3.2631 \(\int \frac {(A+B x) \sqrt {d+e x}}{\sqrt {a+b x+c x^2}} \, dx\)

Optimal. Leaf size=452 \[ \frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (3 A c e-2 b B e+B c d) E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 c^2 e \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {2 \sqrt {2} B \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 c^2 e \sqrt {d+e x} \sqrt {a+b x+c x^2}}+\frac {2 B \sqrt {d+e x} \sqrt {a+b x+c x^2}}{3 c} \]

[Out]

2/3*B*(e*x+d)^(1/2)*(c*x^2+b*x+a)^(1/2)/c+1/3*(3*A*c*e-2*B*b*e+B*c*d)*EllipticE(1/2*((b+2*c*x+(-4*a*c+b^2)^(1/
2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),(-2*e*(-4*a*c+b^2)^(1/2)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2))))^(1/2))*2^(1/2
)*(-4*a*c+b^2)^(1/2)*(e*x+d)^(1/2)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/2)/c^2/e/(c*x^2+b*x+a)^(1/2)/(c*(e*x+d)/
(2*c*d-e*(b+(-4*a*c+b^2)^(1/2))))^(1/2)-2/3*B*(a*e^2-b*d*e+c*d^2)*EllipticF(1/2*((b+2*c*x+(-4*a*c+b^2)^(1/2))/
(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),(-2*e*(-4*a*c+b^2)^(1/2)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2))))^(1/2))*2^(1/2)*(-
4*a*c+b^2)^(1/2)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/2)*(c*(e*x+d)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2))))^(1/2)/c^2/
e/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2)

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Rubi [A]  time = 0.40, antiderivative size = 452, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {832, 843, 718, 424, 419} \[ \frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (3 A c e-2 b B e+B c d) E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 c^2 e \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {2 \sqrt {2} B \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 c^2 e \sqrt {d+e x} \sqrt {a+b x+c x^2}}+\frac {2 B \sqrt {d+e x} \sqrt {a+b x+c x^2}}{3 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*B*Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2])/(3*c) + (Sqrt[2]*Sqrt[b^2 - 4*a*c]*(B*c*d - 2*b*B*e + 3*A*c*e)*Sqrt[
d + e*x]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticE[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sq
rt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(3*c^2*e*Sqrt[(c*(d
+ e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[a + b*x + c*x^2]) - (2*Sqrt[2]*B*Sqrt[b^2 - 4*a*c]*(c*d^2 -
b*d*e + a*e^2)*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a
*c))]*EllipticF[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]
*e)/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(3*c^2*e*Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2])

Rule 419

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1*EllipticF[ArcSin[Rt[-(d/c),
2]*x], (b*c)/(a*d)])/(Sqrt[a]*Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] &
& GtQ[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-(b/a), -(d/c)])

Rule 424

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]*EllipticE[ArcSin[Rt[-(d/c)
, 2]*x], (b*c)/(a*d)])/(Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[
a, 0]

Rule 718

Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[(2*Rt[b^2 - 4*a*c, 2]
*(d + e*x)^m*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))])/(c*Sqrt[a + b*x + c*x^2]*((2*c*(d + e*x))/(2*c*d -
b*e - e*Rt[b^2 - 4*a*c, 2]))^m), Subst[Int[(1 + (2*e*Rt[b^2 - 4*a*c, 2]*x^2)/(2*c*d - b*e - e*Rt[b^2 - 4*a*c,
2]))^m/Sqrt[1 - x^2], x], x, Sqrt[(b + Rt[b^2 - 4*a*c, 2] + 2*c*x)/(2*Rt[b^2 - 4*a*c, 2])]], 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] && EqQ[m^2, 1/4]

