\(\int \frac {A+B x}{\sqrt {d+e x} (b x+c x^2)^{5/2}} \, dx\) [1282]

Optimal result

Integrand size = 28, antiderivative size = 543 \[ \int \frac {A+B x}{\sqrt {d+e x} \left (b x+c x^2\right )^{5/2}} \, dx=-\frac {2 \sqrt {d+e x} (A b (c d-b e)+c (2 A c d-b (B d+A e)) x)}{3 b^2 d (c d-b e) \left (b x+c x^2\right )^{3/2}}+\frac {2 \sqrt {d+e x} \left (b (c d-b e) \left (8 A c^2 d^2+b^2 e (3 B d-2 A e)-b c d (4 B d+5 A e)\right )+c \left (16 A c^3 d^3-b^3 e^2 (3 B d-2 A e)-8 b c^2 d^2 (B d+3 A e)+b^2 c d e (13 B d+4 A e)\right ) x\right )}{3 b^4 d^2 (c d-b e)^2 \sqrt {b x+c x^2}}-\frac {2 \sqrt {c} \left (16 A c^3 d^3-b^3 e^2 (3 B d-2 A e)-8 b c^2 d^2 (B d+3 A e)+b^2 c d e (13 B d+4 A e)\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{7/2} d^2 (c d-b e)^2 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {2 \sqrt {c} \left (16 A c^2 d^2+b^2 e (9 B d-A e)-8 b c d (B d+2 A e)\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{3 (-b)^{7/2} d (c d-b e) \sqrt {d+e x} \sqrt {b x+c x^2}} \]

[Out]

-2/3*(A*b*(-b*e+c*d)+c*(2*A*c*d-b*(A*e+B*d))*x)*(e*x+d)^(1/2)/b^2/d/(-b*e+c*d)/(c*x^2+b*x)^(3/2)+2/3*(b*(-b*e+
c*d)*(8*A*c^2*d^2+b^2*e*(-2*A*e+3*B*d)-b*c*d*(5*A*e+4*B*d))+c*(16*A*c^3*d^3-b^3*e^2*(-2*A*e+3*B*d)-8*b*c^2*d^2
*(3*A*e+B*d)+b^2*c*d*e*(4*A*e+13*B*d))*x)*(e*x+d)^(1/2)/b^4/d^2/(-b*e+c*d)^2/(c*x^2+b*x)^(1/2)-2/3*(16*A*c^3*d
^3-b^3*e^2*(-2*A*e+3*B*d)-8*b*c^2*d^2*(3*A*e+B*d)+b^2*c*d*e*(4*A*e+13*B*d))*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^2/(-b*e+c*d)^2/(1+e*x/d)^(1/2)/
(c*x^2+b*x)^(1/2)+2/3*(16*A*c^2*d^2+b^2*e*(-A*e+9*B*d)-8*b*c*d*(2*A*e+B*d))*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/(-b*e+c*d)/(e*x+d)^(1/2)/(c*x
^2+b*x)^(1/2)

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 543, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {836, 857, 729, 113, 111, 118, 117} \[ \int \frac {A+B x}{\sqrt {d+e x} \left (b x+c x^2\right )^{5/2}} \, dx=\frac {2 \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} \left (b^2 e (9 B d-A e)-8 b c d (2 A e+B d)+16 A c^2 d^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{3 (-b)^{7/2} d \sqrt {b x+c x^2} \sqrt {d+e x} (c d-b e)}-\frac {2 \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (b^3 \left (-e^2\right ) (3 B d-2 A e)+b^2 c d e (4 A e+13 B d)-8 b c^2 d^2 (3 A e+B d)+16 A c^3 d^3\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{7/2} d^2 \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1} (c d-b e)^2}-\frac {2 \sqrt {d+e x} (c x (2 A c d-b (A e+B d))+A b (c d-b e))}{3 b^2 d \left (b x+c x^2\right )^{3/2} (c d-b e)}+\frac {2 \sqrt {d+e x} \left (b (c d-b e) \left (b^2 e (3 B d-2 A e)-b c d (5 A e+4 B d)+8 A c^2 d^2\right )+c x \left (b^3 \left (-e^2\right ) (3 B d-2 A e)+b^2 c d e (4 A e+13 B d)-8 b c^2 d^2 (3 A e+B d)+16 A c^3 d^3\right )\right )}{3 b^4 d^2 \sqrt {b x+c x^2} (c d-b e)^2} \]

