3.14.54 \(\int (b+2 c x) (d+e x)^4 \sqrt {a+b x+c x^2} \, dx\)

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

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Rubi [A]  time = 0.79, 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 = {832, 779, 612, 621, 206} \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 ) (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 {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 {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 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 612

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 621

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 779

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 832

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 ) \operatorname {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 = 0.85, size = 549, normalized size = 1.27 \begin {gather*} \frac {\sqrt {a+x (b+c x)} \left (2048 a^3 c^3 e^4-16 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 )-315 b^6 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 )+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 )-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 )+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 )+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 )\right )}{13440 c^5}+\frac {e \left (b^2-4 a c\right )^2 (b e-2 c d) \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+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))*ArcTanh[
(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/(256*c^(11/2))

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IntegrateAlgebraic [A]  time = 2.05, size = 827, normalized size = 1.92 \begin {gather*} \frac {\sqrt {c x^2+b x+a} \left (-315 e^4 b^6+1470 c d e^3 b^5+210 c e^4 x b^5+2520 a c e^4 b^4-2520 c^2 d^2 e^2 b^4-168 c^2 e^4 x^2 b^4-980 c^2 d e^3 x b^4-10640 a c^2 d e^3 b^3+144 c^3 e^4 x^3 b^3+784 c^3 d e^3 x^2 b^3+1680 c^3 d^3 e b^3-1456 a c^2 e^4 x b^3+1680 c^3 d^2 e^2 x b^3-5488 a^2 c^2 e^4 b^2-128 c^4 e^4 x^4 b^2-672 c^4 d e^3 x^3 b^2+16800 a c^3 d^2 e^2 b^2+992 a c^3 e^4 x^2 b^2-1344 c^4 d^2 e^2 x^2 b^2+6048 a c^3 d