Rule 832

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 843

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*} \int \frac {(A+B x) \sqrt {d+e x}}{\sqrt {a+b x+c x^2}} \, dx &=\frac {2 B \sqrt {d+e x} \sqrt {a+b x+c x^2}}{3 c}+\frac {2 \int \frac {\frac {1}{2} (-b B d+3 A c d-a B e)+\frac {1}{2} (B c d-2 b B e+3 A c e) x}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{3 c}\\ &=\frac {2 B \sqrt {d+e x} \sqrt {a+b x+c x^2}}{3 c}+\frac {(B c d-2 b B e+3 A c e) \int \frac {\sqrt {d+e x}}{\sqrt {a+b x+c x^2}} \, dx}{3 c e}-\frac {\left (B \left (c d^2-b d e+a e^2\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{3 c e}\\ &=\frac {2 B \sqrt {d+e x} \sqrt {a+b x+c x^2}}{3 c}+\frac {\left (\sqrt {2} \sqrt {b^2-4 a c} (B c d-2 b B e+3 A c e) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {2 \sqrt {b^2-4 a c} e x^2}{2 c d-b e-\sqrt {b^2-4 a c} e}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )}{3 c^2 e \sqrt {\frac {c (d+e x)}{2 c d-b e-\sqrt {b^2-4 a c} e}} \sqrt {a+b x+c x^2}}-\frac {\left (2 \sqrt {2} B \sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-b e-\sqrt {b^2-4 a c} e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 \sqrt {b^2-4 a c} e x^2}{2 c d-b e-\sqrt {b^2-4 a c} e}}} \, dx,x,\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )}{3 c^2 e \sqrt {d+e x} \sqrt {a+b x+c x^2}}\\ &=\frac {2 B \sqrt {d+e x} \sqrt {a+b x+c x^2}}{3 c}+\frac {\sqrt {2} \sqrt {b^2-4 a c} (B c d-2 b B e+3 A c e) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 c^2 e \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}-\frac {2 \sqrt {2} B \sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 c^2 e \sqrt {d+e x} \sqrt {a+b x+c x^2}}\\ \end {align*}

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Mathematica [C]  time = 7.12, size = 653, normalized size = 1.44 \[ \frac {2 \sqrt {d+e x} \left (B c (a+x (b+c x))+\frac {(d+e x) \left (\frac {e^2 (a+x (b+c x)) (3 A c e-2 b B e+B c d)}{(d+e x)^2}+\frac {i \sqrt {1-\frac {2 \left (e (a e-b d)+c d^2\right )}{(d+e x) \left (\sqrt {e^2 \left (b^2-4 a c\right )}-b e+2 c d\right )}} \sqrt {\frac {2 \left (e (a e-b d)+c d^2\right )}{(d+e x) \left (\sqrt {e^2 \left (b^2-4 a c\right )}+b e-2 c d\right )}+1} \left (\left (-b e \left (2 B \sqrt {e^2 \left (b^2-4 a c\right )}+3 A c e+3 B c d\right )+c \left (3 A e \left (\sqrt {e^2 \left (b^2-4 a c\right )}+2 c d\right )+B d \sqrt {e^2 \left (b^2-4 a c\right )}-2 a B e^2\right )+2 b^2 B e^2\right ) F\left (i \sinh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {c d^2-b e d+a e^2}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}{\sqrt {d+e x}}\right )|-\frac {-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}\right )+\left (\sqrt {e^2 \left (b^2-4 a c\right )}-b e+2 c d\right ) (-3 A c e+2 b B e-B c d) E\left (i \sinh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {c d^2-b e d+a e^2}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}{\sqrt {d+e x}}\right )|-\frac {-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}\right )\right )}{2 \sqrt {2} \sqrt {d+e x} \sqrt {\frac {e (a e-b d)+c d^2}{\sqrt {e^2 \left (b^2-4 a c\right )}+b e-2 c d}}}\right )}{e^2}\right )}{3 c^2 \sqrt {a+x (b+c x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[d + e*x]*(B*c*(a + x*(b + c*x)) + ((d + e*x)*((e^2*(B*c*d - 2*b*B*e + 3*A*c*e)*(a + x*(b + c*x)))/(d +
e*x)^2 + ((I/2)*Sqrt[1 - (2*(c*d^2 + e*(-(b*d) + a*e)))/((2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*
Sqrt[1 + (2*(c*d^2 + e*(-(b*d) + a*e)))/((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*((-(B*c*d) + 2*b
*B*e - 3*A*c*e)*(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e
^2)/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*c
*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))] + (2*b^2*B*e^2 - b*e*(3*B*c*d + 3*A*c*e + 2*B*Sqrt[(b^2 - 4*a*c)*e^2]) +
c*(-2*a*B*e^2 + B*d*Sqrt[(b^2 - 4*a*c)*e^2] + 3*A*e*(2*c*d + Sqrt[(b^2 - 4*a*c)*e^2])))*EllipticF[I*ArcSinh[(
Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*c*d + b*
e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))]))/(Sqrt[2]*Sqrt[(c*d^2 + e*(-(b*d) + a*
e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*Sqrt[d + e*x])))/e^2))/(3*c^2*Sqrt[a + x*(b + c*x)])

