\(\int (d+e x)^3 (a+b x+c x^2)^{3/2} \, dx\) [2343]

Optimal result

Integrand size = 22, antiderivative size = 321 \[ \int (d+e x)^3 \left (a+b x+c x^2\right )^{3/2} \, dx=-\frac {3 \left (b^2-4 a c\right ) (2 c d-b e) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{1024 c^5}+\frac {(2 c d-b e) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{128 c^4}+\frac {e (d+e x)^2 \left (a+b x+c x^2\right )^{5/2}}{7 c}+\frac {e \left (128 c^2 d^2+21 b^2 e^2-2 c e (49 b d+8 a e)+30 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{5/2}}{280 c^3}+\frac {3 \left (b^2-4 a c\right )^2 (2 c d-b e) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2048 c^{11/2}} \]

[Out]

1/128*(-b*e+2*c*d)*(8*c^2*d^2+3*b^2*e^2-4*c*e*(a*e+2*b*d))*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)/c^4+1/7*e*(e*x+d)^2*(
c*x^2+b*x+a)^(5/2)/c+1/280*e*(128*c^2*d^2+21*b^2*e^2-2*c*e*(8*a*e+49*b*d)+30*c*e*(-b*e+2*c*d)*x)*(c*x^2+b*x+a)
^(5/2)/c^3+3/2048*(-4*a*c+b^2)^2*(-b*e+2*c*d)*(8*c^2*d^2+3*b^2*e^2-4*c*e*(a*e+2*b*d))*arctanh(1/2*(2*c*x+b)/c^
(1/2)/(c*x^2+b*x+a)^(1/2))/c^(11/2)-3/1024*(-4*a*c+b^2)*(-b*e+2*c*d)*(8*c^2*d^2+3*b^2*e^2-4*c*e*(a*e+2*b*d))*(
2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c^5

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 321, 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, 793, 626, 635, 212} \[ \int (d+e x)^3 \left (a+b x+c x^2\right )^{3/2} \, dx=\frac {3 \left (b^2-4 a c\right )^2 (2 c d-b e) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right )}{2048 c^{11/2}}-\frac {3 \left (b^2-4 a c\right ) (b+2 c x) \sqrt {a+b x+c x^2} (2 c d-b e) \left (-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right )}{1024 c^5}+\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2} (2 c d-b e) \left (-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right )}{128 c^4}+\frac {e \left (a+b x+c x^2\right )^{5/2} \left (-2 c e (8 a e+49 b d)+21 b^2 e^2+30 c e x (2 c d-b e)+128 c^2 d^2\right )}{280 c^3}+\frac {e (d+e x)^2 \left (a+b x+c x^2\right )^{5/2}}{7 c} \]

[In]

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

[Out]

(-3*(b^2 - 4*a*c)*(2*c*d - b*e)*(8*c^2*d^2 + 3*b^2*e^2 - 4*c*e*(2*b*d + a*e))*(b + 2*c*x)*Sqrt[a + b*x + c*x^2
])/(1024*c^5) + ((2*c*d - b*e)*(8*c^2*d^2 + 3*b^2*e^2 - 4*c*e*(2*b*d + a*e))*(b + 2*c*x)*(a + b*x + c*x^2)^(3/
2))/(128*c^4) + (e*(d + e*x)^2*(a + b*x + c*x^2)^(5/2))/(7*c) + (e*(128*c^2*d^2 + 21*b^2*e^2 - 2*c*e*(49*b*d +
8*a*e) + 30*c*e*(2*c*d - b*e)*x)*(a + b*x + c*x^2)^(5/2))/(280*c^3) + (3*(b^2 - 4*a*c)^2*(2*c*d - b*e)*(8*c^2
*d^2 + 3*b^2*e^2 - 4*c*e*(2*b*d + a*e))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(2048*c^(11/2)
)

