\(\int \frac {(d+e x)^3}{\sqrt {a d e+(c d^2+a e^2) x+c d e x^2}} \, dx\) [1946]

Optimal result

Integrand size = 37, antiderivative size = 255 \[ \int \frac {(d+e x)^3}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {5 \left (c d^2-a e^2\right )^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 c^3 d^3}+\frac {5 \left (c d^2-a e^2\right ) (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 c^2 d^2}+\frac {(d+e x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c d}+\frac {5 \left (c d^2-a e^2\right )^3 \text {arctanh}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{16 c^{7/2} d^{7/2} \sqrt {e}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.108, Rules used = {684, 654, 635, 212} \[ \int \frac {(d+e x)^3}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {5 \left (c d^2-a e^2\right )^3 \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{16 c^{7/2} d^{7/2} \sqrt {e}}+\frac {5 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 c^3 d^3}+\frac {5 (d+e x) \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{12 c^2 d^2}+\frac {(d+e x)^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 c d} \]

[In]

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

[Out]

(5*(c*d^2 - a*e^2)^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(8*c^3*d^3) + (5*(c*d^2 - a*e^2)*(d + e*x)*S
qrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(12*c^2*d^2) + ((d + e*x)^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e
*x^2])/(3*c*d) + (5*(c*d^2 - a*e^2)^3*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])])/(16*c^(7/2)*d^(7/2)*Sqrt[e])

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 654

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

Rule 684

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

Rubi steps \begin{align*} \text {integral}& = \frac {(d+e x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c d}+\frac {\left (5 \left (d^2-\frac {a e^2}{c}\right )\right ) \int \frac {(d+e x)^2}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{6 d} \\ & = \frac {5 \left (c d^2-a e^2\right ) (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 c^2 d^2}+\frac {(d+e x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c d}+\frac {\left (5 \left (d^2-\frac {a e^2}{c}\right )^2\right ) \int \frac {d+e x}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{8 d^2} \\ & = \frac {5 \left (c d^2-a e^2\right )^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 c^3 d^3}+\frac {5 \left (c d^2-a e^2\right ) (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 c^2 d^2}+\frac {(d+e x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c d}+\frac {\left (5 \left (d^2-\frac {a e^2}{c}\right )^3\right ) \int \frac {1}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{16 d^3} \\ & = \frac {5 \left (c d^2-a e^2\right )^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 c^3 d^3}+\frac {5 \left (c d^2-a e^2\right ) (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 c^2 d^2}+\frac {(d+e x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c d}+\frac {\left (5 \left (d^2-\frac {a e^2}{c}\right )^3\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 )}{8 d^3} \\ & = \frac {5 \left (c d^2-a e^2\right )^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 c^3 d^3}+\frac {5 \left (c d^2-a e^2\right ) (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 c^2 d^2}+\frac {(d+e x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c d}+\frac {5 \left (c d^2-a e^2\right )^3 \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 )}{16 c^{7/2} d^{7/2} \sqrt {e}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.73 \[ \int \frac {(d+e x)^3}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {\sqrt {c} \sqrt {d} (a e+c d x) (d+e x) \left (15 a^2 e^4-10 a c d e^2 (4 d+e x)+c^2 d^2 \left (33 d^2+26 d e x+8 e^2 x^2\right )\right )+\frac {15 \left (c d^2-a e^2\right )^3 \sqrt {a e+c d x} \sqrt {d+e x} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \sqrt {a e+c d x}}\right )}{\sqrt {e}}}{24 c^{7/2} d^{7/2} \sqrt {(a e+c d x) (d+e x)}} \]

[In]

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

[Out]

(Sqrt[c]*Sqrt[d]*(a*e + c*d*x)*(d + e*x)*(15*a^2*e^4 - 10*a*c*d*e^2*(4*d + e*x) + c^2*d^2*(33*d^2 + 26*d*e*x +
8*e^2*x^2)) + (15*(c*d^2 - a*e^2)^3*Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/(
Sqrt[e]*Sqrt[a*e + c*d*x])])/Sqrt[e])/(24*c^(7/2)*d^(7/2)*Sqrt[(a*e + c*d*x)*(d + e*x)])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(887\) vs. \(2(225)=450\).

