### 3.2344 $$\int (d+e x)^2 (a+b x+c x^2)^{3/2} \, dx$$

Optimal. Leaf size=257 $\frac{(b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right )}{192 c^3}-\frac{\left (b^2-4 a c\right ) (b+2 c x) \sqrt{a+b x+c x^2} \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right )}{512 c^4}+\frac{\left (b^2-4 a c\right )^2 \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{1024 c^{9/2}}+\frac{7 e \left (a+b x+c x^2\right )^{5/2} (2 c d-b e)}{60 c^2}+\frac{e (d+e x) \left (a+b x+c x^2\right )^{5/2}}{6 c}$

[Out]

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

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Rubi [A]  time = 0.332935, antiderivative size = 257, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.227, Rules used = {742, 640, 612, 621, 206} $\frac{(b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right )}{192 c^3}-\frac{\left (b^2-4 a c\right ) (b+2 c x) \sqrt{a+b x+c x^2} \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right )}{512 c^4}+\frac{\left (b^2-4 a c\right )^2 \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{1024 c^{9/2}}+\frac{7 e \left (a+b x+c x^2\right )^{5/2} (2 c d-b e)}{60 c^2}+\frac{e (d+e x) \left (a+b x+c x^2\right )^{5/2}}{6 c}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Rule 742

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)
*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 1)), x] + Dist[1/(c*(m + 2*p + 1)), Int[(d + e*x)^(m - 2)*Simp[c*d^2
*(m + 2*p + 1) - e*(a*e*(m - 1) + b*d*(p + 1)) + e*(2*c*d - b*e)*(m + p)*x, x]*(a + b*x + c*x^2)^p, x], x] /;
FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0]
&& If[RationalQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadraticQ[a, b, c, d, e, m,
p, x]

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 (d+e x)^2 \left (a+b x+c x^2\right )^{3/2} \, dx &=\frac{e (d+e x) \left (a+b x+c x^2\right )^{5/2}}{6 c}+\frac{\int \left (\frac{1}{2} \left (12 c d^2-2 e \left (\frac{5 b d}{2}+a e\right )\right )+\frac{7}{2} e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2} \, dx}{6 c}\\ &=\frac{7 e (2 c d-b e) \left (a+b x+c x^2\right )^{5/2}}{60 c^2}+\frac{e (d+e x) \left (a+b x+c x^2\right )^{5/2}}{6 c}+\frac{\left (-\frac{7}{2} b e (2 c d-b e)+c \left (12 c d^2-2 e \left (\frac{5 b d}{2}+a e\right )\right )\right ) \int \left (a+b x+c x^2\right )^{3/2} \, dx}{12 c^2}\\ &=\frac{\left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{192 c^3}+\frac{7 e (2 c d-b e) \left (a+b x+c x^2\right )^{5/2}}{60 c^2}+\frac{e (d+e 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^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}{128 c^3}\\ &=-\frac{\left (b^2-4 a c\right ) \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}}{512 c^4}+\frac{\left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{192 c^3}+\frac{7 e (2 c d-b e) \left (a+b x+c x^2\right )^{5/2}}{60 c^2}+\frac{e (d+e 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^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}{1024 c^4}\\ &=-\frac{\left (b^2-4 a c\right ) \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}}{512 c^4}+\frac{\left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{192 c^3}+\frac{7 e (2 c d-b e) \left (a+b x+c x^2\right )^{5/2}}{60 c^2}+\frac{e (d+e 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^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 )}{512 c^4}\\ &=-\frac{\left (b^2-4 a c\right ) \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}}{512 c^4}+\frac{\left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{192 c^3}+\frac{7 e (2 c d-b e) \left (a+b x+c x^2\right )^{5/2}}{60 c^2}+\frac{e (d+e 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^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 )}{1024 c^{9/2}}\\ \end{align*}

Mathematica [A]  time = 0.338076, size = 188, normalized size = 0.73 $\frac{\frac{\left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right ) \left (2 \sqrt{c} (b+2 c x) \sqrt{a+x (b+c x)} \left (4 c \left (5 a+2 c x^2\right )-3 b^2+8 b c x\right )+3 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )\right )}{512 c^{7/2}}+\frac{7 e (a+x (b+c x))^{5/2} (2 c d-b e)}{10 c}+e (d+e x) (a+x (b+c x))^{5/2}}{6 c}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((7*e*(2*c*d - b*e)*(a + x*(b + c*x))^(5/2))/(10*c) + e*(d + e*x)*(a + x*(b + c*x))^(5/2) + ((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)]*(-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)])]))/(512*c^(7/2)))/(6*c)