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 )^{5/2}}{7 c}+\frac {\int (d+e x) \left (\frac {1}{2} \left (14 c d^2-e (5 b d+4 a e)\right )+\frac {9}{2} e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2} \, dx}{7 c} \\ & = \frac {e (d+e x)^2 \left (a+b x+c x^2\right )^{5/2}}{7 c}+\frac {e \left (128 c^2 d^2+21 b^2 e^2-2 c e (49 b d+8 a e)+30 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{5/2}}{280 c^3}+\frac {\left ((2 c d-b e) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right )\right ) \int \left (a+b x+c x^2\right )^{3/2} \, dx}{16 c^3} \\ & = \frac {(2 c d-b e) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{128 c^4}+\frac {e (d+e x)^2 \left (a+b x+c x^2\right )^{5/2}}{7 c}+\frac {e \left (128 c^2 d^2+21 b^2 e^2-2 c e (49 b d+8 a e)+30 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{5/2}}{280 c^3}-\frac {\left (3 \left (b^2-4 a c\right ) (2 c d-b e) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right )\right ) \int \sqrt {a+b x+c x^2} \, dx}{256 c^4} \\ & = -\frac {3 \left (b^2-4 a c\right ) (2 c d-b e) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{1024 c^5}+\frac {(2 c d-b e) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{128 c^4}+\frac {e (d+e x)^2 \left (a+b x+c x^2\right )^{5/2}}{7 c}+\frac {e \left (128 c^2 d^2+21 b^2 e^2-2 c e (49 b d+8 a e)+30 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{5/2}}{280 c^3}+\frac {\left (3 \left (b^2-4 a c\right )^2 (2 c d-b e) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{2048 c^5} \\ & = -\frac {3 \left (b^2-4 a c\right ) (2 c d-b e) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{1024 c^5}+\frac {(2 c d-b e) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{128 c^4}+\frac {e (d+e x)^2 \left (a+b x+c x^2\right )^{5/2}}{7 c}+\frac {e \left (128 c^2 d^2+21 b^2 e^2-2 c e (49 b d+8 a e)+30 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{5/2}}{280 c^3}+\frac {\left (3 \left (b^2-4 a c\right )^2 (2 c d-b e) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 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 )}{1024 c^5} \\ & = -\frac {3 \left (b^2-4 a c\right ) (2 c d-b e) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{1024 c^5}+\frac {(2 c d-b e) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{128 c^4}+\frac {e (d+e x)^2 \left (a+b x+c x^2\right )^{5/2}}{7 c}+\frac {e \left (128 c^2 d^2+21 b^2 e^2-2 c e (49 b d+8 a e)+30 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{5/2}}{280 c^3}+\frac {3 \left (b^2-4 a c\right )^2 (2 c d-b e) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2048 c^{11/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 4.49 (sec) , antiderivative size = 509, normalized size of antiderivative = 1.59 \[ \int (d+e x)^3 \left (a+b x+c x^2\right )^{3/2} \, dx=\frac {\sqrt {c} \sqrt {a+x (b+c x)} \left (315 b^6 e^3-210 b^5 c e^2 (7 d+e x)+28 b^4 c e \left (-90 a e^2+c \left (90 d^2+35 d e x+6 e^2 x^2\right )\right )+16 b^2 c^2 \left (343 a^2 e^3-2 a c e \left (525 d^2+189 d e x+31 e^2 x^2\right )+2 c^2 x \left (35 d^3+42 d^2 e x+21 d e^2 x^2+4 e^3 x^3\right )\right )-16 b^3 c^2 \left (-7 a e^2 (95 d+13 e x)+c \left (105 d^3+105 d^2 e x+49 d e^2 x^2+9 e^3 x^3\right )\right )-32 b c^3 \left (a^2 e^2 (567 d+73 e x)-2 a c \left (175 d^3+147 d^2 e x+63 d e^2 x^2+11 e^3 x^3\right )-4 c^2 x^2 \left (105 d^3+231 d^2 e x+182 d e^2 x^2+50 e^3 x^3\right )\right )+64 c^3 \left (-32 a^3 e^3+a^2 c e \left (336 d^2+105 d e x+16 e^2 x^2\right )+4 c^3 x^3 \left (35 d^3+84 d^2 e x+70 d e^2 x^2+20 e^3 x^3\right )+2 a c^2 x \left (175 d^3+336 d^2 e x+245 d e^2 x^2+64 e^3 x^3\right )\right )\right )-105 \left (b^2-4 a c\right )^2 (-2 c d+b e) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{-\sqrt {a}+\sqrt {a+x (b+c x)}}\right )}{35840 c^{11/2}} \]

[In]

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

[Out]

(Sqrt[c]*Sqrt[a + x*(b + c*x)]*(315*b^6*e^3 - 210*b^5*c*e^2*(7*d + e*x) + 28*b^4*c*e*(-90*a*e^2 + c*(90*d^2 +
35*d*e*x + 6*e^2*x^2)) + 16*b^2*c^2*(343*a^2*e^3 - 2*a*c*e*(525*d^2 + 189*d*e*x + 31*e^2*x^2) + 2*c^2*x*(35*d^
3 + 42*d^2*e*x + 21*d*e^2*x^2 + 4*e^3*x^3)) - 16*b^3*c^2*(-7*a*e^2*(95*d + 13*e*x) + c*(105*d^3 + 105*d^2*e*x
+ 49*d*e^2*x^2 + 9*e^3*x^3)) - 32*b*c^3*(a^2*e^2*(567*d + 73*e*x) - 2*a*c*(175*d^3 + 147*d^2*e*x + 63*d*e^2*x^
2 + 11*e^3*x^3) - 4*c^2*x^2*(105*d^3 + 231*d^2*e*x + 182*d*e^2*x^2 + 50*e^3*x^3)) + 64*c^3*(-32*a^3*e^3 + a^2*
c*e*(336*d^2 + 105*d*e*x + 16*e^2*x^2) + 4*c^3*x^3*(35*d^3 + 84*d^2*e*x + 70*d*e^2*x^2 + 20*e^3*x^3) + 2*a*c^2
*x*(175*d^3 + 336*d^2*e*x + 245*d*e^2*x^2 + 64*e^3*x^3))) - 105*(b^2 - 4*a*c)^2*(-2*c*d + b*e)*(8*c^2*d^2 + 3*
b^2*e^2 - 4*c*e*(2*b*d + a*e))*ArcTanh[(Sqrt[c]*x)/(-Sqrt[a] + Sqrt[a + x*(b + c*x)])])/(35840*c^(11/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(753\) vs. \(2(295)=590\).

