### 3.301 $$\int \frac{(b x+c x^2)^{3/2}}{(d+e x)^3} \, dx$$

Optimal. Leaf size=205 $\frac{3 \left (b^2 e^2-8 b c d e+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{8 \sqrt{d} e^4 \sqrt{c d-b e}}+\frac{3 \sqrt{b x+c x^2} (-b e+4 c d+2 c e x)}{4 e^3 (d+e x)}-\frac{3 \sqrt{c} (2 c d-b e) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{e^4}-\frac{\left (b x+c x^2\right )^{3/2}}{2 e (d+e x)^2}$

[Out]

(3*(4*c*d - b*e + 2*c*e*x)*Sqrt[b*x + c*x^2])/(4*e^3*(d + e*x)) - (b*x + c*x^2)^(3/2)/(2*e*(d + e*x)^2) - (3*S
qrt[c]*(2*c*d - b*e)*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/e^4 + (3*(8*c^2*d^2 - 8*b*c*d*e + b^2*e^2)*ArcTan
h[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/(8*Sqrt[d]*e^4*Sqrt[c*d - b*e])

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Rubi [A]  time = 0.197579, antiderivative size = 205, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 21, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.286, Rules used = {732, 812, 843, 620, 206, 724} $\frac{3 \left (b^2 e^2-8 b c d e+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{8 \sqrt{d} e^4 \sqrt{c d-b e}}+\frac{3 \sqrt{b x+c x^2} (-b e+4 c d+2 c e x)}{4 e^3 (d+e x)}-\frac{3 \sqrt{c} (2 c d-b e) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{e^4}-\frac{\left (b x+c x^2\right )^{3/2}}{2 e (d+e x)^2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(3*(4*c*d - b*e + 2*c*e*x)*Sqrt[b*x + c*x^2])/(4*e^3*(d + e*x)) - (b*x + c*x^2)^(3/2)/(2*e*(d + e*x)^2) - (3*S
qrt[c]*(2*c*d - b*e)*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/e^4 + (3*(8*c^2*d^2 - 8*b*c*d*e + b^2*e^2)*ArcTan
h[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/(8*Sqrt[d]*e^4*Sqrt[c*d - b*e])

Rule 732

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(
a + b*x + c*x^2)^p)/(e*(m + 1)), x] - Dist[p/(e*(m + 1)), Int[(d + e*x)^(m + 1)*(b + 2*c*x)*(a + b*x + c*x^2)^
(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, 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] && GtQ[p, 0] && (IntegerQ[p] || LtQ[m, -1]) && NeQ[m, -1] &&  !ILtQ[m + 2*p + 1, 0] && IntQua
draticQ[a, b, c, d, e, m, p, x]

Rule 812

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[((d + e*x)^(m + 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*(a + b*x + c*x^2)^p)/(e^2*(m + 1)*(m
+ 2*p + 2)), x] + Dist[p/(e^2*(m + 1)*(m + 2*p + 2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p - 1)*Simp[g*(
b*d + 2*a*e + 2*a*e*m + 2*b*d*p) - f*b*e*(m + 2*p + 2) + (g*(2*c*d + b*e + b*e*m + 4*c*d*p) - 2*c*e*f*(m + 2*p
+ 2))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2
, 0] && RationalQ[p] && p > 0 && (LtQ[m, -1] || EqQ[p, 1] || (IntegerQ[p] &&  !RationalQ[m])) && NeQ[m, -1] &&
!ILtQ[m + 2*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 843

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + b*x + c*x^
2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
&&  !IGtQ[m, 0]

Rule 620

Int[1/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(1 - c*x^2), x], x, x/Sqrt[b*x + c*x^2
]], x] /; FreeQ[{b, c}, x]

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])

