3.1547 \(\int (b+2 c x) (d+e x)^3 \sqrt{a+b x+c x^2} \, dx\)

Optimal. Leaf size=312 \[ \frac{\left (a+b x+c x^2\right )^{3/2} \left (6 c e x \left (-4 c e (5 a e+2 b d)+7 b^2 e^2+8 c^2 d^2\right )-24 c^2 d e (16 a e+3 b d)+12 b c e^2 (11 a e+10 b d)-35 b^3 e^3+64 c^3 d^3\right )}{480 c^3}+\frac{e \left (b^2-4 a c\right ) (b+2 c x) \sqrt{a+b x+c x^2} \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right )}{256 c^4}-\frac{e \left (b^2-4 a c\right )^2 \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{512 c^{9/2}}+\frac{1}{3} (d+e x)^3 \left (a+b x+c x^2\right )^{3/2}+\frac{(d+e x)^2 \left (a+b x+c x^2\right )^{3/2} (2 c d-b e)}{10 c} \]

[Out]

((b^2 - 4*a*c)*e*(24*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(6*b*d + a*e))*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(256*c^4) +
((2*c*d - b*e)*(d + e*x)^2*(a + b*x + c*x^2)^(3/2))/(10*c) + ((d + e*x)^3*(a + b*x + c*x^2)^(3/2))/3 + ((64*c
^3*d^3 - 35*b^3*e^3 + 12*b*c*e^2*(10*b*d + 11*a*e) - 24*c^2*d*e*(3*b*d + 16*a*e) + 6*c*e*(8*c^2*d^2 + 7*b^2*e^
2 - 4*c*e*(2*b*d + 5*a*e))*x)*(a + b*x + c*x^2)^(3/2))/(480*c^3) - ((b^2 - 4*a*c)^2*e*(24*c^2*d^2 + 7*b^2*e^2
- 4*c*e*(6*b*d + a*e))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(512*c^(9/2))

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Rubi [A]  time = 0.41921, antiderivative size = 312, normalized size of antiderivative = 1., number of steps used = 6, 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} \[ \frac{\left (a+b x+c x^2\right )^{3/2} \left (6 c e x \left (-4 c e (5 a e+2 b d)+7 b^2 e^2+8 c^2 d^2\right )-24 c^2 d e (16 a e+3 b d)+12 b c e^2 (11 a e+10 b d)-35 b^3 e^3+64 c^3 d^3\right )}{480 c^3}+\frac{e \left (b^2-4 a c\right ) (b+2 c x) \sqrt{a+b x+c x^2} \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right )}{256 c^4}-\frac{e \left (b^2-4 a c\right )^2 \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{512 c^{9/2}}+\frac{1}{3} (d+e x)^3 \left (a+b x+c x^2\right )^{3/2}+\frac{(d+e x)^2 \left (a+b x+c x^2\right )^{3/2} (2 c d-b e)}{10 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((b^2 - 4*a*c)*e*(24*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(6*b*d + a*e))*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(256*c^4) +
((2*c*d - b*e)*(d + e*x)^2*(a + b*x + c*x^2)^(3/2))/(10*c) + ((d + e*x)^3*(a + b*x + c*x^2)^(3/2))/3 + ((64*c
^3*d^3 - 35*b^3*e^3 + 12*b*c*e^2*(10*b*d + 11*a*e) - 24*c^2*d*e*(3*b*d + 16*a*e) + 6*c*e*(8*c^2*d^2 + 7*b^2*e^
2 - 4*c*e*(2*b*d + 5*a*e))*x)*(a + b*x + c*x^2)^(3/2))/(480*c^3) - ((b^2 - 4*a*c)^2*e*(24*c^2*d^2 + 7*b^2*e^2
- 4*c*e*(6*b*d + a*e))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(512*c^(9/2))

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])

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 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 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])

