\(\int (d+e x)^3 \sqrt {a+b x+c x^2} \, dx\) [2333]

Optimal result

Integrand size = 22, antiderivative size = 248 \[ \int (d+e x)^3 \sqrt {a+b x+c x^2} \, dx=\frac {(2 c d-b e) \left (16 c^2 d^2+7 b^2 e^2-4 c e (4 b d+3 a e)\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{128 c^4}+\frac {e (d+e x)^2 \left (a+b x+c x^2\right )^{3/2}}{5 c}+\frac {e \left (192 c^2 d^2+35 b^2 e^2-2 c e (75 b d+16 a e)+42 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{240 c^3}-\frac {\left (b^2-4 a c\right ) (2 c d-b e) \left (16 c^2 d^2+7 b^2 e^2-4 c e (4 b d+3 a e)\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{256 c^{9/2}} \]

[Out]

1/5*e*(e*x+d)^2*(c*x^2+b*x+a)^(3/2)/c+1/240*e*(192*c^2*d^2+35*b^2*e^2-2*c*e*(16*a*e+75*b*d)+42*c*e*(-b*e+2*c*d
)*x)*(c*x^2+b*x+a)^(3/2)/c^3-1/256*(-4*a*c+b^2)*(-b*e+2*c*d)*(16*c^2*d^2+7*b^2*e^2-4*c*e*(3*a*e+4*b*d))*arctan
h(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^(9/2)+1/128*(-b*e+2*c*d)*(16*c^2*d^2+7*b^2*e^2-4*c*e*(3*a*e+4*b
*d))*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c^4

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {756, 793, 626, 635, 212} \[ \int (d+e x)^3 \sqrt {a+b x+c x^2} \, dx=-\frac {\left (b^2-4 a c\right ) (2 c d-b e) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-4 c e (3 a e+4 b d)+7 b^2 e^2+16 c^2 d^2\right )}{256 c^{9/2}}+\frac {(b+2 c x) \sqrt {a+b x+c x^2} (2 c d-b e) \left (-4 c e (3 a e+4 b d)+7 b^2 e^2+16 c^2 d^2\right )}{128 c^4}+\frac {e \left (a+b x+c x^2\right )^{3/2} \left (-2 c e (16 a e+75 b d)+35 b^2 e^2+42 c e x (2 c d-b e)+192 c^2 d^2\right )}{240 c^3}+\frac {e (d+e x)^2 \left (a+b x+c x^2\right )^{3/2}}{5 c} \]

[In]

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

[Out]

((2*c*d - b*e)*(16*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(4*b*d + 3*a*e))*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(128*c^4) +
(e*(d + e*x)^2*(a + b*x + c*x^2)^(3/2))/(5*c) + (e*(192*c^2*d^2 + 35*b^2*e^2 - 2*c*e*(75*b*d + 16*a*e) + 42*c
*e*(2*c*d - b*e)*x)*(a + b*x + c*x^2)^(3/2))/(240*c^3) - ((b^2 - 4*a*c)*(2*c*d - b*e)*(16*c^2*d^2 + 7*b^2*e^2
- 4*c*e*(4*b*d + 3*a*e))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(256*c^(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 756

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[1/(c*(m + 2*p + 1)), Int[(d + e*x)^(m - 2)*Simp[c*d^2
*(m + 2*p + 1) - e*(a*e*(m - 1) + b*d*(p + 1)) + e*(2*c*d - b*e)*(m + p)*x, x]*(a + b*x + c*x^2)^p, x], x] /;
FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0]
&& If[RationalQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadraticQ[a, b, c, d, e, m,
p, x]

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]

