\(\int \frac {A+B x}{\sqrt {d+e x} (b x+c x^2)^{3/2}} \, dx\) [1275]

Optimal result

Integrand size = 28, antiderivative size = 295 \[ \int \frac {A+B x}{\sqrt {d+e x} \left (b x+c x^2\right )^{3/2}} \, dx=-\frac {2 \sqrt {d+e x} (A b (c d-b e)+c (2 A c d-b (B d+A e)) x)}{b^2 d (c d-b e) \sqrt {b x+c x^2}}-\frac {2 \sqrt {c} (b B d-2 A c d+A b e) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{(-b)^{3/2} d (c d-b e) \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {2 (b B-2 A c) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{(-b)^{3/2} \sqrt {c} \sqrt {d+e x} \sqrt {b x+c x^2}} \]

[Out]

-2*(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)^(1/2)-2*(A*b*e-2*A*c*
d+B*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)^(3/2)/d/(-b*e+c*d)/(1+e*x/d)^(1/2)/(c*x^2+b*x)^(1/2)+2*(-2*A*c+B*b)*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)^(3/2)/c^(1/2)/(e*x+d)^(1/2)/(c*x^2+b*x)^(1/2)

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {836, 857, 729, 113, 111, 118, 117} \[ \int \frac {A+B x}{\sqrt {d+e x} \left (b x+c x^2\right )^{3/2}} \, dx=\frac {2 \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (b B-2 A c) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{(-b)^{3/2} \sqrt {c} \sqrt {b x+c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (A b e-2 A c d+b B d) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{(-b)^{3/2} d \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1} (c d-b e)}-\frac {2 \sqrt {d+e x} (c x (2 A c d-b (A e+B d))+A b (c d-b e))}{b^2 d \sqrt {b x+c x^2} (c d-b e)} \]

[In]

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

[Out]

(-2*Sqrt[d + e*x]*(A*b*(c*d - b*e) + c*(2*A*c*d - b*(B*d + A*e))*x))/(b^2*d*(c*d - b*e)*Sqrt[b*x + c*x^2]) - (
2*Sqrt[c]*(b*B*d - 2*A*c*d + A*b*e)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[x])
/Sqrt[-b]], (b*e)/(c*d)])/((-b)^(3/2)*d*(c*d - b*e)*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) + (2*(b*B - 2*A*c)*Sq
rt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/((-b)^(3
/2)*Sqrt[c]*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)}{b^2 d (c d-b e) \sqrt {b x+c x^2}}-\frac {2 \int \frac {\frac {1}{2} b (b B-A c) d e+\frac {1}{2} c e (b B d-2 A c d+A b e) x}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{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)}{b^2 d (c d-b e) \sqrt {b x+c x^2}}+\frac {(b B-2 A c) \int \frac {1}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{b^2}-\frac {(c (b B d-2 A c d+A b e)) \int \frac {\sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx}{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)}{b^2 d (c d-b e) \sqrt {b x+c x^2}}+\frac {\left ((b B-2 A c) \sqrt {x} \sqrt {b+c x}\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}} \, dx}{b^2 \sqrt {b x+c x^2}}-\frac {\left (c (b B d-2 A c d+A b e) \sqrt {x} \sqrt {b+c x}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {x} \sqrt {b+c x}} \, dx}{b^2 d (c d-b e) \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)}{b^2 d (c d-b e) \sqrt {b x+c x^2}}-\frac {\left (c (b B d-2 A c d+A b e) \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}{b^2 d (c d-b e) \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {\left ((b B-2 A c) \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}{b^2 \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)}{b^2 d (c d-b e) \sqrt {b x+c x^2}}-\frac {2 \sqrt {c} (b B d-2 A c d+A b e) \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 )}{(-b)^{3/2} d (c d-b e) \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {2 (b B-2 A c) \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 )}{(-b)^{3/2} \sqrt {c} \sqrt {d+e x} \sqrt {b x+c x^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 18.80 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.79 \[ \int \frac {A+B x}{\sqrt {d+e x} \left (b x+c x^2\right )^{3/2}} \, dx=\frac {2 \sqrt {\frac {b}{c}} (b B-A c) d (d+e x)-2 i e (2 A c d-b (B d+A e)) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )+2 i A e (c d-b e) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right ),\frac {c d}{b e}\right )}{b \sqrt {\frac {b}{c}} d (-c d+b e) \sqrt {x (b+c x)} \sqrt {d+e x}} \]

[In]

Integrate[(A + B*x)/(Sqrt[d + e*x]*(b*x + c*x^2)^(3/2)),x]

[Out]

(2*Sqrt[b/c]*(b*B - A*c)*d*(d + e*x) - (2*I)*e*(2*A*c*d - b*(B*d + A*e))*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)] + (2*I)*A*e*(c*d - b*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)])/(b*Sqrt[b/c]*d*(-(c*d) + b*e)*Sqrt[x*(
b + c*x)]*Sqrt[d + e*x])

Maple [A] (verified)

