\(\int \frac {(A+B x) \sqrt {b x+c x^2}}{d+e x} \, dx\) [1166]

Optimal result

Integrand size = 26, antiderivative size = 200 \[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{d+e x} \, dx=-\frac {(4 B c d-b B e-4 A c e-2 B c e x) \sqrt {b x+c x^2}}{4 c e^2}-\frac {\left (4 A c e (2 c d-b e)-B \left (8 c^2 d^2-4 b c d e-b^2 e^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{4 c^{3/2} e^3}-\frac {\sqrt {d} (B d-A e) \sqrt {c d-b e} \text {arctanh}\left (\frac {b d+(2 c d-b e) x}{2 \sqrt {d} \sqrt {c d-b e} \sqrt {b x+c x^2}}\right )}{e^3} \]

[Out]

-1/4*(4*A*c*e*(-b*e+2*c*d)-B*(-b^2*e^2-4*b*c*d*e+8*c^2*d^2))*arctanh(x*c^(1/2)/(c*x^2+b*x)^(1/2))/c^(3/2)/e^3-
(-A*e+B*d)*arctanh(1/2*(b*d+(-b*e+2*c*d)*x)/d^(1/2)/(-b*e+c*d)^(1/2)/(c*x^2+b*x)^(1/2))*d^(1/2)*(-b*e+c*d)^(1/
2)/e^3-1/4*(-2*B*c*e*x-4*A*c*e-B*b*e+4*B*c*d)*(c*x^2+b*x)^(1/2)/c/e^2

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {828, 857, 634, 212, 738} \[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{d+e x} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) \left (4 A c e (2 c d-b e)-B \left (-b^2 e^2-4 b c d e+8 c^2 d^2\right )\right )}{4 c^{3/2} e^3}-\frac {\sqrt {d} (B d-A e) \sqrt {c d-b e} \text {arctanh}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{e^3}-\frac {\sqrt {b x+c x^2} (-4 A c e-b B e+4 B c d-2 B c e x)}{4 c e^2} \]

[In]

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

[Out]

-1/4*((4*B*c*d - b*B*e - 4*A*c*e - 2*B*c*e*x)*Sqrt[b*x + c*x^2])/(c*e^2) - ((4*A*c*e*(2*c*d - b*e) - B*(8*c^2*
d^2 - 4*b*c*d*e - b^2*e^2))*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(4*c^(3/2)*e^3) - (Sqrt[d]*(B*d - A*e)*Sqr
t[c*d - b*e]*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/e^3

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 634

Int[1/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(1 - c*x^2), x], x, x/Sqrt[b*x + c*x^2
]], x] /; FreeQ[{b, c}, x]

