3.16.46 \(\int (b+2 c x) (d+e x)^4 \sqrt {a+b x+c x^2} \, dx\) [1546]

Optimal. Leaf size=431 \[ \frac {\left (b^2-4 a c\right ) e (2 c d-b e) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{128 c^5}+\frac {\left (4 c^2 d^2+3 b^2 e^2-4 c e (b d+2 a e)\right ) (d+e x)^2 \left (a+b x+c x^2\right )^{3/2}}{35 c^2}+\frac {2 (2 c d-b e) (d+e x)^3 \left (a+b x+c x^2\right )^{3/2}}{21 c}+\frac {2}{7} (d+e x)^4 \left (a+b x+c x^2\right )^{3/2}+\frac {\left (128 c^4 d^4+105 b^4 e^4-14 b^2 c e^3 (35 b d+34 a e)-16 c^3 d^2 e (13 b d+144 a e)+8 c^2 e^2 \left (87 b^2 d^2+231 a b d e+32 a^2 e^2\right )+6 c e (2 c d-b e) \left (8 c^2 d^2+21 b^2 e^2-4 c e (2 b d+19 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{1680 c^4}-\frac {\left (b^2-4 a c\right )^2 e (2 c d-b e) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{256 c^{11/2}} \]

[Out]

1/35*(4*c^2*d^2+3*b^2*e^2-4*c*e*(2*a*e+b*d))*(e*x+d)^2*(c*x^2+b*x+a)^(3/2)/c^2+2/21*(-b*e+2*c*d)*(e*x+d)^3*(c*
x^2+b*x+a)^(3/2)/c+2/7*(e*x+d)^4*(c*x^2+b*x+a)^(3/2)+1/1680*(128*c^4*d^4+105*b^4*e^4-14*b^2*c*e^3*(34*a*e+35*b
*d)-16*c^3*d^2*e*(144*a*e+13*b*d)+8*c^2*e^2*(32*a^2*e^2+231*a*b*d*e+87*b^2*d^2)+6*c*e*(-b*e+2*c*d)*(8*c^2*d^2+
21*b^2*e^2-4*c*e*(19*a*e+2*b*d))*x)*(c*x^2+b*x+a)^(3/2)/c^4-1/256*(-4*a*c+b^2)^2*e*(-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)+1/128*(-4*a*c+b^2)*e*(-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

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Rubi [A]
time = 0.51, antiderivative size = 431, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {846, 793, 626, 635, 212} \begin {gather*} \frac {\left (a+b x+c x^2\right )^{3/2} \left (8 c^2 e^2 \left (32 a^2 e^2+231 a b d e+87 b^2 d^2\right )+6 c e x (2 c d-b e) \left (-4 c e (19 a e+2 b d)+21 b^2 e^2+8 c^2 d^2\right )-14 b^2 c e^3 (34 a e+35 b d)-16 c^3 d^2 e (144 a e+13 b d)+105 b^4 e^4+128 c^4 d^4\right )}{1680 c^4}+\frac {(d+e x)^2 \left (a+b x+c x^2\right )^{3/2} \left (-4 c e (2 a e+b d)+3 b^2 e^2+4 c^2 d^2\right )}{35 c^2}-\frac {e \left (b^2-4 a c\right )^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 ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{256 c^{11/2}}+\frac {e \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 )}{128 c^5}+\frac {2}{7} (d+e x)^4 \left (a+b x+c x^2\right )^{3/2}+\frac {2 (d+e x)^3 \left (a+b x+c x^2\right )^{3/2} (2 c d-b e)}{21 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b + 2*c*x)*(d + e*x)^4*Sqrt[a + b*x + c*x^2],x]

[Out]

((b^2 - 4*a*c)*e*(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]
)/(128*c^5) + ((4*c^2*d^2 + 3*b^2*e^2 - 4*c*e*(b*d + 2*a*e))*(d + e*x)^2*(a + b*x + c*x^2)^(3/2))/(35*c^2) + (
2*(2*c*d - b*e)*(d + e*x)^3*(a + b*x + c*x^2)^(3/2))/(21*c) + (2*(d + e*x)^4*(a + b*x + c*x^2)^(3/2))/7 + ((12
8*c^4*d^4 + 105*b^4*e^4 - 14*b^2*c*e^3*(35*b*d + 34*a*e) - 16*c^3*d^2*e*(13*b*d + 144*a*e) + 8*c^2*e^2*(87*b^2
*d^2 + 231*a*b*d*e + 32*a^2*e^2) + 6*c*e*(2*c*d - b*e)*(8*c^2*d^2 + 21*b^2*e^2 - 4*c*e*(2*b*d + 19*a*e))*x)*(a
+ b*x + c*x^2)^(3/2))/(1680*c^4) - ((b^2 - 4*a*c)^2*e*(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])])/(256*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 793

