### 3.386 $$\int (d+e x)^{3/2} \sqrt{b x+c x^2} \, dx$$

Optimal. Leaf size=362 $\frac{4 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) \left (2 b^2 e^2-3 b c d e+3 c^2 d^2\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right ),\frac{b e}{c d}\right )}{105 c^{5/2} e^2 \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{2 \sqrt{b x+c x^2} \sqrt{d+e x} \left (-4 b^2 e^2+12 c e x (2 c d-b e)+9 b c d e+3 c^2 d^2\right )}{105 c^2 e}-\frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (2 c d-b e) \left (8 b^2 e^2-3 b c d e+3 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{105 c^{5/2} e^2 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}+\frac{2 e \left (b x+c x^2\right )^{3/2} \sqrt{d+e x}}{7 c}$

[Out]

(2*Sqrt[d + e*x]*(3*c^2*d^2 + 9*b*c*d*e - 4*b^2*e^2 + 12*c*e*(2*c*d - b*e)*x)*Sqrt[b*x + c*x^2])/(105*c^2*e) +
(2*e*Sqrt[d + e*x]*(b*x + c*x^2)^(3/2))/(7*c) - (2*Sqrt[-b]*(2*c*d - b*e)*(3*c^2*d^2 - 3*b*c*d*e + 8*b^2*e^2)
*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(105*c^(5
/2)*e^2*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) + (4*Sqrt[-b]*d*(c*d - b*e)*(3*c^2*d^2 - 3*b*c*d*e + 2*b^2*e^2)*S
qrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(105*c^
(5/2)*e^2*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])

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Rubi [A]  time = 0.394124, antiderivative size = 362, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 23, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.348, Rules used = {742, 814, 843, 715, 112, 110, 117, 116} $\frac{2 \sqrt{b x+c x^2} \sqrt{d+e x} \left (-4 b^2 e^2+12 c e x (2 c d-b e)+9 b c d e+3 c^2 d^2\right )}{105 c^2 e}+\frac{4 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) \left (2 b^2 e^2-3 b c d e+3 c^2 d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{105 c^{5/2} e^2 \sqrt{b x+c x^2} \sqrt{d+e x}}-\frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (2 c d-b e) \left (8 b^2 e^2-3 b c d e+3 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{105 c^{5/2} e^2 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}+\frac{2 e \left (b x+c x^2\right )^{3/2} \sqrt{d+e x}}{7 c}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(2*Sqrt[d + e*x]*(3*c^2*d^2 + 9*b*c*d*e - 4*b^2*e^2 + 12*c*e*(2*c*d - b*e)*x)*Sqrt[b*x + c*x^2])/(105*c^2*e) +
(2*e*Sqrt[d + e*x]*(b*x + c*x^2)^(3/2))/(7*c) - (2*Sqrt[-b]*(2*c*d - b*e)*(3*c^2*d^2 - 3*b*c*d*e + 8*b^2*e^2)
*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(105*c^(5
/2)*e^2*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) + (4*Sqrt[-b]*d*(c*d - b*e)*(3*c^2*d^2 - 3*b*c*d*e + 2*b^2*e^2)*S
qrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(105*c^
(5/2)*e^2*Sqrt[d + e*x]*Sqrt[b*x + c*x^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 814

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[((d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*(a + b*x + c*x^
2)^p)/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), x] - Dist[p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), Int[(d + e*x)^m*(a
+ b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2*a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p -
c*d - 2*c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c^2*d^2*(1 + 2*p) - c*e*(b*
d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*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] && GtQ[p, 0] && (IntegerQ[p] ||  !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])
) &&  !ILtQ[m + 2*p, 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 715

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

Rule 112

Int[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Dist[(Sqrt[e + f*x]*Sqrt[
1 + (d*x)/c])/(Sqrt[c + d*x]*Sqrt[1 + (f*x)/e]), Int[Sqrt[1 + (f*x)/e]/(Sqrt[b*x]*Sqrt[1 + (d*x)/c]), x], x] /
; FreeQ[{b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] &&  !(GtQ[c, 0] && GtQ[e, 0])

Rule 110

Int[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[(2*Sqrt[e]*Rt[-(b/d)
, 2]*EllipticE[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[-(b/d), 2])], (c*f)/(d*e)])/b, x] /; FreeQ[{b, c, d, e, f}, x] &&
NeQ[d*e - c*f, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !LtQ[-(b/d), 0]

Rule 117

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Dist[(Sqrt[1 + (d*x)/c]
*Sqrt[1 + (f*x)/e])/(Sqrt[c + d*x]*Sqrt[e + f*x]), Int[1/(Sqrt[b*x]*Sqrt[1 + (d*x)/c]*Sqrt[1 + (f*x)/e]), x],
x] /; FreeQ[{b, c, d, e, f}, x] &&  !(GtQ[c, 0] && GtQ[e, 0])

