\(\int (d+e x)^2 (a+b x+c x^2)^{3/2} \, dx\) [2344]

Optimal result

Integrand size = 22, antiderivative size = 257 \[ \int (d+e x)^2 \left (a+b x+c x^2\right )^{3/2} \, dx=-\frac {\left (b^2-4 a c\right ) \left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{512 c^4}+\frac {\left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{192 c^3}+\frac {7 e (2 c d-b e) \left (a+b x+c x^2\right )^{5/2}}{60 c^2}+\frac {e (d+e x) \left (a+b x+c x^2\right )^{5/2}}{6 c}+\frac {\left (b^2-4 a c\right )^2 \left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{1024 c^{9/2}} \]

[Out]

1/192*(24*c^2*d^2+7*b^2*e^2-4*c*e*(a*e+6*b*d))*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)/c^3+7/60*e*(-b*e+2*c*d)*(c*x^2+b*
x+a)^(5/2)/c^2+1/6*e*(e*x+d)*(c*x^2+b*x+a)^(5/2)/c+1/1024*(-4*a*c+b^2)^2*(24*c^2*d^2+7*b^2*e^2-4*c*e*(a*e+6*b*
d))*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^(9/2)-1/512*(-4*a*c+b^2)*(24*c^2*d^2+7*b^2*e^2-4*c*e*
(a*e+6*b*d))*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c^4

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {756, 654, 626, 635, 212} \[ \int (d+e x)^2 \left (a+b x+c x^2\right )^{3/2} \, dx=\frac {\left (b^2-4 a c\right )^2 \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right )}{1024 c^{9/2}}-\frac {\left (b^2-4 a c\right ) (b+2 c x) \sqrt {a+b x+c x^2} \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right )}{512 c^4}+\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right )}{192 c^3}+\frac {7 e \left (a+b x+c x^2\right )^{5/2} (2 c d-b e)}{60 c^2}+\frac {e (d+e x) \left (a+b x+c x^2\right )^{5/2}}{6 c} \]

[In]

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

[Out]

-1/512*((b^2 - 4*a*c)*(24*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(6*b*d + a*e))*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/c^4 +
((24*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(6*b*d + a*e))*(b + 2*c*x)*(a + b*x + c*x^2)^(3/2))/(192*c^3) + (7*e*(2*c*d -
b*e)*(a + b*x + c*x^2)^(5/2))/(60*c^2) + (e*(d + e*x)*(a + b*x + c*x^2)^(5/2))/(6*c) + ((b^2 - 4*a*c)^2*(24*c
^2*d^2 + 7*b^2*e^2 - 4*c*e*(6*b*d + a*e))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(1024*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 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 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]