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

Rule 846

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

Rubi steps

\begin {align*} \int (b+2 c x) (d+e x)^4 \sqrt {a+b x+c x^2} \, dx &=\frac {2}{7} (d+e x)^4 \left (a+b x+c x^2\right )^{3/2}+\frac {\int (d+e x)^3 (4 c (b d-2 a e)+4 c (2 c d-b e) x) \sqrt {a+b x+c x^2} \, dx}{7 c}\\ &=\frac {2 (2 c d-b e) (d+e x)^3 \left (a+b x+c x^2\right )^{3/2}}{21 c}+\frac {2}{7} (d+e x)^4 \left (a+b x+c x^2\right )^{3/2}+\frac {\int (d+e x)^2 \left (6 c \left (b^2 d e-12 a c d e+2 b \left (c d^2+a e^2\right )\right )+6 c \left (4 c^2 d^2+3 b^2 e^2-4 c e (b d+2 a e)\right ) x\right ) \sqrt {a+b x+c x^2} \, dx}{42 c^2}\\ &=\frac {\left (4 c^2 d^2+3 b^2 e^2-4 c e (b d+2 a e)\right ) (d+e x)^2 \left (a+b x+c x^2\right )^{3/2}}{35 c^2}+\frac {2 (2 c d-b e) (d+e x)^3 \left (a+b x+c x^2\right )^{3/2}}{21 c}+\frac {2}{7} (d+e x)^4 \left (a+b x+c x^2\right )^{3/2}+\frac {\int (d+e x) \left (-3 c \left (9 b^3 d e^2+8 a c e \left (17 c d^2-4 a e^2\right )-4 b c d \left (2 c d^2+15 a e^2\right )-2 b^2 \left (11 c d^2 e-6 a e^3\right )\right )+3 c (2 c d-b e) \left (8 c^2 d^2+21 b^2 e^2-4 c e (2 b d+19 a e)\right ) x\right ) \sqrt {a+b x+c x^2} \, dx}{210 c^3}\\ &=\frac {\left (4 c^2 d^2+3 b^2 e^2-4 c e (b d+2 a e)\right ) (d+e x)^2 \left (a+b x+c x^2\right )^{3/2}}{35 c^2}+\frac {2 (2 c d-b e) (d+e x)^3 \left (a+b x+c x^2\right )^{3/2}}{21 c}+\frac {2}{7} (d+e x)^4 \left (a+b x+c x^2\right )^{3/2}+\frac {\left (128 c^4 d^4+105 b^4 e^4-14 b^2 c e^3 (35 b d+34 a e)-16 c^3 d^2 e (13 b d+144 a e)+8 c^2 e^2 \left (87 b^2 d^2+231 a b d e+32 a^2 e^2\right )+6 c e (2 c d-b e) \left (8 c^2 d^2+21 b^2 e^2-4 c e (2 b d+19 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{1680 c^4}+\frac {\left (\left (b^2-4 a c\right ) e (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}{32 c^4}\\ &=\frac {\left (b^2-4 a c\right ) e (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}}{128 c^5}+\frac {\left (4 c^2 d^2+3 b^2 e^2-4 c e (b d+2 a e)\right ) (d+e x)^2 \left (a+b x+c x^2\right )^{3/2}}{35 c^2}+\frac {2 (2 c d-b e) (d+e x)^3 \left (a+b x+c x^2\right )^{3/2}}{21 c}+\frac {2}{7} (d+e x)^4 \left (a+b x+c x^2\right )^{3/2}+\frac {\left (128 c^4 d^4+105 b^4 e^4-14 b^2 c e^3 (35 b d+34 a e)-16 c^3 d^2 e (13 b d+144 a e)+8 c^2 e^2 \left (87 b^2 d^2+231 a b d e+32 a^2 e^2\right )+6 c e (2 c d-b e) \left (8 c^2 d^2+21 b^2 e^2-4 c e (2 b d+19 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{1680 c^4}-\frac {\left (\left (b^2-4 a c\right )^2 e (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}{256 c^5}\\ &=\frac {\left (b^2-4 a c\right ) e (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}}{128 c^5}+\frac {\left (4 c^2 d^2+3 b^2 e^2-4 c e (b d+2 a e)\right ) (d+e x)^2 \left (a+b x+c x^2\right )^{3/2}}{35 c^2}+\frac {2 (2 c d-b e) (d+e x)^3 \left (a+b x+c x^2\right )^{3/2}}{21 c}+\frac {2}{7} (d+e x)^4 \left (a+b x+c x^2\right )^{3/2}+\frac {\left (128 c^4 d^4+105 b^4 e^4-14 b^2 c e^3 (35 b d+34 a e)-16 c^3 d^2 e (13 b d+144 a e)+8 c^2 e^2 \left (87 b^2 d^2+231 a b d e+32 a^2 e^2\right )+6 c e (2 c d-b e) \left (8 c^2 d^2+21 b^2 e^2-4 c e (2 b d+19 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{1680 c^4}-\frac {\left (\left (b^2-4 a c\right )^2 e (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 )}{128 c^5}\\ &=\frac {\left (b^2-4 a c\right ) e (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}}{128 c^5}+\frac {\left (4 c^2 d^2+3 b^2 e^2-4 c e (b d+2 a e)\right ) (d+e x)^2 \left (a+b x+c x^2\right )^{3/2}}{35 c^2}+\frac {2 (2 c d-b e) (d+e x)^3 \left (a+b x+c x^2\right )^{3/2}}{21 c}+\frac {2}{7} (d+e x)^4 \left (a+b x+c x^2\right )^{3/2}+\frac {\left (128 c^4 d^4+105 b^4 e^4-14 b^2 c e^3 (35 b d+34 a e)-16 c^3 d^2 e (13 b d+144 a e)+8 c^2 e^2 \left (87 b^2 d^2+231 a b d e+32 a^2 e^2\right )+6 c e (2 c d-b e) \left (8 c^2 d^2+21 b^2 e^2-4 c e (2 b d+19 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{1680 c^4}-\frac {\left (b^2-4 a c\right )^2 e (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 )}{256 c^{11/2}}\\ \end {align*}

