\(\int \frac {1}{(d+e x)^{3/2} (b x+c x^2)^{5/2}} \, dx\) [427]

Optimal result

Integrand size = 23, antiderivative size = 567 \[ \int \frac {1}{(d+e x)^{3/2} \left (b x+c x^2\right )^{5/2}} \, dx=-\frac {2 (b (c d-b e)+c (2 c d-b e) x)}{3 b^2 d (c d-b e) \sqrt {d+e x} \left (b x+c x^2\right )^{3/2}}+\frac {2 \left (b (c d-b e) \left (8 c^2 d^2-3 b c d e-4 b^2 e^2\right )+4 c \left (4 c^3 d^3-6 b c^2 d^2 e+b^3 e^3\right ) x\right )}{3 b^4 d^2 (c d-b e)^2 \sqrt {d+e x} \sqrt {b x+c x^2}}+\frac {2 e \left (16 c^4 d^4-32 b c^3 d^3 e+9 b^2 c^2 d^2 e^2+7 b^3 c d e^3-8 b^4 e^4\right ) \sqrt {b x+c x^2}}{3 b^4 d^3 (c d-b e)^3 \sqrt {d+e x}}-\frac {2 \sqrt {c} \left (16 c^4 d^4-32 b c^3 d^3 e+9 b^2 c^2 d^2 e^2+7 b^3 c d e^3-8 b^4 e^4\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{7/2} d^3 (c d-b e)^3 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {8 \sqrt {c} (2 c d-b e) \left (2 c^2 d^2-2 b c d e-b^2 e^2\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{3 (-b)^{7/2} d^2 (c d-b e)^2 \sqrt {d+e x} \sqrt {b x+c x^2}} \]

[Out]

-2/3*(b*(-b*e+c*d)+c*(-b*e+2*c*d)*x)/b^2/d/(-b*e+c*d)/(c*x^2+b*x)^(3/2)/(e*x+d)^(1/2)+2/3*(b*(-b*e+c*d)*(-4*b^
2*e^2-3*b*c*d*e+8*c^2*d^2)+4*c*(b^3*e^3-6*b*c^2*d^2*e+4*c^3*d^3)*x)/b^4/d^2/(-b*e+c*d)^2/(e*x+d)^(1/2)/(c*x^2+
b*x)^(1/2)-2/3*(-8*b^4*e^4+7*b^3*c*d*e^3+9*b^2*c^2*d^2*e^2-32*b*c^3*d^3*e+16*c^4*d^4)*EllipticE(c^(1/2)*x^(1/2
)/(-b)^(1/2),(b*e/c/d)^(1/2))*c^(1/2)*x^(1/2)*(1+c*x/b)^(1/2)*(e*x+d)^(1/2)/(-b)^(7/2)/d^3/(-b*e+c*d)^3/(1+e*x
/d)^(1/2)/(c*x^2+b*x)^(1/2)+8/3*(-b*e+2*c*d)*(-b^2*e^2-2*b*c*d*e+2*c^2*d^2)*EllipticF(c^(1/2)*x^(1/2)/(-b)^(1/
2),(b*e/c/d)^(1/2))*c^(1/2)*x^(1/2)*(1+c*x/b)^(1/2)*(1+e*x/d)^(1/2)/(-b)^(7/2)/d^2/(-b*e+c*d)^2/(e*x+d)^(1/2)/
(c*x^2+b*x)^(1/2)+2/3*e*(-8*b^4*e^4+7*b^3*c*d*e^3+9*b^2*c^2*d^2*e^2-32*b*c^3*d^3*e+16*c^4*d^4)*(c*x^2+b*x)^(1/
2)/b^4/d^3/(-b*e+c*d)^3/(e*x+d)^(1/2)

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 567, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {754, 836, 848, 857, 729, 113, 111, 118, 117} \[ \int \frac {1}{(d+e x)^{3/2} \left (b x+c x^2\right )^{5/2}} \, dx=\frac {8 \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (2 c d-b e) \left (-b^2 e^2-2 b c d e+2 c^2 d^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{3 (-b)^{7/2} d^2 \sqrt {b x+c x^2} \sqrt {d+e x} (c d-b e)^2}-\frac {2 \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (-8 b^4 e^4+7 b^3 c d e^3+9 b^2 c^2 d^2 e^2-32 b c^3 d^3 e+16 c^4 d^4\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{7/2} d^3 \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1} (c d-b e)^3}-\frac {2 (c x (2 c d-b e)+b (c d-b e))}{3 b^2 d \left (b x+c x^2\right )^{3/2} \sqrt {d+e x} (c d-b e)}+\frac {2 \left (4 c x \left (b^3 e^3-6 b c^2 d^2 e+4 c^3 d^3\right )+b (c d-b e) \left (-4 b^2 e^2-3 b c d e+8 c^2 d^2\right )\right )}{3 b^4 d^2 \sqrt {b x+c x^2} \sqrt {d+e x} (c d-b e)^2}+\frac {2 e \sqrt {b x+c x^2} \left (-8 b^4 e^4+7 b^3 c d e^3+9 b^2 c^2 d^2 e^2-32 b c^3 d^3 e+16 c^4 d^4\right )}{3 b^4 d^3 \sqrt {d+e x} (c d-b e)^3} \]

