\(\int \frac {(d+e x)^{9/2}}{(b x+c x^2)^{5/2}} \, dx\) [421]

Optimal result

Integrand size = 23, antiderivative size = 470 \[ \int \frac {(d+e x)^{9/2}}{\left (b x+c x^2\right )^{5/2}} \, dx=-\frac {2 (d+e x)^{7/2} (b d+(2 c d-b e) x)}{3 b^2 \left (b x+c x^2\right )^{3/2}}+\frac {2 (d+e x)^{3/2} \left (b c d^2 (8 c d-11 b e)+(2 c d-b e) \left (8 c^2 d^2-8 b c d e-3 b^2 e^2\right ) x\right )}{3 b^4 c \sqrt {b x+c x^2}}-\frac {8 e \left (4 c^3 d^3-6 b c^2 d^2 e+b^3 e^3\right ) \sqrt {d+e x} \sqrt {b x+c x^2}}{3 b^4 c^2}-\frac {2 \left (16 c^4 d^4-32 b c^3 d^3 e+9 b^2 c^2 d^2 e^2+7 b^3 c d e^3-8 b^4 e^4\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{7/2} c^{5/2} \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {8 d (c d-b e) (2 c d-b e) \left (2 c^2 d^2-2 b c d e-b^2 e^2\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{3 (-b)^{7/2} c^{5/2} \sqrt {d+e x} \sqrt {b x+c x^2}} \]

[Out]

-2/3*(e*x+d)^(7/2)*(b*d+(-b*e+2*c*d)*x)/b^2/(c*x^2+b*x)^(3/2)+2/3*(e*x+d)^(3/2)*(b*c*d^2*(-11*b*e+8*c*d)+(-b*e
+2*c*d)*(-3*b^2*e^2-8*b*c*d*e+8*c^2*d^2)*x)/b^4/c/(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))*x^(1/2)*(1+c*x/b)^(1/2)
*(e*x+d)^(1/2)/(-b)^(7/2)/c^(5/2)/(1+e*x/d)^(1/2)/(c*x^2+b*x)^(1/2)+8/3*d*(-b*e+c*d)*(-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))*x^(1/2)*(1+c*x/b)^(1/2)*(1+e*x/d)^(1/
2)/(-b)^(7/2)/c^(5/2)/(e*x+d)^(1/2)/(c*x^2+b*x)^(1/2)-8/3*e*(b^3*e^3-6*b*c^2*d^2*e+4*c^3*d^3)*(e*x+d)^(1/2)*(c
*x^2+b*x)^(1/2)/b^4/c^2

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 470, 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 = {752, 832, 846, 857, 729, 113, 111, 118, 117} \[ \int \frac {(d+e x)^{9/2}}{\left (b x+c x^2\right )^{5/2}} \, dx=\frac {8 d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) (2 c d-b e) \left (-b^2 e^2-2 b c d e+2 c^2 d^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{3 (-b)^{7/2} c^{5/2} \sqrt {b x+c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (-8 b^4 e^4+7 b^3 c d e^3+9 b^2 c^2 d^2 e^2-32 b c^3 d^3 e+16 c^4 d^4\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{7/2} c^{5/2} \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {2 (d+e x)^{7/2} (x (2 c d-b e)+b d)}{3 b^2 \left (b x+c x^2\right )^{3/2}}-\frac {8 e \sqrt {b x+c x^2} \sqrt {d+e x} \left (b^3 e^3-6 b c^2 d^2 e+4 c^3 d^3\right )}{3 b^4 c^2}+\frac {2 (d+e x)^{3/2} \left (x (2 c d-b e) \left (-3 b^2 e^2-8 b c d e+8 c^2 d^2\right )+b c d^2 (8 c d-11 b e)\right )}{3 b^4 c \sqrt {b x+c x^2}} \]

[In]

Int[(d + e*x)^(9/2)/(b*x + c*x^2)^(5/2),x]

[Out]