Rubi steps \begin{align*} \text {integral}& = \frac {e (d+e x) \left (a+b x+c x^2\right )^{5/2}}{6 c}+\frac {\int \left (\frac {1}{2} \left (12 c d^2-2 e \left (\frac {5 b d}{2}+a e\right )\right )+\frac {7}{2} e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2} \, dx}{6 c} \\ & = \frac {7 e (2 c d-b e) \left (a+b x+c x^2\right )^{5/2}}{60 c^2}+\frac {e (d+e x) \left (a+b x+c x^2\right )^{5/2}}{6 c}+\frac {\left (-\frac {7}{2} b e (2 c d-b e)+c \left (12 c d^2-2 e \left (\frac {5 b d}{2}+a e\right )\right )\right ) \int \left (a+b x+c x^2\right )^{3/2} \, dx}{12 c^2} \\ & = \frac {\left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{192 c^3}+\frac {7 e (2 c d-b e) \left (a+b x+c x^2\right )^{5/2}}{60 c^2}+\frac {e (d+e x) \left (a+b x+c x^2\right )^{5/2}}{6 c}-\frac {\left (\left (b^2-4 a c\right ) \left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right )\right ) \int \sqrt {a+b x+c x^2} \, dx}{128 c^3} \\ & = -\frac {\left (b^2-4 a c\right ) \left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{512 c^4}+\frac {\left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{192 c^3}+\frac {7 e (2 c d-b e) \left (a+b x+c x^2\right )^{5/2}}{60 c^2}+\frac {e (d+e x) \left (a+b x+c x^2\right )^{5/2}}{6 c}+\frac {\left (\left (b^2-4 a c\right )^2 \left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{1024 c^4} \\ & = -\frac {\left (b^2-4 a c\right ) \left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{512 c^4}+\frac {\left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{192 c^3}+\frac {7 e (2 c d-b e) \left (a+b x+c x^2\right )^{5/2}}{60 c^2}+\frac {e (d+e x) \left (a+b x+c x^2\right )^{5/2}}{6 c}+\frac {\left (\left (b^2-4 a c\right )^2 \left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+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 )}{512 c^4} \\ & = -\frac {\left (b^2-4 a c\right ) \left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{512 c^4}+\frac {\left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{192 c^3}+\frac {7 e (2 c d-b e) \left (a+b x+c x^2\right )^{5/2}}{60 c^2}+\frac {e (d+e x) \left (a+b x+c x^2\right )^{5/2}}{6 c}+\frac {\left (b^2-4 a c\right )^2 \left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{1024 c^{9/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.68 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.31 \[ \int (d+e x)^2 \left (a+b x+c x^2\right )^{3/2} \, dx=\frac {\sqrt {c} \sqrt {a+x (b+c x)} \left (-105 b^5 e^2+10 b^4 c e (36 d+7 e x)-48 b^2 c^2 \left (a e (50 d+9 e x)-c x \left (5 d^2+4 d e x+e^2 x^2\right )\right )-8 b^3 c \left (-95 a e^2+c \left (45 d^2+30 d e x+7 e^2 x^2\right )\right )+16 b c^2 \left (-81 a^2 e^2+6 a c \left (25 d^2+14 d e x+3 e^2 x^2\right )+4 c^2 x^2 \left (45 d^2+66 d e x+26 e^2 x^2\right )\right )+32 c^3 \left (3 a^2 e (32 d+5 e x)+4 c^2 x^3 \left (15 d^2+24 d e x+10 e^2 x^2\right )+2 a c x \left (75 d^2+96 d e x+35 e^2 x^2\right )\right )\right )+15 \left (b^2-4 a c\right )^2 \left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{-\sqrt {a}+\sqrt {a+x (b+c x)}}\right )}{7680 c^{9/2}} \]

[In]

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

[Out]

(Sqrt[c]*Sqrt[a + x*(b + c*x)]*(-105*b^5*e^2 + 10*b^4*c*e*(36*d + 7*e*x) - 48*b^2*c^2*(a*e*(50*d + 9*e*x) - c*
x*(5*d^2 + 4*d*e*x + e^2*x^2)) - 8*b^3*c*(-95*a*e^2 + c*(45*d^2 + 30*d*e*x + 7*e^2*x^2)) + 16*b*c^2*(-81*a^2*e
^2 + 6*a*c*(25*d^2 + 14*d*e*x + 3*e^2*x^2) + 4*c^2*x^2*(45*d^2 + 66*d*e*x + 26*e^2*x^2)) + 32*c^3*(3*a^2*e*(32
*d + 5*e*x) + 4*c^2*x^3*(15*d^2 + 24*d*e*x + 10*e^2*x^2) + 2*a*c*x*(75*d^2 + 96*d*e*x + 35*e^2*x^2))) + 15*(b^
2 - 4*a*c)^2*(24*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(6*b*d + a*e))*ArcTanh[(Sqrt[c]*x)/(-Sqrt[a] + Sqrt[a + x*(b + c*
x)])])/(7680*c^(9/2))

Maple [A] (verified)

