3.392 \(\int (d+e x)^{3/2} (b x+c x^2)^{3/2} \, dx\)

Optimal. Leaf size=521 \[ \frac{2 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) \left (3 b^2 c^2 d^2 e^2+13 b^3 c d e^3-8 b^4 e^4-32 b c^3 d^3 e+16 c^4 d^4\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right ),\frac{b e}{c d}\right )}{1155 c^{7/2} e^4 \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{2 \left (b x+c x^2\right )^{3/2} \sqrt{d+e x} \left (-6 b^2 e^2+14 c e x (2 c d-b e)+13 b c d e+c^2 d^2\right )}{231 c^2 e}+\frac{2 \sqrt{b x+c x^2} \sqrt{d+e x} \left (-3 c e x (2 c d-b e) \left (8 b^2 e^2-b c d e+c^2 d^2\right )+6 b^2 c^2 d^2 e^2-19 b^3 c d e^3+8 b^4 e^4-19 b c^3 d^3 e+8 c^4 d^4\right )}{1155 c^3 e^3}-\frac{16 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (c d-2 b e) (2 c d-b e) (b e+c d) \left (b^2 e^2-b c d e+c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{1155 c^{7/2} e^4 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}+\frac{2 e \left (b x+c x^2\right )^{5/2} \sqrt{d+e x}}{11 c} \]

[Out]