Rubi steps

\begin{align*} \int (b+2 c x) (d+e x)^3 \sqrt{a+b x+c x^2} \, dx &=\frac{1}{3} (d+e x)^3 \left (a+b x+c x^2\right )^{3/2}+\frac{\int (d+e x)^2 (3 c (b d-2 a e)+3 c (2 c d-b e) x) \sqrt{a+b x+c x^2} \, dx}{6 c}\\ &=\frac{(2 c d-b e) (d+e x)^2 \left (a+b x+c x^2\right )^{3/2}}{10 c}+\frac{1}{3} (d+e x)^3 \left (a+b x+c x^2\right )^{3/2}+\frac{\int (d+e x) \left (\frac{3}{2} c \left (3 b^2 d e-28 a c d e+4 b \left (c d^2+a e^2\right )\right )+\frac{3}{2} c \left (8 c^2 d^2+7 b^2 e^2-4 c e (2 b d+5 a e)\right ) x\right ) \sqrt{a+b x+c x^2} \, dx}{30 c^2}\\ &=\frac{(2 c d-b e) (d+e x)^2 \left (a+b x+c x^2\right )^{3/2}}{10 c}+\frac{1}{3} (d+e x)^3 \left (a+b x+c x^2\right )^{3/2}+\frac{\left (64 c^3 d^3-35 b^3 e^3+12 b c e^2 (10 b d+11 a e)-24 c^2 d e (3 b d+16 a e)+6 c e \left (8 c^2 d^2+7 b^2 e^2-4 c e (2 b d+5 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{480 c^3}+\frac{\left (\left (b^2-4 a c\right ) e \left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right )\right ) \int \sqrt{a+b x+c x^2} \, dx}{64 c^3}\\ &=\frac{\left (b^2-4 a c\right ) e \left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{256 c^4}+\frac{(2 c d-b e) (d+e x)^2 \left (a+b x+c x^2\right )^{3/2}}{10 c}+\frac{1}{3} (d+e x)^3 \left (a+b x+c x^2\right )^{3/2}+\frac{\left (64 c^3 d^3-35 b^3 e^3+12 b c e^2 (10 b d+11 a e)-24 c^2 d e (3 b d+16 a e)+6 c e \left (8 c^2 d^2+7 b^2 e^2-4 c e (2 b d+5 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{480 c^3}-\frac{\left (\left (b^2-4 a c\right )^2 e \left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right )\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{512 c^4}\\ &=\frac{\left (b^2-4 a c\right ) e \left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{256 c^4}+\frac{(2 c d-b e) (d+e x)^2 \left (a+b x+c x^2\right )^{3/2}}{10 c}+\frac{1}{3} (d+e x)^3 \left (a+b x+c x^2\right )^{3/2}+\frac{\left (64 c^3 d^3-35 b^3 e^3+12 b c e^2 (10 b d+11 a e)-24 c^2 d e (3 b d+16 a e)+6 c e \left (8 c^2 d^2+7 b^2 e^2-4 c e (2 b d+5 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{480 c^3}-\frac{\left (\left (b^2-4 a c\right )^2 e \left (24 c^2 d^2+7 b^2 e^2-4 c e (6 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 )}{256 c^4}\\ &=\frac{\left (b^2-4 a c\right ) e \left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{256 c^4}+\frac{(2 c d-b e) (d+e x)^2 \left (a+b x+c x^2\right )^{3/2}}{10 c}+\frac{1}{3} (d+e x)^3 \left (a+b x+c x^2\right )^{3/2}+\frac{\left (64 c^3 d^3-35 b^3 e^3+12 b c e^2 (10 b d+11 a e)-24 c^2 d e (3 b d+16 a e)+6 c e \left (8 c^2 d^2+7 b^2 e^2-4 c e (2 b d+5 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{480 c^3}-\frac{\left (b^2-4 a c\right )^2 e \left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{512 c^{9/2}}\\ \end{align*}