Rubi steps \begin{align*} \text {integral}& = \frac {e (d+e x)^2 \left (a+b x+c x^2\right )^{3/2}}{5 c}+\frac {\int (d+e x) \left (\frac {1}{2} \left (10 c d^2-e (3 b d+4 a e)\right )+\frac {7}{2} e (2 c d-b e) x\right ) \sqrt {a+b x+c x^2} \, dx}{5 c} \\ & = \frac {e (d+e x)^2 \left (a+b x+c x^2\right )^{3/2}}{5 c}+\frac {e \left (192 c^2 d^2+35 b^2 e^2-2 c e (75 b d+16 a e)+42 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{240 c^3}+\frac {\left ((2 c d-b e) \left (16 c^2 d^2+7 b^2 e^2-4 c e (4 b d+3 a e)\right )\right ) \int \sqrt {a+b x+c x^2} \, dx}{32 c^3} \\ & = \frac {(2 c d-b e) \left (16 c^2 d^2+7 b^2 e^2-4 c e (4 b d+3 a e)\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{128 c^4}+\frac {e (d+e x)^2 \left (a+b x+c x^2\right )^{3/2}}{5 c}+\frac {e \left (192 c^2 d^2+35 b^2 e^2-2 c e (75 b d+16 a e)+42 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{240 c^3}-\frac {\left (\left (b^2-4 a c\right ) (2 c d-b e) \left (16 c^2 d^2+7 b^2 e^2-4 c e (4 b d+3 a e)\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{256 c^4} \\ & = \frac {(2 c d-b e) \left (16 c^2 d^2+7 b^2 e^2-4 c e (4 b d+3 a e)\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{128 c^4}+\frac {e (d+e x)^2 \left (a+b x+c x^2\right )^{3/2}}{5 c}+\frac {e \left (192 c^2 d^2+35 b^2 e^2-2 c e (75 b d+16 a e)+42 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{240 c^3}-\frac {\left (\left (b^2-4 a c\right ) (2 c d-b e) \left (16 c^2 d^2+7 b^2 e^2-4 c e (4 b d+3 a e)\right )\right ) \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )}{128 c^4} \\ & = \frac {(2 c d-b e) \left (16 c^2 d^2+7 b^2 e^2-4 c e (4 b d+3 a e)\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{128 c^4}+\frac {e (d+e x)^2 \left (a+b x+c x^2\right )^{3/2}}{5 c}+\frac {e \left (192 c^2 d^2+35 b^2 e^2-2 c e (75 b d+16 a e)+42 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{240 c^3}-\frac {\left (b^2-4 a c\right ) (2 c d-b e) \left (16 c^2 d^2+7 b^2 e^2-4 c e (4 b d+3 a e)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{256 c^{9/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.93 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.19 \[ \int (d+e x)^3 \sqrt {a+b x+c x^2} \, dx=\frac {\sqrt {c} \sqrt {a+x (b+c x)} \left (-105 b^4 e^3+10 b^3 c e^2 (45 d+7 e x)-4 b^2 c e \left (-115 a e^2+c \left (180 d^2+75 d e x+14 e^2 x^2\right )\right )+8 b c^2 \left (-a e^2 (195 d+29 e x)+6 c \left (10 d^3+10 d^2 e x+5 d e^2 x^2+e^3 x^3\right )\right )+16 c^2 \left (-16 a^2 e^3+a c e \left (120 d^2+45 d e x+8 e^2 x^2\right )+6 c^2 x \left (10 d^3+20 d^2 e x+15 d e^2 x^2+4 e^3 x^3\right )\right )\right )+15 \left (b^2-4 a c\right ) (-2 c d+b e) \left (16 c^2 d^2+7 b^2 e^2-4 c e (4 b d+3 a e)\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{-\sqrt {a}+\sqrt {a+x (b+c x)}}\right )}{1920 c^{9/2}} \]

[In]

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

[Out]

(Sqrt[c]*Sqrt[a + x*(b + c*x)]*(-105*b^4*e^3 + 10*b^3*c*e^2*(45*d + 7*e*x) - 4*b^2*c*e*(-115*a*e^2 + c*(180*d^
2 + 75*d*e*x + 14*e^2*x^2)) + 8*b*c^2*(-(a*e^2*(195*d + 29*e*x)) + 6*c*(10*d^3 + 10*d^2*e*x + 5*d*e^2*x^2 + e^
3*x^3)) + 16*c^2*(-16*a^2*e^3 + a*c*e*(120*d^2 + 45*d*e*x + 8*e^2*x^2) + 6*c^2*x*(10*d^3 + 20*d^2*e*x + 15*d*e
^2*x^2 + 4*e^3*x^3))) + 15*(b^2 - 4*a*c)*(-2*c*d + b*e)*(16*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(4*b*d + 3*a*e))*ArcTa
nh[(Sqrt[c]*x)/(-Sqrt[a] + Sqrt[a + x*(b + c*x)])])/(1920*c^(9/2))

Maple [A] (verified)

