### 3.2343 $$\int (d+e x)^3 (a+b x+c x^2)^{3/2} \, dx$$

Optimal. Leaf size=321 $\frac{e \left (a+b x+c x^2\right )^{5/2} \left (-2 c e (8 a e+49 b d)+21 b^2 e^2+30 c e x (2 c d-b e)+128 c^2 d^2\right )}{280 c^3}+\frac{(b+2 c x) \left (a+b x+c x^2\right )^{3/2} (2 c d-b e) \left (-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right )}{128 c^4}-\frac{3 \left (b^2-4 a c\right ) (b+2 c x) \sqrt{a+b x+c x^2} (2 c d-b e) \left (-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right )}{1024 c^5}+\frac{3 \left (b^2-4 a c\right )^2 (2 c d-b e) \left (-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2048 c^{11/2}}+\frac{e (d+e x)^2 \left (a+b x+c x^2\right )^{5/2}}{7 c}$

[Out]

(-3*(b^2 - 4*a*c)*(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
])/(1024*c^5) + ((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)*(a + b*x + c*x^2)^(3/
2))/(128*c^4) + (e*(d + e*x)^2*(a + b*x + c*x^2)^(5/2))/(7*c) + (e*(128*c^2*d^2 + 21*b^2*e^2 - 2*c*e*(49*b*d +
8*a*e) + 30*c*e*(2*c*d - b*e)*x)*(a + b*x + c*x^2)^(5/2))/(280*c^3) + (3*(b^2 - 4*a*c)^2*(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])])/(2048*c^(11/2)
)

________________________________________________________________________________________

Rubi [A]  time = 0.34696, antiderivative size = 321, 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, 779, 612, 621, 206} $\frac{e \left (a+b x+c x^2\right )^{5/2} \left (-2 c e (8 a e+49 b d)+21 b^2 e^2+30 c e x (2 c d-b e)+128 c^2 d^2\right )}{280 c^3}+\frac{(b+2 c x) \left (a+b x+c x^2\right )^{3/2} (2 c d-b e) \left (-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right )}{128 c^4}-\frac{3 \left (b^2-4 a c\right ) (b+2 c x) \sqrt{a+b x+c x^2} (2 c d-b e) \left (-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right )}{1024 c^5}+\frac{3 \left (b^2-4 a c\right )^2 (2 c d-b e) \left (-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2048 c^{11/2}}+\frac{e (d+e x)^2 \left (a+b x+c x^2\right )^{5/2}}{7 c}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-3*(b^2 - 4*a*c)*(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
])/(1024*c^5) + ((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)*(a + b*x + c*x^2)^(3/
2))/(128*c^4) + (e*(d + e*x)^2*(a + b*x + c*x^2)^(5/2))/(7*c) + (e*(128*c^2*d^2 + 21*b^2*e^2 - 2*c*e*(49*b*d +
8*a*e) + 30*c*e*(2*c*d - b*e)*x)*(a + b*x + c*x^2)^(5/2))/(280*c^3) + (3*(b^2 - 4*a*c)^2*(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])])/(2048*c^(11/2)
)