e^3 x b^2-1120 c^4 d^3 e x b^2+2560 c^5 e^4 x^5 b+12544 c^5 d e^3 x^4 b+18144 a^2 c^3 d e^3 b-704 a c^4 e^4 x^3 b+24192 c^5 d^2 e^2 x^3 b-4032 a c^4 d e^3 x^2 b+22400 c^5 d^3 e x^2 b-11200 a c^4 d^3 e b+8960 c^5 d^4 x b+2336 a^2 c^3 e^4 x b-9408 a c^4 d^2 e^2 x b+3840 c^6 e^4 x^6+17920 c^6 d e^3 x^5+8960 a c^5 d^4+2048 a^3 c^3 e^4+768 a c^5 e^4 x^4+32256 c^6 d^2 e^2 x^4+4480 a c^5 d e^3 x^3+26880 c^6 d^3 e x^3-21504 a^2 c^4 d^2 e^2+8960 c^6 d^4 x^2-1024 a^2 c^4 e^4 x^2+10752 a c^5 d^2 e^2 x^2-6720 a^2 c^4 d e^3 x+13440 a c^5 d^3 e x\right )}{13440 c^5}+\frac {\left (-3 e^4 b^7+14 c d e^3 b^6+28 a c e^4 b^5-24 c^2 d^2 e^2 b^5-120 a c^2 d e^3 b^4+16 c^3 d^3 e b^4-80 a^2 c^2 e^4 b^3+192 a c^3 d^2 e^2 b^3+288 a^2 c^3 d e^3 b^2-128 a c^4 d^3 e b^2+64 a^3 c^3 e^4 b-384 a^2 c^4 d^2 e^2 b-128 a^3 c^4 d e^3+256 a^2 c^5 d^3 e\right ) \log \left (b+2 c x-2 \sqrt {c} \sqrt {c x^2+b x+a}\right )}{256 c^{11/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a + b*x + c*x^2]*(8960*a*c^5*d^4 + 1680*b^3*c^3*d^3*e - 11200*a*b*c^4*d^3*e - 2520*b^4*c^2*d^2*e^2 + 168
00*a*b^2*c^3*d^2*e^2 - 21504*a^2*c^4*d^2*e^2 + 1470*b^5*c*d*e^3 - 10640*a*b^3*c^2*d*e^3 + 18144*a^2*b*c^3*d*e^
3 - 315*b^6*e^4 + 2520*a*b^4*c*e^4 - 5488*a^2*b^2*c^2*e^4 + 2048*a^3*c^3*e^4 + 8960*b*c^5*d^4*x - 1120*b^2*c^4
*d^3*e*x + 13440*a*c^5*d^3*e*x + 1680*b^3*c^3*d^2*e^2*x - 9408*a*b*c^4*d^2*e^2*x - 980*b^4*c^2*d*e^3*x + 6048*
a*b^2*c^3*d*e^3*x - 6720*a^2*c^4*d*e^3*x + 210*b^5*c*e^4*x - 1456*a*b^3*c^2*e^4*x + 2336*a^2*b*c^3*e^4*x + 896
0*c^6*d^4*x^2 + 22400*b*c^5*d^3*e*x^2 - 1344*b^2*c^4*d^2*e^2*x^2 + 10752*a*c^5*d^2*e^2*x^2 + 784*b^3*c^3*d*e^3
*x^2 - 4032*a*b*c^4*d*e^3*x^2 - 168*b^4*c^2*e^4*x^2 + 992*a*b^2*c^3*e^4*x^2 - 1024*a^2*c^4*e^4*x^2 + 26880*c^6
*d^3*e*x^3 + 24192*b*c^5*d^2*e^2*x^3 - 672*b^2*c^4*d*e^3*x^3 + 4480*a*c^5*d*e^3*x^3 + 144*b^3*c^3*e^4*x^3 - 70
4*a*b*c^4*e^4*x^3 + 32256*c^6*d^2*e^2*x^4 + 12544*b*c^5*d*e^3*x^4 - 128*b^2*c^4*e^4*x^4 + 768*a*c^5*e^4*x^4 +
17920*c^6*d*e^3*x^5 + 2560*b*c^5*e^4*x^5 + 3840*c^6*e^4*x^6))/(13440*c^5) + ((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)*Log[b + 2*c*x - 2*Sqrt[c]*Sqrt[a + b*x + c*x^2]])/(256*c^(11/2))