Time = 0.30 (sec) , antiderivative size = 460, normalized size of antiderivative = 1.79

method	result	size
risch	\(-\frac {\left (-1280 c^{5} e^{2} x^{5}-1664 b \,c^{4} e^{2} x^{4}-3072 c^{5} d e \,x^{4}-2240 a \,c^{4} e^{2} x^{3}-48 b^{2} c^{3} e^{2} x^{3}-4224 b \,c^{4} d e \,x^{3}-1920 c^{5} d^{2} x^{3}-288 a b \,c^{3} e^{2} x^{2}-6144 a \,c^{4} d e \,x^{2}+56 b^{3} c^{2} e^{2} x^{2}-192 b^{2} c^{3} d e \,x^{2}-2880 b \,c^{4} d^{2} x^{2}-480 a^{2} c^{3} e^{2} x +432 a \,b^{2} c^{2} e^{2} x -1344 a b \,c^{3} d e x -4800 a \,c^{4} d^{2} x -70 b^{4} c \,e^{2} x +240 b^{3} c^{2} d e x -240 b^{2} c^{3} d^{2} x +1296 a^{2} b \,c^{2} e^{2}-3072 a^{2} c^{3} d e -760 a \,b^{3} c \,e^{2}+2400 a \,b^{2} c^{2} d e -2400 a b \,c^{3} d^{2}+105 b^{5} e^{2}-360 b^{4} c d e +360 b^{3} c^{2} d^{2}\right ) \sqrt {c \,x^{2}+b x +a}}{7680 c^{4}}-\frac {\left (64 a^{3} c^{3} e^{2}-144 a^{2} b^{2} c^{2} e^{2}+384 a^{2} b \,c^{3} d e -384 a^{2} c^{4} d^{2}+60 a \,b^{4} c \,e^{2}-192 a \,b^{3} c^{2} d e +192 a \,b^{2} c^{3} d^{2}-7 b^{6} e^{2}+24 b^{5} c d e -24 b^{4} c^{2} d^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{1024 c^{\frac {9}{2}}}\)	\(460\)
default	\(d^{2} \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )+e^{2} \left (\frac {x \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{6 c}-\frac {7 b \left (\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{5 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{2 c}\right )}{12 c}-\frac {a \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{6 c}\right )+2 d e \left (\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{5 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{2 c}\right )\)	\(505\)

[In]

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

[Out]