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 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 (d+e x)^3 \left (a+b x+c x^2\right )^{3/2} \, dx &=\frac{e (d+e x)^2 \left (a+b x+c x^2\right )^{5/2}}{7 c}+\frac{\int (d+e x) \left (\frac{1}{2} \left (14 c d^2-e (5 b d+4 a e)\right )+\frac{9}{2} e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2} \, dx}{7 c}\\ &=\frac{e (d+e x)^2 \left (a+b x+c x^2\right )^{5/2}}{7 c}+\frac{e \left (128 c^2 d^2+21 b^2 e^2-2 c e (49 b d+8 a e)+30 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{5/2}}{280 c^3}+\frac{\left ((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 \left (a+b x+c x^2\right )^{3/2} \, dx}{16 c^3}\\ &=\frac{(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) \left (a+b x+c x^2\right )^{3/2}}{128 c^4}+\frac{e (d+e x)^2 \left (a+b x+c x^2\right )^{5/2}}{7 c}+\frac{e \left (128 c^2 d^2+21 b^2 e^2-2 c e (49 b d+8 a e)+30 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{5/2}}{280 c^3}-\frac{\left (3 \left (b^2-4 a c\right ) (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}{256 c^4}\\ &=-\frac{3 \left (b^2-4 a c\right ) (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}}{1024 c^5}+\frac{(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) \left (a+b x+c x^2\right )^{3/2}}{128 c^4}+\frac{e (d+e x)^2 \left (a+b x+c x^2\right )^{5/2}}{7 c}+\frac{e \left (128 c^2 d^2+21 b^2 e^2-2 c e (49 b d+8 a e)+30 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{5/2}}{280 c^3}+\frac{\left (3 \left (b^2-4 a c\right )^2 (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}{2048 c^5}\\ &=-\frac{3 \left (b^2-4 a c\right ) (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}}{1024 c^5}+\frac{(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) \left (a+b x+c x^2\right )^{3/2}}{128 c^4}+\frac{e (d+e x)^2 \left (a+b x+c x^2\right )^{5/2}}{7 c}+\frac{e \left (128 c^2 d^2+21 b^2 e^2-2 c e (49 b d+8 a e)+30 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{5/2}}{280 c^3}+\frac{\left (3 \left (b^2-4 a c\right )^2 (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 )}{1024 c^5}\\ &=-\frac{3 \left (b^2-4 a c\right ) (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}}{1024 c^5}+\frac{(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) \left (a+b x+c x^2\right )^{3/2}}{128 c^4}+\frac{e (d+e x)^2 \left (a+b x+c x^2\right )^{5/2}}{7 c}+\frac{e \left (128 c^2 d^2+21 b^2 e^2-2 c e (49 b d+8 a e)+30 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{5/2}}{280 c^3}+\frac{3 \left (b^2-4 a c\right )^2 (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 )}{2048 c^{11/2}}\\ \end{align*}

Mathematica [A]  time = 0.527782, size = 231, normalized size = 0.72 $\frac{\frac{7 (2 c d-b e) \left (-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right ) \left (2 \sqrt{c} (b+2 c x) \sqrt{a+x (b+c x)} \left (4 c \left (5 a+2 c x^2\right )-3 b^2+8 b c x\right )+3 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )\right )}{2048 c^{9/2}}+\frac{e (a+x (b+c x))^{5/2} \left (-2 c e (8 a e+49 b d+15 b e x)+21 b^2 e^2+4 c^2 d (32 d+15 e x)\right )}{40 c^2}+e (d+e x)^2 (a+x (b+c x))^{5/2}}{7 c}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(e*(d + e*x)^2*(a + x*(b + c*x))^(5/2) + (e*(a + x*(b + c*x))^(5/2)*(21*b^2*e^2 + 4*c^2*d*(32*d + 15*e*x) - 2*
c*e*(49*b*d + 8*a*e + 15*b*e*x)))/(40*c^2) + (7*(2*c*d - b*e)*(8*c^2*d^2 + 3*b^2*e^2 - 4*c*e*(2*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*ArcTan
h[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])]))/(2048*c^(9/2)))/(7*c)

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Maple [B]  time = 0.052, size = 1437, normalized size = 4.5 \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)^3*(c*x^2+b*x+a)^(3/2),x)

[Out]