Rule 724

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rubi steps

\begin{align*} \int \frac{\left (b x+c x^2\right )^{3/2}}{(d+e x)^3} \, dx &=-\frac{\left (b x+c x^2\right )^{3/2}}{2 e (d+e x)^2}+\frac{3 \int \frac{(b+2 c x) \sqrt{b x+c x^2}}{(d+e x)^2} \, dx}{4 e}\\ &=\frac{3 (4 c d-b e+2 c e x) \sqrt{b x+c x^2}}{4 e^3 (d+e x)}-\frac{\left (b x+c x^2\right )^{3/2}}{2 e (d+e x)^2}-\frac{3 \int \frac{b (4 c d-b e)+4 c (2 c d-b e) x}{(d+e x) \sqrt{b x+c x^2}} \, dx}{8 e^3}\\ &=\frac{3 (4 c d-b e+2 c e x) \sqrt{b x+c x^2}}{4 e^3 (d+e x)}-\frac{\left (b x+c x^2\right )^{3/2}}{2 e (d+e x)^2}-\frac{(3 c (2 c d-b e)) \int \frac{1}{\sqrt{b x+c x^2}} \, dx}{2 e^4}+\frac{\left (3 \left (8 c^2 d^2-8 b c d e+b^2 e^2\right )\right ) \int \frac{1}{(d+e x) \sqrt{b x+c x^2}} \, dx}{8 e^4}\\ &=\frac{3 (4 c d-b e+2 c e x) \sqrt{b x+c x^2}}{4 e^3 (d+e x)}-\frac{\left (b x+c x^2\right )^{3/2}}{2 e (d+e x)^2}-\frac{(3 c (2 c d-b e)) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{b x+c x^2}}\right )}{e^4}-\frac{\left (3 \left (8 c^2 d^2-8 b c d e+b^2 e^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{4 c d^2-4 b d e-x^2} \, dx,x,\frac{-b d-(2 c d-b e) x}{\sqrt{b x+c x^2}}\right )}{4 e^4}\\ &=\frac{3 (4 c d-b e+2 c e x) \sqrt{b x+c x^2}}{4 e^3 (d+e x)}-\frac{\left (b x+c x^2\right )^{3/2}}{2 e (d+e x)^2}-\frac{3 \sqrt{c} (2 c d-b e) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{e^4}+\frac{3 \left (8 c^2 d^2-8 b c d e+b^2 e^2\right ) \tanh ^{-1}\left (\frac{b d+(2 c d-b e) x}{2 \sqrt{d} \sqrt{c d-b e} \sqrt{b x+c x^2}}\right )}{8 \sqrt{d} e^4 \sqrt{c d-b e}}\\ \end{align*}

Mathematica [A]  time = 1.26578, size = 205, normalized size = 1. $\frac{\sqrt{x (b+c x)} \left (\frac{3 \left (b^2 e^2-8 b c d e+8 c^2 d^2\right ) \tan ^{-1}\left (\frac{\sqrt{x} \sqrt{b e-c d}}{\sqrt{d} \sqrt{b+c x}}\right )}{\sqrt{d} \sqrt{b+c x} \sqrt{b e-c d}}+\frac{e \sqrt{x} \left (2 c \left (6 d^2+9 d e x+2 e^2 x^2\right )-b e (3 d+5 e x)\right )}{(d+e x)^2}+\frac{12 \sqrt{c} (b e-2 c d) \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{\sqrt{b} \sqrt{\frac{c x}{b}+1}}\right )}{4 e^4 \sqrt{x}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(Sqrt[x*(b + c*x)]*((e*Sqrt[x]*(-(b*e*(3*d + 5*e*x)) + 2*c*(6*d^2 + 9*d*e*x + 2*e^2*x^2)))/(d + e*x)^2 + (12*S
qrt[c]*(-2*c*d + b*e)*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(Sqrt[b]*Sqrt[1 + (c*x)/b]) + (3*(8*c^2*d^2 - 8*b*c*
d*e + b^2*e^2)*ArcTan[(Sqrt[-(c*d) + b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])])/(Sqrt[d]*Sqrt[-(c*d) + b*e]*Sqrt[
b + c*x])))/(4*e^4*Sqrt[x])