(-2*(d + e*x)^(7/2)*(b*d + (2*c*d - b*e)*x))/(3*b^2*(b*x + c*x^2)^(3/2)) + (2*(d + e*x)^(3/2)*(b*c*d^2*(8*c*d
- 11*b*e) + (2*c*d - b*e)*(8*c^2*d^2 - 8*b*c*d*e - 3*b^2*e^2)*x))/(3*b^4*c*Sqrt[b*x + c*x^2]) - (8*e*(4*c^3*d^
3 - 6*b*c^2*d^2*e + b^3*e^3)*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])/(3*b^4*c^2) - (2*(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)*c^(5/2)*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) + (8*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[Arc
Sin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(3*(-b)^(7/2)*c^(5/2)*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 752

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m - 1)*(d
*b - 2*a*e + (2*c*d - b*e)*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Dist[1/((p + 1)*(b^2 -
4*a*c)), Int[(d + e*x)^(m - 2)*Simp[e*(2*a*e*(m - 1) + b*d*(2*p - m + 4)) - 2*c*d^2*(2*p + 3) + e*(b*e - 2*d*
c)*(m + 2*p + 2)*x, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, 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] && GtQ[m, 1] && 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[(-(d + e*x)^(m - 1))*(a + b*x + c*x^2)^(p + 1)*((2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*
g - c*(b*e*f + b*d*g + 2*a*e*g))*x)/(c*(p + 1)*(b^2 - 4*a*c))), x] - Dist[1/(c*(p + 1)*(b^2 - 4*a*c)), Int[(d
+ e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1)*Simp[2*c^2*d^2*f*(2*p + 3) + b*e*g*(a*e*(m - 1) + b*d*(p + 2)) - c*(2
*a*e*(e*f*(m - 1) + d*g*m) + b*d*(d*g*(2*p + 3) - e*f*(m - 2*p - 4))) + e*(b^2*e*g*(m + p + 1) + 2*c^2*d*f*(m
+ 2*p + 2) - c*(2*a*e*g*m + b*(e*f + d*g)*(m + 2*p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &&
NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 1] && ((EqQ[m, 2] && EqQ[p, -3] &
& RationalQ[a, b, c, d, e, f, g]) ||  !ILtQ[m + 2*p + 3, 0])