7/32*d*e^2*b^2/c^2*x*(c*x^2+b*x+a)^(3/2)-21/256*d*e^2*b^4/c^3*(c*x^2+b*x+a)^(1/2)*x+3/32*e^3*b/c^2*a^2*(c*x^2+
b*x+a)^(1/2)*x-3/32*d*e^2*a^2/c^2*(c*x^2+b*x+a)^(1/2)*b+3/8*d*e^2*b^2/c^2*(c*x^2+b*x+a)^(1/2)*x*a-9/16*d^2*e*b
/c*(c*x^2+b*x+a)^(1/2)*x*a+3/16*d*e^2*b^3/c^3*(c*x^2+b*x+a)^(1/2)*a+27/64*d*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^2-45/256*d*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-1/8*d*e^2
*a/c*x*(c*x^2+b*x+a)^(3/2)+1/4*d^3*x*(c*x^2+b*x+a)^(3/2)+7/64*d*e^2*b^3/c^3*(c*x^2+b*x+a)^(3/2)-21/512*d*e^2*b
^5/c^4*(c*x^2+b*x+a)^(1/2)+21/1024*d*e^2*b^6/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-3/16*d*e^2*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^2*e*b^2/c^2*(c*x^2+b*x+a)^(3/2)+9/128*d^2*e*b^4/c
^3*(c*x^2+b*x+a)^(1/2)-9/256*d^2*e*b^5/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-3/32*d^3/c*(c*x^2+b
*x+a)^(1/2)*x*b^2+3/16*d^3/c*(c*x^2+b*x+a)^(1/2)*b*a-3/16*d^3/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/
2))*b^2*a+3/5*d^2*e*(c*x^2+b*x+a)^(5/2)/c+3/40*e^3*b^2/c^3*(c*x^2+b*x+a)^(5/2)-3/128*e^3*b^4/c^4*(c*x^2+b*x+a)
^(3/2)+9/1024*e^3*b^6/c^5*(c*x^2+b*x+a)^(1/2)-2/35*e^3*a/c^2*(c*x^2+b*x+a)^(5/2)+1/7*e^3*x^2*(c*x^2+b*x+a)^(5/
2)/c-9/2048*e^3*b^7/c^(11/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+1/8*d^3/c*(c*x^2+b*x+a)^(3/2)*b+3/8*d
^3*(c*x^2+b*x+a)^(1/2)*x*a-3/64*d^3/c^2*(c*x^2+b*x+a)^(1/2)*b^3+3/8*d^3/c^(1/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+
b*x+a)^(1/2))*a^2+3/128*d^3/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*b^4+21/512*e^3*b^5/c^(9/2)*ln(
(1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a+3/32*e^3*b/c^(5/2)*a^3*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+
1/32*e^3*b^2/c^3*a*(c*x^2+b*x+a)^(3/2)-3/32*e^3*b^3/c^3*(c*x^2+b*x+a)^(1/2)*x*a+1/16*e^3*b/c^2*a*x*(c*x^2+b*x+
a)^(3/2)-15/128*e^3*b^3/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^2-1/16*d*e^2*a/c^2*(c*x^2+b*x+a)
^(3/2)*b-9/32*d^2*e*b^2/c^2*(c*x^2+b*x+a)^(1/2)*a-9/16*d^2*e*b/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1
/2))*a^2+9/32*d^2*e*b^3/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a-3/16*d*e^2*a^2/c*(c*x^2+b*x+a)^(
1/2)*x-3/8*d^2*e*b/c*x*(c*x^2+b*x+a)^(3/2)+9/64*d^2*e*b^3/c^2*(c*x^2+b*x+a)^(1/2)*x-7/20*d*e^2*b/c^2*(c*x^2+b*
x+a)^(5/2)+1/2*d*e^2*x*(c*x^2+b*x+a)^(5/2)/c+3/64*e^3*b^2/c^3*a^2*(c*x^2+b*x+a)^(1/2)-3/28*e^3*b/c^2*x*(c*x^2+
b*x+a)^(5/2)-3/64*e^3*b^3/c^3*x*(c*x^2+b*x+a)^(3/2)+9/512*e^3*b^5/c^4*(c*x^2+b*x+a)^(1/2)*x-3/64*e^3*b^4/c^4*(
c*x^2+b*x+a)^(1/2)*a

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 4.96766, size = 3075, normalized size = 9.58 \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)^3*(c*x^2+b*x+a)^(3/2),x, algorithm="fricas")

[Out]