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Maple [B]  time = 0.235, size = 3466, normalized size = 16.9 \begin{align*} \text{output too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2/e^3*c*d/(b*e-c*d)/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*(-d*(b*e-c*d)/e^
2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))/(d/e+x))*b^2+1/2/e/(b*e-c*d)^2*(c*(d/e+x)^
2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(3/2)*c^2-9/16/d/(b*e-c*d)^2*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*
e-c*d)/e^2)^(1/2)*b^3-3/16/e/d/(b*e-c*d)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*b^2-9/4/e^3
*c^(3/2)*d/(b*e-c*d)*ln((1/2*(b*e-2*c*d)/e+(d/e+x)*c)/c^(1/2)+(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e
^2)^(1/2))*b+9/8/e/(b*e-c*d)^2*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*x*b*c^2+3/32/e/c^(1/2
)/d/(b*e-c*d)*ln((1/2*(b*e-2*c*d)/e+(d/e+x)*c)/c^(1/2)+(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/
2))*b^3-1/4*e/d^2/(b*e-c*d)^2/(d/e+x)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(5/2)*b+3/2/e^5*c^3*
d^3/(b*e-c*d)/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*(-d*(b*e-c*d)/e^2)^(1/2)
*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))/(d/e+x))-3/8/d/(b*e-c*d)^2*(c*(d/e+x)^2+(b*e-2*c*d
)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*x*b^2*c-3/4/e^2*d/(b*e-c*d)^2*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d
)/e^2)^(1/2)*x*c^3-27/8/e^2*d/(b*e-c*d)^2*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*b*c^2-51/1
6/e^2*d/(b*e-c*d)^2*ln((1/2*(b*e-2*c*d)/e+(d/e+x)*c)/c^(1/2)+(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^
2)^(1/2))*c^(3/2)*b^2+15/4/e^3*d^2/(b*e-c*d)^2*ln((1/2*(b*e-2*c*d)/e+(d/e+x)*c)/c^(1/2)+(c*(d/e+x)^2+(b*e-2*c*
d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))*c^(5/2)*b-3/2/e^5*d^4/(b*e-c*d)^2/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-
c*d)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*(-d*(b*e-c*d)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^
(1/2))/(d/e+x))*c^4+33/32/e/(b*e-c*d)^2*ln((1/2*(b*e-2*c*d)/e+(d/e+x)*c)/c^(1/2)+(c*(d/e+x)^2+(b*e-2*c*d)/e*(d
/e+x)-d*(b*e-c*d)/e^2)^(1/2))*c^(1/2)*b^3-3/32/d/(b*e-c*d)^2*ln((1/2*(b*e-2*c*d)/e+(d/e+x)*c)/c^(1/2)+(c*(d/e+
x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))/c^(1/2)*b^4-3/4/d/(b*e-c*d)^2*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d
/e+x)-d*(b*e-c*d)/e^2)^(3/2)*b*c+1/2/d/(b*e-c*d)^2/(d/e+x)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)
^(5/2)*c-1/2/d/(b*e-c*d)^2*c^2*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(3/2)*x+15/8/e^2*c/(b*e-c*d
)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*b+9/16/e^2*c^(1/2)/(b*e-c*d)*ln((1/2*(b*e-2*c*d)/e
+(d/e+x)*c)/c^(1/2)+(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))*b^2+1/2/e/d/(b*e-c*d)/(d/e+x)^2
*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(5/2)-3/2/e^4*d^3/(b*e-c*d)^2*ln((1/2*(b*e-2*c*d)/e+(d/e+
x)*c)/c^(1/2)+(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))*c^(7/2)+3/4/e^2*c^2/(b*e-c*d)*(c*(d/e
+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*x-3/8/e/(b*e-c*d)^2/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-
c*d)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*(-d*(b*e-c*d)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^
(1/2))/(d/e+x))*b^4+39/16/e/(b*e-c*d)^2*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*b^2*c+3/2/e^
4*c^(5/2)*d^2/(b*e-c*d)*ln((1/2*(b*e-2*c*d)/e+(d/e+x)*c)/c^(1/2)+(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d
)/e^2)^(1/2))-1/2/e*c/d/(b*e-c*d)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(3/2)-3/2/e^3*c^2*d/(b*e
-c*d)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)+1/4*e/d^2/(b*e-c*d)^2*(c*(d/e+x)^2+(b*e-2*c*d)
/e*(d/e+x)-d*(b*e-c*d)/e^2)^(3/2)*b^2+3/2/e^3*d^2/(b*e-c*d)^2*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e
^2)^(1/2)*c^3-3/e^4*c^2*d^2/(b*e-c*d)/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*
(-d*(b*e-c*d)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))/(d/e+x))*b-39/8/e^3*d^2/(b
*e-c*d)^2/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*(-d*(b*e-c*d)/e^2)^(1/2)*(c*
(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))/(d/e+x))*b^2*c^2+1/4*e/d^2/(b*e-c*d)^2*c*(c*(d/e+x)^2+
(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(3/2)*x*b+9/2/e^4*d^3/(b*e-c*d)^2/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*
e-c*d)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*(-d*(b*e-c*d)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2
)^(1/2))/(d/e+x))*c^3*b+9/4/e^2*d/(b*e-c*d)^2/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(d
/e+x)+2*(-d*(b*e-c*d)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))/(d/e+x))*b^3*c-3/8
/e*c/d/(b*e-c*d)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*x*b

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.89429, size = 3626, normalized size = 17.69 \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)^(3/2)/(e*x+d)^3,x, algorithm="fricas")

[Out]