Time = 0.34 (sec) , antiderivative size = 754, normalized size of antiderivative = 2.35

method	result	size
risch	\(-\frac {\left (-5120 c^{6} e^{3} x^{6}-6400 b \,c^{5} e^{3} x^{5}-17920 c^{6} d \,e^{2} x^{5}-8192 a \,c^{5} e^{3} x^{4}-128 b^{2} c^{4} e^{3} x^{4}-23296 b \,c^{5} d \,e^{2} x^{4}-21504 c^{6} d^{2} e \,x^{4}-704 a b \,c^{4} e^{3} x^{3}-31360 a \,c^{5} d \,e^{2} x^{3}+144 b^{3} c^{3} e^{3} x^{3}-672 b^{2} c^{4} d \,e^{2} x^{3}-29568 b \,c^{5} d^{2} e \,x^{3}-8960 c^{6} d^{3} x^{3}-1024 a^{2} c^{4} e^{3} x^{2}+992 a \,b^{2} c^{3} e^{3} x^{2}-4032 a b \,c^{4} d \,e^{2} x^{2}-43008 a \,c^{5} d^{2} e \,x^{2}-168 b^{4} c^{2} e^{3} x^{2}+784 b^{3} c^{3} d \,e^{2} x^{2}-1344 b^{2} c^{4} d^{2} e \,x^{2}-13440 b \,c^{5} d^{3} x^{2}+2336 a^{2} b \,c^{3} e^{3} x -6720 a^{2} c^{4} d \,e^{2} x -1456 a \,b^{3} c^{2} e^{3} x +6048 a \,b^{2} c^{3} d \,e^{2} x -9408 a b \,c^{4} d^{2} e x -22400 a \,c^{5} d^{3} x +210 b^{5} c \,e^{3} x -980 b^{4} c^{2} d \,e^{2} x +1680 b^{3} c^{3} d^{2} e x -1120 b^{2} c^{4} d^{3} x +2048 a^{3} c^{3} e^{3}-5488 a^{2} b^{2} c^{2} e^{3}+18144 a^{2} b \,c^{3} d \,e^{2}-21504 a^{2} c^{4} d^{2} e +2520 a \,b^{4} c \,e^{3}-10640 a \,b^{3} c^{2} d \,e^{2}+16800 a \,b^{2} c^{3} d^{2} e -11200 a b \,c^{4} d^{3}-315 b^{6} e^{3}+1470 b^{5} c d \,e^{2}-2520 b^{4} c^{2} d^{2} e +1680 b^{3} c^{3} d^{3}\right ) \sqrt {c \,x^{2}+b x +a}}{35840 c^{5}}+\frac {3 \left (64 a^{3} b \,c^{3} e^{3}-128 a^{3} c^{4} d \,e^{2}-80 a^{2} b^{3} c^{2} e^{3}+288 a^{2} b^{2} c^{3} d \,e^{2}-384 a^{2} b \,c^{4} d^{2} e +256 a^{2} c^{5} d^{3}+28 a \,b^{5} c \,e^{3}-120 a \,b^{4} c^{2} d \,e^{2}+192 a \,b^{3} c^{3} d^{2} e -128 a \,b^{2} c^{4} d^{3}-3 b^{7} e^{3}+14 b^{6} c d \,e^{2}-24 b^{5} c^{2} d^{2} e +16 b^{4} c^{3} d^{3}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2048 c^{\frac {11}{2}}}\)	\(754\)
default	\(\text {Expression too large to display}\)	\(934\)

[In]

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

[Out]