Rule 116

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[(2*Rt[-(b/d), 2]*E
llipticF[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[-(b/d), 2])], (c*f)/(d*e)])/(b*Sqrt[e]), x] /; FreeQ[{b, c, d, e, f}, x]
&& GtQ[c, 0] && GtQ[e, 0] && (PosQ[-(b/d)] || NegQ[-(b/f)])

Rubi steps

\begin{align*} \int (d+e x)^{3/2} \sqrt{b x+c x^2} \, dx &=\frac{2 e \sqrt{d+e x} \left (b x+c x^2\right )^{3/2}}{7 c}+\frac{2 \int \frac{\left (\frac{1}{2} d (7 c d-3 b e)+2 e (2 c d-b e) x\right ) \sqrt{b x+c x^2}}{\sqrt{d+e x}} \, dx}{7 c}\\ &=\frac{2 \sqrt{d+e x} \left (3 c^2 d^2+9 b c d e-4 b^2 e^2+12 c e (2 c d-b e) x\right ) \sqrt{b x+c x^2}}{105 c^2 e}+\frac{2 e \sqrt{d+e x} \left (b x+c x^2\right )^{3/2}}{7 c}-\frac{4 \int \frac{\frac{1}{4} b d e \left (3 c^2 d^2+9 b c d e-4 b^2 e^2\right )+\frac{1}{4} e (2 c d-b e) \left (3 c^2 d^2-3 b c d e+8 b^2 e^2\right ) x}{\sqrt{d+e x} \sqrt{b x+c x^2}} \, dx}{105 c^2 e^2}\\ &=\frac{2 \sqrt{d+e x} \left (3 c^2 d^2+9 b c d e-4 b^2 e^2+12 c e (2 c d-b e) x\right ) \sqrt{b x+c x^2}}{105 c^2 e}+\frac{2 e \sqrt{d+e x} \left (b x+c x^2\right )^{3/2}}{7 c}+\frac{\left (2 d (c d-b e) \left (3 c^2 d^2-3 b c d e+2 b^2 e^2\right )\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{b x+c x^2}} \, dx}{105 c^2 e^2}-\frac{\left ((2 c d-b e) \left (3 c^2 d^2-3 b c d e+8 b^2 e^2\right )\right ) \int \frac{\sqrt{d+e x}}{\sqrt{b x+c x^2}} \, dx}{105 c^2 e^2}\\ &=\frac{2 \sqrt{d+e x} \left (3 c^2 d^2+9 b c d e-4 b^2 e^2+12 c e (2 c d-b e) x\right ) \sqrt{b x+c x^2}}{105 c^2 e}+\frac{2 e \sqrt{d+e x} \left (b x+c x^2\right )^{3/2}}{7 c}+\frac{\left (2 d (c d-b e) \left (3 c^2 d^2-3 b c d e+2 b^2 e^2\right ) \sqrt{x} \sqrt{b+c x}\right ) \int \frac{1}{\sqrt{x} \sqrt{b+c x} \sqrt{d+e x}} \, dx}{105 c^2 e^2 \sqrt{b x+c x^2}}-\frac{\left ((2 c d-b e) \left (3 c^2 d^2-3 b c d e+8 b^2 e^2\right ) \sqrt{x} \sqrt{b+c x}\right ) \int \frac{\sqrt{d+e x}}{\sqrt{x} \sqrt{b+c x}} \, dx}{105 c^2 e^2 \sqrt{b x+c x^2}}\\ &=\frac{2 \sqrt{d+e x} \left (3 c^2 d^2+9 b c d e-4 b^2 e^2+12 c e (2 c d-b e) x\right ) \sqrt{b x+c x^2}}{105 c^2 e}+\frac{2 e \sqrt{d+e x} \left (b x+c x^2\right )^{3/2}}{7 c}-\frac{\left ((2 c d-b e) \left (3 c^2 d^2-3 b c d e+8 b^2 e^2\right ) \sqrt{x} \sqrt{1+\frac{c x}{b}} \sqrt{d+e x}\right ) \int \frac{\sqrt{1+\frac{e x}{d}}}{\sqrt{x} \sqrt{1+\frac{c x}{b}}} \, dx}{105 c^2 e^2 \sqrt{1+\frac{e x}{d}} \sqrt{b x+c x^2}}+\frac{\left (2 d (c d-b e) \left (3 c^2 d^2-3 b c d e+2 b^2 e^2\right ) \sqrt{x} \sqrt{1+\frac{c x}{b}} \sqrt{1+\frac{e x}{d}}\right ) \int \frac{1}{\sqrt{x} \sqrt{1+\frac{c x}{b}} \sqrt{1+\frac{e x}{d}}} \, dx}{105 c^2 e^2 \sqrt{d+e x} \sqrt{b x+c x^2}}\\ &=\frac{2 \sqrt{d+e x} \left (3 c^2 d^2+9 b c d e-4 b^2 e^2+12 c e (2 c d-b e) x\right ) \sqrt{b x+c x^2}}{105 c^2 e}+\frac{2 e \sqrt{d+e x} \left (b x+c x^2\right )^{3/2}}{7 c}-\frac{2 \sqrt{-b} (2 c d-b e) \left (3 c^2 d^2-3 b c d e+8 b^2 e^2\right ) \sqrt{x} \sqrt{1+\frac{c x}{b}} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{105 c^{5/2} e^2 \sqrt{1+\frac{e x}{d}} \sqrt{b x+c x^2}}+\frac{4 \sqrt{-b} d (c d-b e) \left (3 c^2 d^2-3 b c d e+2 b^2 e^2\right ) \sqrt{x} \sqrt{1+\frac{c x}{b}} \sqrt{1+\frac{e x}{d}} F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{105 c^{5/2} e^2 \sqrt{d+e x} \sqrt{b x+c x^2}}\\ \end{align*}

