### 3.427 $$\int \frac{1}{(d+e x)^{3/2} (b x+c x^2)^{5/2}} \, dx$$

Optimal. Leaf size=567 $\frac{8 \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (2 c d-b e) \left (-b^2 e^2-2 b c d e+2 c^2 d^2\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right ),\frac{b e}{c d}\right )}{3 (-b)^{7/2} d^2 \sqrt{b x+c x^2} \sqrt{d+e x} (c d-b e)^2}+\frac{2 e \sqrt{b x+c x^2} \left (9 b^2 c^2 d^2 e^2+7 b^3 c d e^3-8 b^4 e^4-32 b c^3 d^3 e+16 c^4 d^4\right )}{3 b^4 d^3 \sqrt{d+e x} (c d-b e)^3}+\frac{2 \left (4 c x \left (b^3 e^3-6 b c^2 d^2 e+4 c^3 d^3\right )+b (c d-b e) \left (-4 b^2 e^2-3 b c d e+8 c^2 d^2\right )\right )}{3 b^4 d^2 \sqrt{b x+c x^2} \sqrt{d+e x} (c d-b e)^2}-\frac{2 \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (9 b^2 c^2 d^2 e^2+7 b^3 c d e^3-8 b^4 e^4-32 b c^3 d^3 e+16 c^4 d^4\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 (-b)^{7/2} d^3 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)^3}-\frac{2 (c x (2 c d-b e)+b (c d-b e))}{3 b^2 d \left (b x+c x^2\right )^{3/2} \sqrt{d+e x} (c d-b e)}$

[Out]

(-2*(b*(c*d - b*e) + c*(2*c*d - b*e)*x))/(3*b^2*d*(c*d - b*e)*Sqrt[d + e*x]*(b*x + c*x^2)^(3/2)) + (2*(b*(c*d
- b*e)*(8*c^2*d^2 - 3*b*c*d*e - 4*b^2*e^2) + 4*c*(4*c^3*d^3 - 6*b*c^2*d^2*e + b^3*e^3)*x))/(3*b^4*d^2*(c*d - b
*e)^2*Sqrt[d + e*x]*Sqrt[b*x + c*x^2]) + (2*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[b*x + c*x^2])/(3*b^4*d^3*(c*d - b*e)^3*Sqrt[d + e*x]) - (2*Sqrt[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)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE[ArcSin
[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(3*(-b)^(7/2)*d^3*(c*d - b*e)^3*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2
]) + (8*Sqrt[c]*(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]*El
lipticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(3*(-b)^(7/2)*d^2*(c*d - b*e)^2*Sqrt[d + e*x]*Sqrt[b
*x + c*x^2])

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Rubi [A]  time = 0.645148, antiderivative size = 567, 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 = {740, 822, 834, 843, 715, 112, 110, 117, 116} $\frac{2 e \sqrt{b x+c x^2} \left (9 b^2 c^2 d^2 e^2+7 b^3 c d e^3-8 b^4 e^4-32 b c^3 d^3 e+16 c^4 d^4\right )}{3 b^4 d^3 \sqrt{d+e x} (c d-b e)^3}+\frac{2 \left (4 c x \left (b^3 e^3-6 b c^2 d^2 e+4 c^3 d^3\right )+b (c d-b e) \left (-4 b^2 e^2-3 b c d e+8 c^2 d^2\right )\right )}{3 b^4 d^2 \sqrt{b x+c x^2} \sqrt{d+e x} (c d-b e)^2}+\frac{8 \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (2 c d-b e) \left (-b^2 e^2-2 b c d e+2 c^2 d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 (-b)^{7/2} d^2 \sqrt{b x+c x^2} \sqrt{d+e x} (c d-b e)^2}-\frac{2 \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (9 b^2 c^2 d^2 e^2+7 b^3 c d e^3-8 b^4 e^4-32 b c^3 d^3 e+16 c^4 d^4\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 (-b)^{7/2} d^3 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)^3}-\frac{2 (c x (2 c d-b e)+b (c d-b e))}{3 b^2 d \left (b x+c x^2\right )^{3/2} \sqrt{d+e x} (c d-b e)}$

Antiderivative was successfully veriﬁed.