[In]

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

[Out]

(-2*Sqrt[d + e*x]*(A*b*(c*d - b*e) + c*(2*A*c*d - b*(B*d + A*e))*x))/(3*b^2*d*(c*d - b*e)*(b*x + c*x^2)^(3/2))
+ (2*Sqrt[d + e*x]*(b*(c*d - b*e)*(8*A*c^2*d^2 + b^2*e*(3*B*d - 2*A*e) - b*c*d*(4*B*d + 5*A*e)) + c*(16*A*c^3
*d^3 - b^3*e^2*(3*B*d - 2*A*e) - 8*b*c^2*d^2*(B*d + 3*A*e) + b^2*c*d*e*(13*B*d + 4*A*e))*x))/(3*b^4*d^2*(c*d -
b*e)^2*Sqrt[b*x + c*x^2]) - (2*Sqrt[c]*(16*A*c^3*d^3 - b^3*e^2*(3*B*d - 2*A*e) - 8*b*c^2*d^2*(B*d + 3*A*e) +
b^2*c*d*e*(13*B*d + 4*A*e))*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^2*(c*d - b*e)^2*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) + (2*Sqrt[c]*(16*A*c^2*
d^2 + b^2*e*(9*B*d - A*e) - 8*b*c*d*(B*d + 2*A*e))*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSi
n[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(3*(-b)^(7/2)*d*(c*d - b*e)*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 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 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 \sqrt {d+e x} (A b (c d-b e)+c (2 A c d-b (B d+A e)) x)}{3 b^2 d (c d-b e) \left (b x+c x^2\right )^{3/2}}-\frac {2 \int \frac {\frac {1}{2} \left (8 A c^2 d^2+b^2 e (3 B d-2 A e)-b c d (4 B d+5 A e)\right )-\frac {3}{2} c e (b B d-2 A c d+A b e) x}{\sqrt {d+e x} \left (b x+c x^2\right )^{3/2}} \, dx}{3 b^2 d (c d-b e)} \\ & = -\frac {2 \sqrt {d+e x} (A b (c d-b e)+c (2 A c d-b (B d+A e)) x)}{3 b^2 d (c d-b e) \left (b x+c x^2\right )^{3/2}}+\frac {2 \sqrt {d+e x} \left (b (c d-b e) \left (8 A c^2 d^2+b^2 e (3 B d-2 A e)-b c d (4 B d+5 A e)\right )+c \left (16 A c^3 d^3-b^3 e^2 (3 B d-2 A e)-8 b c^2 d^2 (B d+3 A e)+b^2 c d e (13 B d+4 A e)\right ) x\right )}{3 b^4 d^2 (c d-b e)^2 \sqrt {b x+c x^2}}+\frac {4 \int \frac {-\frac {1}{4} b c d e \left (8 A c^2 d^2+b^2 e (6 B d+A e)-b c d (4 B d+11 A e)\right )-\frac {1}{4} c e \left (16 A c^3 d^3-b^3 e^2 (3 B d-2 A e)-8 b c^2 d^2 (B d+3 A e)+b^2 c d e (13 B d+4 A e)\right ) x}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{3 b^4 d^2 (c d-b e)^2} \\ & = -\frac {2 \sqrt {d+e x} (A b (c d-b e)+c (2 A c d-b (B d+A e)) x)}{3 b^2 d (c d-b e) \left (b x+c x^2\right )^{3/2}}+\frac {2 \sqrt {d+e x} \left (b (c d-b e) \left (8 A c^2 d^2+b^2 e (3 B d-2 A e)-b c d (4 B d+5 A e)\right )+c \left (16 A c^3 d^3-b^3 e^2 (3 B d-2 A e)-8 b c^2 d^2 (B d+3 A e)+b^2 c d e (13 B d+4 A e)\right ) x\right )}{3 b^4 d^2 (c d-b e)^2 \sqrt {b x+c x^2}}+\frac {\left (c \left (16 A c^2 d^2+b^2 e (9 B d-A e)-8 b c d (B d+2 A e)\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{3 b^4 d (c d-b e)}-\frac {\left (c \left (16 A c^3 d^3-b^3 e^2 (3 B d-2 A e)-8 b c^2 d^2 (B d+3 A e)+b^2 c d e (13 B d+4 A e)\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx}{3 b^4 d^2 (c d-b e)^2} \\ & = -\frac {2 \sqrt {d+e x} (A b (c d-b e)+c (2 A c d-b (B d+A e)) x)}{3 b^2 d (c d-b e) \left (b x+c x^2\right )^{3/2}}+\frac {2 \sqrt {d+e x} \left (b (c d-b e) \left (8 A c^2 d^2+b^2 e (3 B d-2 A e)-b c d (4 B d+5 A e)\right )+c \left (16 A c^3 d^3-b^3 e^2 (3 