Mathematica [A]  time = 0.534862, size = 270, normalized size = 0.87 \[ \frac{1}{6} \left (\frac{(a+x (b+c x))^{3/2} \left (-24 c^2 e (a e (16 d+5 e x)+b d (3 d+2 e x))+6 b c e^2 (22 a e+20 b d+7 b e x)-35 b^3 e^3+16 c^3 d^2 (4 d+3 e x)\right )}{80 c^3}-\frac{3 e \left (b^2-4 a c\right ) \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right ) \left (\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )-2 \sqrt{c} (b+2 c x) \sqrt{a+x (b+c x)}\right )}{256 c^{9/2}}+2 (d+e x)^3 (a+x (b+c x))^{3/2}+\frac{3 (d+e x)^2 (a+x (b+c x))^{3/2} (2 c d-b e)}{5 c}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

((3*(2*c*d - b*e)*(d + e*x)^2*(a + x*(b + c*x))^(3/2))/(5*c) + 2*(d + e*x)^3*(a + x*(b + c*x))^(3/2) + ((a + x
*(b + c*x))^(3/2)*(-35*b^3*e^3 + 16*c^3*d^2*(4*d + 3*e*x) + 6*b*c*e^2*(20*b*d + 22*a*e + 7*b*e*x) - 24*c^2*e*(
b*d*(3*d + 2*e*x) + a*e*(16*d + 5*e*x))))/(80*c^3) - (3*(b^2 - 4*a*c)*e*(24*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(6*b*d
+ a*e))*(-2*Sqrt[c]*(b + 2*c*x)*Sqrt[a + x*(b + c*x)] + (b^2 - 4*a*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a +
x*(b + c*x)])]))/(256*c^(9/2)))/6