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Mathematica [A]
time = 1.55, size = 547, normalized size = 1.27 \begin {gather*} \frac {\sqrt {a+x (b+c x)} \left (-315 b^6 e^4+2048 a^3 c^3 e^4+210 b^5 c e^3 (7 d+e x)-28 b^4 c^2 e^2 \left (90 d^2+35 d e x+6 e^2 x^2\right )-32 b^2 c^4 e x \left (35 d^3+42 d^2 e x+21 d e^2 x^2+4 e^3 x^3\right )+16 b^3 c^3 e \left (105 d^3+105 d^2 e x+49 d e^2 x^2+9 e^3 x^3\right )+256 c^6 x^2 \left (35 d^4+105 d^3 e x+126 d^2 e^2 x^2+70 d e^3 x^3+15 e^4 x^4\right )+128 b c^5 x \left (70 d^4+175 d^3 e x+189 d^2 e^2 x^2+98 d e^3 x^3+20 e^4 x^4\right )-16 a^2 c^2 e^2 \left (343 b^2 e^2-2 b c e (567 d+73 e x)+4 c^2 \left (336 d^2+105 d e x+16 e^2 x^2\right )\right )+8 a c \left (315 b^4 e^4-14 b^3 c e^3 (95 d+13 e x)+4 b^2 c^2 e^2 \left (525 d^2+189 d e x+31 e^2 x^2\right )-8 b c^3 e \left (175 d^3+147 d^2 e x+63 d e^2 x^2+11 e^3 x^3\right )+16 c^4 \left (70 d^4+105 d^3 e x+84 d^2 e^2 x^2+35 d e^3 x^3+6 e^4 x^4\right )\right )\right )}{13440 c^5}-\frac {\left (b^2-4 a c\right )^2 e (-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 ) \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )}{256 c^{11/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b + 2*c*x)*(d + e*x)^4*Sqrt[a + b*x + c*x^2],x]

[Out]

(Sqrt[a + x*(b + c*x)]*(-315*b^6*e^4 + 2048*a^3*c^3*e^4 + 210*b^5*c*e^3*(7*d + e*x) - 28*b^4*c^2*e^2*(90*d^2 +
35*d*e*x + 6*e^2*x^2) - 32*b^2*c^4*e*x*(35*d^3 + 42*d^2*e*x + 21*d*e^2*x^2 + 4*e^3*x^3) + 16*b^3*c^3*e*(105*d
^3 + 105*d^2*e*x + 49*d*e^2*x^2 + 9*e^3*x^3) + 256*c^6*x^2*(35*d^4 + 105*d^3*e*x + 126*d^2*e^2*x^2 + 70*d*e^3*
x^3 + 15*e^4*x^4) + 128*b*c^5*x*(70*d^4 + 175*d^3*e*x + 189*d^2*e^2*x^2 + 98*d*e^3*x^3 + 20*e^4*x^4) - 16*a^2*
c^2*e^2*(343*b^2*e^2 - 2*b*c*e*(567*d + 73*e*x) + 4*c^2*(336*d^2 + 105*d*e*x + 16*e^2*x^2)) + 8*a*c*(315*b^4*e
^4 - 14*b^3*c*e^3*(95*d + 13*e*x) + 4*b^2*c^2*e^2*(525*d^2 + 189*d*e*x + 31*e^2*x^2) - 8*b*c^3*e*(175*d^3 + 14
7*d^2*e*x + 63*d*e^2*x^2 + 11*e^3*x^3) + 16*c^4*(70*d^4 + 105*d^3*e*x + 84*d^2*e^2*x^2 + 35*d*e^3*x^3 + 6*e^4*
x^4))))/(13440*c^5) - ((b^2 - 4*a*c)^2*e*(-2*c*d + b*e)*(8*c^2*d^2 + 3*b^2*e^2 - 4*c*e*(2*b*d + a*e))*Log[b +
2*c*x - 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/(256*c^(11/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2093\) vs. \(2(401)=802\).
time = 1.05, size = 2094, normalized size = 4.86

method	result	size
risch	\(\frac {\left (3840 c^{6} e^{4} x^{6}+2560 b \,c^{5} e^{4} x^{5}+17920 c^{6} d \,e^{3} x^{5}+768 a \,c^{5} e^{4} x^{4}-128 b^{2} c^{4} e^{4} x^{4}+12544 b \,c^{5} d \,e^{3} x^{4}+32256 c^{6} d^{2} e^{2} x^{4}-704 a b \,c^{4} e^{4} x^{3}+4480 a \,c^{5} d \,e^{3} x^{3}+144 b^{3} c^{3} e^{4} x^{3}-672 b^{2} c^{4} d \,e^{3} x^{3}+24192 b \,c^{5} d^{2} e^{2} x^{3}+26880 c^{6} d^{3} e \,x^{3}-1024 a^{2} c^{4} e^{4} x^{2}+992 a \,b^{2} c^{3} e^{4} x^{2}-4032 a b \,c^{4} d \,e^{3} x^{2}+10752 a \,c^{5} d^{2} e^{2} x^{2}-168 b^{4} c^{2} e^{4} x^{2}+784 b^{3} c^{3} d \,e^{3} x^{2}-1344 b^{2} c^{4} d^{2} e^{2} x^{2}+22400 b \,c^{5} d^{3} e \,x^{2}+8960 c^{6} d^{4} x^{2}+2336 a^{2} b \,c^{3} e^{4} x -6720 a^{2} c^{4} d \,e^{3} x -1456 a \,b^{3} c^{2} e^{4} x +6048 a \,b^{2} c^{3} d \,e^{3} x -9408 a b \,c^{4} d^{2} e^{2} x +13440 a \,c^{5} d^{3} e x +210 b^{5} c \,e^{4} x -980 b^{4} c^{2} d \,e^{3} x +1680 b^{3} c^{3} d^{2} e^{2} x -1120 b^{2} c^{4} d^{3} e x +8960 b \,c^{5} d^{4} x +2048 a^{3} c^{3} e^{4}-5488 a^{2} b^{2} c^{2} e^{4}+18144 a^{2} b \,c^{3} d \,e^{3}-21504 a^{2} c^{4} d^{2} e^{2}+2520 a \,b^{4} c \,e^{4}-10640 a \,b^{3} c^{2} d \,e^{3}+16800 a \,b^{2} c^{3} d^{2} e^{2}-11200 a b \,c^{4} d^{3} e +8960 a \,c^{5} d^{4}-315 b^{6} e^{4}+1470 b^{5} c d \,e^{3}-2520 b^{4} c^{2} d^{2} e^{2}+1680 b^{3} c^{3} d^{3} e \right ) \sqrt {c \,x^{2}+b x +a}}{13440 c^{5}}-\frac {e^{4} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) a^{3} b}{4 c^{\frac {5}{2}}}+\frac {e^{3} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) a^{3} d}{2 c^{\frac {3}{2}}}+\frac {5 e^{4} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) a^{2} b^{3}}{16 c^{\frac {7}{2}}}-\frac {9 e^{3} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) a^{2} b^{2} d}{8 c^{\frac {5}{2}}}+\frac {3 e^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) a^{2} b \,d^{2}}{2 c^{\frac {3}{2}}}-\frac {e \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) a^{2} d^{3}}{\sqrt {c}}-\frac {7 e^{4} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) a \,b^{5}}{64 c^{\frac {9}{2}}}+\frac {15 e^{3} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) a \,b^{4} d}{32 c^{\frac {7}{2}}}-\frac {3 e^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) a \,b^{3} d^{2}}{4 c^{\frac {5}{2}}}+\frac {e \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) a \,b^{2} d^{3}}{2 c^{\frac {3}{2}}}+\frac {3 e^{4} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) b^{7}}{256 c^{\frac {11}{2}}}-\frac {7 e^{3} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) b^{6} d}{128 c^{\frac {9}{2}}}+\frac {3 e^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) b^{5} d^{2}}{32 c^{\frac {7}{2}}}-\frac {e \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) b^{4} d^{3}}{16 c^{\frac {5}{2}}}\)	\(1143\)
default	\(\text {Expression too large to display}\)	\(2094\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*x+b)*(e*x+d)^4*(c*x^2+b*x+a)^(1/2),x,method=_RETURNVERBOSE)