Time = 2.72 (sec) , antiderivative size = 888, normalized size of antiderivative = 3.48

method	result	size
default	\(\frac {d^{3} \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 )}{\sqrt {c d e}}+e^{3} \left (\frac {x^{2} \sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}}{3 c d e}-\frac {5 \left (e^{2} a +c \,d^{2}\right ) \left (\frac {x \sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}}{2 c d e}-\frac {3 \left (e^{2} a +c \,d^{2}\right ) \left (\frac {\sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}}{c d e}-\frac {\left (e^{2} a +c \,d^{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 )}{2 c d e \sqrt {c d e}}\right )}{4 c d e}-\frac {a \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 )}{2 c \sqrt {c d e}}\right )}{6 c d e}-\frac {2 a \left (\frac {\sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}}{c d e}-\frac {\left (e^{2} a +c \,d^{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 )}{2 c d e \sqrt {c d e}}\right )}{3 c}\right )+3 d \,e^{2} \left (\frac {x \sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}}{2 c d e}-\frac {3 \left (e^{2} a +c \,d^{2}\right ) \left (\frac {\sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}}{c d e}-\frac {\left (e^{2} a +c \,d^{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 )}{2 c d e \sqrt {c d e}}\right )}{4 c d e}-\frac {a \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 )}{2 c \sqrt {c d e}}\right )+3 d^{2} e \left (\frac {\sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}}{c d e}-\frac {\left (e^{2} a +c \,d^{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 )}{2 c d e \sqrt {c d e}}\right )\)	\(888\)

[In]

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

[Out]

d^3*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)+e^3*
(1/3*x^2/c/d/e*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-5/6*(a*e^2+c*d^2)/c/d/e*(1/2*x/c/d/e*(a*d*e+(a*e^2+c*d^
2)*x+c*d*e*x^2)^(1/2)-3/4*(a*e^2+c*d^2)/c/d/e*(1/c/d/e*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-1/2*(a*e^2+c*d^
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)
)-1/2*a/c*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
))-2/3*a/c*(1/c/d/e*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-1/2*(a*e^2+c*d^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)))+3*d*e^2*(1/2*x/c/d/e*(a*d*e+(a*e
^2+c*d^2)*x+c*d*e*x^2)^(1/2)-3/4*(a*e^2+c*d^2)/c/d/e*(1/c/d/e*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-1/2*(a*e
^2+c*d^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))-1/2*a/c*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))+3*d^2*e*(1/c/d/e*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-1/2*(a*e^2+c*d^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 = 0.35 (sec) , antiderivative size = 534, normalized size of antiderivative = 2.09 \[ \int \frac {(d+e x)^3}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\left [\frac {15 \, {\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} \sqrt {c d e} \log \left (8 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4} + 4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {c d e} + 8 \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right ) + 4 \, {\left (8 \, c^{3} d^{3} e^{3} x^{2} + 33 \, c^{3} d^{5} e - 40 \, a c^{2} d^{3} e^{3} + 15 \, a^{2} c d e^{5} + 2 \, {\left (13 \, c^{3} d^{4} e^{2} - 5 \, a c^{2} d^{2} e^{4}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{96 \, c^{4} d^{4} e}, -\frac {15 \, {\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} \sqrt {-c d e} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {-c d e}}{2 \, {\left (c^{2} d^{2} e^{2} x^{2} + a c d^{2} e^{2} + {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right )}}\right ) - 2 \, {\left (8 \, c^{3} d^{3} e^{3} x^{2} + 33 \, c^{3} d^{5} e - 40 \, a c^{2} d^{3} e^{3} + 15 \, a^{2} c d e^{5} + 2 \, {\left (13 \, c^{3} d^{4} e^{2} - 5 \, a c^{2} d^{2} e^{4}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{48 \, c^{4} d^{4} e}\right ] \]

[In]

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

[Out]

[1/96*(15*(c^3*d^6 - 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 - a^3*e^6)*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*(8*c^3*d^3*e^3*x^2 + 33*c^3*d^5*e - 40*a*c^2*d^3*e^3 + 15*a^2*c*d*e^5 +
2*(13*c^3*d^4*e^2 - 5*a*c^2*d^2*e^4)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^4*d^4*e), -1/48*(15*(
c^3*d^6 - 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 - a^3*e^6)*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*(8*c^3*d^3*e^3*x^2 + 33*c^3*d^5*e - 40*a*c^2*d^3*e^3 + 15*a^2*c*d*e^5 + 2*(13*c^3*d^4*e^2 - 5*a*c^2*d
^2*e^4)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^4*d^4*e)]

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 782 vs. \(2 (245) = 490\).