[-1/8*(12*(2*c^2*d^5 - 3*b*c*d^4*e + b^2*d^3*e^2 + (2*c^2*d^3*e^2 - 3*b*c*d^2*e^3 + b^2*d*e^4)*x^2 + 2*(2*c^2*
d^4*e - 3*b*c*d^3*e^2 + b^2*d^2*e^3)*x)*sqrt(c)*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c)) - 3*(8*c^2*d^4 -
8*b*c*d^3*e + b^2*d^2*e^2 + (8*c^2*d^2*e^2 - 8*b*c*d*e^3 + b^2*e^4)*x^2 + 2*(8*c^2*d^3*e - 8*b*c*d^2*e^2 + b^2
*d*e^3)*x)*sqrt(c*d^2 - b*d*e)*log((b*d + (2*c*d - b*e)*x + 2*sqrt(c*d^2 - b*d*e)*sqrt(c*x^2 + b*x))/(e*x + d)
) - 2*(12*c^2*d^4*e - 15*b*c*d^3*e^2 + 3*b^2*d^2*e^3 + 4*(c^2*d^2*e^3 - b*c*d*e^4)*x^2 + (18*c^2*d^3*e^2 - 23*
b*c*d^2*e^3 + 5*b^2*d*e^4)*x)*sqrt(c*x^2 + b*x))/(c*d^4*e^4 - b*d^3*e^5 + (c*d^2*e^6 - b*d*e^7)*x^2 + 2*(c*d^3
*e^5 - b*d^2*e^6)*x), 1/4*(3*(8*c^2*d^4 - 8*b*c*d^3*e + b^2*d^2*e^2 + (8*c^2*d^2*e^2 - 8*b*c*d*e^3 + b^2*e^4)*
x^2 + 2*(8*c^2*d^3*e - 8*b*c*d^2*e^2 + b^2*d*e^3)*x)*sqrt(-c*d^2 + b*d*e)*arctan(-sqrt(-c*d^2 + b*d*e)*sqrt(c*
x^2 + b*x)/((c*d - b*e)*x)) - 6*(2*c^2*d^5 - 3*b*c*d^4*e + b^2*d^3*e^2 + (2*c^2*d^3*e^2 - 3*b*c*d^2*e^3 + b^2*
d*e^4)*x^2 + 2*(2*c^2*d^4*e - 3*b*c*d^3*e^2 + b^2*d^2*e^3)*x)*sqrt(c)*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt
(c)) + (12*c^2*d^4*e - 15*b*c*d^3*e^2 + 3*b^2*d^2*e^3 + 4*(c^2*d^2*e^3 - b*c*d*e^4)*x^2 + (18*c^2*d^3*e^2 - 23
*b*c*d^2*e^3 + 5*b^2*d*e^4)*x)*sqrt(c*x^2 + b*x))/(c*d^4*e^4 - b*d^3*e^5 + (c*d^2*e^6 - b*d*e^7)*x^2 + 2*(c*d^
3*e^5 - b*d^2*e^6)*x), 1/8*(24*(2*c^2*d^5 - 3*b*c*d^4*e + b^2*d^3*e^2 + (2*c^2*d^3*e^2 - 3*b*c*d^2*e^3 + b^2*d
*e^4)*x^2 + 2*(2*c^2*d^4*e - 3*b*c*d^3*e^2 + b^2*d^2*e^3)*x)*sqrt(-c)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x))
+ 3*(8*c^2*d^4 - 8*b*c*d^3*e + b^2*d^2*e^2 + (8*c^2*d^2*e^2 - 8*b*c*d*e^3 + b^2*e^4)*x^2 + 2*(8*c^2*d^3*e - 8
*b*c*d^2*e^2 + b^2*d*e^3)*x)*sqrt(c*d^2 - b*d*e)*log((b*d + (2*c*d - b*e)*x + 2*sqrt(c*d^2 - b*d*e)*sqrt(c*x^2
+ b*x))/(e*x + d)) + 2*(12*c^2*d^4*e - 15*b*c*d^3*e^2 + 3*b^2*d^2*e^3 + 4*(c^2*d^2*e^3 - b*c*d*e^4)*x^2 + (18
*c^2*d^3*e^2 - 23*b*c*d^2*e^3 + 5*b^2*d*e^4)*x)*sqrt(c*x^2 + b*x))/(c*d^4*e^4 - b*d^3*e^5 + (c*d^2*e^6 - b*d*e
^7)*x^2 + 2*(c*d^3*e^5 - b*d^2*e^6)*x), 1/4*(3*(8*c^2*d^4 - 8*b*c*d^3*e + b^2*d^2*e^2 + (8*c^2*d^2*e^2 - 8*b*c
*d*e^3 + b^2*e^4)*x^2 + 2*(8*c^2*d^3*e - 8*b*c*d^2*e^2 + b^2*d*e^3)*x)*sqrt(-c*d^2 + b*d*e)*arctan(-sqrt(-c*d^
2 + b*d*e)*sqrt(c*x^2 + b*x)/((c*d - b*e)*x)) + 12*(2*c^2*d^5 - 3*b*c*d^4*e + b^2*d^3*e^2 + (2*c^2*d^3*e^2 - 3
*b*c*d^2*e^3 + b^2*d*e^4)*x^2 + 2*(2*c^2*d^4*e - 3*b*c*d^3*e^2 + b^2*d^2*e^3)*x)*sqrt(-c)*arctan(sqrt(c*x^2 +
b*x)*sqrt(-c)/(c*x)) + (12*c^2*d^4*e - 15*b*c*d^3*e^2 + 3*b^2*d^2*e^3 + 4*(c^2*d^2*e^3 - b*c*d*e^4)*x^2 + (18*