Rule 846

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 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 (d+e x)^{7/2} (b d+(2 c d-b e) x)}{3 b^2 \left (b x+c x^2\right )^{3/2}}-\frac {2 \int \frac {(d+e x)^{5/2} \left (\frac {1}{2} d (8 c d-11 b e)-\frac {3}{2} e (2 c d-b e) x\right )}{\left (b x+c x^2\right )^{3/2}} \, dx}{3 b^2} \\ & = -\frac {2 (d+e x)^{7/2} (b d+(2 c d-b e) x)}{3 b^2 \left (b x+c x^2\right )^{3/2}}+\frac {2 (d+e x)^{3/2} \left (b c d^2 (8 c d-11 b e)+(2 c d-b e) \left (8 c^2 d^2-8 b c d e-3 b^2 e^2\right ) x\right )}{3 b^4 c \sqrt {b x+c x^2}}-\frac {4 \int \frac {\sqrt {d+e x} \left (\frac {3}{4} b d e \left (8 c^2 d^2-13 b c d e+b^2 e^2\right )+3 e \left (4 c^3 d^3-6 b c^2 d^2 e+b^3 e^3\right ) x\right )}{\sqrt {b x+c x^2}} \, dx}{3 b^4 c} \\ & = -\frac {2 (d+e x)^{7/2} (b d+(2 c d-b e) x)}{3 b^2 \left (b x+c x^2\right )^{3/2}}+\frac {2 (d+e x)^{3/2} \left (b c d^2 (8 c d-11 b e)+(2 c d-b e) \left (8 c^2 d^2-8 b c d e-3 b^2 e^2\right ) x\right )}{3 b^4 c \sqrt {b x+c x^2}}-\frac {8 e \left (4 c^3 d^3-6 b c^2 d^2 e+b^3 e^3\right ) \sqrt {d+e x} \sqrt {b x+c x^2}}{3 b^4 c^2}-\frac {8 \int \frac {\frac {3}{8} b 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 {3}{8} 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}{9 b^4 c^2} \\ & = -\frac {2 (d+e x)^{7/2} (b d+(2 c d-b e) x)}{3 b^2 \left (b x+c x^2\right )^{3/2}}+\frac {2 (d+e x)^{3/2} \left (b c d^2 (8 c d-11 b e)+(2 c d-b e) \left (8 c^2 d^2-8 b c d e-3 b^2 e^2\right ) x\right )}{3 b^4 c \sqrt {b x+c x^2}}-\frac {8 e \left (4 c^3 d^3-6 b c^2 d^2 e+b^3 e^3\right ) \sqrt {d+e x} \sqrt {b x+c x^2}}{3 b^4 c^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 )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{3 b^4 c^2}-\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}{3 b^4 c^2} \\ & = -\frac {2 (d+e x)^{7/2} (b d+(2 c d-b e) x)}{3 b^2 \left (b x+c x^2\right )^{3/2}}+\frac {2 (d+e x)^{3/2} \left (b c d^2 (8 c d-11 b e)+(2 c d-b e) \left (8 c^2 d^2-8 b c d e-3 b^2 e^2\right ) x\right )}{3 b^4 c \sqrt {b x+c x^2}}-\frac {8 e \left (4 c^3 d^3-6 b c^2 d^2 e+b^3 e^3\right ) \sqrt {d+e x} \sqrt {b x+c x^2}}{3 b^4 c^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 {b+c x}\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}} \, dx}{3 b^4 c^2 \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}{3 b^4 c^2 \sqrt {b x+c x^2}} \\ & = -\frac {2 (d+e x)^{7/2} (b d+(2 c d-b e) x)}{3 b^2 \left (b x+c x^2\right )^{3/2}}+\frac {2 (d+e x)^{3/2} \left (b c d^2 (8 c d-11 b e)+(2 c d-b e) \left (8 c^2 d^2-8 b c d e-3 b^2 e^2\right ) x\right )}{3 b^4 c \sqrt {b x+c x^2}}-\frac {8 e \left (4 c^3 d^3-6 b c^2 d^2 e+b^3 e^3\right ) \sqrt {d+e x} \sqrt {b x+c x^2}}{3 b^4 c^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 {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 c^2 \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}{3 b^4 c^2 \sqrt {d+e x} \sqrt {b x+c x^2}} \\ & = -\frac {2 (d+e x)^{7/2} (b d+(2 c d-b e) x)}{3 b^2 \left (b x+c x^2\right )^{3/2}}+\frac {2 (d+e x)^{3/2} \left (b c d^2 (8 c d-11 b e)+(2 c d-b e) \left (8 c^2 d^2-8 b c d e-3 b^2 e^2\right ) x\right )}{3 b^4 c \sqrt {b x+c x^2}}-\frac {8 e \left (4 c^3 d^3-6 b c^2 d^2 e+b^3 e^3\right ) \sqrt {d+e x} \sqrt {b x+c x^2}}{3 b^4 c^2}-\frac {2 \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} c^{5/2} \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {8 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 )}{3 (-b)^{7/2} c^{5/2} \sqrt {d+e x} \sqrt {b x+c x^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 24.93 (sec) , antiderivative size = 451, normalized size of antiderivative = 0.96 \[ \int \frac {(d+e x)^{9/2}}{\left (b x+c x^2\right )^{5/2}} \, dx=\frac {2 \left (b (d+e x) \left (b (c d-b e)^4 x^2+(c d-b e)^3 (8 c d+5 b e) x^2 (b+c x)-b c^2 d^4 (b+c x)^2+c^2 d^3 (8 c d-13 b e) x (b+c x)^2\right )-\sqrt {\frac {b}{c}} x (b+c x) \left (\sqrt {\frac {b}{c}} \left (16 c^4 d^4-32 b c^3 d^3 e+9 b^2 c^2 d^2 e^2+7 b^3 c d e^3-8 b^4 e^4\right ) (b+c x) (d+e x)+i b e \left (16 c^4 d^4-32 b c^3 d^3 e+9 b^2 c^2 d^2 e^2+7 b^3 c d e^3-8 b^4 e^4\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )-i b e \left (8 c^4 d^4-17 b c^3 d^3 e+6 b^2 c^2 d^2 e^2+11 b^3 c d e^3-8 b^4 e^4\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right ),\frac {c d}{b e}\right )\right )\right )}{3 b^5 c^2 (x (b+c x))^{3/2} \sqrt {d+e x}} \]

[In]

Integrate[(d + e*x)^(9/2)/(b*x + c*x^2)^(5/2),x]

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(894\) vs. \(2(410)=820\).