[Out]

2*c*e^4*(1/7*x^4*(c*x^2+b*x+a)^(3/2)/c-11/14*b/c*(1/6*x^3*(c*x^2+b*x+a)^(3/2)/c-3/4*b/c*(1/5*x^2*(c*x^2+b*x+a)
^(3/2)/c-7/10*b/c*(1/4*x*(c*x^2+b*x+a)^(3/2)/c-5/8*b/c*(1/3*(c*x^2+b*x+a)^(3/2)/c-1/2*b/c*(1/4*(2*c*x+b)*(c*x^
2+b*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))))-1/4*a/c*(1/4*(2*c*x+b)*
(c*x^2+b*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))))-2/5*a/c*(1/3*(c*x^
2+b*x+a)^(3/2)/c-1/2*b/c*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(
c*x^2+b*x+a)^(1/2)))))-1/2*a/c*(1/4*x*(c*x^2+b*x+a)^(3/2)/c-5/8*b/c*(1/3*(c*x^2+b*x+a)^(3/2)/c-1/2*b/c*(1/4*(2
*c*x+b)*(c*x^2+b*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))))-1/4*a/c*(1
/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)))))-4/7*
a/c*(1/5*x^2*(c*x^2+b*x+a)^(3/2)/c-7/10*b/c*(1/4*x*(c*x^2+b*x+a)^(3/2)/c-5/8*b/c*(1/3*(c*x^2+b*x+a)^(3/2)/c-1/
2*b/c*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))
))-1/4*a/c*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(
1/2))))-2/5*a/c*(1/3*(c*x^2+b*x+a)^(3/2)/c-1/2*b/c*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/c^(3/2
)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))))))+(b*e^4+8*c*d*e^3)*(1/6*x^3*(c*x^2+b*x+a)^(3/2)/c-3/4*b/c*(1/
5*x^2*(c*x^2+b*x+a)^(3/2)/c-7/10*b/c*(1/4*x*(c*x^2+b*x+a)^(3/2)/c-5/8*b/c*(1/3*(c*x^2+b*x+a)^(3/2)/c-1/2*b/c*(
1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))))-1/4*
a/c*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))))
-2/5*a/c*(1/3*(c*x^2+b*x+a)^(3/2)/c-1/2*b/c*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/c^(3/2)*ln((1
/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)))))-1/2*a/c*(1/4*x*(c*x^2+b*x+a)^(3/2)/c-5/8*b/c*(1/3*(c*x^2+b*x+a)^(3/2
)/c-1/2*b/c*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^
(1/2))))-1/4*a/c*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*
x+a)^(1/2)))))+(4*b*d*e^3+12*c*d^2*e^2)*(1/5*x^2*(c*x^2+b*x+a)^(3/2)/c-7/10*b/c*(1/4*x*(c*x^2+b*x+a)^(3/2)/c-5
/8*b/c*(1/3*(c*x^2+b*x+a)^(3/2)/c-1/2*b/c*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2
*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))))-1/4*a/c*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/c^(3/2)*ln
((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))))-2/5*a/c*(1/3*(c*x^2+b*x+a)^(3/2)/c-1/2*b/c*(1/4*(2*c*x+b)*(c*x^2+b
*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)))))+(6*b*d^2*e^2+8*c*d^3*e)*(
1/4*x*(c*x^2+b*x+a)^(3/2)/c-5/8*b/c*(1/3*(c*x^2+b*x+a)^(3/2)/c-1/2*b/c*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c+1/
8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))))-1/4*a/c*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)
/c+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))))+(4*b*d^3*e+2*c*d^4)*(1/3*(c*x^2+b*x+a
)^(3/2)/c-1/2*b/c*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b
*x+a)^(1/2))))+b*d^4*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^
2+b*x+a)^(1/2)))