Time = 0.41 (sec) , antiderivative size = 391, normalized size of antiderivative = 1.58

method	result	size
risch	\(-\frac {\left (-384 e^{3} c^{4} x^{4}-48 b \,c^{3} e^{3} x^{3}-1440 c^{4} d \,e^{2} x^{3}-128 a \,c^{3} e^{3} x^{2}+56 b^{2} c^{2} e^{3} x^{2}-240 b \,c^{3} d \,e^{2} x^{2}-1920 c^{4} d^{2} e \,x^{2}+232 a b \,c^{2} e^{3} x -720 a \,c^{3} d \,e^{2} x -70 b^{3} c \,e^{3} x +300 b^{2} c^{2} d \,e^{2} x -480 b \,c^{3} d^{2} e x -960 c^{4} d^{3} x +256 a^{2} c^{2} e^{3}-460 a \,b^{2} c \,e^{3}+1560 a b \,c^{2} d \,e^{2}-1920 a \,c^{3} d^{2} e +105 b^{4} e^{3}-450 b^{3} c d \,e^{2}+720 b^{2} c^{2} d^{2} e -480 b \,c^{3} d^{3}\right ) \sqrt {c \,x^{2}+b x +a}}{1920 c^{4}}+\frac {\left (48 a^{2} b \,c^{2} e^{3}-96 a^{2} c^{3} d \,e^{2}-40 a \,b^{3} c \,e^{3}+144 a \,b^{2} c^{2} d \,e^{2}-192 a b \,c^{3} d^{2} e +128 a \,c^{4} d^{3}+7 b^{5} e^{3}-30 b^{4} c d \,e^{2}+48 b^{3} c^{2} d^{2} e -32 b^{2} c^{3} d^{3}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{256 c^{\frac {9}{2}}}\)	\(391\)
default	\(d^{3} \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )+e^{3} \left (\frac {x^{2} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{5 c}-\frac {7 b \left (\frac {x \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{4 c}-\frac {5 b \left (\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{3 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{2 c}\right )}{8 c}-\frac {a \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{4 c}\right )}{10 c}-\frac {2 a \left (\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{3 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{2 c}\right )}{5 c}\right )+3 d \,e^{2} \left (\frac {x \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{4 c}-\frac {5 b \left (\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{3 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{2 c}\right )}{8 c}-\frac {a \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{4 c}\right )+3 d^{2} e \left (\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{3 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{2 c}\right )\)	\(661\)

[In]

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

[Out]

-1/1920*(-384*c^4*e^3*x^4-48*b*c^3*e^3*x^3-1440*c^4*d*e^2*x^3-128*a*c^3*e^3*x^2+56*b^2*c^2*e^3*x^2-240*b*c^3*d
*e^2*x^2-1920*c^4*d^2*e*x^2+232*a*b*c^2*e^3*x-720*a*c^3*d*e^2*x-70*b^3*c*e^3*x+300*b^2*c^2*d*e^2*x-480*b*c^3*d
^2*e*x-960*c^4*d^3*x+256*a^2*c^2*e^3-460*a*b^2*c*e^3+1560*a*b*c^2*d*e^2-1920*a*c^3*d^2*e+105*b^4*e^3-450*b^3*c
*d*e^2+720*b^2*c^2*d^2*e-480*b*c^3*d^3)*(c*x^2+b*x+a)^(1/2)/c^4+1/256*(48*a^2*b*c^2*e^3-96*a^2*c^3*d*e^2-40*a*
b^3*c*e^3+144*a*b^2*c^2*d*e^2-192*a*b*c^3*d^2*e+128*a*c^4*d^3+7*b^5*e^3-30*b^4*c*d*e^2+48*b^3*c^2*d^2*e-32*b^2
*c^3*d^3)/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 787, normalized size of antiderivative = 3.17 \[ \int (d+e x)^3 \sqrt {a+b x+c x^2} \, dx=\left [-\frac {15 \, {\left (32 \, {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} d^{3} - 48 \, {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} d^{2} e + 6 \, {\left (5 \, b^{4} c - 24 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} d e^{2} - {\left (7 \, b^{5} - 40 \, a b^{3} c + 48 \, a^{2} b c^{2}\right )} e^{3}\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} - 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) - 4 \, {\left (384 \, c^{5} e^{3} x^{4} + 480 \, b c^{4} d^{3} - 240 \, {\left (3 \, b^{2} c^{3} - 8 \, a c^{4}\right )} d^{2} e + 30 \, {\left (15 \, b^{3} c^{2} - 52 \, a b c^{3}\right )} d e^{2} - {\left (105 \, b^{4} c - 460 \, a b^{2} c^{2} + 256 \, a^{2} c^{3}\right )} e^{3} + 48 \, {\left (30 \, c^{5} d e^{2} + b c^{4} e^{3}\right )} x^{3} + 8 \, {\left (240 \, c^{5} d^{2} e + 30 \, b c^{4} d e^{2} - {\left (7 \, b^{2} c^{3} - 16 \, a c^{4}\right )} e^{3}\right )} x^{2} + 2 \, {\left (480 \, c^{5} d^{3} + 240 \, b c^{4} d^{2} e - 30 \, {\left (5 \, b^{2} c^{3} - 12 \, a c^{4}\right )} d e^{2} + {\left (35 \, b^{3} c^{2} - 116 \, a b c^{3}\right )} e^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{7680 \, c^{5}}, \frac {15 \, {\left (32 \, {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} d^{3} - 48 \, {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} d^{2} e + 6 \, {\left (5 \, b^{4} c - 24 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} d e^{2} - {\left (7 \, b^{5} - 40 \, a b^{3} c + 48 \, a^{2} b c^{2}\right )} e^{3}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) + 2 \, {\left (384 \, c^{5} e^{3} x^{4} + 480 \, b c^{4} d^{3} - 240 \, {\left (3 \, b^{2} c^{3} - 8 \, a c^{4}\right )} d^{2} e + 30 \, {\left (15 \, b^{3} c^{2} - 52 \, a b c^{3}\right )} d e^{2} - {\left (105 \, b^{4} c - 460 \, a b^{2} c^{2} + 256 \, a^{2} c^{3}\right )} e^{3} + 48 \, {\left (30 \, c^{5} d e^{2} + b c^{4} e^{3}\right )} x^{3} + 8 \, {\left (240 \, c^{5} d^{2} e + 30 \, b c^{4} d e^{2} - {\left (7 \, b^{2} c^{3} - 16 \, a c^{4}\right )} e^{3}\right )} x^{2} + 2 \, {\left (480 \, c^{5} d^{3} + 240 \, b c^{4} d^{2} e - 30 \, {\left (5 \, b^{2} c^{3} - 12 \, a c^{4}\right )} d e^{2} + {\left (35 \, b^{3} c^{2} - 116 \, a b c^{3}\right )} e^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{3840 \, c^{5}}\right ] \]

