### 3.393 $$\int \sqrt{d+e x} (b x+c x^2)^{3/2} \, dx$$

Optimal. Leaf size=457 $\frac{8 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) (2 c d-b e) \left (-b^2 e^2-2 b c d e+2 c^2 d^2\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right ),\frac{b e}{c d}\right )}{315 c^{5/2} e^4 \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{2 \sqrt{b x+c x^2} \sqrt{d+e x} \left (-6 c e x \left (2 b^2 e^2-b c d e+c^2 d^2\right )+3 b^2 c d e^2-4 b^3 e^3-15 b c^2 d^2 e+8 c^3 d^3\right )}{315 c^2 e^3}-\frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (9 b^2 c^2 d^2 e^2+7 b^3 c d e^3-8 b^4 e^4-32 b c^3 d^3 e+16 c^4 d^4\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{315 c^{5/2} e^4 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}+\frac{2 \left (b x+c x^2\right )^{3/2} (d+e x)^{3/2}}{9 e}-\frac{2 \left (b x+c x^2\right )^{3/2} \sqrt{d+e x} (2 c d-b e)}{21 c e}$

[Out]

(2*Sqrt[d + e*x]*(8*c^3*d^3 - 15*b*c^2*d^2*e + 3*b^2*c*d*e^2 - 4*b^3*e^3 - 6*c*e*(c^2*d^2 - b*c*d*e + 2*b^2*e^
2)*x)*Sqrt[b*x + c*x^2])/(315*c^2*e^3) - (2*(2*c*d - b*e)*Sqrt[d + e*x]*(b*x + c*x^2)^(3/2))/(21*c*e) + (2*(d
+ e*x)^(3/2)*(b*x + c*x^2)^(3/2))/(9*e) - (2*Sqrt[-b]*(16*c^4*d^4 - 32*b*c^3*d^3*e + 9*b^2*c^2*d^2*e^2 + 7*b^3
*c*d*e^3 - 8*b^4*e^4)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b
*e)/(c*d)])/(315*c^(5/2)*e^4*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) + (8*Sqrt[-b]*d*(c*d - b*e)*(2*c*d - b*e)*(2
*c^2*d^2 - 2*b*c*d*e - b^2*e^2)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sqrt[c]*Sqrt[x])
/Sqrt[-b]], (b*e)/(c*d)])/(315*c^(5/2)*e^4*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.577751, antiderivative size = 457, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 23, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.391, Rules used = {734, 832, 814, 843, 715, 112, 110, 117, 116} $\frac{2 \sqrt{b x+c x^2} \sqrt{d+e x} \left (-6 c e x \left (2 b^2 e^2-b c d e+c^2 d^2\right )+3 b^2 c d e^2-4 b^3 e^3-15 b c^2 d^2 e+8 c^3 d^3\right )}{315 c^2 e^3}+\frac{8 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) (2 c d-b e) \left (-b^2 e^2-2 b c d e+2 c^2 d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{315 c^{5/2} e^4 \sqrt{b x+c x^2} \sqrt{d+e x}}-\frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (9 b^2 c^2 d^2 e^2+7 b^3 c d e^3-8 b^4 e^4-32 b c^3 d^3 e+16 c^4 d^4\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{315 c^{5/2} e^4 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}+\frac{2 \left (b x+c x^2\right )^{3/2} (d+e x)^{3/2}}{9 e}-\frac{2 \left (b x+c x^2\right )^{3/2} \sqrt{d+e x} (2 c d-b e)}{21 c e}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(2*Sqrt[d + e*x]*(8*c^3*d^3 - 15*b*c^2*d^2*e + 3*b^2*c*d*e^2 - 4*b^3*e^3 - 6*c*e*(c^2*d^2 - b*c*d*e + 2*b^2*e^
2)*x)*Sqrt[b*x + c*x^2])/(315*c^2*e^3) - (2*(2*c*d - b*e)*Sqrt[d + e*x]*(b*x + c*x^2)^(3/2))/(21*c*e) + (2*(d
+ e*x)^(3/2)*(b*x + c*x^2)^(3/2))/(9*e) - (2*Sqrt[-b]*(16*c^4*d^4 - 32*b*c^3*d^3*e + 9*b^2*c^2*d^2*e^2 + 7*b^3
*c*d*e^3 - 8*b^4*e^4)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b
*e)/(c*d)])/(315*c^(5/2)*e^4*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) + (8*Sqrt[-b]*d*(c*d - b*e)*(2*c*d - b*e)*(2
*c^2*d^2 - 2*b*c*d*e - b^2*e^2)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sqrt[c]*Sqrt[x])
/Sqrt[-b]], (b*e)/(c*d)])/(315*c^(5/2)*e^4*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])

