### 3.199 $$\int (a+b x+c x^2)^{3/2} (d+e x+f x^2) \, dx$$

Optimal. Leaf size=236 $\frac{(b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (-4 a c f+7 b^2 f-12 b c e+24 c^2 d\right )}{192 c^3}-\frac{\left (b^2-4 a c\right ) (b+2 c x) \sqrt{a+b x+c x^2} \left (-4 c (a f+3 b e)+7 b^2 f+24 c^2 d\right )}{512 c^4}+\frac{\left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \left (-4 c (a f+3 b e)+7 b^2 f+24 c^2 d\right )}{1024 c^{9/2}}+\frac{\left (a+b x+c x^2\right )^{5/2} (12 c e-7 b f)}{60 c^2}+\frac{f x \left (a+b x+c x^2\right )^{5/2}}{6 c}$

[Out]

-((b^2 - 4*a*c)*(24*c^2*d + 7*b^2*f - 4*c*(3*b*e + a*f))*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(512*c^4) + ((24*c
^2*d - 12*b*c*e + 7*b^2*f - 4*a*c*f)*(b + 2*c*x)*(a + b*x + c*x^2)^(3/2))/(192*c^3) + ((12*c*e - 7*b*f)*(a + b
*x + c*x^2)^(5/2))/(60*c^2) + (f*x*(a + b*x + c*x^2)^(5/2))/(6*c) + ((b^2 - 4*a*c)^2*(24*c^2*d + 7*b^2*f - 4*c
*(3*b*e + a*f))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(1024*c^(9/2))

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Rubi [A]  time = 0.241952, antiderivative size = 236, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 25, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.2, Rules used = {1661, 640, 612, 621, 206} $\frac{(b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (-4 a c f+7 b^2 f-12 b c e+24 c^2 d\right )}{192 c^3}-\frac{\left (b^2-4 a c\right ) (b+2 c x) \sqrt{a+b x+c x^2} \left (-4 c (a f+3 b e)+7 b^2 f+24 c^2 d\right )}{512 c^4}+\frac{\left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \left (-4 c (a f+3 b e)+7 b^2 f+24 c^2 d\right )}{1024 c^{9/2}}+\frac{\left (a+b x+c x^2\right )^{5/2} (12 c e-7 b f)}{60 c^2}+\frac{f x \left (a+b x+c x^2\right )^{5/2}}{6 c}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-((b^2 - 4*a*c)*(24*c^2*d + 7*b^2*f - 4*c*(3*b*e + a*f))*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(512*c^4) + ((24*c
^2*d - 12*b*c*e + 7*b^2*f - 4*a*c*f)*(b + 2*c*x)*(a + b*x + c*x^2)^(3/2))/(192*c^3) + ((12*c*e - 7*b*f)*(a + b
*x + c*x^2)^(5/2))/(60*c^2) + (f*x*(a + b*x + c*x^2)^(5/2))/(6*c) + ((b^2 - 4*a*c)^2*(24*c^2*d + 7*b^2*f - 4*c
*(3*b*e + a*f))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(1024*c^(9/2))

Rule 1661

Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], e = Coeff[Pq, x, Expo
n[Pq, x]]}, Simp[(e*x^(q - 1)*(a + b*x + c*x^2)^(p + 1))/(c*(q + 2*p + 1)), x] + Dist[1/(c*(q + 2*p + 1)), Int
[(a + b*x + c*x^2)^p*ExpandToSum[c*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b*e*(q + p)*x^(q - 1) - c*e*(q +
2*p + 1)*x^q, x], x], x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[p, -1]