Rule 738

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 828

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*((a + b*x + c*x^
2)^p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] - Dist[p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), Int[(d + e*x)^m*(a
+ b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2*a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p -
c*d - 2*c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c^2*d^2*(1 + 2*p) - c*e*(b*
d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*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] && GtQ[p, 0] && (IntegerQ[p] ||  !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])
) &&  !ILtQ[m + 2*p, 0] && (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 {(4 B c d-b B e-4 A c e-2 B c e x) \sqrt {b x+c x^2}}{4 c e^2}-\frac {\int \frac {-\frac {1}{2} b d (4 B c d-b B e-4 A c e)+\frac {1}{2} \left (4 A c e (2 c d-b e)-B \left (8 c^2 d^2-4 b c d e-b^2 e^2\right )\right ) x}{(d+e x) \sqrt {b x+c x^2}} \, dx}{4 c e^2} \\ & = -\frac {(4 B c d-b B e-4 A c e-2 B c e x) \sqrt {b x+c x^2}}{4 c e^2}-\frac {(d (B d-A e) (c d-b e)) \int \frac {1}{(d+e x) \sqrt {b x+c x^2}} \, dx}{e^3}-\frac {\left (4 A c e (2 c d-b e)-B \left (8 c^2 d^2-4 b c d e-b^2 e^2\right )\right ) \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{8 c e^3} \\ & = -\frac {(4 B c d-b B e-4 A c e-2 B c e x) \sqrt {b x+c x^2}}{4 c e^2}+\frac {(2 d (B d-A e) (c d-b e)) \text {Subst}\left (\int \frac {1}{4 c d^2-4 b d e-x^2} \, dx,x,\frac {-b d-(2 c d-b e) x}{\sqrt {b x+c x^2}}\right )}{e^3}-\frac {\left (4 A c e (2 c d-b e)-B \left (8 c^2 d^2-4 b c d e-b^2 e^2\right )\right ) \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right )}{4 c e^3} \\ & = -\frac {(4 B c d-b B e-4 A c e-2 B c e x) \sqrt {b x+c x^2}}{4 c e^2}-\frac {\left (4 A c e (2 c d-b e)-B \left (8 c^2 d^2-4 b c d e-b^2 e^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{4 c^{3/2} e^3}-\frac {\sqrt {d} (B d-A e) \sqrt {c d-b e} \tanh ^{-1}\left (\frac {b d+(2 c d-b e) x}{2 \sqrt {d} \sqrt {c d-b e} \sqrt {b x+c x^2}}\right )}{e^3} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 2.41 (sec) , antiderivative size = 483, normalized size of antiderivative = 2.42 \[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{d+e x} \, dx=\frac {\sqrt {x} \sqrt {b+c x} \left (\sqrt {c} \sqrt {d} e \sqrt {x} \sqrt {b+c x} (4 A c e+B (-4 c d+b e+2 c e x))+8 \sqrt {c} (B d-A e) \left (c d-b e-i \sqrt {b} \sqrt {e} \sqrt {c d-b e}\right ) \sqrt {-c d+2 b e-2 i \sqrt {b} \sqrt {e} \sqrt {c d-b e}} \arctan \left (\frac {\sqrt {-c d+2 b e-2 i \sqrt {b} \sqrt {e} \sqrt {c d-b e}} \sqrt {x}}{\sqrt {d} \left (-\sqrt {b}+\sqrt {b+c x}\right )}\right )+8 \sqrt {c} (B d-A e) \left (c d-b e+i \sqrt {b} \sqrt {e} \sqrt {c d-b e}\right ) \sqrt {-c d+2 b e+2 i \sqrt {b} \sqrt {e} \sqrt {c d-b e}} \arctan \left (\frac {\sqrt {-c d+2 b e+2 i \sqrt {b} \sqrt {e} \sqrt {c d-b e}} \sqrt {x}}{\sqrt {d} \left (-\sqrt {b}+\sqrt {b+c x}\right )}\right )+2 \sqrt {d} \left (4 A c e (-2 c d+b e)+B \left (8 c^2 d^2-4 b c d e-b^2 e^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {x}}{-\sqrt {b}+\sqrt {b+c x}}\right )\right )}{4 c^{3/2} \sqrt {d} e^3 \sqrt {x (b+c x)}} \]

[In]

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

[Out]

(Sqrt[x]*Sqrt[b + c*x]*(Sqrt[c]*Sqrt[d]*e*Sqrt[x]*Sqrt[b + c*x]*(4*A*c*e + B*(-4*c*d + b*e + 2*c*e*x)) + 8*Sqr
t[c]*(B*d - A*e)*(c*d - b*e - I*Sqrt[b]*Sqrt[e]*Sqrt[c*d - b*e])*Sqrt[-(c*d) + 2*b*e - (2*I)*Sqrt[b]*Sqrt[e]*S
qrt[c*d - b*e]]*ArcTan[(Sqrt[-(c*d) + 2*b*e - (2*I)*Sqrt[b]*Sqrt[e]*Sqrt[c*d - b*e]]*Sqrt[x])/(Sqrt[d]*(-Sqrt[
b] + Sqrt[b + c*x]))] + 8*Sqrt[c]*(B*d - A*e)*(c*d - b*e + I*Sqrt[b]*Sqrt[e]*Sqrt[c*d - b*e])*Sqrt[-(c*d) + 2*
b*e + (2*I)*Sqrt[b]*Sqrt[e]*Sqrt[c*d - b*e]]*ArcTan[(Sqrt[-(c*d) + 2*b*e + (2*I)*Sqrt[b]*Sqrt[e]*Sqrt[c*d - b*
e]]*Sqrt[x])/(Sqrt[d]*(-Sqrt[b] + Sqrt[b + c*x]))] + 2*Sqrt[d]*(4*A*c*e*(-2*c*d + b*e) + B*(8*c^2*d^2 - 4*b*c*
d*e - b^2*e^2))*ArcTanh[(Sqrt[c]*Sqrt[x])/(-Sqrt[b] + Sqrt[b + c*x])]))/(4*c^(3/2)*Sqrt[d]*e^3*Sqrt[x*(b + c*x
)])

Maple [A] (verified)

