )}{2 \sqrt {c d e}}\right )}{e^{4}}\)	\(955\)

[In]

int(x^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d),x,method=_RETURNVERBOSE)

[Out]

1/e*(1/4*x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/c/d/e-5/8*(a*e^2+c*d^2)/c/d/e*(1/3*(a*d*e+(a*e^2+c*d^2)*x+c
*d*e*x^2)^(3/2)/c/d/e-1/2*(a*e^2+c*d^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)))-1/4*a/c*(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^2/e^3*(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*(1/3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(
3/2)/c/d/e-1/2*(a*e^2+c*d^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^3/e^4*((c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)+1/2*(a*e^2-c*d^2)*l
n((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 = 0.38 (sec) , antiderivative size = 678, normalized size of antiderivative = 1.97 \[ \int \frac {x^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{d+e x} \, dx=\left [-\frac {3 \, {\left (35 \, c^{4} d^{8} - 20 \, a c^{3} d^{6} e^{2} - 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} - 5 \, a^{4} e^{8}\right )} \sqrt {c d e} \log \left (8 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4} - 4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {c d e} + 8 \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right ) - 4 \, {\left (48 \, c^{4} d^{4} e^{4} x^{3} - 105 \, c^{4} d^{7} e + 25 \, a c^{3} d^{5} e^{3} + 17 \, a^{2} c^{2} d^{3} e^{5} + 15 \, a^{3} c d e^{7} - 8 \, {\left (7 \, c^{4} d^{5} e^{3} - a c^{3} d^{3} e^{5}\right )} x^{2} + 2 \, {\left (35 \, c^{4} d^{6} e^{2} - 6 \, a c^{3} d^{4} e^{4} - 5 \, a^{2} c^{2} d^{2} e^{6}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{768 \, c^{4} d^{4} e^{5}}, -\frac {3 \, {\left (35 \, c^{4} d^{8} - 20 \, a c^{3} d^{6} e^{2} - 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} - 5 \, a^{4} e^{8}\right )} \sqrt {-c d e} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {-c d e}}{2 \, {\left (c^{2} d^{2} e^{2} x^{2} + a c d^{2} e^{2} + {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right )}}\right ) - 2 \, {\left (48 \, c^{4} d^{4} e^{4} x^{3} - 105 \, c^{4} d^{7} e + 25 \, a c^{3} d^{5} e^{3} + 17 \, a^{2} c^{2} d^{3} e^{5} + 15 \, a^{3} c d e^{7} - 8 \, {\left (7 \, c^{4} d^{5} e^{3} - a c^{3} d^{3} e^{5}\right )} x^{2} + 2 \, {\left (35 \, c^{4} d^{6} e^{2} - 6 \, a c^{3} d^{4} e^{4} - 5 \, a^{2} c^{2} d^{2} e^{6}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{384 \, c^{4} d^{4} e^{5}}\right ] \]

[In]

integrate(x^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d),x, algorithm="fricas")

[Out]

[-1/768*(3*(35*c^4*d^8 - 20*a*c^3*d^6*e^2 - 6*a^2*c^2*d^4*e^4 - 4*a^3*c*d^2*e^6 - 5*a^4*e^8)*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*(48*c^4*d^4*e^4*x^3 - 105*c^4*d^7*e + 25*a*c
^3*d^5*e^3 + 17*a^2*c^2*d^3*e^5 + 15*a^3*c*d*e^7 - 8*(7*c^4*d^5*e^3 - a*c^3*d^3*e^5)*x^2 + 2*(35*c^4*d^6*e^2 -
6*a*c^3*d^4*e^4 - 5*a^2*c^2*d^2*e^6)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^4*d^4*e^5), -1/384*(3
*(35*c^4*d^8 - 20*a*c^3*d^6*e^2 - 6*a^2*c^2*d^4*e^4 - 4*a^3*c*d^2*e^6 - 5*a^4*e^8)*sqrt(-c*d*e)*arctan(1/2*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)/(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*(48*c^4*d^4*e^4*x^3 - 105*c^4*d^7*e + 25*a*c^3*d^5*e^3 + 17*a^2*c^2*d^3*e
^5 + 15*a^3*c*d*e^7 - 8*(7*c^4*d^5*e^3 - a*c^3*d^3*e^5)*x^2 + 2*(35*c^4*d^6*e^2 - 6*a*c^3*d^4*e^4 - 5*a^2*c^2*
d^2*e^6)*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]

\[ \int \frac {x^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{d+e x} \, dx=\int \frac {x^{3} \sqrt {\left (d + e x\right ) \left (a e + c d x\right )}}{d + e x}\, dx \]

[In]

integrate(x**3*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2)/(e*x+d),x)

[Out]

Integral(x**3*sqrt((d + e*x)*(a*e + c*d*x))/(d + e*x), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{d+e x} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(x^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(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 = 0.36 (sec) , antiderivative size = 305, normalized size of antiderivative = 0.88 \[ \int \frac {x^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{d+e x} \, dx=\frac {1}{192} \, \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e} {\left (2 \, {\left (4 \, x {\left (\frac {6 \, x}{e} - \frac {7 \, c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}}{c^{3} d^{3} e^{4}}\right )} + \frac {35 \, c^{3} d^{5} e - 6 \, a c^{2} d^{3} e^{3} - 5 \, a^{2} c d e^{5}}{c^{3} d^{3} e^{4}}\right )} x - \frac {105 \, c^{3} d^{6} - 25 \, a c^{2} d^{4} e^{2} - 17 \, a^{2} c d^{2} e^{4} - 15 \, a^{3} e^{6}}{c^{3} d^{3} e^{4}}\right )} - \frac {{\left (35 \, c^{4} d^{8} - 20 \, a c^{3} d^{6} e^{2} - 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} - 5 \, a^{4} e^{8}\right )} \log \left ({\left | -c d^{2} - a e^{2} - 2 \, \sqrt {c d e} {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )} \right |}\right )}{128 \, \sqrt {c d e} c^{3} d^{3} e^{4}} \]

[In]

integrate(x^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d),x, algorithm="giac")

[Out]

1/192*sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e)*(2*(4*x*(6*x/e - (7*c^3*d^4*e^2 - a*c^2*d^2*e^4)/(c^3*d^3*e^
4)) + (35*c^3*d^5*e - 6*a*c^2*d^3*e^3 - 5*a^2*c*d*e^5)/(c^3*d^3*e^4))*x - (105*c^3*d^6 - 25*a*c^2*d^4*e^2 - 17
*a^2*c*d^2*e^4 - 15*a^3*e^6)/(c^3*d^3*e^4)) - 1/128*(35*c^4*d^8 - 20*a*c^3*d^6*e^2 - 6*a^2*c^2*d^4*e^4 - 4*a^3
*c*d^2*e^6 - 5*a^4*e^8)*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^4)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{d+e x} \, dx=\int \frac {x^3\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{d+e\,x} \,d x \]

[In]

int((x^3*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x),x)

[Out]

int((x^3*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x), x)