### 3.2333 $$\int (d+e x)^3 \sqrt{a+b x+c x^2} \, dx$$

Optimal. Leaf size=248 $\frac{e \left (a+b x+c x^2\right )^{3/2} \left (-2 c e (16 a e+75 b d)+35 b^2 e^2+42 c e x (2 c d-b e)+192 c^2 d^2\right )}{240 c^3}+\frac{(b+2 c x) \sqrt{a+b x+c x^2} (2 c d-b e) \left (-4 c e (3 a e+4 b d)+7 b^2 e^2+16 c^2 d^2\right )}{128 c^4}-\frac{\left (b^2-4 a c\right ) (2 c d-b e) \left (-4 c e (3 a e+4 b d)+7 b^2 e^2+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{256 c^{9/2}}+\frac{e (d+e x)^2 \left (a+b x+c x^2\right )^{3/2}}{5 c}$

[Out]

((2*c*d - b*e)*(16*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(4*b*d + 3*a*e))*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(128*c^4) +
(e*(d + e*x)^2*(a + b*x + c*x^2)^(3/2))/(5*c) + (e*(192*c^2*d^2 + 35*b^2*e^2 - 2*c*e*(75*b*d + 16*a*e) + 42*c
*e*(2*c*d - b*e)*x)*(a + b*x + c*x^2)^(3/2))/(240*c^3) - ((b^2 - 4*a*c)*(2*c*d - b*e)*(16*c^2*d^2 + 7*b^2*e^2
- 4*c*e*(4*b*d + 3*a*e))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(256*c^(9/2))

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Rubi [A]  time = 0.254903, antiderivative size = 248, normalized size of antiderivative = 1., number of steps used = 5, 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} $\frac{e \left (a+b x+c x^2\right )^{3/2} \left (-2 c e (16 a e+75 b d)+35 b^2 e^2+42 c e x (2 c d-b e)+192 c^2 d^2\right )}{240 c^3}+\frac{(b+2 c x) \sqrt{a+b x+c x^2} (2 c d-b e) \left (-4 c e (3 a e+4 b d)+7 b^2 e^2+16 c^2 d^2\right )}{128 c^4}-\frac{\left (b^2-4 a c\right ) (2 c d-b e) \left (-4 c e (3 a e+4 b d)+7 b^2 e^2+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{256 c^{9/2}}+\frac{e (d+e x)^2 \left (a+b x+c x^2\right )^{3/2}}{5 c}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((2*c*d - b*e)*(16*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(4*b*d + 3*a*e))*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(128*c^4) +
(e*(d + e*x)^2*(a + b*x + c*x^2)^(3/2))/(5*c) + (e*(192*c^2*d^2 + 35*b^2*e^2 - 2*c*e*(75*b*d + 16*a*e) + 42*c
*e*(2*c*d - b*e)*x)*(a + b*x + c*x^2)^(3/2))/(240*c^3) - ((b^2 - 4*a*c)*(2*c*d - b*e)*(16*c^2*d^2 + 7*b^2*e^2
- 4*c*e*(4*b*d + 3*a*e))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(256*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 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 \sqrt{a+b x+c x^2} \, dx &=\frac{e (d+e x)^2 \left (a+b x+c x^2\right )^{3/2}}{5 c}+\frac{\int (d+e x) \left (\frac{1}{2} \left (10 c d^2-e (3 b d+4 a e)\right )+\frac{7}{2} e (2 c d-b e) x\right ) \sqrt{a+b x+c x^2} \, dx}{5 c}\\ &=\frac{e (d+e x)^2 \left (a+b x+c x^2\right )^{3/2}}{5 c}+\frac{e \left (192 c^2 d^2+35 b^2 e^2-2 c e (75 b d+16 a e)+42 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{240 c^3}+\frac{\left ((2 c d-b e) \left (16 c^2 d^2+7 b^2 e^2-4 c e (4 b d+3 a e)\right )\right ) \int \sqrt{a+b x+c x^2} \, dx}{32 c^3}\\ &=\frac{(2 c d-b e) \left (16 c^2 d^2+7 b^2 e^2-4 c e (4 b d+3 a e)\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{128 c^4}+\frac{e (d+e x)^2 \left (a+b x+c x^2\right )^{3/2}}{5 c}+\frac{e \left (192 c^2 d^2+35 b^2 e^2-2 c e (75 b d+16 a e)+42 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{240 c^3}-\frac{\left (\left (b^2-4 a c\right ) (2 c d-b e) \left (16 c^2 d^2+7 b^2 e^2-4 c e (4 b d+3 a e)\right )\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{256 c^4}\\ &=\frac{(2 c d-b e) \left (16 c^2 d^2+7 b^2 e^2-4 c e (4 b d+3 a e)\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{128 c^4}+\frac{e (d+e x)^2 \left (a+b x+c x^2\right )^{3/2}}{5 c}+\frac{e \left (192 c^2 d^2+35 b^2 e^2-2 c e (75 b d+16 a e)+42 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{240 c^3}-\frac{\left (\left (b^2-4 a c\right ) (2 c d-b e) \left (16 c^2 d^2+7 b^2 e^2-4 c e (4 b d+3 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 )}{128 c^4}\\ &=\frac{(2 c d-b e) \left (16 c^2 d^2+7 b^2 e^2-4 c e (4 b d+3 a e)\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{128 c^4}+\frac{e (d+e x)^2 \left (a+b x+c x^2\right )^{3/2}}{5 c}+\frac{e \left (192 c^2 d^2+35 b^2 e^2-2 c e (75 b d+16 a e)+42 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{240 c^3}-\frac{\left (b^2-4 a c\right ) (2 c d-b e) \left (16 c^2 d^2+7 b^2 e^2-4 c e (4 b d+3 a e)\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{256 c^{9/2}}\\ \end{align*}