Rule 734

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 + 2*p + 1)), x] - Dist[p/(e*(m + 2*p + 1)), Int[(d + e*x)^m*Simp[b*d - 2*a*e + (2*c*
d - b*e)*x, 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] && NeQ[m + 2*p + 1, 0] && ( !RationalQ[m] || Lt
Q[m, 1]) &&  !ILtQ[m + 2*p, 0] && IntQuadraticQ[a, b, c, d, e, m, p, x]

Rule 832

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

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 \sqrt{d+e x} \left (b x+c x^2\right )^{3/2} \, dx &=\frac{2 (d+e x)^{3/2} \left (b x+c x^2\right )^{3/2}}{9 e}-\frac{\int \sqrt{d+e x} (b d+(2 c d-b e) x) \sqrt{b x+c x^2} \, dx}{3 e}\\ &=-\frac{2 (2 c d-b e) \sqrt{d+e x} \left (b x+c x^2\right )^{3/2}}{21 c e}+\frac{2 (d+e x)^{3/2} \left (b x+c x^2\right )^{3/2}}{9 e}-\frac{2 \int \frac{\left (\frac{1}{2} b d (c d+3 b e)+\left (c^2 d^2-b c d e+2 b^2 e^2\right ) x\right ) \sqrt{b x+c x^2}}{\sqrt{d+e x}} \, dx}{21 c e}\\ &=\frac{2 \sqrt{d+e x} \left (8 c^3 d^3-15 b c^2 d^2 e+3 b^2 c d e^2-4 b^3 e^3-6 c e \left (c^2 d^2-b c d e+2 b^2 e^2\right ) x\right ) \sqrt{b x+c x^2}}{315 c^2 e^3}-\frac{2 (2 c d-b e) \sqrt{d+e x} \left (b x+c x^2\right )^{3/2}}{21 c e}+\frac{2 (d+e x)^{3/2} \left (b x+c x^2\right )^{3/2}}{9 e}+\frac{4 \int \frac{-\frac{1}{4} b d \left (8 c^3 d^3-15 b c^2 d^2 e+3 b^2 c d e^2-4 b^3 e^3\right )-\frac{1}{4} \left (16 c^4 d^4-32 b c^3 d^3 e+9 b^2 c^2 d^2 e^2+7 b^3 c d e^3-8 b^4 e^4\right ) x}{\sqrt{d+e x} \sqrt{b x+c x^2}} \, dx}{315 c^2 e^3}\\ &=\frac{2 \sqrt{d+e x} \left (8 c^3 d^3-15 b c^2 d^2 e+3 b^2 c d e^2-4 b^3 e^3-6 c e \left (c^2 d^2-b c d e+2 b^2 e^2\right ) x\right ) \sqrt{b x+c x^2}}{315 c^2 e^3}-\frac{2 (2 c d-b e) \sqrt{d+e x} \left (b x+c x^2\right )^{3/2}}{21 c e}+\frac{2 (d+e x)^{3/2} \left (b x+c x^2\right )^{3/2}}{9 e}+\frac{\left (4 d (c d-b e) (2 c d-b e) \left (2 c^2 d^2-2 b c d e-b^2 e^2\right )\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{b x+c x^2}} \, dx}{315 c^2 e^4}-\frac{\left (16 c^4 d^4-32 b c^3 d^3 e+9 b^2 c^2 d^2 e^2+7 b^3 c d e^3-8 b^4 e^4\right ) \int \frac{\sqrt{d+e