Rule 640

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(a + b*x + c*x^2)^(p +
1))/(2*c*(p + 1)), x] + Dist[(2*c*d - b*e)/(2*c), Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}
, x] && NeQ[2*c*d - b*e, 0] && NeQ[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 \left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right ) \, dx &=\frac{f x \left (a+b x+c x^2\right )^{5/2}}{6 c}+\frac{\int \left (6 c d-a f+\frac{1}{2} (12 c e-7 b f) x\right ) \left (a+b x+c x^2\right )^{3/2} \, dx}{6 c}\\ &=\frac{(12 c e-7 b f) \left (a+b x+c x^2\right )^{5/2}}{60 c^2}+\frac{f x \left (a+b x+c x^2\right )^{5/2}}{6 c}+\frac{\left (2 c (6 c d-a f)-\frac{1}{2} b (12 c e-7 b f)\right ) \int \left (a+b x+c x^2\right )^{3/2} \, dx}{12 c^2}\\ &=\frac{\left (24 c^2 d-12 b c e+7 b^2 f-4 a c f\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{192 c^3}+\frac{(12 c e-7 b f) \left (a+b x+c x^2\right )^{5/2}}{60 c^2}+\frac{f x \left (a+b x+c x^2\right )^{5/2}}{6 c}-\frac{\left (\left (b^2-4 a c\right ) \left (24 c^2 d+7 b^2 f-4 c (3 b e+a f)\right )\right ) \int \sqrt{a+b x+c x^2} \, dx}{128 c^3}\\ &=-\frac{\left (b^2-4 a c\right ) \left (24 c^2 d+7 b^2 f-4 c (3 b e+a f)\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{512 c^4}+\frac{\left (24 c^2 d-12 b c e+7 b^2 f-4 a c f\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{192 c^3}+\frac{(12 c e-7 b f) \left (a+b x+c x^2\right )^{5/2}}{60 c^2}+\frac{f x \left (a+b x+c x^2\right )^{5/2}}{6 c}+\frac{\left (\left (b^2-4 a c\right )^2 \left (24 c^2 d+7 b^2 f-4 c (3 b e+a f)\right )\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{1024 c^4}\\ &=-\frac{\left (b^2-4 a c\right ) \left (24 c^2 d+7 b^2 f-4 c (3 b e+a f)\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{512 c^4}+\frac{\left (24 c^2 d-12 b c e+7 b^2 f-4 a c f\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{192 c^3}+\frac{(12 c e-7 b f) \left (a+b x+c x^2\right )^{5/2}}{60 c^2}+\frac{f x \left (a+b x+c x^2\right )^{5/2}}{6 c}+\frac{\left (\left (b^2-4 a c\right )^2 \left (24 c^2 d+7 b^2 f-4 c (3 b e+a f)\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 )}{512 c^4}\\ &=-\frac{\left (b^2-4 a c\right ) \left (24 c^2 d+7 b^2 f-4 c (3 b e+a f)\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{512 c^4}+\frac{\left (24 c^2 d-12 b c e+7 b^2 f-4 a c f\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{192 c^3}+\frac{(12 c e-7 b f) \left (a+b x+c x^2\right )^{5/2}}{60 c^2}+\frac{f x \left (a+b x+c x^2\right )^{5/2}}{6 c}+\frac{\left (b^2-4 a c\right )^2 \left (24 c^2 d+7 b^2 f-4 c (3 b e+a f)\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{1024 c^{9/2}}\\ \end{align*}

Mathematica [A]  time = 0.696889, size = 392, normalized size = 1.66 $\frac{\frac{360 d \left (b^2-4 a c\right ) \left (\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )-2 \sqrt{c} (b+2 c x) \sqrt{a+x (b+c x)}\right )}{c^{3/2}}-60 b e \left (\frac{3 \left (b^2-4 a c\right ) \left (\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )-2 \sqrt{c} (b+2 c x) \sqrt{a+x (b+c x)}\right )}{c^{5/2}}+\frac{16 (b+2 c x) (a+x (b+c x))^{3/2}}{c}\right )+\frac{f \left (5 \left (7 b^2-4 a c\right ) \left (\frac{3 \left (b^2-4 a c\right ) \left (\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )-2 \sqrt{c} (b+2 c x) \sqrt{a+x (b+c x)}\right )}{c^{5/2}}+\frac{16 (b+2 c x) (a+x (b+c x))^{3/2}}{c}\right )-1792 b (a+x (b+c x))^{5/2}\right )}{c}+1920 d (b+2 c x) (a+x (b+c x))^{3/2}+3072 e (a+x (b+c x))^{5/2}+2560 f x (a+x (b+c x))^{5/2}}{15360 c}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(1920*d*(b + 2*c*x)*(a + x*(b + c*x))^(3/2) + 3072*e*(a + x*(b + c*x))^(5/2) + 2560*f*x*(a + x*(b + c*x))^(5/2
) + (360*(b^2 - 4*a*c)*d*(-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)])]))/c^(3/2) - 60*b*e*((16*(b + 2*c*x)*(a + x*(b + c*x))^(3/2))/c + (3*(b^2 - 4*a
*c)*(-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)])]))/c^(5/2)) + (f*(-1792*b*(a + x*(b + c*x))^(5/2) + 5*(7*b^2 - 4*a*c)*((16*(b + 2*c*x)*(a + x*(b + c
*x))^(3/2))/c + (3*(b^2 - 4*a*c)*(-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)])]))/c^(5/2))))/c)/(15360*c)