[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 111

Int[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[2*(Sqrt[e]/b)*Rt[-b/
d, 2]*EllipticE[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[-b/d, 2])], c*(f/(d*e))], 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 113

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 117

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

Rule 118

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 729

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 754

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 836

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 848

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 857

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]

Rubi steps \begin{align*} \text {integral}& = -\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] (verified)

Result contains complex when optimal does not.

Time = 14.07 (sec) , antiderivative size = 504, normalized size of antiderivative = 0.89 \[ \int \frac {1}{(d+e x)^{3/2} \left (b x+c x^2\right )^{5/2}} \, dx=-\frac {2 \left (b \left (3 b^4 e^5 x^2 (b+c x)^2+b c^4 d^3 (-c d+b e) x^2 (d+e x)-c^4 d^3 (8 c d-13 b e) x^2 (b+c x) (d+e x)+b d (c d-b e)^3 (b+c x)^2 (d+e x)-(c d-b e)^3 (8 c d+5 b e) x (b+c x)^2 (d+e x)\right )+\sqrt {\frac {b}{c}} c x (b+c x) \left (\sqrt {\frac {b}{c}} \left (16 c^4 d^4-32 b c^3 d^3 e+9 b^2 c^2 d^2 e^2+7 b^3 c d e^3-8 b^4 e^4\right ) (b+c x) (d+e x)+i b e \left (16 c^4 d^4-32 b c^3 d^3 e+9 b^2 c^2 d^2 e^2+7 b^3 c d e^3-8 b^4 e^4\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )-i b e \left (8 c^4 d^4-17 b c^3 d^3 e+6 b^2 c^2 d^2 e^2+11 b^3 c d e^3-8 b^4 e^4\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right ),\frac {c d}{b e}\right )\right )\right )}{3 b^5 d^3 (c d-b e)^3 (x (b+c x))^{3/2} \sqrt {d+e x}} \]

[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])

Maple [A] (verified)

Time = 2.72 (sec) , antiderivative size = 762, normalized size of antiderivative = 1.34

method	result	size
elliptic	\(\frac {\sqrt {x \left (e x +d \right ) \left (c x +b \right )}\, \left (-\frac {2 \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{3 d^{2} b^{3} x^{2}}+\frac {2 \left (c e \,x^{2}+b e x +c d x +b d \right ) \left (5 b e +8 c d \right )}{3 b^{4} d^{3} \sqrt {x \left (c e \,x^{2}+b e x +c d x +b d \right )}}+\frac {2 c^{2} \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{3 b^{3} \left (b e -c d \right )^{2} \left (\frac {b}{c}+x \right )^{2}}+\frac {2 \left (c e \,x^{2}+c d x \right ) c^{3} \left (13 b e -8 c d \right )}{3 b^{4} \left (b e -c d \right )^{3} \sqrt {\left (\frac {b}{c}+x \right ) \left (c e \,x^{2}+c d x \right )}}+\frac {2 \left (c e \,x^{2}+b e x \right ) e^{4}}{\left (b e -c d \right )^{3} d^{3} \sqrt {\left (x +\frac {d}{e}\right ) \left (c e \,x^{2}+b e x \right )}}+\frac {2 \left (-\frac {c e}{3 b^{3} d^{2}}+\frac {e \,c^{3}}{3 \left (b e -c d \right )^{2} b^{3}}-\frac {c^{3} \left (13 b e -8 c d \right )}{3 \left (b e -c d \right )^{2} b^{4}}-\frac {c^{4} d \left (13 b e -8 c d \right )}{3 b^{4} \left (b e -c d \right )^{3}}+\frac {e^{4}}{\left (b e -c d \right )^{2} d^{3}}-\frac {b \,e^{5}}{\left (b e -c d \right )^{3} d^{3}}\right ) b \sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, F\left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{c \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}+\frac {2 \left (-\frac {c e \left (5 b e +8 c d \right )}{3 b^{4} d^{3}}-\frac {c^{4} e \left (13 b e -8 c d \right )}{3 \left (b e -c d \right )^{3} b^{4}}-\frac {c \,e^{5}}{d^{3} \left (b e -c d \right )^{3}}\right ) b \sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, \left (\left (-\frac {b}{c}+\frac {d}{e}\right ) E\left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )-\frac {d F\left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{e}\right )}{c \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}\right )}{\sqrt {x \left (c x +b \right )}\, \sqrt {e x +d}}\)	\(762\)
default	\(\text {Expression too large to display}\)	\(2189\)