[In]

Int[1/((d + e*x)^(3/2)*(b*x + c*x^2)^(5/2)),x]

[Out]

(-2*(b*(c*d - b*e) + c*(2*c*d - b*e)*x))/(3*b^2*d*(c*d - b*e)*Sqrt[d + e*x]*(b*x + c*x^2)^(3/2)) + (2*(b*(c*d
- b*e)*(8*c^2*d^2 - 3*b*c*d*e - 4*b^2*e^2) + 4*c*(4*c^3*d^3 - 6*b*c^2*d^2*e + b^3*e^3)*x))/(3*b^4*d^2*(c*d - b
*e)^2*Sqrt[d + e*x]*Sqrt[b*x + c*x^2]) + (2*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[b*x + c*x^2])/(3*b^4*d^3*(c*d - b*e)^3*Sqrt[d + e*x]) - (2*Sqrt[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)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE[ArcSin
[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(3*(-b)^(7/2)*d^3*(c*d - b*e)^3*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2
]) + (8*Sqrt[c]*(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]*El
lipticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(3*(-b)^(7/2)*d^2*(c*d - b*e)^2*Sqrt[d + e*x]*Sqrt[b
*x + c*x^2])

Rule 740

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(
b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e
+ a*e^2)), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*Simp[b*c*d*e*(2*p - m
+ 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*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] && LtQ[p, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]

Rule 822

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[((d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x
)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((p + 1)*(b^2 - 4*a*
c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2
*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d*m + b*e*m) - b*d*(3*c*d -
b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*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] && LtQ[p, -1] && (IntegerQ[m] ||
IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 834