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fricas [F]  time = 0.71, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B x + A\right )} \sqrt {e x + d}}{\sqrt {c x^{2} + b x + a}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (B x + A\right )} \sqrt {e x + d}}{\sqrt {c x^{2} + b x + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.04, size = 3804, normalized size = 8.42 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(e*x+d)^(1/2)*(c*x^2+b*x+a)^(1/2)/c^2*(6*A*2^(1/2)*(-c*(e*x+d)/(e*(-4*a*c+b^2)^(1/2)+e*b-2*c*d))^(1/2)*((-
2*c*x+(-4*a*c+b^2)^(1/2)-b)*e/(e*(-4*a*c+b^2)^(1/2)-e*b+2*c*d))^(1/2)*((2*c*x+(-4*a*c+b^2)^(1/2)+b)*e/(e*(-4*a
*c+b^2)^(1/2)+e*b-2*c*d))^(1/2)*EllipticF(2^(1/2)*(-c*(e*x+d)/(e*(-4*a*c+b^2)^(1/2)+e*b-2*c*d))^(1/2),(-(e*(-4
*a*c+b^2)^(1/2)+e*b-2*c*d)/(e*(-4*a*c+b^2)^(1/2)-e*b+2*c*d))^(1/2))*a*c*e^3-6*A*2^(1/2)*(-c*(e*x+d)/(e*(-4*a*c
+b^2)^(1/2)+e*b-2*c*d))^(1/2)*((-2*c*x+(-4*a*c+b^2)^(1/2)-b)*e/(e*(-4*a*c+b^2)^(1/2)-e*b+2*c*d))^(1/2)*((2*c*x
+(-4*a*c+b^2)^(1/2)+b)*e/(e*(-4*a*c+b^2)^(1/2)+e*b-2*c*d))^(1/2)*EllipticF(2^(1/2)*(-c*(e*x+d)/(e*(-4*a*c+b^2)
^(1/2)+e*b-2*c*d))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)+e*b-2*c*d)/(e*(-4*a*c+b^2)^(1/2)-e*b+2*c*d))^(1/2))*b*c*d*e^2
+6*A*2^(1/2)*(-c*(e*x+d)/(e*(-4*a*c+b^2)^(1/2)+e*b-2*c*d))^(1/2)*((-2*c*x+(-4*a*c+b^2)^(1/2)-b)*e/(e*(-4*a*c+b
^2)^(1/2)-e*b+2*c*d))^(1/2)*((2*c*x+(-4*a*c+b^2)^(1/2)+b)*e/(e*(-4*a*c+b^2)^(1/2)+e*b-2*c*d))^(1/2)*EllipticF(
2^(1/2)*(-c*(e*x+d)/(e*(-4*a*c+b^2)^(1/2)+e*b-2*c*d))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)+e*b-2*c*d)/(e*(-4*a*c+b^2)
^(1/2)-e*b+2*c*d))^(1/2))*c^2*d^2*e-6*A*2^(1/2)*(-c*(e*x+d)/(e*(-4*a*c+b^2)^(1/2)+e*b-2*c*d))^(1/2)*((-2*c*x+(
-4*a*c+b^2)^(1/2)-b)*e/(e*(-4*a*c+b^2)^(1/2)-e*b+2*c*d))^(1/2)*((2*c*x+(-4*a*c+b^2)^(1/2)+b)*e/(e*(-4*a*c+b^2)
^(1/2)+e*b-2*c*d))^(1/2)*EllipticE(2^(1/2)*(-c*(e*x+d)/(e*(-4*a*c+b^2)^(1/2)+e*b-2*c*d))^(1/2),(-(e*(-4*a*c+b^