x}}{\sqrt{b x+c x^2}} \, dx}{315 c^2 e^4}\\ &=\frac{2 \sqrt{d+e x} \left (8 c^3 d^3-15 b c^2 d^2 e+3 b^2 c d e^2-4 b^3 e^3-6 c e \left (c^2 d^2-b c d e+2 b^2 e^2\right ) x\right ) \sqrt{b x+c x^2}}{315 c^2 e^3}-\frac{2 (2 c d-b e) \sqrt{d+e x} \left (b x+c x^2\right )^{3/2}}{21 c e}+\frac{2 (d+e x)^{3/2} \left (b x+c x^2\right )^{3/2}}{9 e}+\frac{\left (4 d (c d-b e) (2 c d-b e) \left (2 c^2 d^2-2 b c d e-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}{315 c^2 e^4 \sqrt{b x+c x^2}}-\frac{\left (\left (16 c^4 d^4-32 b c^3 d^3 e+9 b^2 c^2 d^2 e^2+7 b^3 c d e^3-8 b^4 e^4\right ) \sqrt{x} \sqrt{b+c x}\right ) \int \frac{\sqrt{d+e x}}{\sqrt{x} \sqrt{b+c x}} \, dx}{315 c^2 e^4 \sqrt{b x+c x^2}}\\ &=\frac{2 \sqrt{d+e x} \left (8 c^3 d^3-15 b c^2 d^2 e+3 b^2 c d e^2-4 b^3 e^3-6 c e \left (c^2 d^2-b c d e+2 b^2 e^2\right ) x\right ) \sqrt{b x+c x^2}}{315 c^2 e^3}-\frac{2 (2 c d-b e) \sqrt{d+e x} \left (b x+c x^2\right )^{3/2}}{21 c e}+\frac{2 (d+e x)^{3/2} \left (b x+c x^2\right )^{3/2}}{9 e}-\frac{\left (\left (16 c^4 d^4-32 b c^3 d^3 e+9 b^2 c^2 d^2 e^2+7 b^3 c d e^3-8 b^4 e^4\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}{315 c^2 e^4 \sqrt{1+\frac{e x}{d}} \sqrt{b x+c x^2}}+\frac{\left (4 d (c d-b e) (2 c d-b e) \left (2 c^2 d^2-2 b c d e-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}{315 c^2 e^4 \sqrt{d+e x} \sqrt{b x+c x^2}}\\ &=\frac{2 \sqrt{d+e x} \left (8 c^3 d^3-15 b c^2 d^2 e+3 b^2 c d e^2-4 b^3 e^3-6 c e \left (c^2 d^2-b c d e+2 b^2 e^2\right ) x\right ) \sqrt{b x+c x^2}}{315 c^2 e^3}-\frac{2 (2 c d-b e) \sqrt{d+e x} \left (b x+c x^2\right )^{3/2}}{21 c e}+\frac{2 (d+e x)^{3/2} \left (b x+c x^2\right )^{3/2}}{9 e}-\frac{2 \sqrt{-b} \left (16 c^4 d^4-32 b c^3 d^3 e+9 b^2 c^2 d^2 e^2+7 b^3 c d e^3-8 b^4 e^4\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 )}{315 c^{5/2} e^4 \sqrt{1+\frac{e x}{d}} \sqrt{b x+c x^2}}+\frac{8 \sqrt{-b} d (c d-b e) (2 c d-b e) \left (2 c^2 d^2-2 b c d e-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 )}{315 c^{5/2} e^4 \sqrt{d+e x} \sqrt{b x+c x^2}}\\ \end{align*}