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Maple [B]  time = 0.053, size = 862, 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((c*x^2+b*x+a)^(3/2)*(f*x^2+e*x+d),x)

[Out]

1/16*f*b^3/c^3*(c*x^2+b*x+a)^(1/2)*a-3/32*d/c*(c*x^2+b*x+a)^(1/2)*x*b^2+1/8*f*b^2/c^2*(c*x^2+b*x+a)^(1/2)*x*a-
3/16*e*b/c*(c*x^2+b*x+a)^(1/2)*x*a+7/1024*f*b^6/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-7/512*f*b^
5/c^4*(c*x^2+b*x+a)^(1/2)-7/60*f*b/c^2*(c*x^2+b*x+a)^(5/2)+7/192*f*b^3/c^3*(c*x^2+b*x+a)^(3/2)+3/8*d/c^(1/2)*l
n((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^2-3/256*e*b^5/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))
-1/16*e*b^2/c^2*(c*x^2+b*x+a)^(3/2)+3/128*e*b^4/c^3*(c*x^2+b*x+a)^(1/2)+3/8*d*(c*x^2+b*x+a)^(1/2)*x*a-3/64*d/c
^2*(c*x^2+b*x+a)^(1/2)*b^3+1/8*d/c*(c*x^2+b*x+a)^(3/2)*b+3/128*d/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^
(1/2))*b^4-1/16*f*a^3/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-3/16*d/c^(3/2)*ln((1/2*b+c*x)/c^(1/2
)+(c*x^2+b*x+a)^(1/2))*b^2*a-1/32*f*a^2/c^2*(c*x^2+b*x+a)^(1/2)*b-1/24*f*a/c*x*(c*x^2+b*x+a)^(3/2)-1/48*f*a/c^
2*(c*x^2+b*x+a)^(3/2)*b-1/16*f*a^2/c*(c*x^2+b*x+a)^(1/2)*x+7/96*f*b^2/c^2*x*(c*x^2+b*x+a)^(3/2)-7/256*f*b^4/c^
3*(c*x^2+b*x+a)^(1/2)*x-3/32*e*b^2/c^2*(c*x^2+b*x+a)^(1/2)*a-3/16*e*b/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*
x+a)^(1/2))*a^2+3/32*e*b^3/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a-1/8*e*b/c*x*(c*x^2+b*x+a)^(3/
2)+3/64*e*b^3/c^2*(c*x^2+b*x+a)^(1/2)*x+3/16*d/c*(c*x^2+b*x+a)^(1/2)*b*a+9/64*f*b^2/c^(5/2)*ln((1/2*b+c*x)/c^(
1/2)+(c*x^2+b*x+a)^(1/2))*a^2-15/256*f*b^4/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a+1/5*e*(c*x^2+
b*x+a)^(5/2)/c+1/4*d*x*(c*x^2+b*x+a)^(3/2)+1/6*f*x*(c*x^2+b*x+a)^(5/2)/c