[1/143360*(105*(16*(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*d^3 - 24*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d^2*e
+ 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^2 - (3*b^7 - 28*a*b^5*c + 80*a^2*b^3*c^2 - 64*
a^3*b*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*(5120*c^7*e^3*x^6 + 1280*(14*c^7*d*e^2 + 5*b*c^6*e^3)*x^5 + 128*(168*c^7*d^2*e + 182*b*c^6*d*e^2 + (b^2*c^
5 + 64*a*c^6)*e^3)*x^4 - 560*(3*b^3*c^4 - 20*a*b*c^5)*d^3 + 168*(15*b^4*c^3 - 100*a*b^2*c^4 + 128*a^2*c^5)*d^2
*e - 14*(105*b^5*c^2 - 760*a*b^3*c^3 + 1296*a^2*b*c^4)*d*e^2 + (315*b^6*c - 2520*a*b^4*c^2 + 5488*a^2*b^2*c^3
- 2048*a^3*c^4)*e^3 + 16*(560*c^7*d^3 + 1848*b*c^6*d^2*e + 14*(3*b^2*c^5 + 140*a*c^6)*d*e^2 - (9*b^3*c^4 - 44*
a*b*c^5)*e^3)*x^3 + 8*(1680*b*c^6*d^3 + 168*(b^2*c^5 + 32*a*c^6)*d^2*e - 14*(7*b^3*c^4 - 36*a*b*c^5)*d*e^2 + (
21*b^4*c^3 - 124*a*b^2*c^4 + 128*a^2*c^5)*e^3)*x^2 + 2*(560*(b^2*c^5 + 20*a*c^6)*d^3 - 168*(5*b^3*c^4 - 28*a*b
*c^5)*d^2*e + 14*(35*b^4*c^3 - 216*a*b^2*c^4 + 240*a^2*c^5)*d*e^2 - (105*b^5*c^2 - 728*a*b^3*c^3 + 1168*a^2*b*
c^4)*e^3)*x)*sqrt(c*x^2 + b*x + a))/c^6, -1/71680*(105*(16*(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*d^3 - 24*(b^5*
c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d^2*e + 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^2 - (3
*b^7 - 28*a*b^5*c + 80*a^2*b^3*c^2 - 64*a^3*b*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*(5120*c^7*e^3*x^6 + 1280*(14*c^7*d*e^2 + 5*b*c^6*e^3)*x^5 + 128*(168*c^7
*d^2*e + 182*b*c^6*d*e^2 + (b^2*c^5 + 64*a*c^6)*e^3)*x^4 - 560*(3*b^3*c^4 - 20*a*b*c^5)*d^3 + 168*(15*b^4*c^3
- 100*a*b^2*c^4 + 128*a^2*c^5)*d^2*e - 14*(105*b^5*c^2 - 760*a*b^3*c^3 + 1296*a^2*b*c^4)*d*e^2 + (315*b^6*c -
2520*a*b^4*c^2 + 5488*a^2*b^2*c^3 - 2048*a^3*c^4)*e^3 + 16*(560*c^7*d^3 + 1848*b*c^6*d^2*e + 14*(3*b^2*c^5 + 1
40*a*c^6)*d*e^2 - (9*b^3*c^4 - 44*a*b*c^5)*e^3)*x^3 + 8*(1680*b*c^6*d^3 + 168*(b^2*c^5 + 32*a*c^6)*d^2*e - 14*
(7*b^3*c^4 - 36*a*b*c^5)*d*e^2 + (21*b^4*c^3 - 124*a*b^2*c^4 + 128*a^2*c^5)*e^3)*x^2 + 2*(560*(b^2*c^5 + 20*a*
c^6)*d^3 - 168*(5*b^3*c^4 - 28*a*b*c^5)*d^2*e + 14*(35*b^4*c^3 - 216*a*b^2*c^4 + 240*a^2*c^5)*d*e^2 - (105*b^5
*c^2 - 728*a*b^3*c^3 + 1168*a^2*b*c^4)*e^3)*x)*sqrt(c*x^2 + b*x + a))/c^6]

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

[Out]

