3.1558 \(\int (b+2 c x) (d+e x)^2 (a+b x+c x^2)^{3/2} \, dx\)

Optimal. Leaf size=242 \[ \frac{\left (a+b x+c x^2\right )^{5/2} \left (-2 c e (12 a e+7 b d)+7 b^2 e^2+10 c e x (2 c d-b e)+24 c^2 d^2\right )}{210 c^2}-\frac{e \left (b^2-4 a c\right )^2 (b+2 c x) \sqrt{a+b x+c x^2} (2 c d-b e)}{256 c^4}+\frac{e \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2} (2 c d-b e)}{96 c^3}+\frac{e \left (b^2-4 a c\right )^3 (2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{512 c^{9/2}}+\frac{2}{7} (d+e x)^2 \left (a+b x+c x^2\right )^{5/2} \]

[Out]

-((b^2 - 4*a*c)^2*e*(2*c*d - b*e)*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(256*c^4) + ((b^2 - 4*a*c)*e*(2*c*d - b*e
)*(b + 2*c*x)*(a + b*x + c*x^2)^(3/2))/(96*c^3) + (2*(d + e*x)^2*(a + b*x + c*x^2)^(5/2))/7 + ((24*c^2*d^2 + 7
*b^2*e^2 - 2*c*e*(7*b*d + 12*a*e) + 10*c*e*(2*c*d - b*e)*x)*(a + b*x + c*x^2)^(5/2))/(210*c^2) + ((b^2 - 4*a*c
)^3*e*(2*c*d - b*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 = 0.447347, antiderivative size = 242, 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} \[ \frac{\left (a+b x+c x^2\right )^{5/2} \left (-2 c e (12 a e+7 b d)+7 b^2 e^2+10 c e x (2 c d-b e)+24 c^2 d^2\right )}{210 c^2}-\frac{e \left (b^2-4 a c\right )^2 (b+2 c x) \sqrt{a+b x+c x^2} (2 c d-b e)}{256 c^4}+\frac{e \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2} (2 c d-b e)}{96 c^3}+\frac{e \left (b^2-4 a c\right )^3 (2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{512 c^{9/2}}+\frac{2}{7} (d+e x)^2 \left (a+b x+c x^2\right )^{5/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((b^2 - 4*a*c)^2*e*(2*c*d - b*e)*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(256*c^4) + ((b^2 - 4*a*c)*e*(2*c*d - b*e
)*(b + 2*c*x)*(a + b*x + c*x^2)^(3/2))/(96*c^3) + (2*(d + e*x)^2*(a + b*x + c*x^2)^(5/2))/7 + ((24*c^2*d^2 + 7
*b^2*e^2 - 2*c*e*(7*b*d + 12*a*e) + 10*c*e*(2*c*d - b*e)*x)*(a + b*x + c*x^2)^(5/2))/(210*c^2) + ((b^2 - 4*a*c
)^3*e*(2*c*d - b*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)^2 \left (a+b x+c x^2\right )^{3/2} \, dx &=\frac{2}{7} (d+e x)^2 \left (a+b x+c x^2\right )^{5/2}+\frac{\int (d+e x) (2 c (b d-2 a e)+2 c (2 c d-b e) x) \left (a+b x+c x^2\right )^{3/2} \, dx}{7 c}\\ &=\frac{2}{7} (d+e x)^2 \left (a+b x+c x^2\right )^{5/2}+\frac{\left (24 c^2 d^2+7 b^2 e^2-2 c e (7 b d+12 a e)+10 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{5/2}}{210 c^2}+\frac{\left (\left (b^2-4 a c\right ) e (2 c d-b e)\right ) \int \left (a+b x+c x^2\right )^{3/2} \, dx}{12 c^2}\\ &=\frac{\left (b^2-4 a c\right ) e (2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{96 c^3}+\frac{2}{7} (d+e x)^2 \left (a+b x+c x^2\right )^{5/2}+\frac{\left (24 c^2 d^2+7 b^2 e^2-2 c e (7 b d+12 a e)+10 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{5/2}}{210 c^2}-\frac{\left (\left (b^2-4 a c\right )^2 e (2 c d-b e)\right ) \int \sqrt{a+b x+c x^2} \, dx}{64 c^3}\\ &=-\frac{\left (b^2-4 a c\right )^2 e (2 c d-b e) (b+2 c x) \sqrt{a+b x+c x^2}}{256 c^4}+\frac{\left (b^2-4 a c\right ) e (2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{96 c^3}+\frac{2}{7} (d+e x)^2 \left (a+b x+c x^2\right )^{5/2}+\frac{\left (24 c^2 d^2+7 b^2 e^2-2 c e (7 b d+12 a e)+10 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{5/2}}{210 c^2}+\frac{\left (\left (b^2-4 a c\right )^3 e (2 c d-b e)\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{512 c^4}\\ &=-\frac{\left (b^2-4 a c\right )^2 e (2 c d-b e) (b+2 c x) \sqrt{a+b x+c x^2}}{256 c^4}+\frac{\left (b^2-4 a c\right ) e (2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{96 c^3}+\frac{2}{7} (d+e x)^2 \left (a+b x+c x^2\right )^{5/2}+\frac{\left (24 c^2 d^2+7 b^2 e^2-2 c e (7 b d+12 a e)+10 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{5/2}}{210 c^2}+\frac{\left (\left (b^2-4 a c\right )^3 e (2 c d-b e)\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 )^2 e (2 c d-b e) (b+2 c x) \sqrt{a+b x+c x^2}}{256 c^4}+\frac{\left (b^2-4 a c\right ) e (2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{96 c^3}+\frac{2}{7} (d+e x)^2 \left (a+b x+c x^2\right )^{5/2}+\frac{\left (24 c^2 d^2+7 b^2 e^2-2 c e (7 b d+12 a e)+10 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{5/2}}{210 c^2}+\frac{\left (b^2-4 a c\right )^3 e (2 c d-b e) \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 = 0.354282, size = 204, normalized size = 0.84 \[ \frac{(a+x (b+c x))^{5/2} \left (-2 c e (12 a e+7 b d+5 b e x)+7 b^2 e^2+4 c^2 d (6 d+5 e x)\right )}{210 c^2}-\frac{e \left (b^2-4 a c\right ) (b e-2 c d) \left (2 \sqrt{c} (b+2 c x) \sqrt{a+x (b+c x)} \left (4 c \left (5 a+2 c x^2\right )-3 b^2+8 b c x\right )+3 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )\right )}{1536 c^{9/2}}+\frac{2}{7} (d+e x)^2 (a+x (b+c x))^{5/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^2*(a + x*(b + c*x))^(5/2))/7 + ((a + x*(b + c*x))^(5/2)*(7*b^2*e^2 + 4*c^2*d*(6*d + 5*e*x) - 2*c*
e*(7*b*d + 12*a*e + 5*b*e*x)))/(210*c^2) - ((b^2 - 4*a*c)*e*(-2*c*d + b*e)*(2*Sqrt[c]*(b + 2*c*x)*Sqrt[a + x*(
b + c*x)]*(-3*b^2 + 8*b*c*x + 4*c*(5*a + 2*c*x^2)) + 3*(b^2 - 4*a*c)^2*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a +
x*(b + c*x)])]))/(1536*c^(9/2))