-1/7680/c^4*(-1280*c^5*e^2*x^5-1664*b*c^4*e^2*x^4-3072*c^5*d*e*x^4-2240*a*c^4*e^2*x^3-48*b^2*c^3*e^2*x^3-4224*
b*c^4*d*e*x^3-1920*c^5*d^2*x^3-288*a*b*c^3*e^2*x^2-6144*a*c^4*d*e*x^2+56*b^3*c^2*e^2*x^2-192*b^2*c^3*d*e*x^2-2
880*b*c^4*d^2*x^2-480*a^2*c^3*e^2*x+432*a*b^2*c^2*e^2*x-1344*a*b*c^3*d*e*x-4800*a*c^4*d^2*x-70*b^4*c*e^2*x+240
*b^3*c^2*d*e*x-240*b^2*c^3*d^2*x+1296*a^2*b*c^2*e^2-3072*a^2*c^3*d*e-760*a*b^3*c*e^2+2400*a*b^2*c^2*d*e-2400*a
*b*c^3*d^2+105*b^5*e^2-360*b^4*c*d*e+360*b^3*c^2*d^2)*(c*x^2+b*x+a)^(1/2)-1/1024*(64*a^3*c^3*e^2-144*a^2*b^2*c
^2*e^2+384*a^2*b*c^3*d*e-384*a^2*c^4*d^2+60*a*b^4*c*e^2-192*a*b^3*c^2*d*e+192*a*b^2*c^3*d^2-7*b^6*e^2+24*b^5*c
*d*e-24*b^4*c^2*d^2)/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 = 899, normalized size of antiderivative = 3.50 \[ \int (d+e x)^2 \left (a+b x+c x^2\right )^{3/2} \, dx=\left [-\frac {15 \, {\left (24 \, {\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} d^{2} - 24 \, {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} d e + {\left (7 \, b^{6} - 60 \, a b^{4} c + 144 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} e^{2}\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 (1280 \, c^{6} e^{2} x^{5} + 128 \, {\left (24 \, c^{6} d e + 13 \, b c^{5} e^{2}\right )} x^{4} + 16 \, {\left (120 \, c^{6} d^{2} + 264 \, b c^{5} d e + {\left (3 \, b^{2} c^{4} + 140 \, a c^{5}\right )} e^{2}\right )} x^{3} - 120 \, {\left (3 \, b^{3} c^{3} - 20 \, a b c^{4}\right )} d^{2} + 24 \, {\left (15 \, b^{4} c^{2} - 100 \, a b^{2} c^{3} + 128 \, a^{2} c^{4}\right )} d e - {\left (105 \, b^{5} c - 760 \, a b^{3} c^{2} + 1296 \, a^{2} b c^{3}\right )} e^{2} + 8 \, {\left (360 \, b c^{5} d^{2} + 24 \, {\left (b^{2} c^{4} + 32 \, a c^{5}\right )} d e - {\left (7 \, b^{3} c^{3} - 36 \, a b c^{4}\right )} e^{2}\right )} x^{2} + 2 \, {\left (120 \, {\left (b^{2} c^{4} + 20 \, a c^{5}\right )} d^{2} - 24 \, {\left (5 \, b^{3} c^{3} - 28 \, a b c^{4}\right )} d e + {\left (35 \, b^{4} c^{2} - 216 \, a b^{2} c^{3} + 240 \, a^{2} c^{4}\right )} e^{2}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{30720 \, c^{5}}, -\frac {15 \, {\left (24 \, {\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} d^{2} - 24 \, {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} d e + {\left (7 \, b^{6} - 60 \, a b^{4} c + 144 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} e^{2}\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 (1280 \, c^{6} e^{2} x^{5} + 128 \, {\left (24 \, c^{6} d e + 13 \, b c^{5} e^{2}\right )} x^{4} + 16 \, {\left (120 \, c^{6} d^{2} + 264 \, b c^{5} d e + {\left (3 \, b^{2} c^{4} + 140 \, a c^{5}\right )} e^{2}\right )} x^{3} - 120 \, {\left (3 \, b^{3} c^{3} - 20 \, a b c^{4}\right )} d^{2} + 24 \, {\left (15 \, b^{4} c^{2} - 100 \, a b^{2} c^{3} + 128 \, a^{2} c^{4}\right )} d e - {\left (105 \, b^{5} c - 760 \, a b^{3} c^{2} + 1296 \, a^{2} b c^{3}\right )} e^{2} + 8 \, {\left (360 \, b c^{5} d^{2} + 24 \, {\left (b^{2} c^{4} + 32 \, a c^{5}\right )} d e - {\left (7 \, b^{3} c^{3} - 36 \, a b c^{4}\right )} e^{2}\right )} x^{2} + 2 \, {\left (120 \, {\left (b^{2} c^{4} + 20 \, a c^{5}\right )} d^{2} - 24 \, {\left (5 \, b^{3} c^{3} - 28 \, a b c^{4}\right )} d e + {\left (35 \, b^{4} c^{2} - 216 \, a b^{2} c^{3} + 240 \, a^{2} c^{4}\right )} e^{2}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{15360 \, c^{5}}\right ] \]

[In]

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

[Out]