Time = 2.55 (sec) , antiderivative size = 895, normalized size of antiderivative = 1.90

method	result	size
elliptic	\(\frac {\sqrt {x \left (e x +d \right ) \left (c x +b \right )}\, \left (-\frac {2 d^{4} \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{3 b^{3} x^{2}}-\frac {2 \left (c e \,x^{2}+b e x +c d x +b d \right ) d^{3} \left (13 b e -8 c d \right )}{3 b^{4} \sqrt {x \left (c e \,x^{2}+b e x +c d x +b d \right )}}+\frac {2 \left (b^{4} e^{4}-4 b^{3} c d \,e^{3}+6 b^{2} c^{2} d^{2} e^{2}-4 b \,c^{3} d^{3} e +c^{4} d^{4}\right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{3 c^{4} b^{3} \left (\frac {b}{c}+x \right )^{2}}-\frac {2 \left (c e \,x^{2}+c d x \right ) \left (5 b^{4} e^{4}-7 b^{3} c d \,e^{3}-9 b^{2} c^{2} d^{2} e^{2}+19 b \,c^{3} d^{3} e -8 c^{4} d^{4}\right )}{3 b^{4} c^{3} \sqrt {\left (\frac {b}{c}+x \right ) \left (c e \,x^{2}+c d x \right )}}+\frac {2 \left (-\frac {e^{4} \left (2 b e -5 c d \right )}{c^{3}}-\frac {d^{4} c e}{3 b^{3}}+\frac {\left (b^{4} e^{4}-4 b^{3} c d \,e^{3}+6 b^{2} c^{2} d^{2} e^{2}-4 b \,c^{3} d^{3} e +c^{4} d^{4}\right ) e}{3 c^{3} b^{3}}+\frac {\left (5 b^{4} e^{4}-7 b^{3} c d \,e^{3}-9 b^{2} c^{2} d^{2} e^{2}+19 b \,c^{3} d^{3} e -8 c^{4} d^{4}\right ) \left (b e -c d \right )}{3 c^{3} b^{4}}+\frac {d \left (5 b^{4} e^{4}-7 b^{3} c d \,e^{3}-9 b^{2} c^{2} d^{2} e^{2}+19 b \,c^{3} d^{3} e -8 c^{4} d^{4}\right )}{3 c^{2} b^{4}}\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 {e^{5}}{c^{2}}+\frac {d^{3} e c \left (13 b e -8 c d \right )}{3 b^{4}}+\frac {\left (5 b^{4} e^{4}-7 b^{3} c d \,e^{3}-9 b^{2} c^{2} d^{2} e^{2}+19 b \,c^{3} d^{3} e -8 c^{4} d^{4}\right ) e}{3 c^{2} b^{4}}\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}}\)	\(895\)
default	\(\text {Expression too large to display}\)	\(2086\)

[In]