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fricas [A]  time = 0.59, size = 1445, normalized size = 3.35

result too large to display

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*(3840*c^7*e^4*x^6 + 8960*a*c^6*d^4 + 2560*(7*c^7*d*e^3 + b*c^6*e^4)*x^5 + 560*(3*b^3*c^4 - 20*a*b*c^5)*
d^3*e - 168*(15*b^4*c^3 - 100*a*b^2*c^4 + 128*a^2*c^5)*d^2*e^2 + 14*(105*b^5*c^2 - 760*a*b^3*c^3 + 1296*a^2*b*
c^4)*d*e^3 - (315*b^6*c - 2520*a*b^4*c^2 + 5488*a^2*b^2*c^3 - 2048*a^3*c^4)*e^4 + 128*(252*c^7*d^2*e^2 + 98*b*
c^6*d*e^3 - (b^2*c^5 - 6*a*c^6)*e^4)*x^4 + 16*(1680*c^7*d^3*e + 1512*b*c^6*d^2*e^2 - 14*(3*b^2*c^5 - 20*a*c^6)
*d*e^3 + (9*b^3*c^4 - 44*a*b*c^5)*e^4)*x^3 + 8*(1120*c^7*d^4 + 2800*b*c^6*d^3*e - 168*(b^2*c^5 - 8*a*c^6)*d^2*
e^2 + 14*(7*b^3*c^4 - 36*a*b*c^5)*d*e^3 - (21*b^4*c^3 - 124*a*b^2*c^4 + 128*a^2*c^5)*e^4)*x^2 + 2*(4480*b*c^6*
d^4 - 560*(b^2*c^5 - 12*a*c^6)*d^3*e + 168*(5*b^3*c^4 - 28*a*b*c^5)*d^2*e^2 - 14*(35*b^4*c^3 - 216*a*b^2*c^4 +
240*a^2*c^5)*d*e^3 + (105*b^5*c^2 - 728*a*b^3*c^3 + 1168*a^2*b*c^4)*e^4)*x)*sqrt(c*x^2 + b*x + a))/c^6, 1/268
80*(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*(3840
*c^7*e^4*x^6 + 8960*a*c^6*d^4 + 2560*(7*c^7*d*e^3 + b*c^6*e^4)*x^5 + 560*(3*b^3*c^4 - 20*a*b*c^5)*d^3*e - 168*
(15*b^4*c^3 - 100*a*b^2*c^4 + 128*a^2*c^5)*d^2*e^2 + 14*(105*b^5*c^2 - 760*a*b^3*c^3 + 1296*a^2*b*c^4)*d*e^3 -
(315*b^6*c - 2520*a*b^4*c^2 + 5488*a^2*b^2*c^3 - 2048*a^3*c^4)*e^4 + 128*(252*c^7*d^2*e^2 + 98*b*c^6*d*e^3 -
(b^2*c^5 - 6*a*c^6)*e^4)*x^4 + 16*(1680*c^7*d^3*e + 1512*b*c^6*d^2*e^2 - 14*(3*b^2*c^5 - 20*a*c^6)*d*e^3 + (9*
b^3*c^4 - 44*a*b*c^5)*e^4)*x^3 + 8*(1120*c^7*d^4 + 2800*b*c^6*d^3*e - 168*(b^2*c^5 - 8*a*c^6)*d^2*e^2 + 14*(7*
b^3*c^4 - 36*a*b*c^5)*d*e^3 - (21*b^4*c^3 - 124*a*b^2*c^4 + 128*a^2*c^5)*e^4)*x^2 + 2*(4480*b*c^6*d^4 - 560*(b
^2*c^5 - 12*a*c^6)*d^3*e + 168*(5*b^3*c^4 - 28*a*b*c^5)*d^2*e^2 - 14*(35*b^4*c^3 - 216*a*b^2*c^4 + 240*a^2*c^5
)*d*e^3 + (105*b^5*c^2 - 728*a*b^3*c^3 + 1168*a^2*b*c^4)*e^4)*x)*sqrt(c*x^2 + b*x + a))/c^6]

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giac [A]  time = 0.25, 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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maple [B]  time = 0.07, size = 1537, normalized size = 3.57

result too large to display

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)

[Out]