-1/35840/c^5*(-5120*c^6*e^3*x^6-6400*b*c^5*e^3*x^5-17920*c^6*d*e^2*x^5-8192*a*c^5*e^3*x^4-128*b^2*c^4*e^3*x^4-
23296*b*c^5*d*e^2*x^4-21504*c^6*d^2*e*x^4-704*a*b*c^4*e^3*x^3-31360*a*c^5*d*e^2*x^3+144*b^3*c^3*e^3*x^3-672*b^
2*c^4*d*e^2*x^3-29568*b*c^5*d^2*e*x^3-8960*c^6*d^3*x^3-1024*a^2*c^4*e^3*x^2+992*a*b^2*c^3*e^3*x^2-4032*a*b*c^4
*d*e^2*x^2-43008*a*c^5*d^2*e*x^2-168*b^4*c^2*e^3*x^2+784*b^3*c^3*d*e^2*x^2-1344*b^2*c^4*d^2*e*x^2-13440*b*c^5*
d^3*x^2+2336*a^2*b*c^3*e^3*x-6720*a^2*c^4*d*e^2*x-1456*a*b^3*c^2*e^3*x+6048*a*b^2*c^3*d*e^2*x-9408*a*b*c^4*d^2
*e*x-22400*a*c^5*d^3*x+210*b^5*c*e^3*x-980*b^4*c^2*d*e^2*x+1680*b^3*c^3*d^2*e*x-1120*b^2*c^4*d^3*x+2048*a^3*c^
3*e^3-5488*a^2*b^2*c^2*e^3+18144*a^2*b*c^3*d*e^2-21504*a^2*c^4*d^2*e+2520*a*b^4*c*e^3-10640*a*b^3*c^2*d*e^2+16
800*a*b^2*c^3*d^2*e-11200*a*b*c^4*d^3-315*b^6*e^3+1470*b^5*c*d*e^2-2520*b^4*c^2*d^2*e+1680*b^3*c^3*d^3)*(c*x^2
+b*x+a)^(1/2)+3/2048*(64*a^3*b*c^3*e^3-128*a^3*c^4*d*e^2-80*a^2*b^3*c^2*e^3+288*a^2*b^2*c^3*d*e^2-384*a^2*b*c^
4*d^2*e+256*a^2*c^5*d^3+28*a*b^5*c*e^3-120*a*b^4*c^2*d*e^2+192*a*b^3*c^3*d^2*e-128*a*b^2*c^4*d^3-3*b^7*e^3+14*
b^6*c*d*e^2-24*b^5*c^2*d^2*e+16*b^4*c^3*d^3)/c^(11/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 678 vs. \(2 (295) = 590\).

Time = 0.38 (sec) , antiderivative size = 1359, normalized size of antiderivative = 4.23 \[ \int (d+e x)^3 \left (a+b x+c x^2\right )^{3/2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

[1/143360*(105*(16*(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*d^3 - 24*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d^2*e
+ 2*(7*b^6*c - 60*a*b^4*c^2 + 144*a^2*b^2*c^3 - 64*a^3*c^4)*d*e^2 - (3*b^7 - 28*a*b^5*c + 80*a^2*b^3*c^2 - 64*
a^3*b*c^3)*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*(5120*c^7*e^3*x^6 + 1280*(14*c^7*d*e^2 + 5*b*c^6*e^3)*x^5 + 128*(168*c^7*d^2*e + 182*b*c^6*d*e^2 + (b^2*c^