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Maple [B]  time = 0.01, size = 895, normalized size = 3.7 \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)^2*(c*x^2+b*x+a)^(3/2),x)

[Out]

2/5*(c*x^2+b*x+a)^(5/2)*d^2+1/8*b^2/c*(c*x^2+b*x+a)^(1/2)*x*a*d*e+1/128/c^3*e^2*b^5*(c*x^2+b*x+a)^(1/2)*x+1/16
/c^2*e^2*b^2*a^2*(c*x^2+b*x+a)^(1/2)-1/96/c^3*e^2*b^4*(c*x^2+b*x+a)^(3/2)+1/256/c^4*e^2*b^6*(c*x^2+b*x+a)^(1/2
)+1/30/c^2*e^2*b^2*(c*x^2+b*x+a)^(5/2)+1/8/c*e^2*b*a^2*(c*x^2+b*x+a)^(1/2)*x-1/8*a^2/c*(c*x^2+b*x+a)^(1/2)*b*d
*e-1/16/c^2*e^2*b^3*(c*x^2+b*x+a)^(1/2)*x*a+1/12/c*e^2*b*a*(c*x^2+b*x+a)^(3/2)*x+3/16*b^2/c^(3/2)*ln((1/2*b+c*
x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^2*d*e-3/64*b^4/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a*d*e-1/1
2*a/c*(c*x^2+b*x+a)^(3/2)*b*d*e+1/16*b^3/c^2*(c*x^2+b*x+a)^(1/2)*a*d*e+1/24*b^2/c*(c*x^2+b*x+a)^(3/2)*x*d*e-1/
64*b^4/c^2*(c*x^2+b*x+a)^(1/2)*x*d*e-1/512/c^(9/2)*e^2*b^7*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+2/3*x*(
c*x^2+b*x+a)^(5/2)*d*e-4/35/c*e^2*a*(c*x^2+b*x+a)^(5/2)+2/7*e^2*x^2*(c*x^2+b*x+a)^(5/2)-1/4*a^3/c^(1/2)*ln((1/
2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*d*e+1/256*b^6/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*d*e-3/
32/c^(5/2)*e^2*b^3*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^2+3/128/c^(7/2)*e^2*b^5*ln((1/2*b+c*x)/c^(1/2
)+(c*x^2+b*x+a)^(1/2))*a+1/8/c^(3/2)*e^2*b*a^3*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-1/128*b^5/c^3*(c*x^
2+b*x+a)^(1/2)*d*e-1/4*a^2*(c*x^2+b*x+a)^(1/2)*x*d*e-1/6*a*(c*x^2+b*x+a)^(3/2)*x*d*e-1/15*b/c*(c*x^2+b*x+a)^(5
/2)*d*e+1/48*b^3/c^2*(c*x^2+b*x+a)^(3/2)*d*e-1/32/c^3*e^2*b^4*(c*x^2+b*x+a)^(1/2)*a+1/24/c^2*e^2*b^2*a*(c*x^2+
b*x+a)^(3/2)-1/48/c^2*e^2*b^3*(c*x^2+b*x+a)^(3/2)*x-1/21/c*e^2*b*x*(c*x^2+b*x+a)^(5/2)