int((e*x+d)^(9/2)/(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^4/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*(13*b*e-8*c*d)/(x*(c*e*x^2+b*e*x+c*d*x+b*d))^(1/2)+2/3*(b^4*e^4-4*b^
3*c*d*e^3+6*b^2*c^2*d^2*e^2-4*b*c^3*d^3*e+c^4*d^4)/c^4/b^3*(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)*(5*b^4*e^4-7*b^3*c*d*e^3-9*b^2*c^2*d^2*e^2+19*b*c^3*d^3*e-8*c^4*d^4)/b^4/c^3/((1/c*b+x)*(c*
e*x^2+c*d*x))^(1/2)+2*(-e^4*(2*b*e-5*c*d)/c^3-1/3*d^4/b^3*c*e+1/3*(b^4*e^4-4*b^3*c*d*e^3+6*b^2*c^2*d^2*e^2-4*b
*c^3*d^3*e+c^4*d^4)/c^3*e/b^3+1/3*(5*b^4*e^4-7*b^3*c*d*e^3-9*b^2*c^2*d^2*e^2+19*b*c^3*d^3*e-8*c^4*d^4)/c^3*(b*
e-c*d)/b^4+1/3/c^2*d*(5*b^4*e^4-7*b^3*c*d*e^3-9*b^2*c^2*d^2*e^2+19*b*c^3*d^3*e-8*c^4*d^4)/b^4)/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*(e^5/c^2+1/3*d^3*e*c*(13*b*e-8*c*d)/b^4+1/3*(5*b^4*e^4-7*b^3*c*d
*e^3-9*b^2*c^2*d^2*e^2+19*b*c^3*d^3*e-8*c^4*d^4)/c^2*e/b^4)/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 = 0.31 (sec) , antiderivative size = 933, normalized size of antiderivative = 1.99 \[ \int \frac {(d+e x)^{9/2}}{\left (b x+c x^2\right )^{5/2}} \, dx=\frac {2 \, {\left ({\left ({\left (16 \, c^{7} d^{5} - 40 \, b c^{6} d^{4} e + 22 \, b^{2} c^{5} d^{3} e^{2} + 7 \, b^{3} c^{4} d^{2} e^{3} + 11 \, b^{4} c^{3} d e^{4} - 8 \, b^{5} c^{2} e^{5}\right )} x^{4} + 2 \, {\left (16 \, b c^{6} d^{5} - 40 \, b^{2} c^{5} d^{4} e + 22 \, b^{3} c^{4} d^{3} e^{2} + 7 \, b^{4} c^{3} d^{2} e^{3} + 11 \, b^{5} c^{2} d e^{4} - 8 \, b^{6} c e^{5}\right )} x^{3} + {\left (16 \, b^{2} c^{5} d^{5} - 40 \, b^{3} c^{4} d^{4} e + 22 \, b^{4} c^{3} d^{3} e^{2} + 7 \, b^{5} c^{2} d^{2} e^{3} + 11 \, b^{6} c d e^{4} - 8 \, b^{7} e^{5}\right )} x^{2}\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right ) + 3 \, {\left ({\left (16 \, c^{7} d^{4} e - 32 \, b c^{6} d^{3} e^{2} + 9 \, b^{2} c^{5} d^{2} e^{3} + 7 \, b^{3} c^{4} d e^{4} - 8 \, b^{4} c^{3} e^{5}\right )} x^{4} + 2 \, {\left (16 \, b c^{6} d^{4} e - 32 \, b^{2} c^{5} d^{3} e^{2} + 9 \, b^{3} c^{4} d^{2} e^{3} + 7 \, b^{4} c^{3} d e^{4} - 8 \, b^{5} c^{2} e^{5}\right )} x^{3} + {\left (16 \, b^{2} c^{5} d^{4} e - 32 \, b^{3} c^{4} d^{3} e^{2} + 9 \, b^{4} c^{3} d^{2} e^{3} + 7 \, b^{5} c^{2} d e^{4} - 8 \, b^{6} c e^{5}\right )} x^{2}\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right )\right ) - 3 \, {\left (b^{3} c^{4} d^{4} e - {\left (16 \, c^{7} d^{4} e - 32 \, b c^{6} d^{3} e^{2} + 9 \, b^{2} c^{5} d^{2} e^{3} + 7 \, b^{3} c^{4} d e^{4} - 5 \, b^{4} c^{3} e^{5}\right )} x^{3} - {\left (24 \, b c^{6} d^{4} e - 49 \, b^{2} c^{5} d^{3} e^{2} + 15 \, b^{3} c^{4} d^{2} e^{3} + 3 \, b^{4} c^{3} d e^{4} - 4 \, b^{5} c^{2} e^{5}\right )} x^{2} - {\left (6 \, b^{2} c^{5} d^{4} e - 13 \, b^{3} c^{4} d^{3} e^{2}\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {e x + d}\right )}}{9 \, {\left (b^{4} c^{6} e x^{4} + 2 \, b^{5} c^{5} e x^{3} + b^{6} c^{4} e x^{2}\right )}} \]