(2*Sqrt[d + e*x]*(8*c^4*d^4 - 19*b*c^3*d^3*e + 6*b^2*c^2*d^2*e^2 - 19*b^3*c*d*e^3 + 8*b^4*e^4 - 3*c*e*(2*c*d -
b*e)*(c^2*d^2 - b*c*d*e + 8*b^2*e^2)*x)*Sqrt[b*x + c*x^2])/(1155*c^3*e^3) + (2*Sqrt[d + e*x]*(c^2*d^2 + 13*b*
c*d*e - 6*b^2*e^2 + 14*c*e*(2*c*d - b*e)*x)*(b*x + c*x^2)^(3/2))/(231*c^2*e) + (2*e*Sqrt[d + e*x]*(b*x + c*x^2
)^(5/2))/(11*c) - (16*Sqrt[-b]*(c*d - 2*b*e)*(2*c*d - b*e)*(c*d + b*e)*(c^2*d^2 - b*c*d*e + b^2*e^2)*Sqrt[x]*S
qrt[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(1155*c^(7/2)*e^4*S
qrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) + (2*Sqrt[-b]*d*(c*d - b*e)*(16*c^4*d^4 - 32*b*c^3*d^3*e + 3*b^2*c^2*d^2*e
^2 + 13*b^3*c*d*e^3 - 8*b^4*e^4)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sqrt[c]*Sqrt[x]
)/Sqrt[-b]], (b*e)/(c*d)])/(1155*c^(7/2)*e^4*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.72893, antiderivative size = 521, normalized size of antiderivative = 1., number of steps used = 10, 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} \[ \frac{2 \left (b x+c x^2\right )^{3/2} \sqrt{d+e x} \left (-6 b^2 e^2+14 c e x (2 c d-b e)+13 b c d e+c^2 d^2\right )}{231 c^2 e}+\frac{2 \sqrt{b x+c x^2} \sqrt{d+e x} \left (-3 c e x (2 c d-b e) \left (8 b^2 e^2-b c d e+c^2 d^2\right )+6 b^2 c^2 d^2 e^2-19 b^3 c d e^3+8 b^4 e^4-19 b c^3 d^3 e+8 c^4 d^4\right )}{1155 c^3 e^3}+\frac{2 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) \left (3 b^2 c^2 d^2 e^2+13 b^3 c d e^3-8 b^4 e^4-32 b c^3 d^3 e+16 c^4 d^4\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{1155 c^{7/2} e^4 \sqrt{b x+c x^2} \sqrt{d+e x}}-\frac{16 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (c d-2 b e) (2 c d-b e) (b e+c d) \left (b^2 e^2-b c d e+c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{1155 c^{7/2} e^4 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}+\frac{2 e \left (b x+c x^2\right )^{5/2} \sqrt{d+e x}}{11 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[d + e*x]*(8*c^4*d^4 - 19*b*c^3*d^3*e + 6*b^2*c^2*d^2*e^2 - 19*b^3*c*d*e^3 + 8*b^4*e^4 - 3*c*e*(2*c*d -
b*e)*(c^2*d^2 - b*c*d*e + 8*b^2*e^2)*x)*Sqrt[b*x + c*x^2])/(1155*c^3*e^3) + (2*Sqrt[d + e*x]*(c^2*d^2 + 13*b*
c*d*e - 6*b^2*e^2 + 14*c*e*(2*c*d - b*e)*x)*(b*x + c*x^2)^(3/2))/(231*c^2*e) + (2*e*Sqrt[d + e*x]*(b*x + c*x^2
)^(5/2))/(11*c) - (16*Sqrt[-b]*(c*d - 2*b*e)*(2*c*d - b*e)*(c*d + b*e)*(c^2*d^2 - b*c*d*e + b^2*e^2)*Sqrt[x]*S
qrt[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(1155*c^(7/2)*e^4*S
qrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) + (2*Sqrt[-b]*d*(c*d - b*e)*(16*c^4*d^4 - 32*b*c^3*d^3*e + 3*b^2*c^2*d^2*e
^2 + 13*b^3*c*d*e^3 - 8*b^4*e^4)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sqrt[c]*Sqrt[x]
)/Sqrt[-b]], (b*e)/(c*d)])/(1155*c^(7/2)*e^4*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} \left (b x+c x^2\right )^{3/2} \, dx &=\frac{2 e \sqrt{d+e x} \left (b x+c x^2\right )^{5/2}}{11 c}+\frac{2 \int \frac{\left (\frac{1}{2} d (11 c d-5 b e)+3 e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{3/2}}{\sqrt{d+e x}} \, dx}{11 c}\\ &=\frac{2 \sqrt{d+e x} \left (c^2 d^2+13 b c d e-6 b^2 e^2+14 c e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{3/2}}{231 c^2 e}+\frac{2 e \sqrt{d+e x} \left (b x+c x^2\right )^{5/2}}{11 c}-\frac{4 \int \frac{\left (\frac{3}{4} b d e \left (c^2 d^2+13 b c d e-6 b^2 e^2\right )+\frac{3}{4} e (2 c d-b e) \left (c^2 d^2-b c d e+8 b^2 e^2\right ) x\right ) \sqrt{b x+c x^2}}{\sqrt{d+e x}} \, dx}{231 c^2 e^2}\\ &=\frac{2 \sqrt{d+e x} \left (8 c^4 d^4-19 b c^3 d^3 e+6 b^2 c^2 d^2 e^2-19 b^3 c d e^3+8 b^4 e^4-3 c e (2 c d-b e) \left (c^2 d^2-b c d e+8 b^2 e^2\right ) x\right ) \sqrt{b x+c x^2}}{1155 c^3 e^3}+\frac{2 \sqrt{d+e x} \left (c^2 d^2+13 b c d e-6 b^2 e^2+14 c e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{3/2}}{231 c^2 e}+\frac{2 e \sqrt{d+e x} \left (b x+c x^2\right )^{5/2}}{11 c}+\frac{8 \int \frac{-\frac{3}{8} b d e \left (8 c^4 d^4-19 b c^3 d^3 e+6 b^2 c^2 d^2 e^2-19 b^3 c d e^3+8 b^4 e^4\right )-3 e (c d-2 b e) (2 c d-b e) (c d+b e) \left (c^2 d^2-b c d e+b^2 e^2\right ) x}{\sqrt{d+e x} \sqrt{b x+c x^2}} \, dx}{3465 c^3 e^4}\\ &=\frac{2 \sqrt{d+e x} \left (8 c^4 d^4-19 b c^3 d^3 e+6 b^2 c^2 d^2 e^2-19 b^3 c d e^3+8 b^4 e^4-3 c e (2 c d-b e) \left (c^2 d^2-b c d e+8 b^2 e^2\right ) x\right ) \sqrt{b x+c x^2}}{1155 c^3 e^3}+\frac{2 \sqrt{d+e x} \left (c^2 d^2+13 b c d e-6 b^2 e^2+14 c e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{3/2}}{231 c^2 e}+\frac{2 e \sqrt{d+e x} \left (b x+c x^2\right )^{5/2}}{11 c}-\frac{\left (8 (c d-2 b e) (2 c d-b e) (c d+b e) \left (c^2 d^2-b c d e+b^2 e^2\right )\right ) \int \frac{\sqrt{d+e x}}{\sqrt{b x+c x^2}} \, dx}{1155 c^3 e^4}+\frac{\left (d (c d-b e) \left (16 c^4 d^4-32 b c^3 d^3 e+3 b^2 c^2 d^2 e^2+13 b^3 c d e^3-8 b^4 e^4\right )\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{b x+c x^2}} \, dx}{1155 c^3 e^4}\\ &=\frac{2 \sqrt{d+e x} \left (8 c^4 d^4-19 b c^3 d^3 e+6 b^2 c^2 d^2 e^2-19 b^3 c d e^3+8 b^4 e^4-3 c e (2 c d-b e) \left (c^2 d^2-b c d e+8 b^2 e^2\right ) x\right ) \sqrt{b x+c x^2}}{1155 c^3 e^3}+\frac{2 \sqrt{d+e x} \left (c^2 d^2+13 b c d e-6 b^2 e^2+14 c e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{3/2}}{231 c^2 e}+\frac{2 e \sqrt{d+e x} \left (b x+c x^2\right )^{5/2}}{11 c}-\frac{\left (8 (c d-2 b e) (2 c d-b e) (c d+b e) \left (c^2 d^2-b c d e+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}{1155 c^3 e^4 \sqrt{b x+c x^2}}+\frac{\left (d (c d-b e) \left (16 c^4 d^4-32 b c^3 d^3 e+3 b^2 c^2 d^2 e^2+13 b^3 c d e^3-8 b^4 e^4\right ) \sqrt{x} \sqrt{b+c x}\right ) \int \frac{1}{\sqrt{x} \sqrt{b+c x} \sqrt{d+e x}} \, dx}{1155 c^3 e^4 \sqrt{b x+c x^2}}\\ &=\frac{2 \sqrt{d+e x} \left (8 c^4 d^4-19 b c^3 d^3 e+6 b^2 c^2 d^2 e^2-19 b^3 c d e^3+8 b^4 e^4-3 c e (2 c d-b e) \left (c^2 d^2-b c d e+8 b^2 e^2\right ) x\right ) \sqrt{b x+c x^2}}{1155 c^3 e^3}+\frac{2 \sqrt{d+e x} \left (c^2 d^2+13 b c d e-6 b^2 e^2+14 c e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{3/2}}{231 c^2 e}+\frac{2 e \sqrt{d+e x} \left (b x+c x^2\right )^{5/2}}{11 c}-\frac{\left (8 (c d-2 b e) (2 c d-b e) (c d+b e) \left (c^2 d^2-b c d e+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}{1155 c^3 e^4 \sqrt{1+\frac{e x}{d}} \sqrt{b x+c x^2}}+\frac{\left (d (c d-b e) \left (16 c^4 d^4-32 b c^3 d^3 e+3 b^2 c^2 d^2 e^2+13 b^3 c d e^3-8 b^4 e^4\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}{1155 c^3 e^4 \sqrt{d+e x} \sqrt{b x+c x^2}}\\ &=\frac{2 \sqrt{d+e x} \left (8 c^4 d^4-19 b c^3 d^3 e+6 b^2 c^2 d^2 e^2-19 b^3 c d e^3+8 b^4 e^4-3 c e (2 c d-b e) \left (c^2 d^2-b c d e+8 b^2 e^2\right ) x\right ) \sqrt{b x+c x^2}}{1155 c^3 e^3}+\frac{2 \sqrt{d+e x} \left (c^2 d^2+13 b c d e-6 b^2 e^2+14 c e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{3/2}}{231 c^2 e}+\frac{2 e \sqrt{d+e x} \left (b x+c x^2\right )^{5/2}}{11 c}-\frac{16 \sqrt{-b} (c d-2 b e) (2 c d-b e) (c d+b e) \left (c^2 d^2-b c d e+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 )}{1155 c^{7/2} e^4 \sqrt{1+\frac{e x}{d}} \sqrt{b x+c x^2}}+\frac{2 \sqrt{-b} d (c d-b e) \left (16 c^4 d^4-32 b c^3 d^3 e+3 b^2 c^2 d^2 e^2+13 b^3 c d e^3-8 b^4 e^4\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 )}{1155 c^{7/2} e^4 \sqrt{d+e x} \sqrt{b x+c x^2}}\\ \end{align*}