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.38159, size = 2336, normalized size = 9.65 \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)^2*(c*x^2+b*x+a)^(3/2),x, algorithm="fricas")

[Out]

[1/107520*(105*(2*(b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*d*e - (b^7 - 12*a*b^5*c + 48*a^2*b^3*c^
2 - 64*a^3*b*c^3)*e^2)*sqrt(c)*log(-8*c^2*x^2 - 8*b*c*x - b^2 - 4*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(c) -
4*a*c) + 4*(7680*c^7*e^2*x^6 + 10752*a^2*c^5*d^2 + 1280*(14*c^7*d*e + 11*b*c^6*e^2)*x^5 + 128*(84*c^7*d^2 + 26
6*b*c^6*d*e + (47*b^2*c^5 + 96*a*c^6)*e^2)*x^4 + 16*(1344*b*c^6*d^2 + 14*(69*b^2*c^5 + 140*a*c^6)*d*e - (3*b^3
*c^4 - 556*a*b*c^5)*e^2)*x^3 - 14*(15*b^5*c^2 - 160*a*b^3*c^3 + 528*a^2*b*c^4)*d*e + (105*b^6*c - 1120*a*b^4*c
^2 + 3696*a^2*b^2*c^3 - 3072*a^3*c^4)*e^2 + 8*(1344*(b^2*c^5 + 2*a*c^6)*d^2 - 14*(b^3*c^4 - 228*a*b*c^5)*d*e +
(7*b^4*c^3 - 60*a*b^2*c^4 + 192*a^2*c^5)*e^2)*x^2 + 2*(10752*a*b*c^5*d^2 + 14*(5*b^4*c^3 - 48*a*b^2*c^4 + 240
*a^2*c^5)*d*e - (35*b^5*c^2 - 336*a*b^3*c^3 + 912*a^2*b*c^4)*e^2)*x)*sqrt(c*x^2 + b*x + a))/c^5, -1/53760*(105
*(2*(b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*d*e - (b^7 - 12*a*b^5*c + 48*a^2*b^3*c^2 - 64*a^3*b*c
^3)*e^2)*sqrt(-c)*arctan(1/2*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(-c)/(c^2*x^2 + b*c*x + a*c)) - 2*(7680*c^7
*e^2*x^6 + 10752*a^2*c^5*d^2 + 1280*(14*c^7*d*e + 11*b*c^6*e^2)*x^5 + 128*(84*c^7*d^2 + 266*b*c^6*d*e + (47*b^
2*c^5 + 96*a*c^6)*e^2)*x^4 + 16*(1344*b*c^6*d^2 + 14*(69*b^2*c^5 + 140*a*c^6)*d*e - (3*b^3*c^4 - 556*a*b*c^5)*
e^2)*x^3 - 14*(15*b^5*c^2 - 160*a*b^3*c^3 + 528*a^2*b*c^4)*d*e + (105*b^6*c - 1120*a*b^4*c^2 + 3696*a^2*b^2*c^
3 - 3072*a^3*c^4)*e^2 + 8*(1344*(b^2*c^5 + 2*a*c^6)*d^2 - 14*(b^3*c^4 - 228*a*b*c^5)*d*e + (7*b^4*c^3 - 60*a*b
^2*c^4 + 192*a^2*c^5)*e^2)*x^2 + 2*(10752*a*b*c^5*d^2 + 14*(5*b^4*c^3 - 48*a*b^2*c^4 + 240*a^2*c^5)*d*e - (35*
b^5*c^2 - 336*a*b^3*c^3 + 912*a^2*b*c^4)*e^2)*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 )^{2} \left (a + b x + c x^{2}\right )^{\frac{3}{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.19015, size = 721, normalized size = 2.98 \begin{align*} \frac{1}{26880} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (10 \,{\left (6 \, c^{2} x e^{2} + \frac{14 \, c^{8} d e + 11 \, b c^{7} e^{2}}{c^{6}}\right )} x + \frac{84 \, c^{8} d^{2} + 266 \, b c^{7} d e + 47 \, b^{2} c^{6} e^{2} + 96 \, a c^{7} e^{2}}{c^{6}}\right )} x + \frac{1344 \, b c^{7} d^{2} + 966 \, b^{2} c^{6} d e + 1960 \, a c^{7} d e - 3 \, b^{3} c^{5} e^{2} + 556 \, a b c^{6} e^{2}}{c^{6}}\right )} x + \frac{1344 \, b^{2} c^{6} d^{2} + 2688 \, a c^{7} d^{2} - 14 \, b^{3} c^{5} d e + 3192 \, a b c^{6} d e + 7 \, b^{4} c^{4} e^{2} - 60 \, a b^{2} c^{5} e^{2} + 192 \, a^{2} c^{6} e^{2}}{c^{6}}\right )} x + \frac{10752 \, a b c^{6} d^{2} + 70 \, b^{4} c^{4} d e - 672 \, a b^{2} c^{5} d e + 3360 \, a^{2} c^{6} d e - 35 \, b^{5} c^{3} e^{2} + 336 \, a b^{3} c^{4} e^{2} - 912 \, a^{2} b c^{5} e^{2}}{c^{6}}\right )} x + \frac{10752 \, a^{2} c^{6} d^{2} - 210 \, b^{5} c^{3} d e + 2240 \, a b^{3} c^{4} d e - 7392 \, a^{2} b c^{5} d e + 105 \, b^{6} c^{2} e^{2} - 1120 \, a b^{4} c^{3} e^{2} + 3696 \, a^{2} b^{2} c^{4} e^{2} - 3072 \, a^{3} c^{5} e^{2}}{c^{6}}\right )} - \frac{{\left (2 \, b^{6} c d e - 24 \, a b^{4} c^{2} d e + 96 \, a^{2} b^{2} c^{3} d e - 128 \, a^{3} c^{4} d e - b^{7} e^{2} + 12 \, a b^{5} c e^{2} - 48 \, a^{2} b^{3} c^{2} e^{2} + 64 \, a^{3} b c^{3} e^{2}\right )} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{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)^2*(c*x^2+b*x+a)^(3/2),x, algorithm="giac")