Time = 1.21 (sec) , antiderivative size = 499, normalized size of antiderivative = 1.69

method	result	size
elliptic	\(\frac {\sqrt {\left (e x +d \right ) x \left (c x +b \right )}\, \left (-\frac {2 \left (c e \,x^{2}+b e x +c d x +b d \right ) A}{b^{2} d \sqrt {x \left (c e \,x^{2}+b e x +c d x +b d \right )}}+\frac {2 \left (c e \,x^{2}+c d x \right ) \left (A c -B b \right )}{\left (b e -c d \right ) b^{2} \sqrt {\left (x +\frac {b}{c}\right ) \left (c e \,x^{2}+c d x \right )}}+\frac {2 \left (-\frac {A c -B b}{b^{2}}-\frac {c d \left (A c -B b \right )}{\left (b e -c d \right ) b^{2}}\right ) b \sqrt {\frac {\left (x +\frac {b}{c}\right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, F\left (\sqrt {\frac {\left (x +\frac {b}{c}\right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{c \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}+\frac {2 \left (\frac {A c e}{b^{2} d}-\frac {c e \left (A c -B b \right )}{b^{2} \left (b e -c d \right )}\right ) b \sqrt {\frac {\left (x +\frac {b}{c}\right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, \left (\left (-\frac {b}{c}+\frac {d}{e}\right ) E\left (\sqrt {\frac {\left (x +\frac {b}{c}\right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )-\frac {d F\left (\sqrt {\frac {\left (x +\frac {b}{c}\right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{e}\right )}{c \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}\right )}{\sqrt {x \left (c x +b \right )}\, \sqrt {e x +d}}\)	\(499\)
default	\(-\frac {2 \left (2 A \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, F\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b^{2} c d e -2 A \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, F\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b \,c^{2} d^{2}+A \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, E\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b^{3} e^{2}-3 A \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, E\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b^{2} c d e +2 A \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, E\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b \,c^{2} d^{2}-B \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, F\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b^{3} d e +B \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, F\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b^{2} c \,d^{2}+B \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, E\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b^{3} d e -B \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, E\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b^{2} c \,d^{2}+A \,x^{2} b \,c^{2} e^{2}-2 A \,x^{2} c^{3} d e +B \,x^{2} b \,c^{2} d e +A x \,b^{2} c \,e^{2}-2 A \,c^{3} d^{2} x +B x b \,c^{2} d^{2}+A \,b^{2} c d e -A \,d^{2} b \,c^{2}\right ) \sqrt {x \left (c x +b \right )}}{x \left (c x +b \right ) \left (b e -c d \right ) c \,b^{2} d \sqrt {e x +d}}\)	\(814\)

[In]

int((B*x+A)/(c*x^2+b*x)^(3/2)/(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*(c*e*x^2+b*e*x+c*d*x+b*d)*A/b^2/d/(x*(c*e*x^2+b*
e*x+c*d*x+b*d))^(1/2)+2*(c*e*x^2+c*d*x)/(b*e-c*d)/b^2*(A*c-B*b)/((x+b/c)*(c*e*x^2+c*d*x))^(1/2)+2*(-(A*c-B*b)/
b^2-c*d/(b*e-c*d)/b^2*(A*c-B*b))*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*(A*c*e/b^2/d-c*e*(A*c-B*b)
/b^2/(b*e-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)*((-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 = 0.11 (sec) , antiderivative size = 560, normalized size of antiderivative = 1.90 \[ \int \frac {A+B x}{\sqrt {d+e x} \left (b x+c x^2\right )^{3/2}} \, dx=\frac {2 \, {\left ({\left ({\left (A b^{2} c e^{2} + {\left (B b c^{2} - 2 \, A c^{3}\right )} d^{2} - 2 \, {\left (B b^{2} c - A b c^{2}\right )} d e\right )} x^{2} + {\left (A b^{3} e^{2} + {\left (B b^{2} c - 2 \, A b c^{2}\right )} d^{2} - 2 \, {\left (B b^{3} - A b^{2} c\right )} d e\right )} x\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right ) + 3 \, {\left ({\left (A b c^{2} e^{2} + {\left (B b c^{2} - 2 \, A c^{3}\right )} d e\right )} x^{2} + {\left (A b^{2} c e^{2} + {\left (B b^{2} c - 2 \, A b c^{2}\right )} d e\right )} x\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right )\right ) - 3 \, {\left (A b c^{2} d e - A b^{2} c e^{2} - {\left (A b c^{2} e^{2} + {\left (B b c^{2} - 2 \, A c^{3}\right )} d e\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {e x + d}\right )}}{3 \, {\left ({\left (b^{2} c^{3} d^{2} e - b^{3} c^{2} d e^{2}\right )} x^{2} + {\left (b^{3} c^{2} d^{2} e - b^{4} c d e^{2}\right )} x\right )}} \]

[In]

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

[Out]

2/3*(((A*b^2*c*e^2 + (B*b*c^2 - 2*A*c^3)*d^2 - 2*(B*b^2*c - A*b*c^2)*d*e)*x^2 + (A*b^3*e^2 + (B*b^2*c - 2*A*b*
c^2)*d^2 - 2*(B*b^3 - A*b^2*c)*d*e)*x)*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*((A*b*c^2*e^2 + (B*b*c^2 - 2*A*c^3)*d*e)*x^2 + (A*b^2*c*e^2 + (B*b^2*c - 2*A*b*c^2)*d*e)*x)*sqrt(c*e)*weie
rstrassZeta(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*c^2*d*e - A*b^2*c*e
^2 - (A*b*c^2*e^2 + (B*b*c^2 - 2*A*c^3)*d*e)*x)*sqrt(c*x^2 + b*x)*sqrt(e*x + d))/((b^2*c^3*d^2*e - b^3*c^2*d*e
^2)*x^2 + (b^3*c^2*d^2*e - b^4*c*d*e^2)*x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

integrate((B*x+A)/(c*x^2+b*x)^(3/2)/(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

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

Giac [F]

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

[In]

integrate((B*x+A)/(c*x^2+b*x)^(3/2)/(e*x+d)^(1/2),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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