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Maple [B]  time = 0.013, size = 992, normalized size = 3.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(c*x^2+b*x+a)^(3/2)*d^3+3/4*b/c*a*(c*x^2+b*x+a)^(1/2)*x*d*e^2+1/16/c^2*e^3*a^2*(c*x^2+b*x+a)^(1/2)*b+1/8/c
*e^3*a^2*(c*x^2+b*x+a)^(1/2)*x+1/8/c^(3/2)*e^3*a^3*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-7/512/c^(9/2)*e
^3*b^6*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+3/2*x*(c*x^2+b*x+a)^(3/2)*d^2*e+6/5*x^2*(c*x^2+b*x+a)^(3/2)
*d*e^2+7/256/c^4*e^3*b^5*(c*x^2+b*x+a)^(1/2)-7/96/c^3*e^3*b^3*(c*x^2+b*x+a)^(3/2)-3/8*a/c*(c*x^2+b*x+a)^(1/2)*
b*d^2*e+3/16*b^2/c*(c*x^2+b*x+a)^(1/2)*x*d^2*e+3/8*b^2/c^2*a*(c*x^2+b*x+a)^(1/2)*d*e^2-3/16*b^3/c^2*(c*x^2+b*x
+a)^(1/2)*x*d*e^2-3/10*b/c*x*(c*x^2+b*x+a)^(3/2)*d*e^2-1/4/c^2*e^3*b^2*a*(c*x^2+b*x+a)^(1/2)*x+3/8*b^2/c^(3/2)
*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a*d^2*e-3/8*b^3/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2
))*a*d*e^2+3/4*b/c^(3/2)*a^2*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*d*e^2+1/3*e^3*x^3*(c*x^2+b*x+a)^(3/2)
-3/4*a*(c*x^2+b*x+a)^(1/2)*x*d^2*e-4/5*a/c*(c*x^2+b*x+a)^(3/2)*d*e^2+1/4*b^2/c^2*(c*x^2+b*x+a)^(3/2)*d*e^2-3/3
2*b^4/c^3*(c*x^2+b*x+a)^(1/2)*d*e^2+3/32*b^3/c^2*(c*x^2+b*x+a)^(1/2)*d^2*e-1/4*b/c*(c*x^2+b*x+a)^(3/2)*d^2*e-9
/32/c^(5/2)*e^3*b^2*a^2*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+15/128/c^(7/2)*e^3*b^4*ln((1/2*b+c*x)/c^(1
/2)+(c*x^2+b*x+a)^(1/2))*a+7/80/c^2*e^3*b^2*x*(c*x^2+b*x+a)^(3/2)-1/10/c*e^3*b*x^2*(c*x^2+b*x+a)^(3/2)-3/64*b^
4/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*d^2*e-3/4*a^2/c^(1/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+
a)^(1/2))*d^2*e+3/64*b^5/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*d*e^2-1/4/c*e^3*a*x*(c*x^2+b*x+a)
^(3/2)+7/128/c^3*e^3*b^4*(c*x^2+b*x+a)^(1/2)*x-1/8/c^3*e^3*b^3*a*(c*x^2+b*x+a)^(1/2)+11/40/c^2*e^3*b*a*(c*x^2+
b*x+a)^(3/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.21878, size = 2222, normalized size = 7.12 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/15360*(15*(24*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^2*e - 24*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d*e^2 +
(7*b^6 - 60*a*b^4*c + 144*a^2*b^2*c^2 - 64*a^3*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*(1280*c^6*e^3*x^5 + 2560*a*c^5*d^3 + 128*(36*c^6*d*e^2 + 7*b*c^5
*e^3)*x^4 + 120*(3*b^3*c^3 - 20*a*b*c^4)*d^2*e - 24*(15*b^4*c^2 - 100*a*b^2*c^3 + 128*a^2*c^4)*d*e^2 + (105*b^
5*c - 760*a*b^3*c^2 + 1296*a^2*b*c^3)*e^3 + 16*(360*c^6*d^2*e + 216*b*c^5*d*e^2 - (3*b^2*c^4 - 20*a*c^5)*e^3)*
x^3 + 8*(320*c^6*d^3 + 600*b*c^5*d^2*e - 24*(b^2*c^4 - 8*a*c^5)*d*e^2 + (7*b^3*c^3 - 36*a*b*c^4)*e^3)*x^2 + 2*
(1280*b*c^5*d^3 - 120*(b^2*c^4 - 12*a*c^5)*d^2*e + 24*(5*b^3*c^3 - 28*a*b*c^4)*d*e^2 - (35*b^4*c^2 - 216*a*b^2
*c^3 + 240*a^2*c^4)*e^3)*x)*sqrt(c*x^2 + b*x + a))/c^5, 1/7680*(15*(24*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^
2*e - 24*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d*e^2 + (7*b^6 - 60*a*b^4*c + 144*a^2*b^2*c^2 - 64*a^3*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*(1280*c^6*e^3*x^5
+ 2560*a*c^5*d^3 + 128*(36*c^6*d*e^2 + 7*b*c^5*e^3)*x^4 + 120*(3*b^3*c^3 - 20*a*b*c^4)*d^2*e - 24*(15*b^4*c^2
- 100*a*b^2*c^3 + 128*a^2*c^4)*d*e^2 + (105*b^5*c - 760*a*b^3*c^2 + 1296*a^2*b*c^3)*e^3 + 16*(360*c^6*d^2*e +
216*b*c^5*d*e^2 - (3*b^2*c^4 - 20*a*c^5)*e^3)*x^3 + 8*(320*c^6*d^3 + 600*b*c^5*d^2*e - 24*(b^2*c^4 - 8*a*c^5)
*d*e^2 + (7*b^3*c^3 - 36*a*b*c^4)*e^3)*x^2 + 2*(1280*b*c^5*d^3 - 120*(b^2*c^4 - 12*a*c^5)*d^2*e + 24*(5*b^3*c^
3 - 28*a*b*c^4)*d*e^2 - (35*b^4*c^2 - 216*a*b^2*c^3 + 240*a^2*c^4)*e^3)*x)*sqrt(c*x^2 + b*x + a))/c^5]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b + 2 c x\right ) \left (d + e x\right )^{3} \sqrt{a + b x + c x^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((b + 2*c*x)*(d + e*x)**3*sqrt(a + b*x + c*x**2), x)