[-1/30720*(15*(24*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^2 - 24*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d*e + (7*
b^6 - 60*a*b^4*c + 144*a^2*b^2*c^2 - 64*a^3*c^3)*e^2)*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*(1280*c^6*e^2*x^5 + 128*(24*c^6*d*e + 13*b*c^5*e^2)*x^4 + 16*(120*c^
6*d^2 + 264*b*c^5*d*e + (3*b^2*c^4 + 140*a*c^5)*e^2)*x^3 - 120*(3*b^3*c^3 - 20*a*b*c^4)*d^2 + 24*(15*b^4*c^2 -
100*a*b^2*c^3 + 128*a^2*c^4)*d*e - (105*b^5*c - 760*a*b^3*c^2 + 1296*a^2*b*c^3)*e^2 + 8*(360*b*c^5*d^2 + 24*(
b^2*c^4 + 32*a*c^5)*d*e - (7*b^3*c^3 - 36*a*b*c^4)*e^2)*x^2 + 2*(120*(b^2*c^4 + 20*a*c^5)*d^2 - 24*(5*b^3*c^3
- 28*a*b*c^4)*d*e + (35*b^4*c^2 - 216*a*b^2*c^3 + 240*a^2*c^4)*e^2)*x)*sqrt(c*x^2 + b*x + a))/c^5, -1/15360*(1
5*(24*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^2 - 24*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d*e + (7*b^6 - 60*a*b
^4*c + 144*a^2*b^2*c^2 - 64*a^3*c^3)*e^2)*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*(1280*c^6*e^2*x^5 + 128*(24*c^6*d*e + 13*b*c^5*e^2)*x^4 + 16*(120*c^6*d^2 + 264*b*c^5*
d*e + (3*b^2*c^4 + 140*a*c^5)*e^2)*x^3 - 120*(3*b^3*c^3 - 20*a*b*c^4)*d^2 + 24*(15*b^4*c^2 - 100*a*b^2*c^3 + 1
28*a^2*c^4)*d*e - (105*b^5*c - 760*a*b^3*c^2 + 1296*a^2*b*c^3)*e^2 + 8*(360*b*c^5*d^2 + 24*(b^2*c^4 + 32*a*c^5
)*d*e - (7*b^3*c^3 - 36*a*b*c^4)*e^2)*x^2 + 2*(120*(b^2*c^4 + 20*a*c^5)*d^2 - 24*(5*b^3*c^3 - 28*a*b*c^4)*d*e
+ (35*b^4*c^2 - 216*a*b^2*c^3 + 240*a^2*c^4)*e^2)*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. 1656 vs. \(2 (248) = 496\).