Time = 1.12 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.82

method	result	size
pseudoelliptic	\(-\frac {-\frac {2 \left (b e -c d \right ) \left (A e -B d \right ) d \arctan \left (\frac {\sqrt {x \left (c x +b \right )}\, d}{x \sqrt {d \left (b e -c d \right )}}\right )}{\sqrt {d \left (b e -c d \right )}}-\frac {e \sqrt {x \left (c x +b \right )}\, \left (2 B c e x +4 A c e +B b e -4 B c d \right )}{4 c}-\frac {\left (4 A b c \,e^{2}-8 A \,c^{2} d e -B \,b^{2} e^{2}-4 B b c d e +8 B \,c^{2} d^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {x \left (c x +b \right )}}{x \sqrt {c}}\right )}{4 c^{\frac {3}{2}}}}{e^{3}}\)	\(165\)
risch	\(\frac {\left (2 B c e x +4 A c e +B b e -4 B c d \right ) x \left (c x +b \right )}{4 c \,e^{2} \sqrt {x \left (c x +b \right )}}+\frac {\frac {\left (4 A b c \,e^{2}-8 A \,c^{2} d e -B \,b^{2} e^{2}-4 B b c d e +8 B \,c^{2} d^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{e \sqrt {c}}+\frac {8 d \left (A b \,e^{2}-A c d e -B b d e +B c \,d^{2}\right ) c \ln \left (\frac {-\frac {2 d \left (b e -c d \right )}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}}{8 e^{2} c}\)	\(286\)
default	\(\frac {B \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{e}+\frac {\left (A e -B d \right ) \left (\sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}+\frac {\left (b e -2 c d \right ) \ln \left (\frac {\frac {b e -2 c d}{2 e}+c \left (x +\frac {d}{e}\right )}{\sqrt {c}}+\sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}\right )}{2 e \sqrt {c}}+\frac {d \left (b e -c d \right ) \ln \left (\frac {-\frac {2 d \left (b e -c d \right )}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}\right )}{e^{2}}\)	\(353\)

[In]

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

[Out]

-1/e^3*(-2*(b*e-c*d)*(A*e-B*d)*d/(d*(b*e-c*d))^(1/2)*arctan((x*(c*x+b))^(1/2)/x*d/(d*(b*e-c*d))^(1/2))-1/4*e*(
x*(c*x+b))^(1/2)*(2*B*c*e*x+4*A*c*e+B*b*e-4*B*c*d)/c-1/4*(4*A*b*c*e^2-8*A*c^2*d*e-B*b^2*e^2-4*B*b*c*d*e+8*B*c^
2*d^2)/c^(3/2)*arctanh((x*(c*x+b))^(1/2)/x/c^(1/2)))

Fricas [A] (verification not implemented)

none

Time = 0.77 (sec) , antiderivative size = 800, normalized size of antiderivative = 4.00 \[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{d+e x} \, dx=\left [\frac {{\left (8 \, B c^{2} d^{2} - 4 \, {\left (B b c + 2 \, A c^{2}\right )} d e - {\left (B b^{2} - 4 \, A b c\right )} e^{2}\right )} \sqrt {c} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) - 8 \, {\left (B c^{2} d - A c^{2} e\right )} \sqrt {c d^{2} - b d e} \log \left (\frac {b d + {\left (2 \, c d - b e\right )} x + 2 \, \sqrt {c d^{2} - b d e} \sqrt {c x^{2} + b x}}{e x + d}\right ) + 2 \, {\left (2 \, B c^{2} e^{2} x - 4 \, B c^{2} d e + {\left (B b c + 4 \, A c^{2}\right )} e^{2}\right )} \sqrt {c x^{2} + b x}}{8 \, c^{2} e^{3}}, -\frac {16 \, {\left (B c^{2} d - A c^{2} e\right )} \sqrt {-c d^{2} + b d e} \arctan \left (-\frac {\sqrt {-c d^{2} + b d e} \sqrt {c x^{2} + b x}}{{\left (c d - b e\right )} x}\right ) - {\left (8 \, B c^{2} d^{2} - 4 \, {\left (B b c + 2 \, A c^{2}\right )} d e - {\left (B b^{2} - 4 \, A b c\right )} e^{2}\right )} \sqrt {c} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) - 2 \, {\left (2 \, B c^{2} e^{2} x - 4 \, B c^{2} d e + {\left (B b c + 4 \, A c^{2}\right )} e^{2}\right )} \sqrt {c x^{2} + b x}}{8 \, c^{2} e^{3}}, -\frac {{\left (8 \, B c^{2} d^{2} - 4 \, {\left (B b c + 2 \, A c^{2}\right )} d e - {\left (B b^{2} - 4 \, A b c\right )} e^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) + 4 \, {\left (B c^{2} d - A c^{2} e\right )} \sqrt {c d^{2} - b d e} \log \left (\frac {b d + {\left (2 \, c d - b e\right )} x + 2 \, \sqrt {c d^{2} - b d e} \sqrt {c x^{2} + b x}}{e x + d}\right ) - {\left (2 \, B c^{2} e^{2} x - 4 \, B c^{2} d e + {\left (B b c + 4 \, A c^{2}\right )} e^{2}\right )} \sqrt {c x^{2} + b x}}{4 \, c^{2} e^{3}}, -\frac {8 \, {\left (B c^{2} d - A c^{2} e\right )} \sqrt {-c d^{2} + b d e} \arctan \left (-\frac {\sqrt {-c d^{2} + b d e} \sqrt {c x^{2} + b x}}{{\left (c d - b e\right )} x}\right ) + {\left (8 \, B c^{2} d^{2} - 4 \, {\left (B b c + 2 \, A c^{2}\right )} d e - {\left (B b^{2} - 4 \, A b c\right )} e^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) - {\left (2 \, B c^{2} e^{2} x - 4 \, B c^{2} d e + {\left (B b c + 4 \, A c^{2}\right )} e^{2}\right )} \sqrt {c x^{2} + b x}}{4 \, c^{2} e^{3}}\right ] \]