Mathematica [A]  time = 0.349463, size = 206, normalized size = 0.83 $\frac{\frac{5 (2 c d-b e) \left (-4 c e (3 a e+4 b d)+7 b^2 e^2+16 c^2 d^2\right ) \left (2 \sqrt{c} (b+2 c x) \sqrt{a+x (b+c x)}-\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )\right )}{256 c^{7/2}}+\frac{e (a+x (b+c x))^{3/2} \left (-2 c e (16 a e+75 b d+21 b e x)+35 b^2 e^2+12 c^2 d (16 d+7 e x)\right )}{48 c^2}+e (d+e x)^2 (a+x (b+c x))^{3/2}}{5 c}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(e*(d + e*x)^2*(a + x*(b + c*x))^(3/2) + (e*(a + x*(b + c*x))^(3/2)*(35*b^2*e^2 + 12*c^2*d*(16*d + 7*e*x) - 2*
c*e*(75*b*d + 16*a*e + 21*b*e*x)))/(48*c^2) + (5*(2*c*d - b*e)*(16*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(4*b*d + 3*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^(7/2)))/(5*c)

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Maple [B]  time = 0.052, size = 795, 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((e*x+d)^3*(c*x^2+b*x+a)^(1/2),x)

[Out]

-7/40*e^3*b/c^2*x*(c*x^2+b*x+a)^(3/2)-7/64*e^3*b^3/c^3*x*(c*x^2+b*x+a)^(1/2)+15/64*d*e^2*b^3/c^3*(c*x^2+b*x+a)
^(1/2)-15/128*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))-3/8*d*e^2*a^2/c^(3/2)*ln((1/2*b+c*
x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-3/8*d^2*e*b^2/c^2*(c*x^2+b*x+a)^(1/2)+3/16*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))+1/2*d^3*x*(c*x^2+b*x+a)^(1/2)+3/16*e^3*b/c^2*a*x*(c*x^2+b*x+a)^(1/2)-3/4*d^2*e*b/c*
x*(c*x^2+b*x+a)^(1/2)-3/8*d*e^2*a/c*x*(c*x^2+b*x+a)^(1/2)-3/16*d*e^2*a/c^2*(c*x^2+b*x+a)^(1/2)*b+9/16*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+3/4*d*e^2*x*(c*x^2+b*x+a)^(3/2)/c-5/8*d*e^2*b/c^2*(c*x
^2+b*x+a)^(3/2)-5/32*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+3/32*e^3*b^2/c^3*a*(c*x^2+b
*x+a)^(1/2)+3/16*e^3*b/c^(5/2)*a^2*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+7/48*e^3*b^2/c^3*(c*x^2+b*x+a)^
(3/2)-7/128*e^3*b^4/c^4*(c*x^2+b*x+a)^(1/2)+7/256*e^3*b^5/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-
2/15*e^3*a/c^2*(c*x^2+b*x+a)^(3/2)+d^2*e*(c*x^2+b*x+a)^(3/2)/c+1/5*e^3*x^2*(c*x^2+b*x+a)^(3/2)/c+1/4*d^3/c*(c*
x^2+b*x+a)^(1/2)*b-3/4*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+15/32*d*e^2*b^2/c^2*x*(c*
x^2+b*x+a)^(1/2)+1/2*d^3/c^(1/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a-1/8*d^3/c^(3/2)*ln((1/2*b+c*x)/
c^(1/2)+(c*x^2+b*x+a)^(1/2))*b^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)^3*(c*x^2+b*x+a)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 3.11841, size = 1762, normalized size = 7.1 \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)^(1/2),x, algorithm="fricas")

[Out]