Mathematica [C]  time = 1.76458, size = 372, normalized size = 1.03 $\frac{2 \left (b e x (b+c x) (d+e x) \left (-4 b^2 e^2+3 b c e (3 d+e x)+3 c^2 \left (d^2+8 d e x+5 e^2 x^2\right )\right )-\sqrt{\frac{b}{c}} \left (-i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (23 b^2 c d e^2-8 b^3 e^3-18 b c^2 d^2 e+3 c^3 d^3\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right ),\frac{c d}{b e}\right )+i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (19 b^2 c d e^2-8 b^3 e^3-9 b c^2 d^2 e+6 c^3 d^3\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+\sqrt{\frac{b}{c}} (b+c x) (d+e x) \left (19 b^2 c d e^2-8 b^3 e^3-9 b c^2 d^2 e+6 c^3 d^3\right )\right )\right )}{105 b c^2 e^2 \sqrt{x (b+c x)} \sqrt{d+e x}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(2*(b*e*x*(b + c*x)*(d + e*x)*(-4*b^2*e^2 + 3*b*c*e*(3*d + e*x) + 3*c^2*(d^2 + 8*d*e*x + 5*e^2*x^2)) - Sqrt[b/
c]*(Sqrt[b/c]*(6*c^3*d^3 - 9*b*c^2*d^2*e + 19*b^2*c*d*e^2 - 8*b^3*e^3)*(b + c*x)*(d + e*x) + I*b*e*(6*c^3*d^3
- 9*b*c^2*d^2*e + 19*b^2*c*d*e^2 - 8*b^3*e^3)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticE[I*ArcSinh[
Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] - I*b*e*(3*c^3*d^3 - 18*b*c^2*d^2*e + 23*b^2*c*d*e^2 - 8*b^3*e^3)*Sqrt[1 + b/
(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)])))/(105*b*c^2*e^2*Sqrt[x
*(b + c*x)]*Sqrt[d + e*x])

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Maple [B]  time = 0.343, size = 920, normalized size = 2.5 \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)^(3/2)*(c*x^2+b*x)^(1/2),x)

[Out]

-2/105*(e*x+d)^(1/2)*(x*(c*x+b))^(1/2)*(-15*x^5*c^5*e^4+4*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x
/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^4*c*d*e^3-10*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e
-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*c^2*d^2*e^2+12*((c*x+b)/b)^
(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2*c^3*d
^3*e-6*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d
))^(1/2))*b*c^4*d^4+8*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2
),(b*e/(b*e-c*d))^(1/2))*b^5*e^4-27*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((
c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^4*c*d*e^3+28*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^
(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*c^2*d^2*e^2-15*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e
-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2*c^3*d^3*e+6*((c*x+b)/b)^(1/
2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b*c^4*d^4-18
*x^4*b*c^4*e^4-39*x^4*c^5*d*e^3+x^3*b^2*c^3*e^4-51*x^3*b*c^4*d*e^3-27*x^3*c^5*d^2*e^2+4*x^2*b^3*c^2*e^4-8*x^2*
b^2*c^3*d*e^3-36*x^2*b*c^4*d^2*e^2-3*x^2*c^5*d^3*e+4*x*b^3*c^2*d*e^3-9*x*b^2*c^3*d^2*e^2-3*x*b*c^4*d^3*e)/e^2/
x/(c*e*x^2+b*e*x+c*d*x+b*d)/c^4

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)*(c*x^2+b*x)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{c x^{2} + b x}{\left (e x + d\right )}^{\frac{3}{2}}, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)*(c*x^2+b*x)^(1/2),x, algorithm="fricas")

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(3/2)*(c*x**2+b*x)**(1/2),x)

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)*(c*x^2+b*x)^(1/2),x, algorithm="giac")

[Out]

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