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((m
+ 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[(c*d*f - f*b*e + a*e*g)*(m + 1)
+ b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*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] && LtQ[m, -1] && (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 \frac{1}{(d+e x)^{3/2} \left (b x+c x^2\right )^{5/2}} \, dx &=-\frac{2 (b (c d-b e)+c (2 c d-b e) x)}{3 b^2 d (c d-b e) \sqrt{d+e x} \left (b x+c x^2\right )^{3/2}}-\frac{2 \int \frac{\frac{1}{2} \left (8 c^2 d^2-3 b c d e-4 b^2 e^2\right )+\frac{5}{2} c e (2 c d-b e) x}{(d+e x)^{3/2} \left (b x+c x^2\right )^{3/2}} \, dx}{3 b^2 d (c d-b e)}\\ &=-\frac{2 (b (c d-b e)+c (2 c d-b e) x)}{3 b^2 d (c d-b e) \sqrt{d+e x} \left (b x+c x^2\right )^{3/2}}+\frac{2 \left (b (c d-b e) \left (8 c^2 d^2-3 b c d e-4 b^2 e^2\right )+4 c \left (4 c^3 d^3-6 b c^2 d^2 e+b^3 e^3\right ) x\right )}{3 b^4 d^2 (c d-b e)^2 \sqrt{d+e x} \sqrt{b x+c x^2}}+\frac{4 \int \frac{\frac{1}{4} b e \left (8 c^3 d^3-9 b c^2 d^2 e-3 b^2 c d e^2+8 b^3 e^3\right )+c e \left (4 c^3 d^3-6 b c^2 d^2 e+b^3 e^3\right ) x}{(d+e x)^{3/2} \sqrt{b x+c x^2}} \, dx}{3 b^4 d^2 (c d-b e)^2}\\ &=-\frac{2 (b (c d-b e)+c (2 c d-b e) x)}{3 b^2 d (c d-b e) \sqrt{d+e x} \left (b x+c x^2\right )^{3/2}}+\frac{2 \left (b (c d-b e) \left (8 c^2 d^2-3 b c d e-4 b^2 e^2\right )+4 c \left (4 c^3 d^3-6 b c^2 d^2 e+b^3 e^3\right ) x\right )}{3 b^4 d^2 (c d-b e)^2 \sqrt{d+e x} \sqrt{b x+c x^2}}+\frac{2 e \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{b x+c x^2}}{3 b^4 d^3 (c d-b e)^3 \sqrt{d+e x}}-\frac{8 \int \frac{\frac{1}{8} b c d e \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}{8} c e \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}{3 b^4 d^3 (c d-b e)^3}\\ &=-\frac{2 (b (c d-b e)+c (2 c d-b e) x)}{3 b^2 d (c d-b e) \sqrt{d+e x} \left (b x+c x^2\right )^{3/2}}+\frac{2 \left (b (c d-b e) \left (8 c^2 d^2-3 b c d e-4 b^2 e^2\right )+4 c \left (4 c^3 d^3-6 b c^2 d^2 e+b^3 e^3\right ) x\right )}{3 b^4 d^2 (c d-b e)^2 \sqrt{d+e x} \sqrt{b x+c x^2}}+\frac{2 e \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{b x+c x^2}}{3 b^4 d^3 (c d-b e)^3 \sqrt{d+e x}}+\frac{\left (4 c (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}{3 b^4 d^2 (c d-b e)^2}-\frac{\left (c \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 )\right ) \int \frac{\sqrt{d+e x}}{\sqrt{b x+c x^2}} \, dx}{3 b^4 d^3 (c d-b e)^3}\\ &=-\frac{2 (b (c d-b e)+c (2 c d-b e) x)}{3 b^2 d (c d-b e) \sqrt{d+e x} \left (b x+c x^2\right )^{3/2}}+\frac{2 \left (b (c d-b e) \left (8 c^2 d^2-3 b c d e-4 b^2 e^2\right )+4 c \left (4 c^3 d^3-6 b c^2 d^2 e+b^3 e^3\right ) x\right )}{3 b^4 d^2 (c d-b e)^2 \sqrt{d+e x} \sqrt{b x+c x^2}}+\frac{2 e \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{b x+c x^2}}{3 b^4 d^3 (c d-b e)^3 \sqrt{d+e x}}+\frac{\left (4 c (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}{3 b^4 d^2 (c d-b e)^2 \sqrt{b x+c x^2}}-\frac{\left (c \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}{3 b^4 d^3 (c d-b e)^3 \sqrt{b x+c x^2}}\\ &=-\frac{2 (b (c d-b e)+c (2 c d-b e) x)}{3 b^2 d (c d-b e) \sqrt{d+e x} \left (b x+c x^2\right )^{3/2}}+\frac{2 \left (b (c d-b e) \left (8 c^2 d^2-3 b c d e-4 b^2 e^2\right )+4 c \left (4 c^3 d^3-6 b c^2 d^2 e+b^3 e^3\right ) x\right )}{3 b^4 d^2 (c d-b e)^2 \sqrt{d+e x} \sqrt{b x+c x^2}}+\frac{2 e \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{b x+c x^2}}{3 b^4 d^3 (c d-b e)^3 \sqrt{d+e x}}-\frac{\left (c \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}{3 b^4 d^3 (c d-b e)^3 \sqrt{1+\frac{e x}{d}} \sqrt{b x+c x^2}}+\frac{\left (4 c (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}{3 b^4 d^2 (c d-b e)^2 \sqrt{d+e x} \sqrt{b x+c x^2}}\\ &=-\frac{2 (b (c d-b e)+c (2 c d-b e) x)}{3 b^2 d (c d-b e) \sqrt{d+e x} \left (b x+c x^2\right )^{3/2}}+\frac{2 \left (b (c d-b e) \left (8 c^2 d^2-3 b c d e-4 b^2 e^2\right )+4 c \left (4 c^3 d^3-6 b c^2 d^2 e+b^3 e^3\right ) x\right )}{3 b^4 d^2 (c d-b e)^2 \sqrt{d+e x} \sqrt{b x+c x^2}}+\frac{2 e \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{b x+c x^2}}{3 b^4 d^3 (c d-b e)^3 \sqrt{d+e x}}-\frac{2 \sqrt{c} \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 )}{3 (-b)^{7/2} d^3 (c d-b e)^3 \sqrt{1+\frac{e x}{d}} \sqrt{b x+c x^2}}+\frac{8 \sqrt{c} (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 )}{3 (-b)^{7/2} d^2 (c d-b e)^2 \sqrt{d+e x} \sqrt{b x+c x^2}}\\ \end{align*}