[-1/7680*(15*(32*(b^2*c^3 - 4*a*c^4)*d^3 - 48*(b^3*c^2 - 4*a*b*c^3)*d^2*e + 6*(5*b^4*c - 24*a*b^2*c^2 + 16*a^2
*c^3)*d*e^2 - (7*b^5 - 40*a*b^3*c + 48*a^2*b*c^2)*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*(384*c^5*e^3*x^4 + 480*b*c^4*d^3 - 240*(3*b^2*c^3 - 8*a*c^4)*d^2*e
+ 30*(15*b^3*c^2 - 52*a*b*c^3)*d*e^2 - (105*b^4*c - 460*a*b^2*c^2 + 256*a^2*c^3)*e^3 + 48*(30*c^5*d*e^2 + b*c^
4*e^3)*x^3 + 8*(240*c^5*d^2*e + 30*b*c^4*d*e^2 - (7*b^2*c^3 - 16*a*c^4)*e^3)*x^2 + 2*(480*c^5*d^3 + 240*b*c^4*
d^2*e - 30*(5*b^2*c^3 - 12*a*c^4)*d*e^2 + (35*b^3*c^2 - 116*a*b*c^3)*e^3)*x)*sqrt(c*x^2 + b*x + a))/c^5, 1/384
0*(15*(32*(b^2*c^3 - 4*a*c^4)*d^3 - 48*(b^3*c^2 - 4*a*b*c^3)*d^2*e + 6*(5*b^4*c - 24*a*b^2*c^2 + 16*a^2*c^3)*d
*e^2 - (7*b^5 - 40*a*b^3*c + 48*a^2*b*c^2)*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*(384*c^5*e^3*x^4 + 480*b*c^4*d^3 - 240*(3*b^2*c^3 - 8*a*c^4)*d^2*e + 30*(15*b^3*
c^2 - 52*a*b*c^3)*d*e^2 - (105*b^4*c - 460*a*b^2*c^2 + 256*a^2*c^3)*e^3 + 48*(30*c^5*d*e^2 + b*c^4*e^3)*x^3 +
8*(240*c^5*d^2*e + 30*b*c^4*d*e^2 - (7*b^2*c^3 - 16*a*c^4)*e^3)*x^2 + 2*(480*c^5*d^3 + 240*b*c^4*d^2*e - 30*(5
*b^2*c^3 - 12*a*c^4)*d*e^2 + (35*b^3*c^2 - 116*a*b*c^3)*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 (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((e*x+d)**3*(c*x**2+b*x+a)**(1/2),x)

[Out]

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

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Giac [A]  time = 1.95457, size = 513, normalized size = 2.07 \begin{align*} \frac{1}{1920} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (6 \,{\left (8 \, x e^{3} + \frac{30 \, c^{4} d e^{2} + b c^{3} e^{3}}{c^{4}}\right )} x + \frac{240 \, c^{4} d^{2} e + 30 \, b c^{3} d e^{2} - 7 \, b^{2} c^{2} e^{3} + 16 \, a c^{3} e^{3}}{c^{4}}\right )} x + \frac{480 \, c^{4} d^{3} + 240 \, b c^{3} d^{2} e - 150 \, b^{2} c^{2} d e^{2} + 360 \, a c^{3} d e^{2} + 35 \, b^{3} c e^{3} - 116 \, a b c^{2} e^{3}}{c^{4}}\right )} x + \frac{480 \, b c^{3} d^{3} - 720 \, b^{2} c^{2} d^{2} e + 1920 \, a c^{3} d^{2} e + 450 \, b^{3} c d e^{2} - 1560 \, a b c^{2} d e^{2} - 105 \, b^{4} e^{3} + 460 \, a b^{2} c e^{3} - 256 \, a^{2} c^{2} e^{3}}{c^{4}}\right )} + \frac{{\left (32 \, b^{2} c^{3} d^{3} - 128 \, a c^{4} d^{3} - 48 \, b^{3} c^{2} d^{2} e + 192 \, a b c^{3} d^{2} e + 30 \, b^{4} c d e^{2} - 144 \, a b^{2} c^{2} d e^{2} + 96 \, a^{2} c^{3} d e^{2} - 7 \, b^{5} e^{3} + 40 \, a b^{3} c e^{3} - 48 \, a^{2} b c^{2} e^{3}\right )} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{256 \, c^{\frac{9}{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)^(1/2),x, algorithm="giac")

[Out]

1/1920*sqrt(c*x^2 + b*x + a)*(2*(4*(6*(8*x*e^3 + (30*c^4*d*e^2 + b*c^3*e^3)/c^4)*x + (240*c^4*d^2*e + 30*b*c^3
*d*e^2 - 7*b^2*c^2*e^3 + 16*a*c^3*e^3)/c^4)*x + (480*c^4*d^3 + 240*b*c^3*d^2*e - 150*b^2*c^2*d*e^2 + 360*a*c^3
*d*e^2 + 35*b^3*c*e^3 - 116*a*b*c^2*e^3)/c^4)*x + (480*b*c^3*d^3 - 720*b^2*c^2*d^2*e + 1920*a*c^3*d^2*e + 450*
b^3*c*d*e^2 - 1560*a*b*c^2*d*e^2 - 105*b^4*e^3 + 460*a*b^2*c*e^3 - 256*a^2*c^2*e^3)/c^4) + 1/256*(32*b^2*c^3*d
^3 - 128*a*c^4*d^3 - 48*b^3*c^2*d^2*e + 192*a*b*c^3*d^2*e + 30*b^4*c*d*e^2 - 144*a*b^2*c^2*d*e^2 + 96*a^2*c^3*
d*e^2 - 7*b^5*e^3 + 40*a*b^3*c*e^3 - 48*a^2*b*c^2*e^3)*log(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c)
- b))/c^(9/2)