[In]

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

[Out]

[-1/7680*(15*(32*(b^2*c^3 - 4*a*c^4)*d^3 - 48*(b^3*c^2 - 4*a*b*c^3)*d^2*e + 6*(5*b^4*c - 24*a*b^2*c^2 + 16*a^2
*c^3)*d*e^2 - (7*b^5 - 40*a*b^3*c + 48*a^2*b*c^2)*e^3)*sqrt(c)*log(-8*c^2*x^2 - 8*b*c*x - b^2 - 4*sqrt(c*x^2 +
b*x + a)*(2*c*x + b)*sqrt(c) - 4*a*c) - 4*(384*c^5*e^3*x^4 + 480*b*c^4*d^3 - 240*(3*b^2*c^3 - 8*a*c^4)*d^2*e
+ 30*(15*b^3*c^2 - 52*a*b*c^3)*d*e^2 - (105*b^4*c - 460*a*b^2*c^2 + 256*a^2*c^3)*e^3 + 48*(30*c^5*d*e^2 + b*c^
4*e^3)*x^3 + 8*(240*c^5*d^2*e + 30*b*c^4*d*e^2 - (7*b^2*c^3 - 16*a*c^4)*e^3)*x^2 + 2*(480*c^5*d^3 + 240*b*c^4*
d^2*e - 30*(5*b^2*c^3 - 12*a*c^4)*d*e^2 + (35*b^3*c^2 - 116*a*b*c^3)*e^3)*x)*sqrt(c*x^2 + b*x + a))/c^5, 1/384
0*(15*(32*(b^2*c^3 - 4*a*c^4)*d^3 - 48*(b^3*c^2 - 4*a*b*c^3)*d^2*e + 6*(5*b^4*c - 24*a*b^2*c^2 + 16*a^2*c^3)*d
*e^2 - (7*b^5 - 40*a*b^3*c + 48*a^2*b*c^2)*e^3)*sqrt(-c)*arctan(1/2*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(-c)
/(c^2*x^2 + b*c*x + a*c)) + 2*(384*c^5*e^3*x^4 + 480*b*c^4*d^3 - 240*(3*b^2*c^3 - 8*a*c^4)*d^2*e + 30*(15*b^3*
c^2 - 52*a*b*c^3)*d*e^2 - (105*b^4*c - 460*a*b^2*c^2 + 256*a^2*c^3)*e^3 + 48*(30*c^5*d*e^2 + b*c^4*e^3)*x^3 +
8*(240*c^5*d^2*e + 30*b*c^4*d*e^2 - (7*b^2*c^3 - 16*a*c^4)*e^3)*x^2 + 2*(480*c^5*d^3 + 240*b*c^4*d^2*e - 30*(5
*b^2*c^3 - 12*a*c^4)*d*e^2 + (35*b^3*c^2 - 116*a*b*c^3)*e^3)*x)*sqrt(c*x^2 + b*x + a))/c^5]

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 887 vs. \(2 (241) = 482\).