________________________________________________________________________________________

Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(e*x+d)^4*(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

________________________________________________________________________________________

Fricas [A]
time = 4.48, size = 1397, normalized size = 3.24 \begin {gather*} \left [\frac {105 \, {\left (16 \, {\left (b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} d^{3} e - 24 \, {\left (b^{5} c^{2} - 8 \, a b^{3} c^{3} + 16 \, a^{2} b c^{4}\right )} d^{2} e^{2} + 2 \, {\left (7 \, b^{6} c - 60 \, a b^{4} c^{2} + 144 \, a^{2} b^{2} c^{3} - 64 \, a^{3} c^{4}\right )} d e^{3} - {\left (3 \, b^{7} - 28 \, a b^{5} c + 80 \, a^{2} b^{3} c^{2} - 64 \, a^{3} b c^{3}\right )} e^{4}\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 (8960 \, c^{7} d^{4} x^{2} + 8960 \, b c^{6} d^{4} x + 8960 \, a c^{6} d^{4} + {\left (3840 \, c^{7} x^{6} + 2560 \, b c^{6} x^{5} - 315 \, b^{6} c + 2520 \, a b^{4} c^{2} - 5488 \, a^{2} b^{2} c^{3} + 2048 \, a^{3} c^{4} - 128 \, {\left (b^{2} c^{5} - 6 \, a c^{6}\right )} x^{4} + 16 \, {\left (9 \, b^{3} c^{4} - 44 \, a b c^{5}\right )} x^{3} - 8 \, {\left (21 \, b^{4} c^{3} - 124 \, a b^{2} c^{4} + 128 \, a^{2} c^{5}\right )} x^{2} + 2 \, {\left (105 \, b^{5} c^{2} - 728 \, a b^{3} c^{3} + 1168 \, a^{2} b c^{4}\right )} x\right )} e^{4} + 14 \, {\left (1280 \, c^{7} d x^{5} + 896 \, b c^{6} d x^{4} - 16 \, {\left (3 \, b^{2} c^{5} - 20 \, a c^{6}\right )} d x^{3} + 8 \, {\left (7 \, b^{3} c^{4} - 36 \, a b c^{5}\right )} d x^{2} - 2 \, {\left (35 \, b^{4} c^{3} - 216 \, a b^{2} c^{4} + 240 \, a^{2} c^{5}\right )} d x + {\left (105 \, b^{5} c^{2} - 760 \, a b^{3} c^{3} + 1296 \, a^{2} b c^{4}\right )} d\right )} e^{3} + 168 \, {\left (192 \, c^{7} d^{2} x^{4} + 144 \, b c^{6} d^{2} x^{3} - 8 \, {\left (b^{2} c^{5} - 8 \, a c^{6}\right )} d^{2} x^{2} + 2 \, {\left (5 \, b^{3} c^{4} - 28 \, a b c^{5}\right )} d^{2} x - {\left (15 \, b^{4} c^{3} - 100 \, a b^{2} c^{4} + 128 \, a^{2} c^{5}\right )} d^{2}\right )} e^{2} + 560 \, {\left (48 \, c^{7} d^{3} x^{3} + 40 \, b c^{6} d^{3} x^{2} - 2 \, {\left (b^{2} c^{5} - 12 \, a c^{6}\right )} d^{3} x + {\left (3 \, b^{3} c^{4} - 20 \, a b c^{5}\right )} d^{3}\right )} e\right )} \sqrt {c x^{2} + b x + a}}{53760 \, c^{6}}, \frac {105 \, {\left (16 \, {\left (b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} d^{3} e - 24 \, {\left (b^{5} c^{2} - 8 \, a b^{3} c^{3} + 16 \, a^{2} b c^{4}\right )} d^{2} e^{2} + 2 \, {\left (7 \, b^{6} c - 60 \, a b^{4} c^{2} + 144 \, a^{2} b^{2} c^{3} - 64 \, a^{3} c^{4}\right )} d e^{3} - {\left (3 \, b^{7} - 28 \, a b^{5} c + 80 \, a^{2} b^{3} c^{2} - 64 \, a^{3} b c^{3}\right )} e^{4}\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 (8960 \, c^{7} d^{4} x^{2} + 8960 \, b c^{6} d^{4} x + 8960 \, a c^{6} d^{4} + {\left (3840 \, c^{7} x^{6} + 2560 \, b c^{6} x^{5} - 315 \, b^{6} c + 2520 \, a b^{4} c^{2} - 5488 \, a^{2} b^{2} c^{3} + 2048 \, a^{3} c^{4} - 128 \, {\left (b^{2} c^{5} - 6 \, a c^{6}\right )} x^{4} + 16 \, {\left (9 \, b^{3} c^{4} - 44 \, a b c^{5}\right )} x^{3} - 8 \, {\left (21 \, b^{4} c^{3} - 124 \, a b^{2} c^{4} + 128 \, a^{2} c^{5}\right )} x^{2} + 2 \, {\left (105 \, b^{5} c^{2} - 728 \, a b^{3} c^{3} + 1168 \, a^{2} b c^{4}\right )} x\right )} e^{4} + 14 \, {\left (1280 \, c^{7} d x^{5} + 896 \, b c^{6} d x^{4} - 16 \, {\left (3 \, b^{2} c^{5} - 20 \, a c^{6}\right )} d x^{3} + 8 \, {\left (7 \, b^{3} c^{4} - 36 \, a b c^{5}\right )} d x^{2} - 2 \, {\left (35 \, b^{4} c^{3} - 216 \, a b^{2} c^{4} + 240 \, a^{2} c^{5}\right )} d x + {\left (105 \, b^{5} c^{2} - 760 \, a b^{3} c^{3} + 1296 \, a^{2} b c^{4}\right )} d\right )} e^{3} + 168 \, {\left (192 \, c^{7} d^{2} x^{4} + 144 \, b c^{6} d^{2} x^{3} - 8 \, {\left (b^{2} c^{5} - 8 \, a c^{6}\right )} d^{2} x^{2} + 2 \, {\left (5 \, b^{3} c^{4} - 28 \, a b c^{5}\right )} d^{2} x - {\left (15 \, b^{4} c^{3} - 100 \, a b^{2} c^{4} + 128 \, a^{2} c^{5}\right )} d^{2}\right )} e^{2} + 560 \, {\left (48 \, c^{7} d^{3} x^{3} + 40 \, b c^{6} d^{3} x^{2} - 2 \, {\left (b^{2} c^{5} - 12 \, a c^{6}\right )} d^{3} x + {\left (3 \, b^{3} c^{4} - 20 \, a b c^{5}\right )} d^{3}\right )} e\right )} \sqrt {c x^{2} + b x + a}}{26880 \, c^{6}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(e*x+d)^4*(c*x^2+b*x+a)^(1/2),x, algorithm="fricas")