Mathematica [C]  time = 2.82639, size = 559, normalized size = 1.07 \[ \frac{2 (x (b+c x))^{3/2} \left (b e x (b+c x) (d+e x) \left (b^2 c^2 e^2 \left (6 d^2+14 d e x+5 e^2 x^2\right )-b^3 c e^3 (19 d+6 e x)+8 b^4 e^4+b c^3 e \left (14 d^2 e x-19 d^3+205 d e^2 x^2+140 e^3 x^3\right )+c^4 \left (5 d^2 e^2 x^2-6 d^3 e x+8 d^4+140 d e^3 x^3+105 e^4 x^4\right )\right )+\sqrt{\frac{b}{c}} \left (i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (10 b^2 c^3 d^3 e^2+35 b^3 c^2 d^2 e^3-48 b^4 c d e^4+16 b^5 e^5-21 b c^4 d^4 e+8 c^5 d^5\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right ),\frac{c d}{b e}\right )-8 i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (2 b^2 c^3 d^3 e^2+2 b^3 c^2 d^2 e^3-5 b^4 c d e^4+2 b^5 e^5-5 b c^4 d^4 e+2 c^5 d^5\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )-8 \sqrt{\frac{b}{c}} (b+c x) (d+e x) \left (2 b^2 c^3 d^3 e^2+2 b^3 c^2 d^2 e^3-5 b^4 c d e^4+2 b^5 e^5-5 b c^4 d^4 e+2 c^5 d^5\right )\right )\right )}{1155 b c^3 e^4 x^2 (b+c x)^2 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(3/2)*(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)*(8*b^4*e^4 - b^3*c*e^3*(19*d + 6*e*x) + b^2*c^2*e^2*(6*d^2 +
14*d*e*x + 5*e^2*x^2) + b*c^3*e*(-19*d^3 + 14*d^2*e*x + 205*d*e^2*x^2 + 140*e^3*x^3) + c^4*(8*d^4 - 6*d^3*e*x
+ 5*d^2*e^2*x^2 + 140*d*e^3*x^3 + 105*e^4*x^4)) + Sqrt[b/c]*(-8*Sqrt[b/c]*(2*c^5*d^5 - 5*b*c^4*d^4*e + 2*b^2*
c^3*d^3*e^2 + 2*b^3*c^2*d^2*e^3 - 5*b^4*c*d*e^4 + 2*b^5*e^5)*(b + c*x)*(d + e*x) - (8*I)*b*e*(2*c^5*d^5 - 5*b*
c^4*d^4*e + 2*b^2*c^3*d^3*e^2 + 2*b^3*c^2*d^2*e^3 - 5*b^4*c*d*e^4 + 2*b^5*e^5)*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*(8*c^5*d^5 - 21*b*c^4*d^4*e + 10*b^2
*c^3*d^3*e^2 + 35*b^3*c^2*d^2*e^3 - 48*b^4*c*d*e^4 + 16*b^5*e^5)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*E
llipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)])))/(1155*b*c^3*e^4*x^2*(b + c*x)^2*Sqrt[d + e*x])