Time = 0.94 (sec) , antiderivative size = 782, normalized size of antiderivative = 3.07 \[ \int \frac {(d+e x)^3}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\begin {cases} \left (- \frac {a \left (3 d e^{2} - \frac {e^{2} \cdot \left (\frac {5 a e^{2}}{2} + \frac {5 c d^{2}}{2}\right )}{3 c d}\right )}{2 c} + d^{3} - \frac {\left (a e^{2} + c d^{2}\right ) \left (- \frac {2 a e^{3}}{3 c} + 3 d^{2} e - \frac {\left (\frac {3 a e^{2}}{2} + \frac {3 c d^{2}}{2}\right ) \left (3 d e^{2} - \frac {e^{2} \cdot \left (\frac {5 a e^{2}}{2} + \frac {5 c d^{2}}{2}\right )}{3 c d}\right )}{2 c d e}\right )}{2 c d e}\right ) \left (\begin {cases} \frac {\log {\left (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 \left (a e^{2} + c d^{2}\right )} \right )}}{\sqrt {c d e}} & \text {for}\: a d e - \frac {\left (a e^{2} + c d^{2}\right )^{2}}{4 c d e} \neq 0 \\\frac {\left (x - \frac {- a e^{2} - c d^{2}}{2 c d e}\right ) \log {\left (x - \frac {- a e^{2} - c d^{2}}{2 c d e} \right )}}{\sqrt {c d e \left (x - \frac {- a e^{2} - c d^{2}}{2 c d e}\right )^{2}}} & \text {otherwise} \end {cases}\right ) + \sqrt {a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )} \left (\frac {e^{2} x^{2}}{3 c d} + \frac {x \left (3 d e^{2} - \frac {e^{2} \cdot \left (\frac {5 a e^{2}}{2} + \frac {5 c d^{2}}{2}\right )}{3 c d}\right )}{2 c d e} + \frac {- \frac {2 a e^{3}}{3 c} + 3 d^{2} e - \frac {\left (\frac {3 a e^{2}}{2} + \frac {3 c d^{2}}{2}\right ) \left (3 d e^{2} - \frac {e^{2} \cdot \left (\frac {5 a e^{2}}{2} + \frac {5 c d^{2}}{2}\right )}{3 c d}\right )}{2 c d e}}{c d e}\right ) & \text {for}\: c d e \neq 0 \\\frac {2 \left (\frac {c^{3} d^{9} \sqrt {a d e + x \left (a e^{2} + c d^{2}\right )}}{a^{3} e^{6} + 3 a^{2} c d^{2} e^{4} + 3 a c^{2} d^{4} e^{2} + c^{3} d^{6}} + \frac {c^{2} d^{6} e \left (a d e + x \left (a e^{2} + c d^{2}\right )\right )^{\frac {3}{2}}}{a^{3} e^{6} + 3 a^{2} c d^{2} e^{4} + 3 a c^{2} d^{4} e^{2} + c^{3} d^{6}} + \frac {3 c d^{3} e^{2} \left (a d e + x \left (a e^{2} + c d^{2}\right )\right )^{\frac {5}{2}}}{5 \left (a^{3} e^{6} + 3 a^{2} c d^{2} e^{4} + 3 a c^{2} d^{4} e^{2} + c^{3} d^{6}\right )} + \frac {e^{3} \left (a d e + x \left (a e^{2} + c d^{2}\right )\right )^{\frac {7}{2}}}{7 \left (a^{3} e^{6} + 3 a^{2} c d^{2} e^{4} + 3 a c^{2} d^{4} e^{2} + c^{3} d^{6}\right )}\right )}{a e^{2} + c d^{2}} & \text {for}\: a e^{2} + c d^{2} \neq 0 \\\frac {\begin {cases} d^{3} x & \text {for}\: e = 0 \\\frac {\left (d + e x\right )^{4}}{4 e} & \text {otherwise} \end {cases}}{\sqrt {a d e}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise(((-a*(3*d*e**2 - e**2*(5*a*e**2/2 + 5*c*d**2/2)/(3*c*d))/(2*c) + d**3 - (a*e**2 + c*d**2)*(-2*a*e**3
/(3*c) + 3*d**2*e - (3*a*e**2/2 + 3*c*d**2/2)*(3*d*e**2 - e**2*(5*a*e**2/2 + 5*c*d**2/2)/(3*c*d))/(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))*(e**2*x**2/(3*c*d) + x*(3*d*e**2 - e**2*(5*a*e**2/2 + 5*c*d**2/2)/(3*c*d))/(2*c*d
*e) + (-2*a*e**3/(3*c) + 3*d**2*e - (3*a*e**2/2 + 3*c*d**2/2)*(3*d*e**2 - e**2*(5*a*e**2/2 + 5*c*d**2/2)/(3*c*
d))/(2*c*d*e))/(c*d*e)), Ne(c*d*e, 0)), (2*(c**3*d**9*sqrt(a*d*e + x*(a*e**2 + c*d**2))/(a**3*e**6 + 3*a**2*c*
d**2*e**4 + 3*a*c**2*d**4*e**2 + c**3*d**6) + c**2*d**6*e*(a*d*e + x*(a*e**2 + c*d**2))**(3/2)/(a**3*e**6 + 3*
a**2*c*d**2*e**4 + 3*a*c**2*d**4*e**2 + c**3*d**6) + 3*c*d**3*e**2*(a*d*e + x*(a*e**2 + c*d**2))**(5/2)/(5*(a*
*3*e**6 + 3*a**2*c*d**2*e**4 + 3*a*c**2*d**4*e**2 + c**3*d**6)) + e**3*(a*d*e + x*(a*e**2 + c*d**2))**(7/2)/(7
*(a**3*e**6 + 3*a**2*c*d**2*e**4 + 3*a*c**2*d**4*e**2 + c**3*d**6)))/(a*e**2 + c*d**2), Ne(a*e**2 + c*d**2, 0)
), (Piecewise((d**3*x, Eq(e, 0)), ((d + e*x)**4/(4*e), True))/sqrt(a*d*e), True))