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Maple [B]  time = 0.05, size = 922, normalized size = 3.6 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/16*d*e*b^3/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a-3/8*d*e*b/c*(c*x^2+b*x+a)^(1/2)*x*a+1/4*d^2
*x*(c*x^2+b*x+a)^(3/2)-3/32*d^2/c*(c*x^2+b*x+a)^(1/2)*x*b^2+3/16*d^2/c*(c*x^2+b*x+a)^(1/2)*b*a+9/64*e^2*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*e^2*b^4/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x
+a)^(1/2))*a-7/256*e^2*b^4/c^3*(c*x^2+b*x+a)^(1/2)*x+1/16*e^2*b^3/c^3*(c*x^2+b*x+a)^(1/2)*a+7/96*e^2*b^2/c^2*x
*(c*x^2+b*x+a)^(3/2)-1/48*e^2*a/c^2*(c*x^2+b*x+a)^(3/2)*b-1/16*e^2*a^2/c*(c*x^2+b*x+a)^(1/2)*x-1/32*e^2*a^2/c^
2*(c*x^2+b*x+a)^(1/2)*b-3/16*d^2/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*b^2*a-1/8*d*e*b^2/c^2*(c*
x^2+b*x+a)^(3/2)+3/64*d*e*b^4/c^3*(c*x^2+b*x+a)^(1/2)-3/128*d*e*b^5/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+
a)^(1/2))-1/24*e^2*a/c*x*(c*x^2+b*x+a)^(3/2)+3/32*d*e*b^3/c^2*(c*x^2+b*x+a)^(1/2)*x+3/128*d^2/c^(5/2)*ln((1/2*
b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*b^4-7/512*e^2*b^5/c^4*(c*x^2+b*x+a)^(1/2)+1/8*d^2/c*(c*x^2+b*x+a)^(3/2)*b+
3/8*d^2*(c*x^2+b*x+a)^(1/2)*x*a-3/64*d^2/c^2*(c*x^2+b*x+a)^(1/2)*b^3+3/8*d^2/c^(1/2)*ln((1/2*b+c*x)/c^(1/2)+(c
*x^2+b*x+a)^(1/2))*a^2-3/16*d*e*b^2/c^2*(c*x^2+b*x+a)^(1/2)*a-3/8*d*e*b/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+
b*x+a)^(1/2))*a^2+1/8*e^2*b^2/c^2*(c*x^2+b*x+a)^(1/2)*x*a-1/4*d*e*b/c*x*(c*x^2+b*x+a)^(3/2)+2/5*d*e*(c*x^2+b*x
+a)^(5/2)/c+7/1024*e^2*b^6/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-1/16*e^2*a^3/c^(3/2)*ln((1/2*b+
c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+1/6*e^2*x*(c*x^2+b*x+a)^(5/2)/c-7/60*e^2*b/c^2*(c*x^2+b*x+a)^(5/2)+7/192*e^2
*b^3/c^3*(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((e*x+d)^2*(c*x^2+b*x+a)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 4.05072, size = 2044, normalized size = 7.95 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/30720*(15*(24*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^2 - 24*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d*e + (7*
b^6 - 60*a*b^4*c + 144*a^2*b^2*c^2 - 64*a^3*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*(1280*c^6*e^2*x^5 + 128*(24*c^6*d*e + 13*b*c^5*e^2)*x^4 + 16*(120*c^
6*d^2 + 264*b*c^5*d*e + (3*b^2*c^4 + 140*a*c^5)*e^2)*x^3 - 120*(3*b^3*c^3 - 20*a*b*c^4)*d^2 + 24*(15*b^4*c^2 -
100*a*b^2*c^3 + 128*a^2*c^4)*d*e - (105*b^5*c - 760*a*b^3*c^2 + 1296*a^2*b*c^3)*e^2 + 8*(360*b*c^5*d^2 + 24*(
b^2*c^4 + 32*a*c^5)*d*e - (7*b^3*c^3 - 36*a*b*c^4)*e^2)*x^2 + 2*(120*(b^2*c^4 + 20*a*c^5)*d^2 - 24*(5*b^3*c^3
- 28*a*b*c^4)*d*e + (35*b^4*c^2 - 216*a*b^2*c^3 + 240*a^2*c^4)*e^2)*x)*sqrt(c*x^2 + b*x + a))/c^5, -1/15360*(1
5*(24*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^2 - 24*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d*e + (7*b^6 - 60*a*b
^4*c + 144*a^2*b^2*c^2 - 64*a^3*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*(1280*c^6*e^2*x^5 + 128*(24*c^6*d*e + 13*b*c^5*e^2)*x^4 + 16*(120*c^6*d^2 + 264*b*c^5*
d*e + (3*b^2*c^4 + 140*a*c^5)*e^2)*x^3 - 120*(3*b^3*c^3 - 20*a*b*c^4)*d^2 + 24*(15*b^4*c^2 - 100*a*b^2*c^3 + 1
28*a^2*c^4)*d*e - (105*b^5*c - 760*a*b^3*c^2 + 1296*a^2*b*c^3)*e^2 + 8*(360*b*c^5*d^2 + 24*(b^2*c^4 + 32*a*c^5
)*d*e - (7*b^3*c^3 - 36*a*b*c^4)*e^2)*x^2 + 2*(120*(b^2*c^4 + 20*a*c^5)*d^2 - 24*(5*b^3*c^3 - 28*a*b*c^4)*d*e
+ (35*b^4*c^2 - 216*a*b^2*c^3 + 240*a^2*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 (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((e*x+d)**2*(c*x**2+b*x+a)**(3/2),x)