[In]

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

[Out]

(x*(e*x+d)*(c*x+b))^(1/2)/(x*(c*x+b))^(1/2)/(e*x+d)^(1/2)*(-2/3/d^2/b^3*(c*e*x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)/
x^2+2/3*(c*e*x^2+b*e*x+c*d*x+b*d)/b^4/d^3*(5*b*e+8*c*d)/(x*(c*e*x^2+b*e*x+c*d*x+b*d))^(1/2)+2/3/b^3/(b*e-c*d)^
2*c^2*(c*e*x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)/(1/c*b+x)^2+2/3*(c*e*x^2+c*d*x)/b^4/(b*e-c*d)^3*c^3*(13*b*e-8*c*d)
/((1/c*b+x)*(c*e*x^2+c*d*x))^(1/2)+2*(c*e*x^2+b*e*x)/(b*e-c*d)^3*e^4/d^3/((x+d/e)*(c*e*x^2+b*e*x))^(1/2)+2*(-1
/3/b^3/d^2*c*e+1/3*e*c^3/(b*e-c*d)^2/b^3-1/3*c^3/(b*e-c*d)^2*(13*b*e-8*c*d)/b^4-1/3*c^4*d/b^4/(b*e-c*d)^3*(13*
b*e-8*c*d)+e^4/(b*e-c*d)^2/d^3-b*e^5/(b*e-c*d)^3/d^3)/c*b*((1/c*b+x)*c/b)^(1/2)*((x+d/e)/(-1/c*b+d/e))^(1/2)*(
-c*x/b)^(1/2)/(c*e*x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)*EllipticF(((1/c*b+x)*c/b)^(1/2),(-1/c*b/(-1/c*b+d/e))^(1/2
))+2*(-1/3*c*e*(5*b*e+8*c*d)/b^4/d^3-1/3*c^4*e*(13*b*e-8*c*d)/(b*e-c*d)^3/b^4-c*e^5/d^3/(b*e-c*d)^3)/c*b*((1/c
*b+x)*c/b)^(1/2)*((x+d/e)/(-1/c*b+d/e))^(1/2)*(-c*x/b)^(1/2)/(c*e*x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)*((-1/c*b+d/
e)*EllipticE(((1/c*b+x)*c/b)^(1/2),(-1/c*b/(-1/c*b+d/e))^(1/2))-d/e*EllipticF(((1/c*b+x)*c/b)^(1/2),(-1/c*b/(-
1/c*b+d/e))^(1/2))))

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.17 (sec) , antiderivative size = 1510, normalized size of antiderivative = 2.66 \[ \int \frac {1}{(d+e x)^{3/2} \left (b x+c x^2\right )^{5/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