Time = 0.70 (sec) , antiderivative size = 1656, normalized size of antiderivative = 6.44 \[ \int (d+e x)^2 \left (a+b x+c x^2\right )^{3/2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Piecewise((sqrt(a + b*x + c*x**2)*(c*e**2*x**5/6 + x**4*(13*b*c*e**2/12 + 2*c**2*d*e)/(5*c) + x**3*(7*a*c*e**2
/6 + b**2*e**2 + 4*b*c*d*e - 9*b*(13*b*c*e**2/12 + 2*c**2*d*e)/(10*c) + c**2*d**2)/(4*c) + x**2*(2*a*b*e**2 +
4*a*c*d*e - 4*a*(13*b*c*e**2/12 + 2*c**2*d*e)/(5*c) + 2*b**2*d*e + 2*b*c*d**2 - 7*b*(7*a*c*e**2/6 + b**2*e**2
+ 4*b*c*d*e - 9*b*(13*b*c*e**2/12 + 2*c**2*d*e)/(10*c) + c**2*d**2)/(8*c))/(3*c) + x*(a**2*e**2 + 4*a*b*d*e +
2*a*c*d**2 - 3*a*(7*a*c*e**2/6 + b**2*e**2 + 4*b*c*d*e - 9*b*(13*b*c*e**2/12 + 2*c**2*d*e)/(10*c) + c**2*d**2)
/(4*c) + b**2*d**2 - 5*b*(2*a*b*e**2 + 4*a*c*d*e - 4*a*(13*b*c*e**2/12 + 2*c**2*d*e)/(5*c) + 2*b**2*d*e + 2*b*
c*d**2 - 7*b*(7*a*c*e**2/6 + b**2*e**2 + 4*b*c*d*e - 9*b*(13*b*c*e**2/12 + 2*c**2*d*e)/(10*c) + c**2*d**2)/(8*
c))/(6*c))/(2*c) + (2*a**2*d*e + 2*a*b*d**2 - 2*a*(2*a*b*e**2 + 4*a*c*d*e - 4*a*(13*b*c*e**2/12 + 2*c**2*d*e)/
(5*c) + 2*b**2*d*e + 2*b*c*d**2 - 7*b*(7*a*c*e**2/6 + b**2*e**2 + 4*b*c*d*e - 9*b*(13*b*c*e**2/12 + 2*c**2*d*e
)/(10*c) + c**2*d**2)/(8*c))/(3*c) - 3*b*(a**2*e**2 + 4*a*b*d*e + 2*a*c*d**2 - 3*a*(7*a*c*e**2/6 + b**2*e**2 +
4*b*c*d*e - 9*b*(13*b*c*e**2/12 + 2*c**2*d*e)/(10*c) + c**2*d**2)/(4*c) + b**2*d**2 - 5*b*(2*a*b*e**2 + 4*a*c
*d*e - 4*a*(13*b*c*e**2/12 + 2*c**2*d*e)/(5*c) + 2*b**2*d*e + 2*b*c*d**2 - 7*b*(7*a*c*e**2/6 + b**2*e**2 + 4*b
*c*d*e - 9*b*(13*b*c*e**2/12 + 2*c**2*d*e)/(10*c) + c**2*d**2)/(8*c))/(6*c))/(4*c))/c) + (a**2*d**2 - a*(a**2*
e**2 + 4*a*b*d*e + 2*a*c*d**2 - 3*a*(7*a*c*e**2/6 + b**2*e**2 + 4*b*c*d*e - 9*b*(13*b*c*e**2/12 + 2*c**2*d*e)/
(10*c) + c**2*d**2)/(4*c) + b**2*d**2 - 5*b*(2*a*b*e**2 + 4*a*c*d*e - 4*a*(13*b*c*e**2/12 + 2*c**2*d*e)/(5*c)
+ 2*b**2*d*e + 2*b*c*d**2 - 7*b*(7*a*c*e**2/6 + b**2*e**2 + 4*b*c*d*e - 9*b*(13*b*c*e**2/12 + 2*c**2*d*e)/(10*
c) + c**2*d**2)/(8*c))/(6*c))/(2*c) - b*(2*a**2*d*e + 2*a*b*d**2 - 2*a*(2*a*b*e**2 + 4*a*c*d*e - 4*a*(13*b*c*e
**2/12 + 2*c**2*d*e)/(5*c) + 2*b**2*d*e + 2*b*c*d**2 - 7*b*(7*a*c*e**2/6 + b**2*e**2 + 4*b*c*d*e - 9*b*(13*b*c
*e**2/12 + 2*c**2*d*e)/(10*c) + c**2*d**2)/(8*c))/(3*c) - 3*b*(a**2*e**2 + 4*a*b*d*e + 2*a*c*d**2 - 3*a*(7*a*c
*e**2/6 + b**2*e**2 + 4*b*c*d*e - 9*b*(13*b*c*e**2/12 + 2*c**2*d*e)/(10*c) + c**2*d**2)/(4*c) + b**2*d**2 - 5*
b*(2*a*b*e**2 + 4*a*c*d*e - 4*a*(13*b*c*e**2/12 + 2*c**2*d*e)/(5*c) + 2*b**2*d*e + 2*b*c*d**2 - 7*b*(7*a*c*e**
2/6 + b**2*e**2 + 4*b*c*d*e - 9*b*(13*b*c*e**2/12 + 2*c**2*d*e)/(10*c) + c**2*d**2)/(8*c))/(6*c))/(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**2*(a + b*x)**(9/2)/(9*b**2) + (a + b*x)*
*(7/2)*(-2*a*e**2 + 2*b*d*e)/(7*b**2) + (a + b*x)**(5/2)*(a**2*e**2 - 2*a*b*d*e + b**2*d**2)/(5*b**2))/b, Ne(b
, 0)), (a**(3/2)*Piecewise((d**2*x, Eq(e, 0)), ((d + e*x)**3/(3*e), True)), True))

Maxima [F(-2)]

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

[In]

integrate((e*x+d)^2*(c*x^2+b*x+a)^(3/2),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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 465 vs. \(2 (231) = 462\).