5 + 64*a*c^6)*e^3)*x^4 - 560*(3*b^3*c^4 - 20*a*b*c^5)*d^3 + 168*(15*b^4*c^3 - 100*a*b^2*c^4 + 128*a^2*c^5)*d^2
*e - 14*(105*b^5*c^2 - 760*a*b^3*c^3 + 1296*a^2*b*c^4)*d*e^2 + (315*b^6*c - 2520*a*b^4*c^2 + 5488*a^2*b^2*c^3
- 2048*a^3*c^4)*e^3 + 16*(560*c^7*d^3 + 1848*b*c^6*d^2*e + 14*(3*b^2*c^5 + 140*a*c^6)*d*e^2 - (9*b^3*c^4 - 44*
a*b*c^5)*e^3)*x^3 + 8*(1680*b*c^6*d^3 + 168*(b^2*c^5 + 32*a*c^6)*d^2*e - 14*(7*b^3*c^4 - 36*a*b*c^5)*d*e^2 + (
21*b^4*c^3 - 124*a*b^2*c^4 + 128*a^2*c^5)*e^3)*x^2 + 2*(560*(b^2*c^5 + 20*a*c^6)*d^3 - 168*(5*b^3*c^4 - 28*a*b
*c^5)*d^2*e + 14*(35*b^4*c^3 - 216*a*b^2*c^4 + 240*a^2*c^5)*d*e^2 - (105*b^5*c^2 - 728*a*b^3*c^3 + 1168*a^2*b*
c^4)*e^3)*x)*sqrt(c*x^2 + b*x + a))/c^6, -1/71680*(105*(16*(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*d^3 - 24*(b^5*
c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d^2*e + 2*(7*b^6*c - 60*a*b^4*c^2 + 144*a^2*b^2*c^3 - 64*a^3*c^4)*d*e^2 - (3
*b^7 - 28*a*b^5*c + 80*a^2*b^3*c^2 - 64*a^3*b*c^3)*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*(5120*c^7*e^3*x^6 + 1280*(14*c^7*d*e^2 + 5*b*c^6*e^3)*x^5 + 128*(168*c^7
*d^2*e + 182*b*c^6*d*e^2 + (b^2*c^5 + 64*a*c^6)*e^3)*x^4 - 560*(3*b^3*c^4 - 20*a*b*c^5)*d^3 + 168*(15*b^4*c^3
- 100*a*b^2*c^4 + 128*a^2*c^5)*d^2*e - 14*(105*b^5*c^2 - 760*a*b^3*c^3 + 1296*a^2*b*c^4)*d*e^2 + (315*b^6*c -
2520*a*b^4*c^2 + 5488*a^2*b^2*c^3 - 2048*a^3*c^4)*e^3 + 16*(560*c^7*d^3 + 1848*b*c^6*d^2*e + 14*(3*b^2*c^5 + 1
40*a*c^6)*d*e^2 - (9*b^3*c^4 - 44*a*b*c^5)*e^3)*x^3 + 8*(1680*b*c^6*d^3 + 168*(b^2*c^5 + 32*a*c^6)*d^2*e - 14*
(7*b^3*c^4 - 36*a*b*c^5)*d*e^2 + (21*b^4*c^3 - 124*a*b^2*c^4 + 128*a^2*c^5)*e^3)*x^2 + 2*(560*(b^2*c^5 + 20*a*
c^6)*d^3 - 168*(5*b^3*c^4 - 28*a*b*c^5)*d^2*e + 14*(35*b^4*c^3 - 216*a*b^2*c^4 + 240*a^2*c^5)*d*e^2 - (105*b^5
*c^2 - 728*a*b^3*c^3 + 1168*a^2*b*c^4)*e^3)*x)*sqrt(c*x^2 + b*x + a))/c^6]