c^2*d^3*e^2 - 23*b*c*d^2*e^3 + 5*b^2*d*e^4)*x)*sqrt(c*x^2 + b*x))/(c*d^4*e^4 - b*d^3*e^5 + (c*d^2*e^6 - b*d*e^
7)*x^2 + 2*(c*d^3*e^5 - b*d^2*e^6)*x)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B]  time = 2.32572, size = 675, normalized size = 3.29 \begin{align*} \sqrt{c x^{2} + b x} c e^{\left (-3\right )} + \frac{3 \,{\left (8 \, c^{2} d^{2} - 8 \, b c d e + b^{2} e^{2}\right )} \arctan \left (-\frac{{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} e + \sqrt{c} d}{\sqrt{-c d^{2} + b d e}}\right ) e^{\left (-4\right )}}{4 \, \sqrt{-c d^{2} + b d e}} + \frac{3 \,{\left (2 \, c^{2} d - b c e\right )} e^{\left (-4\right )} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{2 \, \sqrt{c}} + \frac{{\left (24 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} c^{2} d^{2} e + 40 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} c^{\frac{5}{2}} d^{3} - 24 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} b c^{\frac{3}{2}} d^{2} e + 40 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} b c^{2} d^{3} - 24 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} b c d e^{2} - 28 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} b^{2} c d^{2} e + 10 \, b^{2} c^{\frac{3}{2}} d^{3} -{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} b^{2} \sqrt{c} d e^{2} - 5 \, b^{3} \sqrt{c} d^{2} e + 5 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} b^{2} e^{3} + 3 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} b^{3} d e^{2}\right )} e^{\left (-4\right )}}{4 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} e + 2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} d + b d\right )}^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(c*x^2 + b*x)*c*e^(-3) + 3/4*(8*c^2*d^2 - 8*b*c*d*e + b^2*e^2)*arctan(-((sqrt(c)*x - sqrt(c*x^2 + b*x))*e
+ sqrt(c)*d)/sqrt(-c*d^2 + b*d*e))*e^(-4)/sqrt(-c*d^2 + b*d*e) + 3/2*(2*c^2*d - b*c*e)*e^(-4)*log(abs(-2*(sqrt
(c)*x - sqrt(c*x^2 + b*x))*sqrt(c) - b))/sqrt(c) + 1/4*(24*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*c^2*d^2*e + 40*(s
qrt(c)*x - sqrt(c*x^2 + b*x))^2*c^(5/2)*d^3 - 24*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*b*c^(3/2)*d^2*e + 40*(sqrt(
c)*x - sqrt(c*x^2 + b*x))*b*c^2*d^3 - 24*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*b*c*d*e^2 - 28*(sqrt(c)*x - sqrt(c*
x^2 + b*x))*b^2*c*d^2*e + 10*b^2*c^(3/2)*d^3 - (sqrt(c)*x - sqrt(c*x^2 + b*x))^2*b^2*sqrt(c)*d*e^2 - 5*b^3*sqr
t(c)*d^2*e + 5*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*b^2*e^3 + 3*(sqrt(c)*x - sqrt(c*x^2 + b*x))*b^3*d*e^2)*e^(-4)
/((sqrt(c)*x - sqrt(c*x^2 + b*x))^2*e + 2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c)*d + b*d)^2