[In]

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

[Out]

[1/8*((8*B*c^2*d^2 - 4*(B*b*c + 2*A*c^2)*d*e - (B*b^2 - 4*A*b*c)*e^2)*sqrt(c)*log(2*c*x + b + 2*sqrt(c*x^2 + b
*x)*sqrt(c)) - 8*(B*c^2*d - A*c^2*e)*sqrt(c*d^2 - b*d*e)*log((b*d + (2*c*d - b*e)*x + 2*sqrt(c*d^2 - b*d*e)*sq
rt(c*x^2 + b*x))/(e*x + d)) + 2*(2*B*c^2*e^2*x - 4*B*c^2*d*e + (B*b*c + 4*A*c^2)*e^2)*sqrt(c*x^2 + b*x))/(c^2*
e^3), -1/8*(16*(B*c^2*d - A*c^2*e)*sqrt(-c*d^2 + b*d*e)*arctan(-sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x)/((c*d -
b*e)*x)) - (8*B*c^2*d^2 - 4*(B*b*c + 2*A*c^2)*d*e - (B*b^2 - 4*A*b*c)*e^2)*sqrt(c)*log(2*c*x + b + 2*sqrt(c*x
^2 + b*x)*sqrt(c)) - 2*(2*B*c^2*e^2*x - 4*B*c^2*d*e + (B*b*c + 4*A*c^2)*e^2)*sqrt(c*x^2 + b*x))/(c^2*e^3), -1/
4*((8*B*c^2*d^2 - 4*(B*b*c + 2*A*c^2)*d*e - (B*b^2 - 4*A*b*c)*e^2)*sqrt(-c)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/
(c*x)) + 4*(B*c^2*d - A*c^2*e)*sqrt(c*d^2 - b*d*e)*log((b*d + (2*c*d - b*e)*x + 2*sqrt(c*d^2 - b*d*e)*sqrt(c*x
^2 + b*x))/(e*x + d)) - (2*B*c^2*e^2*x - 4*B*c^2*d*e + (B*b*c + 4*A*c^2)*e^2)*sqrt(c*x^2 + b*x))/(c^2*e^3), -1
/4*(8*(B*c^2*d - A*c^2*e)*sqrt(-c*d^2 + b*d*e)*arctan(-sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x)/((c*d - b*e)*x))
+ (8*B*c^2*d^2 - 4*(B*b*c + 2*A*c^2)*d*e - (B*b^2 - 4*A*b*c)*e^2)*sqrt(-c)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/
(c*x)) - (2*B*c^2*e^2*x - 4*B*c^2*d*e + (B*b*c + 4*A*c^2)*e^2)*sqrt(c*x^2 + b*x))/(c^2*e^3)]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{d+e x} \, dx=\text {Exception raised: ValueError} \]

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more
details)Is e

Giac [F(-2)]

Exception generated. \[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{d+e x} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Error: Bad Argument Type

Mupad [F(-1)]

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

[In]

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

[Out]

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