\(\int \frac {A+B x}{(d+e x)^{3/2} \sqrt {b x+c x^2}} \, dx\) [1268]

Optimal result

Integrand size = 28, antiderivative size = 262 \[ \int \frac {A+B x}{(d+e x)^{3/2} \sqrt {b x+c x^2}} \, dx=\frac {2 (B d-A e) \sqrt {b x+c x^2}}{d (c d-b e) \sqrt {d+e x}}-\frac {2 \sqrt {-b} \sqrt {c} (B d-A e) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{d e (c d-b e) \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {2 \sqrt {-b} B \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {d+e x} \sqrt {b x+c x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 262, 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 = {848, 857, 729, 113, 111, 118, 117} \[ \int \frac {A+B x}{(d+e x)^{3/2} \sqrt {b x+c x^2}} \, dx=-\frac {2 \sqrt {-b} \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (B d-A e) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{d e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1} (c d-b e)}+\frac {2 \sqrt {b x+c x^2} (B d-A e)}{d \sqrt {d+e x} (c d-b e)}+\frac {2 \sqrt {-b} B \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {d+e x}} \]

[In]

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

[Out]

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 16.73 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.86 \[ \int \frac {A+B x}{(d+e x)^{3/2} \sqrt {b x+c x^2}} \, dx=\frac {-2 \sqrt {\frac {b}{c}} d (B d-A e) (b+c x)+2 i b e (-B d+A e) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )+2 i A e (c d-b e) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right ),\frac {c d}{b e}\right )}{\sqrt {\frac {b}{c}} d e (c d-b e) \sqrt {x (b+c x)} \sqrt {d+e x}} \]

[In]

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

[Out]

(-2*Sqrt[b/c]*d*(B*d - A*e)*(b + c*x) + (2*I)*b*e*(-(B*d) + A*e)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*E
llipticE[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)])/(Sqrt[b/c]*d*e*(c*d - b*e)*Sqrt[x*(b + c*x)]*S
qrt[d + e*x])

Maple [A] (verified)

Time = 1.18 (sec) , antiderivative size = 441, normalized size of antiderivative = 1.68

method	result	size
elliptic	\(\frac {\sqrt {\left (e x +d \right ) x \left (c x +b \right )}\, \left (\frac {2 \left (c e \,x^{2}+b e x \right ) \left (A e -B d \right )}{e d \left (b e -c d \right ) \sqrt {\left (x +\frac {d}{e}\right ) \left (c e \,x^{2}+b e x \right )}}+\frac {2 \left (\frac {B}{e}+\frac {A e -B d}{e d}-\frac {\left (A e -B d \right ) b}{d \left (b e -c d \right )}\right ) b \sqrt {\frac {\left (x +\frac {b}{c}\right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, F\left (\sqrt {\frac {\left (x +\frac {b}{c}\right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{c \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}-\frac {2 \left (A e -B d \right ) b \sqrt {\frac {\left (x +\frac {b}{c}\right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, \left (\left (-\frac {b}{c}+\frac {d}{e}\right ) E\left (\sqrt {\frac {\left (x +\frac {b}{c}\right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )-\frac {d F\left (\sqrt {\frac {\left (x +\frac {b}{c}\right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{e}\right )}{d \left (b e -c d \right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}\right )}{\sqrt {x \left (c x +b \right )}\, \sqrt {e x +d}}\)	\(441\)
default	\(\frac {2 \left (A \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, E\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b^{2} e^{2}-A \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, E\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b c d e +B \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, F\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b^{2} d e -B \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, F\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b c \,d^{2}-B \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, E\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b^{2} d e +B \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, E\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b c \,d^{2}+A \,c^{2} e^{2} x^{2}-B \,c^{2} d e \,x^{2}+A b c \,e^{2} x -B b c d e x \right ) \sqrt {x \left (c x +b \right )}\, \sqrt {e x +d}}{d e c \left (b e -c d \right ) x \left (c e \,x^{2}+b e x +c d x +b d \right )}\)	\(547\)

[In]

int((B*x+A)/(e*x+d)^(3/2)/(c*x^2+b*x)^(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)/e/d/(b*e-c*d)*(A*e-B*d)/((x+d/e)*
(c*e*x^2+b*e*x))^(1/2)+2*(B/e+1/e*(A*e-B*d)/d-(A*e-B*d)*b/d/(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)*EllipticF(((x+b/c)/b*c)^(1/2),(-b/c/(-b/c+d/e
))^(1/2))-2*(A*e-B*d)/d/(b*e-c*d)*b*((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 = 0.09 (sec) , antiderivative size = 453, normalized size of antiderivative = 1.73 \[ \int \frac {A+B x}{(d+e x)^{3/2} \sqrt {b x+c x^2}} \, dx=\frac {2 \, {\left ({\left (B c d^{3} - A b d e^{2} - 2 \, {\left (B b - A c\right )} d^{2} e + {\left (B c d^{2} e - A b e^{3} - 2 \, {\left (B b - A c\right )} d e^{2}\right )} x\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right ) + 3 \, {\left (B c d^{2} e - A c d e^{2} + {\left (B c d e^{2} - A c e^{3}\right )} x\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right )\right ) + 3 \, {\left (B c d e^{2} - A c e^{3}\right )} \sqrt {c x^{2} + b x} \sqrt {e x + d}\right )}}{3 \, {\left (c^{2} d^{3} e^{2} - b c d^{2} e^{3} + {\left (c^{2} d^{2} e^{3} - b c d e^{4}\right )} x\right )}} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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