________________________________________________________________________________________

Maple [B]  time = 0.288, size = 1359, 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((e*x+d)^(3/2)*(c*x^2+b*x)^(3/2),x)

[Out]

2/1155*(e*x+d)^(1/2)*(x*(c*x+b))^(1/2)*(245*x^6*b*c^6*e^6+245*x^6*c^7*d*e^5+145*x^5*b^2*c^5*e^6+145*x^5*c^7*d^
2*e^4-x^4*b^3*c^4*e^6-x^4*c^7*d^3*e^3+2*x^3*b^4*c^3*e^6+2*x^3*c^7*d^4*e^2+8*x^2*b^5*c^2*e^6+8*x^2*c^7*d^5*e-17
*x^2*b*c^6*d^4*e^2+8*x*b^5*c^2*d*e^5-19*x*b^4*c^3*d^2*e^4+6*x*b^3*c^4*d^3*e^3-19*x*b^2*c^5*d^4*e^2+8*x*b*c^6*d
^5*e+590*x^5*b*c^6*d*e^5+364*x^4*b^2*c^5*d*e^5+364*x^4*b*c^6*d^2*e^4-6*x^3*b^3*c^4*d*e^5+239*x^3*b^2*c^5*d^2*e
^4-6*x^3*b*c^6*d^3*e^3-17*x^2*b^4*c^3*d*e^5+x^2*b^3*c^4*d^2*e^4+16*((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^7*e^6+16*((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^6*d^6-56*((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^6*c*d*e^
5+105*x^7*c^7*e^6+x^2*b^2*c^5*d^3*e^3-16*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*Ellipti
cE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b*c^6*d^6-48*((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^5*d^5*e+35*((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^4*d^4*e^2+8*((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^6*c*d*e
^5-21*((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^5*c^2*d^2*e^4+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^4*c^3*d^3*e^3+56*((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*c^2*d^2*e^4-56*((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^4*d^4*e^2+56*((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^5*d^5*e
)/c^5/e^4/x/(c*e*x^2+b*e*x+c*d*x+b*d)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (c x^{2} + b x\right )}^{\frac{3}{2}}{\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)^(3/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

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

[Out]

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

[Out]

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

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (c x^{2} + b x\right )}^{\frac{3}{2}}{\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)^(3/2),x, algorithm="giac")

[Out]

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