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.12967, size = 1947, normalized size = 8.25 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/30720*(15*(24*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d - 12*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*e + (7*b^6
- 60*a*b^4*c + 144*a^2*b^2*c^2 - 64*a^3*c^3)*f)*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*f*x^5 + 128*(12*c^6*e + 13*b*c^5*f)*x^4 + 16*(120*c^6*d + 132*b*
c^5*e + (3*b^2*c^4 + 140*a*c^5)*f)*x^3 + 8*(360*b*c^5*d + 12*(b^2*c^4 + 32*a*c^5)*e - (7*b^3*c^3 - 36*a*b*c^4)
*f)*x^2 - 120*(3*b^3*c^3 - 20*a*b*c^4)*d + 12*(15*b^4*c^2 - 100*a*b^2*c^3 + 128*a^2*c^4)*e - (105*b^5*c - 760*
a*b^3*c^2 + 1296*a^2*b*c^3)*f + 2*(120*(b^2*c^4 + 20*a*c^5)*d - 12*(5*b^3*c^3 - 28*a*b*c^4)*e + (35*b^4*c^2 -
216*a*b^2*c^3 + 240*a^2*c^4)*f)*x)*sqrt(c*x^2 + b*x + a))/c^5, -1/15360*(15*(24*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^
2*c^4)*d - 12*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*e + (7*b^6 - 60*a*b^4*c + 144*a^2*b^2*c^2 - 64*a^3*c^3)*f)*
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*f*x^5 +
128*(12*c^6*e + 13*b*c^5*f)*x^4 + 16*(120*c^6*d + 132*b*c^5*e + (3*b^2*c^4 + 140*a*c^5)*f)*x^3 + 8*(360*b*c^5*
d + 12*(b^2*c^4 + 32*a*c^5)*e - (7*b^3*c^3 - 36*a*b*c^4)*f)*x^2 - 120*(3*b^3*c^3 - 20*a*b*c^4)*d + 12*(15*b^4*
c^2 - 100*a*b^2*c^3 + 128*a^2*c^4)*e - (105*b^5*c - 760*a*b^3*c^2 + 1296*a^2*b*c^3)*f + 2*(120*(b^2*c^4 + 20*a
*c^5)*d - 12*(5*b^3*c^3 - 28*a*b*c^4)*e + (35*b^4*c^2 - 216*a*b^2*c^3 + 240*a^2*c^4)*f)*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 (a + b x + c x^{2}\right )^{\frac{3}{2}} \left (d + e x + f x^{2}\right )\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.19662, size = 563, normalized size = 2.39 \begin{align*} \frac{1}{7680} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (10 \, c f x + \frac{13 \, b c^{5} f + 12 \, c^{6} e}{c^{5}}\right )} x + \frac{120 \, c^{6} d + 3 \, b^{2} c^{4} f + 140 \, a c^{5} f + 132 \, b c^{5} e}{c^{5}}\right )} x + \frac{360 \, b c^{5} d - 7 \, b^{3} c^{3} f + 36 \, a b c^{4} f + 12 \, b^{2} c^{4} e + 384 \, a c^{5} e}{c^{5}}\right )} x + \frac{120 \, b^{2} c^{4} d + 2400 \, a c^{5} d + 35 \, b^{4} c^{2} f - 216 \, a b^{2} c^{3} f + 240 \, a^{2} c^{4} f - 60 \, b^{3} c^{3} e + 336 \, a b c^{4} e}{c^{5}}\right )} x - \frac{360 \, b^{3} c^{3} d - 2400 \, a b c^{4} d + 105 \, b^{5} c f - 760 \, a b^{3} c^{2} f + 1296 \, a^{2} b c^{3} f - 180 \, b^{4} c^{2} e + 1200 \, a b^{2} c^{3} e - 1536 \, a^{2} c^{4} e}{c^{5}}\right )} - \frac{{\left (24 \, b^{4} c^{2} d - 192 \, a b^{2} c^{3} d + 384 \, a^{2} c^{4} d + 7 \, b^{6} f - 60 \, a b^{4} c f + 144 \, a^{2} b^{2} c^{2} f - 64 \, a^{3} c^{3} f - 12 \, b^{5} c e + 96 \, a b^{3} c^{2} e - 192 \, a^{2} b c^{3} e\right )} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{1024 \, c^{\frac{9}{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7680*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(10*c*f*x + (13*b*c^5*f + 12*c^6*e)/c^5)*x + (120*c^6*d + 3*b^2*c^4*f
+ 140*a*c^5*f + 132*b*c^5*e)/c^5)*x + (360*b*c^5*d - 7*b^3*c^3*f + 36*a*b*c^4*f + 12*b^2*c^4*e + 384*a*c^5*e)
/c^5)*x + (120*b^2*c^4*d + 2400*a*c^5*d + 35*b^4*c^2*f - 216*a*b^2*c^3*f + 240*a^2*c^4*f - 60*b^3*c^3*e + 336*
a*b*c^4*e)/c^5)*x - (360*b^3*c^3*d - 2400*a*b*c^4*d + 105*b^5*c*f - 760*a*b^3*c^2*f + 1296*a^2*b*c^3*f - 180*b
^4*c^2*e + 1200*a*b^2*c^3*e - 1536*a^2*c^4*e)/c^5) - 1/1024*(24*b^4*c^2*d - 192*a*b^2*c^3*d + 384*a^2*c^4*d +
7*b^6*f - 60*a*b^4*c*f + 144*a^2*b^2*c^2*f - 64*a^3*c^3*f - 12*b^5*c*e + 96*a*b^3*c^2*e - 192*a^2*b*c^3*e)*log
(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/c^(9/2)