Time = 0.74 (sec) , antiderivative size = 887, normalized size of antiderivative = 3.58 \[ \int (d+e x)^3 \sqrt {a+b x+c x^2} \, dx=\begin {cases} \sqrt {a + b x + c x^{2}} \left (\frac {e^{3} x^{4}}{5} + \frac {x^{3} \left (\frac {b e^{3}}{10} + 3 c d e^{2}\right )}{4 c} + \frac {x^{2} \left (\frac {a e^{3}}{5} + 3 b d e^{2} - \frac {7 b \left (\frac {b e^{3}}{10} + 3 c d e^{2}\right )}{8 c} + 3 c d^{2} e\right )}{3 c} + \frac {x \left (3 a d e^{2} - \frac {3 a \left (\frac {b e^{3}}{10} + 3 c d e^{2}\right )}{4 c} + 3 b d^{2} e - \frac {5 b \left (\frac {a e^{3}}{5} + 3 b d e^{2} - \frac {7 b \left (\frac {b e^{3}}{10} + 3 c d e^{2}\right )}{8 c} + 3 c d^{2} e\right )}{6 c} + c d^{3}\right )}{2 c} + \frac {3 a d^{2} e - \frac {2 a \left (\frac {a e^{3}}{5} + 3 b d e^{2} - \frac {7 b \left (\frac {b e^{3}}{10} + 3 c d e^{2}\right )}{8 c} + 3 c d^{2} e\right )}{3 c} + b d^{3} - \frac {3 b \left (3 a d e^{2} - \frac {3 a \left (\frac {b e^{3}}{10} + 3 c d e^{2}\right )}{4 c} + 3 b d^{2} e - \frac {5 b \left (\frac {a e^{3}}{5} + 3 b d e^{2} - \frac {7 b \left (\frac {b e^{3}}{10} + 3 c d e^{2}\right )}{8 c} + 3 c d^{2} e\right )}{6 c} + c d^{3}\right )}{4 c}}{c}\right ) + \left (a d^{3} - \frac {a \left (3 a d e^{2} - \frac {3 a \left (\frac {b e^{3}}{10} + 3 c d e^{2}\right )}{4 c} + 3 b d^{2} e - \frac {5 b \left (\frac {a e^{3}}{5} + 3 b d e^{2} - \frac {7 b \left (\frac {b e^{3}}{10} + 3 c d e^{2}\right )}{8 c} + 3 c d^{2} e\right )}{6 c} + c d^{3}\right )}{2 c} - \frac {b \left (3 a d^{2} e - \frac {2 a \left (\frac {a e^{3}}{5} + 3 b d e^{2} - \frac {7 b \left (\frac {b e^{3}}{10} + 3 c d e^{2}\right )}{8 c} + 3 c d^{2} e\right )}{3 c} + b d^{3} - \frac {3 b \left (3 a d e^{2} - \frac {3 a \left (\frac {b e^{3}}{10} + 3 c d e^{2}\right )}{4 c} + 3 b d^{2} e - \frac {5 b \left (\frac {a e^{3}}{5} + 3 b d e^{2} - \frac {7 b \left (\frac {b e^{3}}{10} + 3 c d e^{2}\right )}{8 c} + 3 c d^{2} e\right )}{6 c} + c d^{3}\right )}{4 c}\right )}{2 c}\right ) \left (\begin {cases} \frac {\log {\left (b + 2 \sqrt {c} \sqrt {a + b x + c x^{2}} + 2 c x \right )}}{\sqrt {c}} & \text {for}\: a - \frac {b^{2}}{4 c} \neq 0 \\\frac {\left (\frac {b}{2 c} + x\right ) \log {\left (\frac {b}{2 c} + x \right )}}{\sqrt {c \left (\frac {b}{2 c} + x\right )^{2}}} & \text {otherwise} \end {cases}\right ) & \text {for}\: c \neq 0 \\\frac {2 \left (\frac {e^{3} \left (a + b x\right )^{\frac {9}{2}}}{9 b^{3}} + \frac {\left (a + b x\right )^{\frac {7}{2}} \left (- 3 a e^{3} + 3 b d e^{2}\right )}{7 b^{3}} + \frac {\left (a + b x\right )^{\frac {5}{2}} \cdot \left (3 a^{2} e^{3} - 6 a b d e^{2} + 3 b^{2} d^{2} e\right )}{5 b^{3}} + \frac {\left (a + b x\right )^{\frac {3}{2}} \left (- a^{3} e^{3} + 3 a^{2} b d e^{2} - 3 a b^{2} d^{2} e + b^{3} d^{3}\right )}{3 b^{3}}\right )}{b} & \text {for}\: b \neq 0 \\\sqrt {a} \left (\begin {cases} d^{3} x & \text {for}\: e = 0 \\\frac {\left (d + e x\right )^{4}}{4 e} & \text {otherwise} \end {cases}\right ) & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((sqrt(a + b*x + c*x**2)*(e**3*x**4/5 + x**3*(b*e**3/10 + 3*c*d*e**2)/(4*c) + x**2*(a*e**3/5 + 3*b*d*
e**2 - 7*b*(b*e**3/10 + 3*c*d*e**2)/(8*c) + 3*c*d**2*e)/(3*c) + x*(3*a*d*e**2 - 3*a*(b*e**3/10 + 3*c*d*e**2)/(
4*c) + 3*b*d**2*e - 5*b*(a*e**3/5 + 3*b*d*e**2 - 7*b*(b*e**3/10 + 3*c*d*e**2)/(8*c) + 3*c*d**2*e)/(6*c) + c*d*
*3)/(2*c) + (3*a*d**2*e - 2*a*(a*e**3/5 + 3*b*d*e**2 - 7*b*(b*e**3/10 + 3*c*d*e**2)/(8*c) + 3*c*d**2*e)/(3*c)
+ b*d**3 - 3*b*(3*a*d*e**2 - 3*a*(b*e**3/10 + 3*c*d*e**2)/(4*c) + 3*b*d**2*e - 5*b*(a*e**3/5 + 3*b*d*e**2 - 7*
b*(b*e**3/10 + 3*c*d*e**2)/(8*c) + 3*c*d**2*e)/(6*c) + c*d**3)/(4*c))/c) + (a*d**3 - a*(3*a*d*e**2 - 3*a*(b*e*
*3/10 + 3*c*d*e**2)/(4*c) + 3*b*d**2*e - 5*b*(a*e**3/5 + 3*b*d*e**2 - 7*b*(b*e**3/10 + 3*c*d*e**2)/(8*c) + 3*c
*d**2*e)/(6*c) + c*d**3)/(2*c) - b*(3*a*d**2*e - 2*a*(a*e**3/5 + 3*b*d*e**2 - 7*b*(b*e**3/10 + 3*c*d*e**2)/(8*
c) + 3*c*d**2*e)/(3*c) + b*d**3 - 3*b*(3*a*d*e**2 - 3*a*(b*e**3/10 + 3*c*d*e**2)/(4*c) + 3*b*d**2*e - 5*b*(a*e
**3/5 + 3*b*d*e**2 - 7*b*(b*e**3/10 + 3*c*d*e**2)/(8*c) + 3*c*d**2*e)/(6*c) + c*d**3)/(4*c))/(2*c))*Piecewise(
(log(b + 2*sqrt(c)*sqrt(a + b*x + c*x**2) + 2*c*x)/sqrt(c), Ne(a - b**2/(4*c), 0)), ((b/(2*c) + x)*log(b/(2*c)
+ x)/sqrt(c*(b/(2*c) + x)**2), True)), Ne(c, 0)), (2*(e**3*(a + b*x)**(9/2)/(9*b**3) + (a + b*x)**(7/2)*(-3*a
*e**3 + 3*b*d*e**2)/(7*b**3) + (a + b*x)**(5/2)*(3*a**2*e**3 - 6*a*b*d*e**2 + 3*b**2*d**2*e)/(5*b**3) + (a + b
*x)**(3/2)*(-a**3*e**3 + 3*a**2*b*d*e**2 - 3*a*b**2*d**2*e + b**3*d**3)/(3*b**3))/b, Ne(b, 0)), (sqrt(a)*Piece
wise((d**3*x, Eq(e, 0)), ((d + e*x)**4/(4*e), True)), True))