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

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Giac [B]  time = 1.15233, size = 965, normalized size = 3.01 \begin{align*} \frac{1}{35840} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (10 \,{\left (4 \, c x e^{3} + \frac{14 \, c^{7} d e^{2} + 5 \, b c^{6} e^{3}}{c^{6}}\right )} x + \frac{168 \, c^{7} d^{2} e + 182 \, b c^{6} d e^{2} + b^{2} c^{5} e^{3} + 64 \, a c^{6} e^{3}}{c^{6}}\right )} x + \frac{560 \, c^{7} d^{3} + 1848 \, b c^{6} d^{2} e + 42 \, b^{2} c^{5} d e^{2} + 1960 \, a c^{6} d e^{2} - 9 \, b^{3} c^{4} e^{3} + 44 \, a b c^{5} e^{3}}{c^{6}}\right )} x + \frac{1680 \, b c^{6} d^{3} + 168 \, b^{2} c^{5} d^{2} e + 5376 \, a c^{6} d^{2} e - 98 \, b^{3} c^{4} d e^{2} + 504 \, a b c^{5} d e^{2} + 21 \, b^{4} c^{3} e^{3} - 124 \, a b^{2} c^{4} e^{3} + 128 \, a^{2} c^{5} e^{3}}{c^{6}}\right )} x + \frac{560 \, b^{2} c^{5} d^{3} + 11200 \, a c^{6} d^{3} - 840 \, b^{3} c^{4} d^{2} e + 4704 \, a b c^{5} d^{2} e + 490 \, b^{4} c^{3} d e^{2} - 3024 \, a b^{2} c^{4} d e^{2} + 3360 \, a^{2} c^{5} d e^{2} - 105 \, b^{5} c^{2} e^{3} + 728 \, a b^{3} c^{3} e^{3} - 1168 \, a^{2} b c^{4} e^{3}}{c^{6}}\right )} x - \frac{1680 \, b^{3} c^{4} d^{3} - 11200 \, a b c^{5} d^{3} - 2520 \, b^{4} c^{3} d^{2} e + 16800 \, a b^{2} c^{4} d^{2} e - 21504 \, a^{2} c^{5} d^{2} e + 1470 \, b^{5} c^{2} d e^{2} - 10640 \, a b^{3} c^{3} d e^{2} + 18144 \, a^{2} b c^{4} d e^{2} - 315 \, b^{6} c e^{3} + 2520 \, a b^{4} c^{2} e^{3} - 5488 \, a^{2} b^{2} c^{3} e^{3} + 2048 \, a^{3} c^{4} e^{3}}{c^{6}}\right )} - \frac{3 \,{\left (16 \, b^{4} c^{3} d^{3} - 128 \, a b^{2} c^{4} d^{3} + 256 \, a^{2} c^{5} d^{3} - 24 \, b^{5} c^{2} d^{2} e + 192 \, a b^{3} c^{3} d^{2} e - 384 \, a^{2} b c^{4} d^{2} e + 14 \, b^{6} c d e^{2} - 120 \, a b^{4} c^{2} d e^{2} + 288 \, a^{2} b^{2} c^{3} d e^{2} - 128 \, a^{3} c^{4} d e^{2} - 3 \, b^{7} e^{3} + 28 \, a b^{5} c e^{3} - 80 \, a^{2} b^{3} c^{2} e^{3} + 64 \, a^{3} b 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 )}{2048 \, c^{\frac{11}{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/35840*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(10*(4*c*x*e^3 + (14*c^7*d*e^2 + 5*b*c^6*e^3)/c^6)*x + (168*c^7*d^2*
e + 182*b*c^6*d*e^2 + b^2*c^5*e^3 + 64*a*c^6*e^3)/c^6)*x + (560*c^7*d^3 + 1848*b*c^6*d^2*e + 42*b^2*c^5*d*e^2
+ 1960*a*c^6*d*e^2 - 9*b^3*c^4*e^3 + 44*a*b*c^5*e^3)/c^6)*x + (1680*b*c^6*d^3 + 168*b^2*c^5*d^2*e + 5376*a*c^6
*d^2*e - 98*b^3*c^4*d*e^2 + 504*a*b*c^5*d*e^2 + 21*b^4*c^3*e^3 - 124*a*b^2*c^4*e^3 + 128*a^2*c^5*e^3)/c^6)*x +
(560*b^2*c^5*d^3 + 11200*a*c^6*d^3 - 840*b^3*c^4*d^2*e + 4704*a*b*c^5*d^2*e + 490*b^4*c^3*d*e^2 - 3024*a*b^2*
c^4*d*e^2 + 3360*a^2*c^5*d*e^2 - 105*b^5*c^2*e^3 + 728*a*b^3*c^3*e^3 - 1168*a^2*b*c^4*e^3)/c^6)*x - (1680*b^3*
c^4*d^3 - 11200*a*b*c^5*d^3 - 2520*b^4*c^3*d^2*e + 16800*a*b^2*c^4*d^2*e - 21504*a^2*c^5*d^2*e + 1470*b^5*c^2*
d*e^2 - 10640*a*b^3*c^3*d*e^2 + 18144*a^2*b*c^4*d*e^2 - 315*b^6*c*e^3 + 2520*a*b^4*c^2*e^3 - 5488*a^2*b^2*c^3*
e^3 + 2048*a^3*c^4*e^3)/c^6) - 3/2048*(16*b^4*c^3*d^3 - 128*a*b^2*c^4*d^3 + 256*a^2*c^5*d^3 - 24*b^5*c^2*d^2*e
+ 192*a*b^3*c^3*d^2*e - 384*a^2*b*c^4*d^2*e + 14*b^6*c*d*e^2 - 120*a*b^4*c^2*d*e^2 + 288*a^2*b^2*c^3*d*e^2 -
128*a^3*c^4*d*e^2 - 3*b^7*e^3 + 28*a*b^5*c*e^3 - 80*a^2*b^3*c^2*e^3 + 64*a^3*b*c^3*e^3)*log(abs(-2*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/c^(11/2)