[Out]

[1/53760*(105*(16*(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*d^3*e - 24*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d^2*e
^2 + 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^3 - (3*b^7 - 28*a*b^5*c + 80*a^2*b^3*c^2 -
64*a^3*b*c^3)*e^4)*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*(8960*c^7*d^4*x^2 + 8960*b*c^6*d^4*x + 8960*a*c^6*d^4 + (3840*c^7*x^6 + 2560*b*c^6*x^5 - 315*b^6*c + 25
20*a*b^4*c^2 - 5488*a^2*b^2*c^3 + 2048*a^3*c^4 - 128*(b^2*c^5 - 6*a*c^6)*x^4 + 16*(9*b^3*c^4 - 44*a*b*c^5)*x^3
- 8*(21*b^4*c^3 - 124*a*b^2*c^4 + 128*a^2*c^5)*x^2 + 2*(105*b^5*c^2 - 728*a*b^3*c^3 + 1168*a^2*b*c^4)*x)*e^4
+ 14*(1280*c^7*d*x^5 + 896*b*c^6*d*x^4 - 16*(3*b^2*c^5 - 20*a*c^6)*d*x^3 + 8*(7*b^3*c^4 - 36*a*b*c^5)*d*x^2 -
2*(35*b^4*c^3 - 216*a*b^2*c^4 + 240*a^2*c^5)*d*x + (105*b^5*c^2 - 760*a*b^3*c^3 + 1296*a^2*b*c^4)*d)*e^3 + 168
*(192*c^7*d^2*x^4 + 144*b*c^6*d^2*x^3 - 8*(b^2*c^5 - 8*a*c^6)*d^2*x^2 + 2*(5*b^3*c^4 - 28*a*b*c^5)*d^2*x - (15
*b^4*c^3 - 100*a*b^2*c^4 + 128*a^2*c^5)*d^2)*e^2 + 560*(48*c^7*d^3*x^3 + 40*b*c^6*d^3*x^2 - 2*(b^2*c^5 - 12*a*
c^6)*d^3*x + (3*b^3*c^4 - 20*a*b*c^5)*d^3)*e)*sqrt(c*x^2 + b*x + a))/c^6, 1/26880*(105*(16*(b^4*c^3 - 8*a*b^2*
c^4 + 16*a^2*c^5)*d^3*e - 24*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d^2*e^2 + 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^3 - (3*b^7 - 28*a*b^5*c + 80*a^2*b^3*c^2 - 64*a^3*b*c^3)*e^4)*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*(8960*c^7*d^4*x^2 + 8960*b*c^6*d^4*x
+ 8960*a*c^6*d^4 + (3840*c^7*x^6 + 2560*b*c^6*x^5 - 315*b^6*c + 2520*a*b^4*c^2 - 5488*a^2*b^2*c^3 + 2048*a^3*
c^4 - 128*(b^2*c^5 - 6*a*c^6)*x^4 + 16*(9*b^3*c^4 - 44*a*b*c^5)*x^3 - 8*(21*b^4*c^3 - 124*a*b^2*c^4 + 128*a^2*
c^5)*x^2 + 2*(105*b^5*c^2 - 728*a*b^3*c^3 + 1168*a^2*b*c^4)*x)*e^4 + 14*(1280*c^7*d*x^5 + 896*b*c^6*d*x^4 - 16
*(3*b^2*c^5 - 20*a*c^6)*d*x^3 + 8*(7*b^3*c^4 - 36*a*b*c^5)*d*x^2 - 2*(35*b^4*c^3 - 216*a*b^2*c^4 + 240*a^2*c^5
)*d*x + (105*b^5*c^2 - 760*a*b^3*c^3 + 1296*a^2*b*c^4)*d)*e^3 + 168*(192*c^7*d^2*x^4 + 144*b*c^6*d^2*x^3 - 8*(
b^2*c^5 - 8*a*c^6)*d^2*x^2 + 2*(5*b^3*c^4 - 28*a*b*c^5)*d^2*x - (15*b^4*c^3 - 100*a*b^2*c^4 + 128*a^2*c^5)*d^2
)*e^2 + 560*(48*c^7*d^3*x^3 + 40*b*c^6*d^3*x^2 - 2*(b^2*c^5 - 12*a*c^6)*d^3*x + (3*b^3*c^4 - 20*a*b*c^5)*d^3)*
e)*sqrt(c*x^2 + b*x + a))/c^6]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (b + 2 c x\right ) \left (d + e x\right )^{4} \sqrt {a + b x + c x^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(e*x+d)**4*(c*x**2+b*x+a)**(1/2),x)