[In]

integrate((e*x+d)^(9/2)/(c*x^2+b*x)^(5/2),x, algorithm="fricas")

[Out]

2/9*(((16*c^7*d^5 - 40*b*c^6*d^4*e + 22*b^2*c^5*d^3*e^2 + 7*b^3*c^4*d^2*e^3 + 11*b^4*c^3*d*e^4 - 8*b^5*c^2*e^5
)*x^4 + 2*(16*b*c^6*d^5 - 40*b^2*c^5*d^4*e + 22*b^3*c^4*d^3*e^2 + 7*b^4*c^3*d^2*e^3 + 11*b^5*c^2*d*e^4 - 8*b^6
*c*e^5)*x^3 + (16*b^2*c^5*d^5 - 40*b^3*c^4*d^4*e + 22*b^4*c^3*d^3*e^2 + 7*b^5*c^2*d^2*e^3 + 11*b^6*c*d*e^4 - 8
*b^7*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 - 32*
b*c^6*d^3*e^2 + 9*b^2*c^5*d^2*e^3 + 7*b^3*c^4*d*e^4 - 8*b^4*c^3*e^5)*x^4 + 2*(16*b*c^6*d^4*e - 32*b^2*c^5*d^3*
e^2 + 9*b^3*c^4*d^2*e^3 + 7*b^4*c^3*d*e^4 - 8*b^5*c^2*e^5)*x^3 + (16*b^2*c^5*d^4*e - 32*b^3*c^4*d^3*e^2 + 9*b^
4*c^3*d^2*e^3 + 7*b^5*c^2*d*e^4 - 8*b^6*c*e^5)*x^2)*sqrt(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^4*e - (16*c^7*d^4*e - 32*b*c^6*d^3*e^2 + 9*b^2*c^5*d^2*e^
3 + 7*b^3*c^4*d*e^4 - 5*b^4*c^3*e^5)*x^3 - (24*b*c^6*d^4*e - 49*b^2*c^5*d^3*e^2 + 15*b^3*c^4*d^2*e^3 + 3*b^4*c
^3*d*e^4 - 4*b^5*c^2*e^5)*x^2 - (6*b^2*c^5*d^4*e - 13*b^3*c^4*d^3*e^2)*x)*sqrt(c*x^2 + b*x)*sqrt(e*x + d))/(b^
4*c^6*e*x^4 + 2*b^5*c^5*e*x^3 + b^6*c^4*e*x^2)

Sympy [F(-1)]

Timed out. \[ \int \frac {(d+e x)^{9/2}}{\left (b x+c x^2\right )^{5/2}} \, dx=\text {Timed out} \]

[In]

integrate((e*x+d)**(9/2)/(c*x**2+b*x)**(5/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {(d+e x)^{9/2}}{\left (b x+c x^2\right )^{5/2}} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {9}{2}}}{{\left (c x^{2} + b x\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate((e*x+d)^(9/2)/(c*x^2+b*x)^(5/2),x, algorithm="maxima")

[Out]

integrate((e*x + d)^(9/2)/(c*x^2 + b*x)^(5/2), x)

Giac [F]

\[ \int \frac {(d+e x)^{9/2}}{\left (b x+c x^2\right )^{5/2}} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {9}{2}}}{{\left (c x^{2} + b x\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate((e*x+d)^(9/2)/(c*x^2+b*x)^(5/2),x, algorithm="giac")

[Out]

integrate((e*x + d)^(9/2)/(c*x^2 + b*x)^(5/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(d+e x)^{9/2}}{\left (b x+c x^2\right )^{5/2}} \, dx=\int \frac {{\left (d+e\,x\right )}^{9/2}}{{\left (c\,x^2+b\,x\right )}^{5/2}} \,d x \]

[In]

int((d + e*x)^(9/2)/(b*x + c*x^2)^(5/2),x)

[Out]

int((d + e*x)^(9/2)/(b*x + c*x^2)^(5/2), x)