B d-2 A e)-8 b c^2 d^2 (B d+3 A e)+b^2 c d e (13 B d+4 A e)\right ) x\right )}{3 b^4 d^2 (c d-b e)^2 \sqrt {b x+c x^2}}+\frac {\left (c \left (16 A c^2 d^2+b^2 e (9 B d-A e)-8 b c d (B d+2 A e)\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 (c d-b e) \sqrt {b x+c x^2}}-\frac {\left (c \left (16 A c^3 d^3-b^3 e^2 (3 B d-2 A e)-8 b c^2 d^2 (B d+3 A e)+b^2 c d e (13 B d+4 A e)\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^2 (c d-b e)^2 \sqrt {b x+c x^2}} \\ & = -\frac {2 \sqrt {d+e x} (A b (c d-b e)+c (2 A c d-b (B d+A e)) x)}{3 b^2 d (c d-b e) \left (b x+c x^2\right )^{3/2}}+\frac {2 \sqrt {d+e x} \left (b (c d-b e) \left (8 A c^2 d^2+b^2 e (3 B d-2 A e)-b c d (4 B d+5 A e)\right )+c \left (16 A c^3 d^3-b^3 e^2 (3 B d-2 A e)-8 b c^2 d^2 (B d+3 A e)+b^2 c d e (13 B d+4 A e)\right ) x\right )}{3 b^4 d^2 (c d-b e)^2 \sqrt {b x+c x^2}}-\frac {\left (c \left (16 A c^3 d^3-b^3 e^2 (3 B d-2 A e)-8 b c^2 d^2 (B d+3 A e)+b^2 c d e (13 B d+4 A e)\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^2 (c d-b e)^2 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {\left (c \left (16 A c^2 d^2+b^2 e (9 B d-A e)-8 b c d (B d+2 A e)\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 (c d-b e) \sqrt {d+e x} \sqrt {b x+c x^2}} \\ & = -\frac {2 \sqrt {d+e x} (A b (c d-b e)+c (2 A c d-b (B d+A e)) x)}{3 b^2 d (c d-b e) \left (b x+c x^2\right )^{3/2}}+\frac {2 \sqrt {d+e x} \left (b (c d-b e) \left (8 A c^2 d^2+b^2 e (3 B d-2 A e)-b c d (4 B d+5 A e)\right )+c \left (16 A c^3 d^3-b^3 e^2 (3 B d-2 A e)-8 b c^2 d^2 (B d+3 A e)+b^2 c d e (13 B d+4 A e)\right ) x\right )}{3 b^4 d^2 (c d-b e)^2 \sqrt {b x+c x^2}}-\frac {2 \sqrt {c} \left (16 A c^3 d^3-b^3 e^2 (3 B d-2 A e)-8 b c^2 d^2 (B d+3 A e)+b^2 c d e (13 B d+4 A e)\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^2 (c d-b e)^2 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {2 \sqrt {c} \left (16 A c^2 d^2+b^2 e (9 B d-A e)-8 b c d (B d+2 A e)\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 (c d-b e) \sqrt {d+e x} \sqrt {b x+c x^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 24.73 (sec) , antiderivative size = 514, normalized size of antiderivative = 0.95 \[ \int \frac {A+B x}{\sqrt {d+e x} \left (b x+c x^2\right )^{5/2}} \, dx=-\frac {2 \left (b (d+e x) \left (b c^2 (b B-A c) d^2 (c d-b e) x^2+c^2 d^2 \left (-8 A c^2 d-7 b^2 B e+5 b c (B d+2 A e)\right ) x^2 (b+c x)+A b d (c d-b e)^2 (b+c x)^2+(c d-b e)^2 (3 b B d-8 A c d-2 A b e) x (b+c x)^2\right )+\sqrt {\frac {b}{c}} c x (b+c x) \left (\sqrt {\frac {b}{c}} \left (16 A c^3 d^3+b^3 e^2 (-3 B d+2 A e)-8 b c^2 d^2 (B d+3 A e)+b^2 c d e (13 B d+4 A e)\right ) (b+c x) (d+e x)+i b e \left (16 A c^3 d^3+b^3 e^2 (-3 B d+2 A e)-8 b c^2 d^2 (B d+3 A e)+b^2 c d e (13 B d+4 A e)\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )-i b e (c d-b e) \left (8 A c^2 d^2+b^2 e (3 B d-2 A e)-b c d (4 B d+5 A e)\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right ),\frac {c d}{b e}\right )\right )\right )}{3 b^5 d^2 (c d-b e)^2 (x (b+c x))^{3/2} \sqrt {d+e x}} \]