[Out]

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

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Giac [B]  time = 1.13443, size = 625, normalized size = 2.43 \begin{align*} \frac{1}{7680} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (10 \, c x e^{2} + \frac{24 \, c^{6} d e + 13 \, b c^{5} e^{2}}{c^{5}}\right )} x + \frac{120 \, c^{6} d^{2} + 264 \, b c^{5} d e + 3 \, b^{2} c^{4} e^{2} + 140 \, a c^{5} e^{2}}{c^{5}}\right )} x + \frac{360 \, b c^{5} d^{2} + 24 \, b^{2} c^{4} d e + 768 \, a c^{5} d e - 7 \, b^{3} c^{3} e^{2} + 36 \, a b c^{4} e^{2}}{c^{5}}\right )} x + \frac{120 \, b^{2} c^{4} d^{2} + 2400 \, a c^{5} d^{2} - 120 \, b^{3} c^{3} d e + 672 \, a b c^{4} d e + 35 \, b^{4} c^{2} e^{2} - 216 \, a b^{2} c^{3} e^{2} + 240 \, a^{2} c^{4} e^{2}}{c^{5}}\right )} x - \frac{360 \, b^{3} c^{3} d^{2} - 2400 \, a b c^{4} d^{2} - 360 \, b^{4} c^{2} d e + 2400 \, a b^{2} c^{3} d e - 3072 \, a^{2} c^{4} d e + 105 \, b^{5} c e^{2} - 760 \, a b^{3} c^{2} e^{2} + 1296 \, a^{2} b c^{3} e^{2}}{c^{5}}\right )} - \frac{{\left (24 \, b^{4} c^{2} d^{2} - 192 \, a b^{2} c^{3} d^{2} + 384 \, a^{2} c^{4} d^{2} - 24 \, b^{5} c d e + 192 \, a b^{3} c^{2} d e - 384 \, a^{2} b c^{3} d e + 7 \, b^{6} e^{2} - 60 \, a b^{4} c e^{2} + 144 \, a^{2} b^{2} c^{2} e^{2} - 64 \, a^{3} 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 )}{1024 \, c^{\frac{9}{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7680*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(10*c*x*e^2 + (24*c^6*d*e + 13*b*c^5*e^2)/c^5)*x + (120*c^6*d^2 + 264
*b*c^5*d*e + 3*b^2*c^4*e^2 + 140*a*c^5*e^2)/c^5)*x + (360*b*c^5*d^2 + 24*b^2*c^4*d*e + 768*a*c^5*d*e - 7*b^3*c
^3*e^2 + 36*a*b*c^4*e^2)/c^5)*x + (120*b^2*c^4*d^2 + 2400*a*c^5*d^2 - 120*b^3*c^3*d*e + 672*a*b*c^4*d*e + 35*b
^4*c^2*e^2 - 216*a*b^2*c^3*e^2 + 240*a^2*c^4*e^2)/c^5)*x - (360*b^3*c^3*d^2 - 2400*a*b*c^4*d^2 - 360*b^4*c^2*d
*e + 2400*a*b^2*c^3*d*e - 3072*a^2*c^4*d*e + 105*b^5*c*e^2 - 760*a*b^3*c^2*e^2 + 1296*a^2*b*c^3*e^2)/c^5) - 1/
1024*(24*b^4*c^2*d^2 - 192*a*b^2*c^3*d^2 + 384*a^2*c^4*d^2 - 24*b^5*c*d*e + 192*a*b^3*c^2*d*e - 384*a^2*b*c^3*
d*e + 7*b^6*e^2 - 60*a*b^4*c*e^2 + 144*a^2*b^2*c^2*e^2 - 64*a^3*c^3*e^2)*log(abs(-2*(sqrt(c)*x - sqrt(c*x^2 +
b*x + a))*sqrt(c) - b))/c^(9/2)