[Out]

Integral((b + 2*c*x)*(d + e*x)**4*sqrt(a + b*x + c*x**2), x)

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Giac [A]
time = 0.98, size = 759, normalized size = 1.76 \begin {gather*} \frac {1}{13440} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, {\left (3 \, c x e^{4} + \frac {2 \, {\left (7 \, c^{7} d e^{3} + b c^{6} e^{4}\right )}}{c^{6}}\right )} x + \frac {252 \, c^{7} d^{2} e^{2} + 98 \, b c^{6} d e^{3} - b^{2} c^{5} e^{4} + 6 \, a c^{6} e^{4}}{c^{6}}\right )} x + \frac {1680 \, c^{7} d^{3} e + 1512 \, b c^{6} d^{2} e^{2} - 42 \, b^{2} c^{5} d e^{3} + 280 \, a c^{6} d e^{3} + 9 \, b^{3} c^{4} e^{4} - 44 \, a b c^{5} e^{4}}{c^{6}}\right )} x + \frac {1120 \, c^{7} d^{4} + 2800 \, b c^{6} d^{3} e - 168 \, b^{2} c^{5} d^{2} e^{2} + 1344 \, a c^{6} d^{2} e^{2} + 98 \, b^{3} c^{4} d e^{3} - 504 \, a b c^{5} d e^{3} - 21 \, b^{4} c^{3} e^{4} + 124 \, a b^{2} c^{4} e^{4} - 128 \, a^{2} c^{5} e^{4}}{c^{6}}\right )} x + \frac {4480 \, b c^{6} d^{4} - 560 \, b^{2} c^{5} d^{3} e + 6720 \, a c^{6} d^{3} e + 840 \, b^{3} c^{4} d^{2} e^{2} - 4704 \, a b c^{5} d^{2} e^{2} - 490 \, b^{4} c^{3} d e^{3} + 3024 \, a b^{2} c^{4} d e^{3} - 3360 \, a^{2} c^{5} d e^{3} + 105 \, b^{5} c^{2} e^{4} - 728 \, a b^{3} c^{3} e^{4} + 1168 \, a^{2} b c^{4} e^{4}}{c^{6}}\right )} x + \frac {8960 \, a c^{6} d^{4} + 1680 \, b^{3} c^{4} d^{3} e - 11200 \, a b c^{5} d^{3} e - 2520 \, b^{4} c^{3} d^{2} e^{2} + 16800 \, a b^{2} c^{4} d^{2} e^{2} - 21504 \, a^{2} c^{5} d^{2} e^{2} + 1470 \, b^{5} c^{2} d e^{3} - 10640 \, a b^{3} c^{3} d e^{3} + 18144 \, a^{2} b c^{4} d e^{3} - 315 \, b^{6} c e^{4} + 2520 \, a b^{4} c^{2} e^{4} - 5488 \, a^{2} b^{2} c^{3} e^{4} + 2048 \, a^{3} c^{4} e^{4}}{c^{6}}\right )} + \frac {{\left (16 \, b^{4} c^{3} d^{3} e - 128 \, a b^{2} c^{4} d^{3} e + 256 \, a^{2} c^{5} d^{3} e - 24 \, b^{5} c^{2} d^{2} e^{2} + 192 \, a b^{3} c^{3} d^{2} e^{2} - 384 \, a^{2} b c^{4} d^{2} e^{2} + 14 \, b^{6} c d e^{3} - 120 \, a b^{4} c^{2} d e^{3} + 288 \, a^{2} b^{2} c^{3} d e^{3} - 128 \, a^{3} c^{4} d e^{3} - 3 \, b^{7} e^{4} + 28 \, a b^{5} c e^{4} - 80 \, a^{2} b^{3} c^{2} e^{4} + 64 \, a^{3} b c^{3} e^{4}\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 {11}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(e*x+d)^4*(c*x^2+b*x+a)^(1/2),x, algorithm="giac")

[Out]

1/13440*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(10*(3*c*x*e^4 + 2*(7*c^7*d*e^3 + b*c^6*e^4)/c^6)*x + (252*c^7*d^2*e
^2 + 98*b*c^6*d*e^3 - b^2*c^5*e^4 + 6*a*c^6*e^4)/c^6)*x + (1680*c^7*d^3*e + 1512*b*c^6*d^2*e^2 - 42*b^2*c^5*d*
e^3 + 280*a*c^6*d*e^3 + 9*b^3*c^4*e^4 - 44*a*b*c^5*e^4)/c^6)*x + (1120*c^7*d^4 + 2800*b*c^6*d^3*e - 168*b^2*c^
5*d^2*e^2 + 1344*a*c^6*d^2*e^2 + 98*b^3*c^4*d*e^3 - 504*a*b*c^5*d*e^3 - 21*b^4*c^3*e^4 + 124*a*b^2*c^4*e^4 - 1
28*a^2*c^5*e^4)/c^6)*x + (4480*b*c^6*d^4 - 560*b^2*c^5*d^3*e + 6720*a*c^6*d^3*e + 840*b^3*c^4*d^2*e^2 - 4704*a
*b*c^5*d^2*e^2 - 490*b^4*c^3*d*e^3 + 3024*a*b^2*c^4*d*e^3 - 3360*a^2*c^5*d*e^3 + 105*b^5*c^2*e^4 - 728*a*b^3*c
^3*e^4 + 1168*a^2*b*c^4*e^4)/c^6)*x + (8960*a*c^6*d^4 + 1680*b^3*c^4*d^3*e - 11200*a*b*c^5*d^3*e - 2520*b^4*c^
3*d^2*e^2 + 16800*a*b^2*c^4*d^2*e^2 - 21504*a^2*c^5*d^2*e^2 + 1470*b^5*c^2*d*e^3 - 10640*a*b^3*c^3*d*e^3 + 181
44*a^2*b*c^4*d*e^3 - 315*b^6*c*e^4 + 2520*a*b^4*c^2*e^4 - 5488*a^2*b^2*c^3*e^4 + 2048*a^3*c^4*e^4)/c^6) + 1/25
6*(16*b^4*c^3*d^3*e - 128*a*b^2*c^4*d^3*e + 256*a^2*c^5*d^3*e - 24*b^5*c^2*d^2*e^2 + 192*a*b^3*c^3*d^2*e^2 - 3
84*a^2*b*c^4*d^2*e^2 + 14*b^6*c*d*e^3 - 120*a*b^4*c^2*d*e^3 + 288*a^2*b^2*c^3*d*e^3 - 128*a^3*c^4*d*e^3 - 3*b^
7*e^4 + 28*a*b^5*c*e^4 - 80*a^2*b^3*c^2*e^4 + 64*a^3*b*c^3*e^4)*log(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))
*sqrt(c) - b))/c^(11/2)