Maxima [F(-2)]

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

[In]

integrate((e*x+d)^3*(c*x^2+b*x+a)^(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(4*a*c-b^2>0)', see `assume?` f
or more deta

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 393, normalized size of antiderivative = 1.58 \[ \int (d+e x)^3 \sqrt {a+b x+c x^2} \, dx=\frac {1}{1920} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (6 \, {\left (8 \, e^{3} x + \frac {30 \, c^{4} d e^{2} + b c^{3} e^{3}}{c^{4}}\right )} x + \frac {240 \, c^{4} d^{2} e + 30 \, b c^{3} d e^{2} - 7 \, b^{2} c^{2} e^{3} + 16 \, a c^{3} e^{3}}{c^{4}}\right )} x + \frac {480 \, c^{4} d^{3} + 240 \, b c^{3} d^{2} e - 150 \, b^{2} c^{2} d e^{2} + 360 \, a c^{3} d e^{2} + 35 \, b^{3} c e^{3} - 116 \, a b c^{2} e^{3}}{c^{4}}\right )} x + \frac {480 \, b c^{3} d^{3} - 720 \, b^{2} c^{2} d^{2} e + 1920 \, a c^{3} d^{2} e + 450 \, b^{3} c d e^{2} - 1560 \, a b c^{2} d e^{2} - 105 \, b^{4} e^{3} + 460 \, a b^{2} c e^{3} - 256 \, a^{2} c^{2} e^{3}}{c^{4}}\right )} + \frac {{\left (32 \, b^{2} c^{3} d^{3} - 128 \, a c^{4} d^{3} - 48 \, b^{3} c^{2} d^{2} e + 192 \, a b c^{3} d^{2} e + 30 \, b^{4} c d e^{2} - 144 \, a b^{2} c^{2} d e^{2} + 96 \, a^{2} c^{3} d e^{2} - 7 \, b^{5} e^{3} + 40 \, a b^{3} c e^{3} - 48 \, a^{2} b c^{2} e^{3}\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{256 \, c^{\frac {9}{2}}} \]

[In]

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

[Out]

1/1920*sqrt(c*x^2 + b*x + a)*(2*(4*(6*(8*e^3*x + (30*c^4*d*e^2 + b*c^3*e^3)/c^4)*x + (240*c^4*d^2*e + 30*b*c^3
*d*e^2 - 7*b^2*c^2*e^3 + 16*a*c^3*e^3)/c^4)*x + (480*c^4*d^3 + 240*b*c^3*d^2*e - 150*b^2*c^2*d*e^2 + 360*a*c^3
*d*e^2 + 35*b^3*c*e^3 - 116*a*b*c^2*e^3)/c^4)*x + (480*b*c^3*d^3 - 720*b^2*c^2*d^2*e + 1920*a*c^3*d^2*e + 450*
b^3*c*d*e^2 - 1560*a*b*c^2*d*e^2 - 105*b^4*e^3 + 460*a*b^2*c*e^3 - 256*a^2*c^2*e^3)/c^4) + 1/256*(32*b^2*c^3*d
^3 - 128*a*c^4*d^3 - 48*b^3*c^2*d^2*e + 192*a*b*c^3*d^2*e + 30*b^4*c*d*e^2 - 144*a*b^2*c^2*d*e^2 + 96*a^2*c^3*
d*e^2 - 7*b^5*e^3 + 40*a*b^3*c*e^3 - 48*a^2*b*c^2*e^3)*log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) +
b))/c^(9/2)

Mupad [B] (verification not implemented)