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3259 vs. \(2 (318) = 636\).

Time = 0.84 (sec) , antiderivative size = 3259, normalized size of antiderivative = 10.15 \[ \int (d+e x)^3 \left (a+b x+c x^2\right )^{3/2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Piecewise((sqrt(a + b*x + c*x**2)*(c*e**3*x**6/7 + x**5*(15*b*c*e**3/14 + 3*c**2*d*e**2)/(6*c) + x**4*(8*a*c*e
**3/7 + b**2*e**3 + 6*b*c*d*e**2 - 11*b*(15*b*c*e**3/14 + 3*c**2*d*e**2)/(12*c) + 3*c**2*d**2*e)/(5*c) + x**3*
(2*a*b*e**3 + 6*a*c*d*e**2 - 5*a*(15*b*c*e**3/14 + 3*c**2*d*e**2)/(6*c) + 3*b**2*d*e**2 + 6*b*c*d**2*e - 9*b*(
8*a*c*e**3/7 + b**2*e**3 + 6*b*c*d*e**2 - 11*b*(15*b*c*e**3/14 + 3*c**2*d*e**2)/(12*c) + 3*c**2*d**2*e)/(10*c)
+ c**2*d**3)/(4*c) + x**2*(a**2*e**3 + 6*a*b*d*e**2 + 6*a*c*d**2*e - 4*a*(8*a*c*e**3/7 + b**2*e**3 + 6*b*c*d*
e**2 - 11*b*(15*b*c*e**3/14 + 3*c**2*d*e**2)/(12*c) + 3*c**2*d**2*e)/(5*c) + 3*b**2*d**2*e + 2*b*c*d**3 - 7*b*
(2*a*b*e**3 + 6*a*c*d*e**2 - 5*a*(15*b*c*e**3/14 + 3*c**2*d*e**2)/(6*c) + 3*b**2*d*e**2 + 6*b*c*d**2*e - 9*b*(
8*a*c*e**3/7 + b**2*e**3 + 6*b*c*d*e**2 - 11*b*(15*b*c*e**3/14 + 3*c**2*d*e**2)/(12*c) + 3*c**2*d**2*e)/(10*c)
+ c**2*d**3)/(8*c))/(3*c) + x*(3*a**2*d*e**2 + 6*a*b*d**2*e + 2*a*c*d**3 - 3*a*(2*a*b*e**3 + 6*a*c*d*e**2 - 5
*a*(15*b*c*e**3/14 + 3*c**2*d*e**2)/(6*c) + 3*b**2*d*e**2 + 6*b*c*d**2*e - 9*b*(8*a*c*e**3/7 + b**2*e**3 + 6*b
*c*d*e**2 - 11*b*(15*b*c*e**3/14 + 3*c**2*d*e**2)/(12*c) + 3*c**2*d**2*e)/(10*c) + c**2*d**3)/(4*c) + b**2*d**
3 - 5*b*(a**2*e**3 + 6*a*b*d*e**2 + 6*a*c*d**2*e - 4*a*(8*a*c*e**3/7 + b**2*e**3 + 6*b*c*d*e**2 - 11*b*(15*b*c
*e**3/14 + 3*c**2*d*e**2)/(12*c) + 3*c**2*d**2*e)/(5*c) + 3*b**2*d**2*e + 2*b*c*d**3 - 7*b*(2*a*b*e**3 + 6*a*c
*d*e**2 - 5*a*(15*b*c*e**3/14 + 3*c**2*d*e**2)/(6*c) + 3*b**2*d*e**2 + 6*b*c*d**2*e - 9*b*(8*a*c*e**3/7 + b**2
*e**3 + 6*b*c*d*e**2 - 11*b*(15*b*c*e**3/14 + 3*c**2*d*e**2)/(12*c) + 3*c**2*d**2*e)/(10*c) + c**2*d**3)/(8*c)
)/(6*c))/(2*c) + (3*a**2*d**2*e + 2*a*b*d**3 - 2*a*(a**2*e**3 + 6*a*b*d*e**2 + 6*a*c*d**2*e - 4*a*(8*a*c*e**3/
7 + b**2*e**3 + 6*b*c*d*e**2 - 11*b*(15*b*c*e**3/14 + 3*c**2*d*e**2)/(12*c) + 3*c**2*d**2*e)/(5*c) + 3*b**2*d*
*2*e + 2*b*c*d**3 - 7*b*(2*a*b*e**3 + 6*a*c*d*e**2 - 5*a*(15*b*c*e**3/14 + 3*c**2*d*e**2)/(6*c) + 3*b**2*d*e**
2 + 6*b*c*d**2*e - 9*b*(8*a*c*e**3/7 + b**2*e**3 + 6*b*c*d*e**2 - 11*b*(15*b*c*e**3/14 + 3*c**2*d*e**2)/(12*c)
+ 3*c**2*d**2*e)/(10*c) + c**2*d**3)/(8*c))/(3*c) - 3*b*(3*a**2*d*e**2 + 6*a*b*d**2*e + 2*a*c*d**3 - 3*a*(2*a
*b*e**3 + 6*a*c*d*e**2 - 5*a*(15*b*c*e**3/14 + 3*c**2*d*e**2)/(6*c) + 3*b**2*d*e**2 + 6*b*c*d**2*e - 9*b*(8*a*
c*e**3/7 + b**2*e**3 + 6*b*c*d*e**2 - 11*b*(15*b*c*e**3/14 + 3*c**2*d*e**2)/(12*c) + 3*c**2*d**2*e)/(10*c) + c