Mathematica [C]  time = 2.36044, size = 463, normalized size = 1.01 $\frac{2 (x (b+c x))^{3/2} \left (b e x (b+c x) (d+e x) \left (3 b^2 c e^2 (d+e x)-4 b^3 e^3+b c^2 e \left (-15 d^2+11 d e x+50 e^2 x^2\right )+c^3 \left (-6 d^2 e x+8 d^3+5 d e^2 x^2+35 e^3 x^3\right )\right )-\sqrt{\frac{b}{c}} \left (-i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (6 b^2 c^2 d^2 e^2+11 b^3 c d e^3-8 b^4 e^4-17 b c^3 d^3 e+8 c^4 d^4\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right ),\frac{c d}{b e}\right )+i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (9 b^2 c^2 d^2 e^2+7 b^3 c d e^3-8 b^4 e^4-32 b c^3 d^3 e+16 c^4 d^4\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+\sqrt{\frac{b}{c}} (b+c x) (d+e x) \left (9 b^2 c^2 d^2 e^2+7 b^3 c d e^3-8 b^4 e^4-32 b c^3 d^3 e+16 c^4 d^4\right )\right )\right )}{315 b c^2 e^4 x^2 (b+c x)^2 \sqrt{d+e x}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(2*(x*(b + c*x))^(3/2)*(b*e*x*(b + c*x)*(d + e*x)*(-4*b^3*e^3 + 3*b^2*c*e^2*(d + e*x) + b*c^2*e*(-15*d^2 + 11*
d*e*x + 50*e^2*x^2) + c^3*(8*d^3 - 6*d^2*e*x + 5*d*e^2*x^2 + 35*e^3*x^3)) - Sqrt[b/c]*(Sqrt[b/c]*(16*c^4*d^4 -
32*b*c^3*d^3*e + 9*b^2*c^2*d^2*e^2 + 7*b^3*c*d*e^3 - 8*b^4*e^4)*(b + c*x)*(d + e*x) + I*b*e*(16*c^4*d^4 - 32*
b*c^3*d^3*e + 9*b^2*c^2*d^2*e^2 + 7*b^3*c*d*e^3 - 8*b^4*e^4)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*Ellip
ticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] - I*b*e*(8*c^4*d^4 - 17*b*c^3*d^3*e + 6*b^2*c^2*d^2*e^2 + 11*b
^3*c*d*e^3 - 8*b^4*e^4)*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)])))/(315*b*c^2*e^4*x^2*(b + c*x)^2*Sqrt[d + e*x])

________________________________________________________________________________________

Maple [B]  time = 0.29, size = 1170, normalized size = 2.6 \begin{align*} \text{result 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)^(1/2),x)

[Out]

-2/315*(x*(c*x+b))^(1/2)*(e*x+d)^(1/2)*(-85*x^5*b*c^5*e^5-40*x^5*c^6*d*e^4-53*x^4*b^2*c^4*e^5+x^4*c^6*d^2*e^3+
x^3*b^3*c^3*e^5-2*x^3*c^6*d^3*e^2+4*x^2*b^4*c^2*e^5-8*x^2*c^6*d^4*e-16*(-(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))*((c*x+b)/b)^(1/2)*b*c^5*d^5+16*(-(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))*((c*x+b)/b)^(1/2)*b*c^5*d^5+8*(-(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))*((c*x+b)/b)^(1/2)*b^6*e
^5+41*(-(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))*((c*x+b)/
b)^(1/2)*b^3*c^3*d^3*e^2-2*(-(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))*((c*x+b)/b)^(1/2)*b^4*c^2*d^2*e^3+40*(-(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))*((c*x+b)/b)^(1/2)*b^2*c^4*d^4*e-15*(-(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))*((c*x+b)/b)^(1/2)*b^5*c*d*e^4-24*(-(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))*((c*x+b)/b)^(1/2)*b^3*c^3*d^3*e^2-4*(-
(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))*((c*x+b)/b)^(1/2)
*b^4*c^2*d^2*e^3+4*(-(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))*((c*x+b)/b)^(1/2)*b^5*c*d*e^4-101*x^4*b*c^5*d*e^4-35*x^6*c^6*e^5-67*x^3*b^2*c^4*d*e^4+5*x^3*b*c^5*d^2*e^3-
2*x^2*b^3*c^3*d*e^4+x^2*b^2*c^4*d^2*e^3+13*x^2*b*c^5*d^3*e^2+4*x*b^4*c^2*d*e^4-3*x*b^3*c^3*d^2*e^3+15*x*b^2*c^
4*d^3*e^2-8*x*b*c^5*d^4*e-48*(-(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))*((c*x+b)/b)^(1/2)*b^2*c^4*d^4*e)/c^4/x/(c*e*x^2+b*e*x+c*d*x+b*d)/e^4

________________________________________________________________________________________

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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