Mathematica [C]  time = 2.95524, size = 504, normalized size = 0.89 $-\frac{2 \left (b \left (3 b^4 e^5 x^2 (b+c x)^2+b c^4 d^3 x^2 (d+e x) (b e-c d)-c^4 d^3 x^2 (b+c x) (d+e x) (8 c d-13 b e)+b d (b+c x)^2 (d+e x) (c d-b e)^3-x (b+c x)^2 (d+e x) (c d-b e)^3 (5 b e+8 c d)\right )+c x \sqrt{\frac{b}{c}} (b+c x) \left (-i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (6 b^2 c^2 d^2 e^2+11 b^3 c d e^3-8 b^4 e^4-17 b c^3 d^3 e+8 c^4 d^4\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right ),\frac{c d}{b e}\right )+i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (9 b^2 c^2 d^2 e^2+7 b^3 c d e^3-8 b^4 e^4-32 b c^3 d^3 e+16 c^4 d^4\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+\sqrt{\frac{b}{c}} (b+c x) (d+e x) \left (9 b^2 c^2 d^2 e^2+7 b^3 c d e^3-8 b^4 e^4-32 b c^3 d^3 e+16 c^4 d^4\right )\right )\right )}{3 b^5 d^3 (x (b+c x))^{3/2} \sqrt{d+e x} (c d-b e)^3}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[1/((d + e*x)^(3/2)*(b*x + c*x^2)^(5/2)),x]

[Out]

(-2*(b*(3*b^4*e^5*x^2*(b + c*x)^2 + b*c^4*d^3*(-(c*d) + b*e)*x^2*(d + e*x) - c^4*d^3*(8*c*d - 13*b*e)*x^2*(b +
c*x)*(d + e*x) + b*d*(c*d - b*e)^3*(b + c*x)^2*(d + e*x) - (c*d - b*e)^3*(8*c*d + 5*b*e)*x*(b + c*x)^2*(d + e
*x)) + Sqrt[b/c]*c*x*(b + c*x)*(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)*EllipticE[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)])))/(3*b^5*d^3*(c*d - b*e)^3*(x*(b + c*x))^(3
/2)*Sqrt[d + e*x])

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Maple [B]  time = 0.315, size = 2189, normalized size = 3.9 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^(3/2)/(c*x^2+b*x)^(5/2),x)

[Out]