**2*d**3)/(4*c) + b**2*d**3 - 5*b*(a**2*e**3 + 6*a*b*d*e**2 + 6*a*c*d**2*e - 4*a*(8*a*c*e**3/7 + b**2*e**3 + 6
*b*c*d*e**2 - 11*b*(15*b*c*e**3/14 + 3*c**2*d*e**2)/(12*c) + 3*c**2*d**2*e)/(5*c) + 3*b**2*d**2*e + 2*b*c*d**3
- 7*b*(2*a*b*e**3 + 6*a*c*d*e**2 - 5*a*(15*b*c*e**3/14 + 3*c**2*d*e**2)/(6*c) + 3*b**2*d*e**2 + 6*b*c*d**2*e
- 9*b*(8*a*c*e**3/7 + b**2*e**3 + 6*b*c*d*e**2 - 11*b*(15*b*c*e**3/14 + 3*c**2*d*e**2)/(12*c) + 3*c**2*d**2*e)
/(10*c) + c**2*d**3)/(8*c))/(6*c))/(4*c))/c) + (a**2*d**3 - a*(3*a**2*d*e**2 + 6*a*b*d**2*e + 2*a*c*d**3 - 3*a
*(2*a*b*e**3 + 6*a*c*d*e**2 - 5*a*(15*b*c*e**3/14 + 3*c**2*d*e**2)/(6*c) + 3*b**2*d*e**2 + 6*b*c*d**2*e - 9*b*
(8*a*c*e**3/7 + b**2*e**3 + 6*b*c*d*e**2 - 11*b*(15*b*c*e**3/14 + 3*c**2*d*e**2)/(12*c) + 3*c**2*d**2*e)/(10*c
) + c**2*d**3)/(4*c) + b**2*d**3 - 5*b*(a**2*e**3 + 6*a*b*d*e**2 + 6*a*c*d**2*e - 4*a*(8*a*c*e**3/7 + b**2*e**
3 + 6*b*c*d*e**2 - 11*b*(15*b*c*e**3/14 + 3*c**2*d*e**2)/(12*c) + 3*c**2*d**2*e)/(5*c) + 3*b**2*d**2*e + 2*b*c
*d**3 - 7*b*(2*a*b*e**3 + 6*a*c*d*e**2 - 5*a*(15*b*c*e**3/14 + 3*c**2*d*e**2)/(6*c) + 3*b**2*d*e**2 + 6*b*c*d*
*2*e - 9*b*(8*a*c*e**3/7 + b**2*e**3 + 6*b*c*d*e**2 - 11*b*(15*b*c*e**3/14 + 3*c**2*d*e**2)/(12*c) + 3*c**2*d*
*2*e)/(10*c) + c**2*d**3)/(8*c))/(6*c))/(2*c) - b*(3*a**2*d**2*e + 2*a*b*d**3 - 2*a*(a**2*e**3 + 6*a*b*d*e**2
+ 6*a*c*d**2*e - 4*a*(8*a*c*e**3/7 + b**2*e**3 + 6*b*c*d*e**2 - 11*b*(15*b*c*e**3/14 + 3*c**2*d*e**2)/(12*c) +
3*c**2*d**2*e)/(5*c) + 3*b**2*d**2*e + 2*b*c*d**3 - 7*b*(2*a*b*e**3 + 6*a*c*d*e**2 - 5*a*(15*b*c*e**3/14 + 3*
c**2*d*e**2)/(6*c) + 3*b**2*d*e**2 + 6*b*c*d**2*e - 9*b*(8*a*c*e**3/7 + b**2*e**3 + 6*b*c*d*e**2 - 11*b*(15*b*
c*e**3/14 + 3*c**2*d*e**2)/(12*c) + 3*c**2*d**2*e)/(10*c) + c**2*d**3)/(8*c))/(3*c) - 3*b*(3*a**2*d*e**2 + 6*a
*b*d**2*e + 2*a*c*d**3 - 3*a*(2*a*b*e**3 + 6*a*c*d*e**2 - 5*a*(15*b*c*e**3/14 + 3*c**2*d*e**2)/(6*c) + 3*b**2*
d*e**2 + 6*b*c*d**2*e - 9*b*(8*a*c*e**3/7 + b**2*e**3 + 6*b*c*d*e**2 - 11*b*(15*b*c*e**3/14 + 3*c**2*d*e**2)/(
12*c) + 3*c**2*d**2*e)/(10*c) + c**2*d**3)/(4*c) + b**2*d**3 - 5*b*(a**2*e**3 + 6*a*b*d*e**2 + 6*a*c*d**2*e -
4*a*(8*a*c*e**3/7 + b**2*e**3 + 6*b*c*d*e**2 - 11*b*(15*b*c*e**3/14 + 3*c**2*d*e**2)/(12*c) + 3*c**2*d**2*e)/(
5*c) + 3*b**2*d**2*e + 2*b*c*d**3 - 7*b*(2*a*b*e**3 + 6*a*c*d*e**2 - 5*a*(15*b*c*e**3/14 + 3*c**2*d*e**2)/(6*c
) + 3*b**2*d*e**2 + 6*b*c*d**2*e - 9*b*(8*a*c*e**3/7 + b**2*e**3 + 6*b*c*d*e**2 - 11*b*(15*b*c*e**3/14 + 3*c**
2*d*e**2)/(12*c) + 3*c**2*d**2*e)/(10*c) + c**2*d**3)/(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**3*(a + b*x)**(11/2)/(11*b**3) + (a + b*x)**(9/2)*(-3*a*e**3 + 3*b*d*e*
*2)/(9*b**3) + (a + b*x)**(7/2)*(3*a**2*e**3 - 6*a*b*d*e**2 + 3*b**2*d**2*e)/(7*b**3) + (a + b*x)**(5/2)*(-a**
3*e**3 + 3*a**2*b*d*e**2 - 3*a*b**2*d**2*e + b**3*d**3)/(5*b**3))/b, Ne(b, 0)), (a**(3/2)*Piecewise((d**3*x, E
q(e, 0)), ((d + e*x)**4/(4*e), True)), True))