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Mupad [B]
time = 13.55, size = 2500, normalized size = 5.80 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b + 2*c*x)*(d + e*x)^4*(a + b*x + c*x^2)^(1/2),x)

[Out]

(2*e^4*x^4*(a + b*x + c*x^2)^(3/2))/7 - 6*b*d*e^3*((7*b*((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) - (2*a*((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) + (x^2*(a + b*x + c*x^2)^(3/2))/(5*c)) + (d^4*(8*c*(a + c*x^2) - 3*b^2 + 2*b*c*
x)*(a + b*x + c*x^2)^(1/2))/(12*c) + (33*b^7*e^4*log(b + 2*c^(1/2)*(a + b*x + c*x^2)^(1/2) + 2*c*x))/(1024*c^(
11/2)) + b*d^4*(x/2 + b/(4*c))*(a + b*x + c*x^2)^(1/2) + (4*d*e^3*x^3*(a + b*x + c*x^2)^(3/2))/3 - 2*a*d^3*e*(
(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))) + (42*b*d^2*e^2*((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)))/5 + (16*a^3*e^4*(a + b*x + c*x^2)^(1/2))/(105*c^2) - (33*b^6*e^4*(a
+ b*x + c*x^2)^(1/2))/(512*c^5) - 5*b*d^3*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)) + ((8*c*(a + c*x^2) - 3*b^2 + 2*b*c*x)*(a + b*x + c*x^2)^(1/2))/(24*c^2)) + (12*d^2*e^2*x^
2*(a + b*x + c*x^2)^(3/2))/5 - (3*b^2*e^4*((7*b*((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) - (2*a*((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) + (x^2*(a + b*x + c*x^2)^(3/2))/(5*c)))/(4*c) + (d^4*log((b + 2*c*x)/c^(1/2) + 2*(a + b
*x + c*x^2)^(1/2))*(b^3 - 4*a*b*c))/(8*c^(3/2)) + 2*d^3*e*x*(a + b*x + c*x^2)^(3/2) - (24*a*d^2*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)))/5 + 4*a*d*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)) - (8*a*e^4*x^2*(a + b*x + c*x^2)^(3/2))/(3
5*c) - (2*b*e^4*x^3*(a + b*x + c*x^2)^(3/2))/(21*c) - (33*b^3*e^4*x*(a + b*x + c*x^2)^(3/2))/(160*c^3) + (11*b
^5*e^4*x*(a + b*x + c*x^2)^(1/2))/(256*c^4) - (15*b^2*d^2*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))
)/(4*c) + (14*b^2*d*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)))/(5*c) + (b*d^4*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)) - (103*a^2*b^2*e^4*(a + b*x + c*x^2)^(1/2))/(160*c^3) + (16*a^2*e^4*x^2*(a + b*x + c*x^2
)^(1/2))/(105*c) + (33*b^2*e^4*x^2*(a + b*x + c*x^2)^(3/2))/(140*c^2) + (11*b^4*e^4*x^2*(a + b*x + c*x^2)^(1/2
))/(64*c^3) - (5*a^3*b*e^4*log(b + 2*c^(1/2)*(a + b*x + c*x^2)^(1/2) + 2*c*x))/(16*c^(5/2)) - (63*a*b^5*e^4*lo
g(b + 2*c^(1/2)*(a + b*x + c*x^2)^(1/2) + 2*c*x))/(256*c^(9/2)) + (a*b*e^4*((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)))/(2*c) + (35*a^2*b^3*
e^4*log(b + 2*c^(1/2)*(a + b*x + c*x^2)^(1/2) + 2*c*x))/(64*c^(7/2)) + (13*a*b^4*e^4*(a + b*x + c*x^2)^(1/2))/
(32*c^4) + (111*a*b*e^4*x*(a + b*x + c*x^2)^(3/2))/(280*c^2) - (269*a^2*b*e^4*x*(a + b*x + c*x^2)^(1/2))/(1680
*c^2) - (3*a*b^3*e^4*x*(a + b*x + c*x^2)^(1/2))/(160*c^3) + (3*b*d^2*e^2*x*(a + b*x + c*x^2)^(3/2))/(2*c) + (4
*b*d*e^3*x^2*(a + b*x + c*x^2)^(3/2))/(5*c) + (b*d^3*e*log((b + 2*c*x)/c^(1/2) + 2*(a + b*x + c*x^2)^(1/2))*(b
^3 - 4*a*b*c))/(4*c^(5/2)) - (8*a*b*d*e^3*((log...

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