[In]

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

[Out]

(-2*(b*(d + e*x)*(b*c^2*(b*B - A*c)*d^2*(c*d - b*e)*x^2 + c^2*d^2*(-8*A*c^2*d - 7*b^2*B*e + 5*b*c*(B*d + 2*A*e
))*x^2*(b + c*x) + A*b*d*(c*d - b*e)^2*(b + c*x)^2 + (c*d - b*e)^2*(3*b*B*d - 8*A*c*d - 2*A*b*e)*x*(b + c*x)^2
) + Sqrt[b/c]*c*x*(b + c*x)*(Sqrt[b/c]*(16*A*c^3*d^3 + b^3*e^2*(-3*B*d + 2*A*e) - 8*b*c^2*d^2*(B*d + 3*A*e) +
b^2*c*d*e*(13*B*d + 4*A*e))*(b + c*x)*(d + e*x) + I*b*e*(16*A*c^3*d^3 + b^3*e^2*(-3*B*d + 2*A*e) - 8*b*c^2*d^2
*(B*d + 3*A*e) + b^2*c*d*e*(13*B*d + 4*A*e))*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticE[I*ArcSinh[S
qrt[b/c]/Sqrt[x]], (c*d)/(b*e)] - I*b*e*(c*d - b*e)*(8*A*c^2*d^2 + b^2*e*(3*B*d - 2*A*e) - b*c*d*(4*B*d + 5*A*
e))*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^2*(c*d - b*e)^2*(x*(b + c*x))^(3/2)*Sqrt[d + e*x])

Maple [A] (verified)

Time = 1.23 (sec) , antiderivative size = 752, normalized size of antiderivative = 1.38