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Giac [A]  time = 1.28384, size = 683, normalized size = 2.19 \begin{align*} \frac{1}{3840} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (10 \, c x e^{3} + \frac{36 \, c^{6} d e^{2} + 7 \, b c^{5} e^{3}}{c^{5}}\right )} x + \frac{360 \, c^{6} d^{2} e + 216 \, b c^{5} d e^{2} - 3 \, b^{2} c^{4} e^{3} + 20 \, a c^{5} e^{3}}{c^{5}}\right )} x + \frac{320 \, c^{6} d^{3} + 600 \, b c^{5} d^{2} e - 24 \, b^{2} c^{4} d e^{2} + 192 \, a c^{5} d e^{2} + 7 \, b^{3} c^{3} e^{3} - 36 \, a b c^{4} e^{3}}{c^{5}}\right )} x + \frac{1280 \, b c^{5} d^{3} - 120 \, b^{2} c^{4} d^{2} e + 1440 \, a c^{5} d^{2} e + 120 \, b^{3} c^{3} d e^{2} - 672 \, a b c^{4} d e^{2} - 35 \, b^{4} c^{2} e^{3} + 216 \, a b^{2} c^{3} e^{3} - 240 \, a^{2} c^{4} e^{3}}{c^{5}}\right )} x + \frac{2560 \, a c^{5} d^{3} + 360 \, b^{3} c^{3} d^{2} e - 2400 \, a b c^{4} d^{2} e - 360 \, b^{4} c^{2} d e^{2} + 2400 \, a b^{2} c^{3} d e^{2} - 3072 \, a^{2} c^{4} d e^{2} + 105 \, b^{5} c e^{3} - 760 \, a b^{3} c^{2} e^{3} + 1296 \, a^{2} b c^{3} e^{3}}{c^{5}}\right )} + \frac{{\left (24 \, b^{4} c^{2} d^{2} e - 192 \, a b^{2} c^{3} d^{2} e + 384 \, a^{2} c^{4} d^{2} e - 24 \, b^{5} c d e^{2} + 192 \, a b^{3} c^{2} d e^{2} - 384 \, a^{2} b c^{3} d e^{2} + 7 \, b^{6} e^{3} - 60 \, a b^{4} c e^{3} + 144 \, a^{2} b^{2} c^{2} e^{3} - 64 \, a^{3} 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 )}{512 \, c^{\frac{9}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3840*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(10*c*x*e^3 + (36*c^6*d*e^2 + 7*b*c^5*e^3)/c^5)*x + (360*c^6*d^2*e +
216*b*c^5*d*e^2 - 3*b^2*c^4*e^3 + 20*a*c^5*e^3)/c^5)*x + (320*c^6*d^3 + 600*b*c^5*d^2*e - 24*b^2*c^4*d*e^2 + 1
92*a*c^5*d*e^2 + 7*b^3*c^3*e^3 - 36*a*b*c^4*e^3)/c^5)*x + (1280*b*c^5*d^3 - 120*b^2*c^4*d^2*e + 1440*a*c^5*d^2
*e + 120*b^3*c^3*d*e^2 - 672*a*b*c^4*d*e^2 - 35*b^4*c^2*e^3 + 216*a*b^2*c^3*e^3 - 240*a^2*c^4*e^3)/c^5)*x + (2
560*a*c^5*d^3 + 360*b^3*c^3*d^2*e - 2400*a*b*c^4*d^2*e - 360*b^4*c^2*d*e^2 + 2400*a*b^2*c^3*d*e^2 - 3072*a^2*c
^4*d*e^2 + 105*b^5*c*e^3 - 760*a*b^3*c^2*e^3 + 1296*a^2*b*c^3*e^3)/c^5) + 1/512*(24*b^4*c^2*d^2*e - 192*a*b^2*
c^3*d^2*e + 384*a^2*c^4*d^2*e - 24*b^5*c*d*e^2 + 192*a*b^3*c^2*d*e^2 - 384*a^2*b*c^3*d*e^2 + 7*b^6*e^3 - 60*a*
b^4*c*e^3 + 144*a^2*b^2*c^2*e^3 - 64*a^3*c^3*e^3)*log(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b))
/c^(9/2)