Maxima [F(-2)]

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

[In]

integrate((e*x+d)^3*(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. 737 vs. \(2 (295) = 590\).

Time = 0.29 (sec) , antiderivative size = 737, normalized size of antiderivative = 2.30 \[ \int (d+e x)^3 \left (a+b x+c x^2\right )^{3/2} \, dx=\frac {1}{35840} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, {\left (4 \, c e^{3} x + \frac {14 \, c^{7} d e^{2} + 5 \, b c^{6} e^{3}}{c^{6}}\right )} x + \frac {168 \, c^{7} d^{2} e + 182 \, b c^{6} d e^{2} + b^{2} c^{5} e^{3} + 64 \, a c^{6} e^{3}}{c^{6}}\right )} x + \frac {560 \, c^{7} d^{3} + 1848 \, b c^{6} d^{2} e + 42 \, b^{2} c^{5} d e^{2} + 1960 \, a c^{6} d e^{2} - 9 \, b^{3} c^{4} e^{3} + 44 \, a b c^{5} e^{3}}{c^{6}}\right )} x + \frac {1680 \, b c^{6} d^{3} + 168 \, b^{2} c^{5} d^{2} e + 5376 \, a c^{6} d^{2} e - 98 \, b^{3} c^{4} d e^{2} + 504 \, a b c^{5} d e^{2} + 21 \, b^{4} c^{3} e^{3} - 124 \, a b^{2} c^{4} e^{3} + 128 \, a^{2} c^{5} e^{3}}{c^{6}}\right )} x + \frac {560 \, b^{2} c^{5} d^{3} + 11200 \, a c^{6} d^{3} - 840 \, b^{3} c^{4} d^{2} e + 4704 \, a b c^{5} d^{2} e + 490 \, b^{4} c^{3} d e^{2} - 3024 \, a b^{2} c^{4} d e^{2} + 3360 \, a^{2} c^{5} d e^{2} - 105 \, b^{5} c^{2} e^{3} + 728 \, a b^{3} c^{3} e^{3} - 1168 \, a^{2} b c^{4} e^{3}}{c^{6}}\right )} x - \frac {1680 \, b^{3} c^{4} d^{3} - 11200 \, a b c^{5} d^{3} - 2520 \, b^{4} c^{3} d^{2} e + 16800 \, a b^{2} c^{4} d^{2} e - 21504 \, a^{2} c^{5} d^{2} e + 1470 \, b^{5} c^{2} d e^{2} - 10640 \, a b^{3} c^{3} d e^{2} + 18144 \, a^{2} b c^{4} d e^{2} - 315 \, b^{6} c e^{3} + 2520 \, a b^{4} c^{2} e^{3} - 5488 \, a^{2} b^{2} c^{3} e^{3} + 2048 \, a^{3} c^{4} e^{3}}{c^{6}}\right )} - \frac {3 \, {\left (16 \, b^{4} c^{3} d^{3} - 128 \, a b^{2} c^{4} d^{3} + 256 \, a^{2} c^{5} d^{3} - 24 \, b^{5} c^{2} d^{2} e + 192 \, a b^{3} c^{3} d^{2} e - 384 \, a^{2} b c^{4} d^{2} e + 14 \, b^{6} c d e^{2} - 120 \, a b^{4} c^{2} d e^{2} + 288 \, a^{2} b^{2} c^{3} d e^{2} - 128 \, a^{3} c^{4} d e^{2} - 3 \, b^{7} e^{3} + 28 \, a b^{5} c e^{3} - 80 \, a^{2} b^{3} c^{2} e^{3} + 64 \, a^{3} b c^{3} e^{3}\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{2048 \, c^{\frac {11}{2}}} \]

[In]

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

[Out]

1/35840*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(10*(4*c*e^3*x + (14*c^7*d*e^2 + 5*b*c^6*e^3)/c^6)*x + (168*c^7*d^2*
e + 182*b*c^6*d*e^2 + b^2*c^5*e^3 + 64*a*c^6*e^3)/c^6)*x + (560*c^7*d^3 + 1848*b*c^6*d^2*e + 42*b^2*c^5*d*e^2
+ 1960*a*c^6*d*e^2 - 9*b^3*c^4*e^3 + 44*a*b*c^5*e^3)/c^6)*x + (1680*b*c^6*d^3 + 168*b^2*c^5*d^2*e + 5376*a*c^6
*d^2*e - 98*b^3*c^4*d*e^2 + 504*a*b*c^5*d*e^2 + 21*b^4*c^3*e^3 - 124*a*b^2*c^4*e^3 + 128*a^2*c^5*e^3)/c^6)*x +
(560*b^2*c^5*d^3 + 11200*a*c^6*d^3 - 840*b^3*c^4*d^2*e + 4704*a*b*c^5*d^2*e + 490*b^4*c^3*d*e^2 - 3024*a*b^2*
c^4*d*e^2 + 3360*a^2*c^5*d*e^2 - 105*b^5*c^2*e^3 + 728*a*b^3*c^3*e^3 - 1168*a^2*b*c^4*e^3)/c^6)*x - (1680*b^3*
c^4*d^3 - 11200*a*b*c^5*d^3 - 2520*b^4*c^3*d^2*e + 16800*a*b^2*c^4*d^2*e - 21504*a^2*c^5*d^2*e + 1470*b^5*c^2*
d*e^2 - 10640*a*b^3*c^3*d*e^2 + 18144*a^2*b*c^4*d*e^2 - 315*b^6*c*e^3 + 2520*a*b^4*c^2*e^3 - 5488*a^2*b^2*c^3*
e^3 + 2048*a^3*c^4*e^3)/c^6) - 3/2048*(16*b^4*c^3*d^3 - 128*a*b^2*c^4*d^3 + 256*a^2*c^5*d^3 - 24*b^5*c^2*d^2*e
+ 192*a*b^3*c^3*d^2*e - 384*a^2*b*c^4*d^2*e + 14*b^6*c*d*e^2 - 120*a*b^4*c^2*d*e^2 + 288*a^2*b^2*c^3*d*e^2 -
128*a^3*c^4*d*e^2 - 3*b^7*e^3 + 28*a*b^5*c*e^3 - 80*a^2*b^3*c^2*e^3 + 64*a^3*b*c^3*e^3)*log(abs(2*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))*sqrt(c) + b))/c^(11/2)

Mupad [F(-1)]

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

[In]

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

[Out]

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