2/3*(-18*x^2*b^4*c^3*d^2*e^3-2*x^2*b^3*c^4*d^3*e^2+43*x^2*b^2*c^5*d^4*e+14*x*b^3*c^4*d^4*e-7*x^4*b^3*c^4*d*e^4
-9*x^4*b^2*c^5*d^2*e^3+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))*x^2*b^6*c*e^5+4*x*b^6*c*d*e^4-6*x*b^5*c^2*d^2*e^3-6*x*b^4*c^3*d^3*e^2+8*x^3*b*c^6*
d^4*e-10*x^3*b^4*c^3*d*e^4-22*x^3*b^3*c^4*d^2*e^3+40*x^3*b^2*c^5*d^3*e^2+x^2*b^5*c^2*d*e^4+8*x^2*b^6*c*e^5-2*(
(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)
)*x^2*b^4*c^3*d^2*e^3-4*((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))*x*b^5*c^2*d^2*e^3-24*((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))*x*b^4*c^3*d^3*e^2+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))*x^2*b^5*c^2*d*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))*x^2*b^4*c^3*d
^2*e^3-24*((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))*x^2*b^3*c^4*d^3*e^2+40*((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))*x^2*b^2*c^5*d^4*e-16*x^4*c^7*d^4*e+16*x^3*b^5*c^2*e^5+8*x^4*b^4*c^3*e^5
-6*x*b^2*c^5*d^5-24*x^2*b*c^6*d^5+32*x^4*b*c^6*d^3*e^2-16*x^3*c^7*d^5-15*((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))*x*b^6*c*d*e^4-48*((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))*x*b^3*c^4*d^4*
e+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))*x*b^6*c*d*e^4-15*((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))*x^2*b^5*c^2*d*e^4+40*((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))*x*b^3*c^4*d^4*e+41*((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))*x^2*b^3*c^4*d^3*e^2-48*((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))*x^2*b^2*c^5
*d^4*e-2*((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))*x*b^5*c^2*d^2*e^3+41*((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))*x*b^4*c^3*d^3*e^2+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))*x*b^7*e^5+b^3*c^4*d^5+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))*x^2*b*c^6*d^5-16*((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))*x^2
*b*c^6*d^5+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))*x*b^2*c^5*d^5-16*((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))*x*b^2*c^5*d^5-3*b^4*c^3*d^4*e-b^6*c*d^2*e^3+3*b^5*c^2*d^3*e^2)/x^2*(x*(c*
x+b))^(1/2)/b^4/d^3/c/(c*x+b)^2/(b*e-c*d)^3/(e*x+d)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (c x^{2} + b x\right )}^{\frac{5}{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(1/(e*x+d)^(3/2)/(c*x^2+b*x)^(5/2),x, algorithm="maxima")

[Out]

integrate(1/((c*x^2 + b*x)^(5/2)*(e*x + d)^(3/2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{c x^{2} + b x} \sqrt{e x + d}}{c^{3} e^{2} x^{8} + b^{3} d^{2} x^{3} +{\left (2 \, c^{3} d e + 3 \, b c^{2} e^{2}\right )} x^{7} +{\left (c^{3} d^{2} + 6 \, b c^{2} d e + 3 \, b^{2} c e^{2}\right )} x^{6} +{\left (3 \, b c^{2} d^{2} + 6 \, b^{2} c d e + b^{3} e^{2}\right )} x^{5} +{\left (3 \, b^{2} c d^{2} + 2 \, b^{3} d e\right )} x^{4}}, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(3/2)/(c*x^2+b*x)^(5/2),x, algorithm="fricas")

[Out]

integral(sqrt(c*x^2 + b*x)*sqrt(e*x + d)/(c^3*e^2*x^8 + b^3*d^2*x^3 + (2*c^3*d*e + 3*b*c^2*e^2)*x^7 + (c^3*d^2
+ 6*b*c^2*d*e + 3*b^2*c*e^2)*x^6 + (3*b*c^2*d^2 + 6*b^2*c*d*e + b^3*e^2)*x^5 + (3*b^2*c*d^2 + 2*b^3*d*e)*x^4)
, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (x \left (b + c x\right )\right )^{\frac{5}{2}} \left (d + e x\right )^{\frac{3}{2}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)**(3/2)/(c*x**2+b*x)**(5/2),x)

[Out]

Integral(1/((x*(b + c*x))**(5/2)*(d + e*x)**(3/2)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (c x^{2} + b x\right )}^{\frac{5}{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(1/(e*x+d)^(3/2)/(c*x^2+b*x)^(5/2),x, algorithm="giac")

[Out]

integrate(1/((c*x^2 + b*x)^(5/2)*(e*x + d)^(3/2)), x)