Time = 11.29 (sec) , antiderivative size = 632, normalized size of antiderivative = 2.55 \[ \int (d+e x)^3 \sqrt {a+b x+c x^2} \, dx=d^3\,\left (\frac {x}{2}+\frac {b}{4\,c}\right )\,\sqrt {c\,x^2+b\,x+a}+\frac {7\,b\,e^3\,\left (\frac {5\,b\,\left (\frac {\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x+a}\right )\,\left (b^3-4\,a\,b\,c\right )}{16\,c^{5/2}}+\frac {\left (-3\,b^2+2\,c\,x\,b+8\,c\,\left (c\,x^2+a\right )\right )\,\sqrt {c\,x^2+b\,x+a}}{24\,c^2}\right )}{8\,c}-\frac {x\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{4\,c}+\frac {a\,\left (\left (\frac {x}{2}+\frac {b}{4\,c}\right )\,\sqrt {c\,x^2+b\,x+a}+\frac {\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x+a}\right )\,\left (a\,c-\frac {b^2}{4}\right )}{2\,c^{3/2}}\right )}{4\,c}\right )}{10\,c}+\frac {e^3\,x^2\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{5\,c}+\frac {d^3\,\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x+a}\right )\,\left (a\,c-\frac {b^2}{4}\right )}{2\,c^{3/2}}-\frac {2\,a\,e^3\,\left (\frac {\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x+a}\right )\,\left (b^3-4\,a\,b\,c\right )}{16\,c^{5/2}}+\frac {\left (-3\,b^2+2\,c\,x\,b+8\,c\,\left (c\,x^2+a\right )\right )\,\sqrt {c\,x^2+b\,x+a}}{24\,c^2}\right )}{5\,c}-\frac {3\,a\,d\,e^2\,\left (\left (\frac {x}{2}+\frac {b}{4\,c}\right )\,\sqrt {c\,x^2+b\,x+a}+\frac {\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x+a}\right )\,\left (a\,c-\frac {b^2}{4}\right )}{2\,c^{3/2}}\right )}{4\,c}+\frac {3\,d^2\,e\,\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x+a}\right )\,\left (b^3-4\,a\,b\,c\right )}{16\,c^{5/2}}-\frac {15\,b\,d\,e^2\,\left (\frac {\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x+a}\right )\,\left (b^3-4\,a\,b\,c\right )}{16\,c^{5/2}}+\frac {\left (-3\,b^2+2\,c\,x\,b+8\,c\,\left (c\,x^2+a\right )\right )\,\sqrt {c\,x^2+b\,x+a}}{24\,c^2}\right )}{8\,c}+\frac {d^2\,e\,\left (-3\,b^2+2\,c\,x\,b+8\,c\,\left (c\,x^2+a\right )\right )\,\sqrt {c\,x^2+b\,x+a}}{8\,c^2}+\frac {3\,d\,e^2\,x\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{4\,c} \]

[In]

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

[Out]

d^3*(x/2 + b/(4*c))*(a + b*x + c*x^2)^(1/2) + (7*b*e^3*((5*b*((log((b + 2*c*x)/c^(1/2) + 2*(a + b*x + c*x^2)^(
1/2))*(b^3 - 4*a*b*c))/(16*c^(5/2)) + ((8*c*(a + c*x^2) - 3*b^2 + 2*b*c*x)*(a + b*x + c*x^2)^(1/2))/(24*c^2)))
/(8*c) - (x*(a + b*x + c*x^2)^(3/2))/(4*c) + (a*((x/2 + b/(4*c))*(a + b*x + c*x^2)^(1/2) + (log((b/2 + c*x)/c^
(1/2) + (a + b*x + c*x^2)^(1/2))*(a*c - b^2/4))/(2*c^(3/2))))/(4*c)))/(10*c) + (e^3*x^2*(a + b*x + c*x^2)^(3/2
))/(5*c) + (d^3*log((b/2 + c*x)/c^(1/2) + (a + b*x + c*x^2)^(1/2))*(a*c - b^2/4))/(2*c^(3/2)) - (2*a*e^3*((log
((b + 2*c*x)/c^(1/2) + 2*(a + b*x + c*x^2)^(1/2))*(b^3 - 4*a*b*c))/(16*c^(5/2)) + ((8*c*(a + c*x^2) - 3*b^2 +
2*b*c*x)*(a + b*x + c*x^2)^(1/2))/(24*c^2)))/(5*c) - (3*a*d*e^2*((x/2 + b/(4*c))*(a + b*x + c*x^2)^(1/2) + (lo
g((b/2 + c*x)/c^(1/2) + (a + b*x + c*x^2)^(1/2))*(a*c - b^2/4))/(2*c^(3/2))))/(4*c) + (3*d^2*e*log((b + 2*c*x)
/c^(1/2) + 2*(a + b*x + c*x^2)^(1/2))*(b^3 - 4*a*b*c))/(16*c^(5/2)) - (15*b*d*e^2*((log((b + 2*c*x)/c^(1/2) +
2*(a + b*x + c*x^2)^(1/2))*(b^3 - 4*a*b*c))/(16*c^(5/2)) + ((8*c*(a + c*x^2) - 3*b^2 + 2*b*c*x)*(a + b*x + c*x
^2)^(1/2))/(24*c^2)))/(8*c) + (d^2*e*(8*c*(a + c*x^2) - 3*b^2 + 2*b*c*x)*(a + b*x + c*x^2)^(1/2))/(8*c^2) + (3
*d*e^2*x*(a + b*x + c*x^2)^(3/2))/(4*c)