-1/16*b^4/c^(5/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*d^3*e-a^2/c^(1/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+
b*x+a)^(1/2))*d^3*e+12/5*x^2*(c*x^2+b*x+a)^(3/2)*d^2*e^2+3/256/c^(11/2)*e^4*b^7*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+
b*x+a)^(1/2))+4/3*x^3*(c*x^2+b*x+a)^(3/2)*d*e^3+2*x*(c*x^2+b*x+a)^(3/2)*d^3*e+1/16/c^4*e^4*b^4*(c*x^2+b*x+a)^(
3/2)-3/128/c^5*e^4*b^6*(c*x^2+b*x+a)^(1/2)+16/105/c^2*e^4*a^2*(c*x^2+b*x+a)^(3/2)+19/70/c^2*e^4*b*a*x*(c*x^2+b
*x+a)^(3/2)-1/4/c^2*e^4*b*a^2*x*(c*x^2+b*x+a)^(1/2)+1/4/c^3*e^4*b^3*a*x*(c*x^2+b*x+a)^(1/2)+1/2*b^2/c^(3/2)*ln
((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a*d^3*e+1/4*b^2/c*x*(c*x^2+b*x+a)^(1/2)*d^3*e+3/32*b^5/c^(7/2)*ln((c
*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*d^2*e^2-7/128*b^6/c^(9/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*d
*e^3+1/2*a^3/c^(3/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*d*e^3-1/4/c^(5/2)*e^4*b*a^3*ln((c*x+1/2*b)/c^
(1/2)+(c*x^2+b*x+a)^(1/2))+5/16/c^(7/2)*e^4*b^3*a^2*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-7/64/c^(9/2)*e
^4*b^5*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a-17/60/c^3*e^4*b^2*a*(c*x^2+b*x+a)^(3/2)-a*x*(c*x^2+b*x+a)
^(1/2)*d^3*e+1/2*b^2/c^2*(c*x^2+b*x+a)^(3/2)*d^2*e^2+3/2*b/c*a*x*(c*x^2+b*x+a)^(1/2)*d^2*e^2-b^2/c^2*a*x*(c*x^
2+b*x+a)^(1/2)*d*e^3+2/7*e^4*x^4*(c*x^2+b*x+a)^(3/2)+2/3*(c*x^2+b*x+a)^(3/2)*d^4-1/2*a/c*(c*x^2+b*x+a)^(1/2)*b
*d^3*e+3/4*b^2/c^2*a*(c*x^2+b*x+a)^(1/2)*d^2*e^2-3/5*b/c*x*(c*x^2+b*x+a)^(3/2)*d^2*e^2-3/8*b^3/c^2*x*(c*x^2+b*
x+a)^(1/2)*d^2*e^2+3/2*b/c^(3/2)*a^2*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*d^2*e^2-3/4*b^3/c^(5/2)*ln((c
*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a*d^2*e^2-9/8*b^2/c^(5/2)*a^2*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2
))*d*e^3+15/32*b^4/c^(7/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a*d*e^3-2/5*b/c*x^2*(c*x^2+b*x+a)^(3/2)
*d*e^3+7/20*b^2/c^2*x*(c*x^2+b*x+a)^(3/2)*d*e^3+7/32*b^4/c^3*x*(c*x^2+b*x+a)^(1/2)*d*e^3+11/10*b/c^2*a*(c*x^2+
b*x+a)^(3/2)*d*e^3+1/2*a^2/c*x*(c*x^2+b*x+a)^(1/2)*d*e^3-a/c*x*(c*x^2+b*x+a)^(3/2)*d*e^3-1/2*b^3/c^3*a*(c*x^2+
b*x+a)^(1/2)*d*e^3+1/4*a^2/c^2*(c*x^2+b*x+a)^(1/2)*b*d*e^3+1/8*b^3/c^2*(c*x^2+b*x+a)^(1/2)*d^3*e+1/8/c^4*e^4*b
^4*a*(c*x^2+b*x+a)^(1/2)-3/16*b^4/c^3*(c*x^2+b*x+a)^(1/2)*d^2*e^2-8/5*a/c*(c*x^2+b*x+a)^(3/2)*d^2*e^2-1/3*b/c*
(c*x^2+b*x+a)^(3/2)*d^3*e-8/35/c*e^4*a*x^2*(c*x^2+b*x+a)^(3/2)-2/21/c*e^4*b*x^3*(c*x^2+b*x+a)^(3/2)+3/35/c^2*e
^4*b^2*x^2*(c*x^2+b*x+a)^(3/2)-3/40/c^3*e^4*b^3*x*(c*x^2+b*x+a)^(3/2)-1/8/c^3*e^4*b^2*a^2*(c*x^2+b*x+a)^(1/2)-
3/64/c^4*e^4*b^5*x*(c*x^2+b*x+a)^(1/2)-7/24*b^3/c^3*(c*x^2+b*x+a)^(3/2)*d*e^3+7/64*b^5/c^4*(c*x^2+b*x+a)^(1/2)
*d*e^3

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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 details)Is 4*a*c-b^2 positive, negative or zero?

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mupad [B]  time = 13.55, size = 2712, normalized size = 6.29

result too large to display

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((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) - (39*a*b^2*
e^4*x^2*(a + b*x + c*x^2)^(1/2))/(80*c^2) + (b*d^3*e*(8*c*(a + c*x^2) - 3*b^2 + 2*b*c*x)*(a + b*x + c*x^2)^(1/
2))/(6*c^2) - (3*a*b*d^2*e^2*((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))))/(2*c)

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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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