method	result	size
elliptic	\(\frac {\sqrt {\left (e x +d \right ) x \left (c x +b \right )}\, \left (-\frac {2 A \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{3 b^{3} d \,x^{2}}+\frac {2 \left (c e \,x^{2}+b e x +c d x +b d \right ) \left (2 A b e +8 A c d -3 B b d \right )}{3 b^{4} d^{2} \sqrt {x \left (c e \,x^{2}+b e x +c d x +b d \right )}}-\frac {2 \left (A c -B b \right ) \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 ) \left (x +\frac {b}{c}\right )^{2}}-\frac {2 \left (c e \,x^{2}+c d x \right ) c \left (10 A b c e -8 A \,c^{2} d -7 b^{2} B e +5 B b c d \right )}{3 b^{4} \left (b e -c d \right )^{2} \sqrt {\left (x +\frac {b}{c}\right ) \left (c e \,x^{2}+c d x \right )}}+\frac {2 \left (-\frac {A c e}{3 b^{3} d}-\frac {c e \left (A c -B b \right )}{3 b^{3} \left (b e -c d \right )}+\frac {c \left (10 A b c e -8 A \,c^{2} d -7 b^{2} B e +5 B b c d \right )}{3 \left (b e -c d \right ) b^{4}}+\frac {c^{2} d \left (10 A b c e -8 A \,c^{2} d -7 b^{2} B e +5 B b c d \right )}{3 b^{4} \left (b e -c d \right )^{2}}\right ) b \sqrt {\frac {\left (x +\frac {b}{c}\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 (x +\frac {b}{c}\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 (2 A b e +8 A c d -3 B b d \right )}{3 d^{2} b^{4}}+\frac {e \,c^{2} \left (10 A b c e -8 A \,c^{2} d -7 b^{2} B e +5 B b c d \right )}{3 b^{4} \left (b e -c d \right )^{2}}\right ) b \sqrt {\frac {\left (x +\frac {b}{c}\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 (x +\frac {b}{c}\right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )-\frac {d F\left (\sqrt {\frac {\left (x +\frac {b}{c}\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}}\)	\(752\)
default	\(\text {Expression too large to display}\)	\(3140\)

[In]

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

[Out]

((e*x+d)*x*(c*x+b))^(1/2)/(x*(c*x+b))^(1/2)/(e*x+d)^(1/2)*(-2/3*A/b^3/d*(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^2*(2*A*b*e+8*A*c*d-3*B*b*d)/(x*(c*e*x^2+b*e*x+c*d*x+b*d))^(1/2)-2/3/b^
3/(b*e-c*d)*(A*c-B*b)*(c*e*x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)/(x+b/c)^2-2/3*(c*e*x^2+c*d*x)/b^4/(b*e-c*d)^2*c*(1
0*A*b*c*e-8*A*c^2*d-7*B*b^2*e+5*B*b*c*d)/((x+b/c)*(c*e*x^2+c*d*x))^(1/2)+2*(-1/3/b^3/d*A*c*e-1/3*c*e*(A*c-B*b)
/b^3/(b*e-c*d)+1/3/(b*e-c*d)*c*(10*A*b*c*e-8*A*c^2*d-7*B*b^2*e+5*B*b*c*d)/b^4+1/3*c^2*d/b^4/(b*e-c*d)^2*(10*A*
b*c*e-8*A*c^2*d-7*B*b^2*e+5*B*b*c*d))*b/c*((x+b/c)/b*c)^(1/2)*((x+d/e)/(-b/c+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(((x+b/c)/b*c)^(1/2),(-b/c/(-b/c+d/e))^(1/2))+2*(-1/3*c*e*(2*A*b*e+8*
A*c*d-3*B*b*d)/d^2/b^4+1/3*e*c^2*(10*A*b*c*e-8*A*c^2*d-7*B*b^2*e+5*B*b*c*d)/b^4/(b*e-c*d)^2)*b/c*((x+b/c)/b*c)
^(1/2)*((x+d/e)/(-b/c+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)*((-b/c+d/e)*EllipticE((
(x+b/c)/b*c)^(1/2),(-b/c/(-b/c+d/e))^(1/2))-d/e*EllipticF(((x+b/c)/b*c)^(1/2),(-b/c/(-b/c+d/e))^(1/2))))

Fricas [C] (verification not implemented)

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

Time = 0.19 (sec) , antiderivative size = 1324, normalized size of antiderivative = 2.44 \[ \int \frac {A+B x}{\sqrt {d+e x} \left (b x+c x^2\right )^{5/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