Time = 0.29 (sec) , antiderivative size = 465, normalized size of antiderivative = 1.81 \[ \int (d+e x)^2 \left (a+b x+c x^2\right )^{3/2} \, dx=\frac {1}{7680} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, c e^{2} x + \frac {24 \, c^{6} d e + 13 \, b c^{5} e^{2}}{c^{5}}\right )} x + \frac {120 \, c^{6} d^{2} + 264 \, b c^{5} d e + 3 \, b^{2} c^{4} e^{2} + 140 \, a c^{5} e^{2}}{c^{5}}\right )} x + \frac {360 \, b c^{5} d^{2} + 24 \, b^{2} c^{4} d e + 768 \, a c^{5} d e - 7 \, b^{3} c^{3} e^{2} + 36 \, a b c^{4} e^{2}}{c^{5}}\right )} x + \frac {120 \, b^{2} c^{4} d^{2} + 2400 \, a c^{5} d^{2} - 120 \, b^{3} c^{3} d e + 672 \, a b c^{4} d e + 35 \, b^{4} c^{2} e^{2} - 216 \, a b^{2} c^{3} e^{2} + 240 \, a^{2} c^{4} e^{2}}{c^{5}}\right )} x - \frac {360 \, b^{3} c^{3} d^{2} - 2400 \, a b c^{4} d^{2} - 360 \, b^{4} c^{2} d e + 2400 \, a b^{2} c^{3} d e - 3072 \, a^{2} c^{4} d e + 105 \, b^{5} c e^{2} - 760 \, a b^{3} c^{2} e^{2} + 1296 \, a^{2} b c^{3} e^{2}}{c^{5}}\right )} - \frac {{\left (24 \, b^{4} c^{2} d^{2} - 192 \, a b^{2} c^{3} d^{2} + 384 \, a^{2} c^{4} d^{2} - 24 \, b^{5} c d e + 192 \, a b^{3} c^{2} d e - 384 \, a^{2} b c^{3} d e + 7 \, b^{6} e^{2} - 60 \, a b^{4} c e^{2} + 144 \, a^{2} b^{2} c^{2} e^{2} - 64 \, a^{3} c^{3} e^{2}\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{1024 \, c^{\frac {9}{2}}} \]

[In]

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

[Out]

1/7680*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(10*c*e^2*x + (24*c^6*d*e + 13*b*c^5*e^2)/c^5)*x + (120*c^6*d^2 + 264
*b*c^5*d*e + 3*b^2*c^4*e^2 + 140*a*c^5*e^2)/c^5)*x + (360*b*c^5*d^2 + 24*b^2*c^4*d*e + 768*a*c^5*d*e - 7*b^3*c
^3*e^2 + 36*a*b*c^4*e^2)/c^5)*x + (120*b^2*c^4*d^2 + 2400*a*c^5*d^2 - 120*b^3*c^3*d*e + 672*a*b*c^4*d*e + 35*b
^4*c^2*e^2 - 216*a*b^2*c^3*e^2 + 240*a^2*c^4*e^2)/c^5)*x - (360*b^3*c^3*d^2 - 2400*a*b*c^4*d^2 - 360*b^4*c^2*d
*e + 2400*a*b^2*c^3*d*e - 3072*a^2*c^4*d*e + 105*b^5*c*e^2 - 760*a*b^3*c^2*e^2 + 1296*a^2*b*c^3*e^2)/c^5) - 1/
1024*(24*b^4*c^2*d^2 - 192*a*b^2*c^3*d^2 + 384*a^2*c^4*d^2 - 24*b^5*c*d*e + 192*a*b^3*c^2*d*e - 384*a^2*b*c^3*
d*e + 7*b^6*e^2 - 60*a*b^4*c*e^2 + 144*a^2*b^2*c^2*e^2 - 64*a^3*c^3*e^2)*log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b
*x + a))*sqrt(c) + b))/c^(9/2)

Mupad [F(-1)]

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

[In]

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

[Out]

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