2/9*(((16*c^7*d^5*e - 40*b*c^6*d^4*e^2 + 22*b^2*c^5*d^3*e^3 + 7*b^3*c^4*d^2*e^4 + 11*b^4*c^3*d*e^5 - 8*b^5*c^2
*e^6)*x^5 + (16*c^7*d^6 - 8*b*c^6*d^5*e - 58*b^2*c^5*d^4*e^2 + 51*b^3*c^4*d^3*e^3 + 25*b^4*c^3*d^2*e^4 + 14*b^
5*c^2*d*e^5 - 16*b^6*c*e^6)*x^4 + (32*b*c^6*d^6 - 64*b^2*c^5*d^5*e + 4*b^3*c^4*d^4*e^2 + 36*b^4*c^3*d^3*e^3 +
29*b^5*c^2*d^2*e^4 - 5*b^6*c*d*e^5 - 8*b^7*e^6)*x^3 + (16*b^2*c^5*d^6 - 40*b^3*c^4*d^5*e + 22*b^4*c^3*d^4*e^2
+ 7*b^5*c^2*d^3*e^3 + 11*b^6*c*d^2*e^4 - 8*b^7*d*e^5)*x^2)*sqrt(c*e)*weierstrassPInverse(4/3*(c^2*d^2 - b*c*d*
e + b^2*e^2)/(c^2*e^2), -4/27*(2*c^3*d^3 - 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 + 2*b^3*e^3)/(c^3*e^3), 1/3*(3*c*e*x
+ c*d + b*e)/(c*e)) + 3*((16*c^7*d^4*e^2 - 32*b*c^6*d^3*e^3 + 9*b^2*c^5*d^2*e^4 + 7*b^3*c^4*d*e^5 - 8*b^4*c^3*
e^6)*x^5 + (16*c^7*d^5*e - 55*b^2*c^5*d^3*e^3 + 25*b^3*c^4*d^2*e^4 + 6*b^4*c^3*d*e^5 - 16*b^5*c^2*e^6)*x^4 + (
32*b*c^6*d^5*e - 48*b^2*c^5*d^4*e^2 - 14*b^3*c^4*d^3*e^3 + 23*b^4*c^3*d^2*e^4 - 9*b^5*c^2*d*e^5 - 8*b^6*c*e^6)
*x^3 + (16*b^2*c^5*d^5*e - 32*b^3*c^4*d^4*e^2 + 9*b^4*c^3*d^3*e^3 + 7*b^5*c^2*d^2*e^4 - 8*b^6*c*d*e^5)*x^2)*sq
rt(c*e)*weierstrassZeta(4/3*(c^2*d^2 - b*c*d*e + b^2*e^2)/(c^2*e^2), -4/27*(2*c^3*d^3 - 3*b*c^2*d^2*e - 3*b^2*
c*d*e^2 + 2*b^3*e^3)/(c^3*e^3), weierstrassPInverse(4/3*(c^2*d^2 - b*c*d*e + b^2*e^2)/(c^2*e^2), -4/27*(2*c^3*
d^3 - 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 + 2*b^3*e^3)/(c^3*e^3), 1/3*(3*c*e*x + c*d + b*e)/(c*e))) - 3*(b^3*c^4*d^5
*e - 3*b^4*c^3*d^4*e^2 + 3*b^5*c^2*d^3*e^3 - b^6*c*d^2*e^4 - (16*c^7*d^4*e^2 - 32*b*c^6*d^3*e^3 + 9*b^2*c^5*d^
2*e^4 + 7*b^3*c^4*d*e^5 - 8*b^4*c^3*e^6)*x^4 - 2*(8*c^7*d^5*e - 4*b*c^6*d^4*e^2 - 20*b^2*c^5*d^3*e^3 + 11*b^3*
c^4*d^2*e^4 + 5*b^4*c^3*d*e^5 - 8*b^5*c^2*e^6)*x^3 - (24*b*c^6*d^5*e - 43*b^2*c^5*d^4*e^2 + 2*b^3*c^4*d^3*e^3
+ 18*b^4*c^3*d^2*e^4 - b^5*c^2*d*e^5 - 8*b^6*c*e^6)*x^2 - 2*(3*b^2*c^5*d^5*e - 7*b^3*c^4*d^4*e^2 + 3*b^4*c^3*d
^3*e^3 + 3*b^5*c^2*d^2*e^4 - 2*b^6*c*d*e^5)*x)*sqrt(c*x^2 + b*x)*sqrt(e*x + d))/((b^4*c^6*d^6*e^2 - 3*b^5*c^5*
d^5*e^3 + 3*b^6*c^4*d^4*e^4 - b^7*c^3*d^3*e^5)*x^5 + (b^4*c^6*d^7*e - b^5*c^5*d^6*e^2 - 3*b^6*c^4*d^5*e^3 + 5*
b^7*c^3*d^4*e^4 - 2*b^8*c^2*d^3*e^5)*x^4 + (2*b^5*c^5*d^7*e - 5*b^6*c^4*d^6*e^2 + 3*b^7*c^3*d^5*e^3 + b^8*c^2*
d^4*e^4 - b^9*c*d^3*e^5)*x^3 + (b^6*c^4*d^7*e - 3*b^7*c^3*d^6*e^2 + 3*b^8*c^2*d^5*e^3 - b^9*c*d^4*e^4)*x^2)

Sympy [F]

\[ \int \frac {1}{(d+e x)^{3/2} \left (b x+c x^2\right )^{5/2}} \, dx=\int \frac {1}{\left (x \left (b + c x\right )\right )^{\frac {5}{2}} \left (d + e x\right )^{\frac {3}{2}}}\, dx \]

[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)

Maxima [F]

\[ \int \frac {1}{(d+e x)^{3/2} \left (b x+c x^2\right )^{5/2}} \, dx=\int { \frac {1}{{\left (c x^{2} + b x\right )}^{\frac {5}{2}} {\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]

[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)

Giac [F]

\[ \int \frac {1}{(d+e x)^{3/2} \left (b x+c x^2\right )^{5/2}} \, dx=\int { \frac {1}{{\left (c x^{2} + b x\right )}^{\frac {5}{2}} {\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x)^{3/2} \left (b x+c x^2\right )^{5/2}} \, dx=\int \frac {1}{{\left (c\,x^2+b\,x\right )}^{5/2}\,{\left (d+e\,x\right )}^{3/2}} \,d x \]

[In]

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

[Out]

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