2/9*(((2*A*b^4*c^2*e^4 - 8*(B*b*c^5 - 2*A*c^6)*d^4 + (17*B*b^2*c^4 - 32*A*b*c^5)*d^3*e - (8*B*b^3*c^3 - 13*A*b
^2*c^4)*d^2*e^2 - 3*(B*b^4*c^2 - A*b^3*c^3)*d*e^3)*x^4 + 2*(2*A*b^5*c*e^4 - 8*(B*b^2*c^4 - 2*A*b*c^5)*d^4 + (1
7*B*b^3*c^3 - 32*A*b^2*c^4)*d^3*e - (8*B*b^4*c^2 - 13*A*b^3*c^3)*d^2*e^2 - 3*(B*b^5*c - A*b^4*c^2)*d*e^3)*x^3
+ (2*A*b^6*e^4 - 8*(B*b^3*c^3 - 2*A*b^2*c^4)*d^4 + (17*B*b^4*c^2 - 32*A*b^3*c^3)*d^3*e - (8*B*b^5*c - 13*A*b^4
*c^2)*d^2*e^2 - 3*(B*b^6 - A*b^5*c)*d*e^3)*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*((2*A*b^3*c^3*e^4 - 8*(B*b*c^5 - 2*A*c^6)*d^3*e + (13*B*b^2*c^4 - 24*A*b*c^5)*d^2*e^2 - (3*B*b^3*
c^3 - 4*A*b^2*c^4)*d*e^3)*x^4 + 2*(2*A*b^4*c^2*e^4 - 8*(B*b^2*c^4 - 2*A*b*c^5)*d^3*e + (13*B*b^3*c^3 - 24*A*b^
2*c^4)*d^2*e^2 - (3*B*b^4*c^2 - 4*A*b^3*c^3)*d*e^3)*x^3 + (2*A*b^5*c*e^4 - 8*(B*b^3*c^3 - 2*A*b^2*c^4)*d^3*e +
(13*B*b^4*c^2 - 24*A*b^3*c^3)*d^2*e^2 - (3*B*b^5*c - 4*A*b^4*c^2)*d*e^3)*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*(A*b^3*c^3*d^3*e - 2*A*b^4*c^2*d^2*e^2 +
A*b^5*c*d*e^3 - (2*A*b^3*c^3*e^4 - 8*(B*b*c^5 - 2*A*c^6)*d^3*e + (13*B*b^2*c^4 - 24*A*b*c^5)*d^2*e^2 - (3*B*b^
3*c^3 - 4*A*b^2*c^4)*d*e^3)*x^3 - (4*A*b^4*c^2*e^4 - 12*(B*b^2*c^4 - 2*A*b*c^5)*d^3*e + (20*B*b^3*c^3 - 37*A*b
^2*c^4)*d^2*e^2 - (6*B*b^4*c^2 - 7*A*b^3*c^3)*d*e^3)*x^2 - (2*A*b^5*c*e^4 - 3*(B*b^3*c^3 - 2*A*b^2*c^4)*d^3*e
+ 2*(3*B*b^4*c^2 - 5*A*b^3*c^3)*d^2*e^2 - (3*B*b^5*c - 2*A*b^4*c^2)*d*e^3)*x)*sqrt(c*x^2 + b*x)*sqrt(e*x + d))
/((b^4*c^5*d^4*e - 2*b^5*c^4*d^3*e^2 + b^6*c^3*d^2*e^3)*x^4 + 2*(b^5*c^4*d^4*e - 2*b^6*c^3*d^3*e^2 + b^7*c^2*d
^2*e^3)*x^3 + (b^6*c^3*d^4*e - 2*b^7*c^2*d^3*e^2 + b^8*c*d^2*e^3)*x^2)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

integrate((B*x+A)/(c*x^2+b*x)^(5/2)/(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

integrate((B*x + A)/((c*x^2 + b*x)^(5/2)*sqrt(e*x + d)), x)

Giac [F]

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

[In]

integrate((B*x+A)/(c*x^2+b*x)^(5/2)/(e*x+d)^(1/2),x, algorithm="giac")

[Out]

integrate((B*x + A)/((c*x^2 + b*x)^(5/2)*sqrt(e*x + d)), x)

Mupad [F(-1)]

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

[In]

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

[Out]

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