[Out]

1/26880*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(10*(6*c^2*x*e^2 + (14*c^8*d*e + 11*b*c^7*e^2)/c^6)*x + (84*c^8*d^2
+ 266*b*c^7*d*e + 47*b^2*c^6*e^2 + 96*a*c^7*e^2)/c^6)*x + (1344*b*c^7*d^2 + 966*b^2*c^6*d*e + 1960*a*c^7*d*e -
3*b^3*c^5*e^2 + 556*a*b*c^6*e^2)/c^6)*x + (1344*b^2*c^6*d^2 + 2688*a*c^7*d^2 - 14*b^3*c^5*d*e + 3192*a*b*c^6*
d*e + 7*b^4*c^4*e^2 - 60*a*b^2*c^5*e^2 + 192*a^2*c^6*e^2)/c^6)*x + (10752*a*b*c^6*d^2 + 70*b^4*c^4*d*e - 672*a
*b^2*c^5*d*e + 3360*a^2*c^6*d*e - 35*b^5*c^3*e^2 + 336*a*b^3*c^4*e^2 - 912*a^2*b*c^5*e^2)/c^6)*x + (10752*a^2*
c^6*d^2 - 210*b^5*c^3*d*e + 2240*a*b^3*c^4*d*e - 7392*a^2*b*c^5*d*e + 105*b^6*c^2*e^2 - 1120*a*b^4*c^3*e^2 + 3
696*a^2*b^2*c^4*e^2 - 3072*a^3*c^5*e^2)/c^6) - 1/512*(2*b^6*c*d*e - 24*a*b^4*c^2*d*e + 96*a^2*b^2*c^3*d*e - 12
8*a^3*c^4*d*e - b^7*e^2 + 12*a*b^5*c*e^2 - 48*a^2*b^3*c^2*e^2 + 64*a^3*b*c^3*e^2)*log(abs(-2*(sqrt(c)*x - sqrt
(c*x^2 + b*x + a))*sqrt(c) - b))/c^(9/2)