Maxima [F(-2)]

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

[In]

integrate((e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),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(a*e^2-c*d^2>0)', see `assume?`
for more de

Giac [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 230, normalized size of antiderivative = 0.90 \[ \int \frac {(d+e x)^3}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {1}{24} \, \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e} {\left (2 \, {\left (\frac {4 \, e^{2} x}{c d} + \frac {13 \, c^{2} d^{3} e^{3} - 5 \, a c d e^{5}}{c^{3} d^{3} e^{2}}\right )} x + \frac {33 \, c^{2} d^{4} e^{2} - 40 \, a c d^{2} e^{4} + 15 \, a^{2} e^{6}}{c^{3} d^{3} e^{2}}\right )} - \frac {5 \, {\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} \log \left ({\left | -c d^{2} - a e^{2} - 2 \, \sqrt {c d e} {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )} \right |}\right )}{16 \, \sqrt {c d e} c^{3} d^{3}} \]

[In]

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

[Out]

1/24*sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e)*(2*(4*e^2*x/(c*d) + (13*c^2*d^3*e^3 - 5*a*c*d*e^5)/(c^3*d^3*e
^2))*x + (33*c^2*d^4*e^2 - 40*a*c*d^2*e^4 + 15*a^2*e^6)/(c^3*d^3*e^2)) - 5/16*(c^3*d^6 - 3*a*c^2*d^4*e^2 + 3*a
^2*c*d^2*e^4 - a^3*e^6)*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)

Mupad [F(-1)]

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

[In]

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

[Out]

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