2)^(1/2)+e*b-2*c*d)/(e*(-4*a*c+b^2)^(1/2)-e*b+2*c*d))^(1/2))*a*c*e^3+6*A*2^(1/2)*(-c*(e*x+d)/(e*(-4*a*c+b^2)^(
1/2)+e*b-2*c*d))^(1/2)*((-2*c*x+(-4*a*c+b^2)^(1/2)-b)*e/(e*(-4*a*c+b^2)^(1/2)-e*b+2*c*d))^(1/2)*((2*c*x+(-4*a*
c+b^2)^(1/2)+b)*e/(e*(-4*a*c+b^2)^(1/2)+e*b-2*c*d))^(1/2)*EllipticE(2^(1/2)*(-c*(e*x+d)/(e*(-4*a*c+b^2)^(1/2)+
e*b-2*c*d))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)+e*b-2*c*d)/(e*(-4*a*c+b^2)^(1/2)-e*b+2*c*d))^(1/2))*b*c*d*e^2-6*A*2^
(1/2)*(-c*(e*x+d)/(e*(-4*a*c+b^2)^(1/2)+e*b-2*c*d))^(1/2)*((-2*c*x+(-4*a*c+b^2)^(1/2)-b)*e/(e*(-4*a*c+b^2)^(1/
2)-e*b+2*c*d))^(1/2)*((2*c*x+(-4*a*c+b^2)^(1/2)+b)*e/(e*(-4*a*c+b^2)^(1/2)+e*b-2*c*d))^(1/2)*EllipticE(2^(1/2)
*(-c*(e*x+d)/(e*(-4*a*c+b^2)^(1/2)+e*b-2*c*d))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)+e*b-2*c*d)/(e*(-4*a*c+b^2)^(1/2)-
e*b+2*c*d))^(1/2))*c^2*d^2*e+B*2^(1/2)*(-c*(e*x+d)/(e*(-4*a*c+b^2)^(1/2)+e*b-2*c*d))^(1/2)*((-2*c*x+(-4*a*c+b^
2)^(1/2)-b)*e/(e*(-4*a*c+b^2)^(1/2)-e*b+2*c*d))^(1/2)*((2*c*x+(-4*a*c+b^2)^(1/2)+b)*e/(e*(-4*a*c+b^2)^(1/2)+e*
b-2*c*d))^(1/2)*EllipticF(2^(1/2)*(-c*(e*x+d)/(e*(-4*a*c+b^2)^(1/2)+e*b-2*c*d))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)+
e*b-2*c*d)/(e*(-4*a*c+b^2)^(1/2)-e*b+2*c*d))^(1/2))*(-4*a*c+b^2)^(1/2)*a*e^3-B*2^(1/2)*(-c*(e*x+d)/(e*(-4*a*c+
b^2)^(1/2)+e*b-2*c*d))^(1/2)*((-2*c*x+(-4*a*c+b^2)^(1/2)-b)*e/(e*(-4*a*c+b^2)^(1/2)-e*b+2*c*d))^(1/2)*((2*c*x+
(-4*a*c+b^2)^(1/2)+b)*e/(e*(-4*a*c+b^2)^(1/2)+e*b-2*c*d))^(1/2)*EllipticF(2^(1/2)*(-c*(e*x+d)/(e*(-4*a*c+b^2)^
(1/2)+e*b-2*c*d))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)+e*b-2*c*d)/(e*(-4*a*c+b^2)^(1/2)-e*b+2*c*d))^(1/2))*(-4*a*c+b^
2)^(1/2)*b*d*e^2+B*2^(1/2)*(-c*(e*x+d)/(e*(-4*a*c+b^2)^(1/2)+e*b-2*c*d))^(1/2)*((-2*c*x+(-4*a*c+b^2)^(1/2)-b)*
e/(e*(-4*a*c+b^2)^(1/2)-e*b+2*c*d))^(1/2)*((2*c*x+(-4*a*c+b^2)^(1/2)+b)*e/(e*(-4*a*c+b^2)^(1/2)+e*b-2*c*d))^(1
/2)*EllipticF(2^(1/2)*(-c*(e*x+d)/(e*(-4*a*c+b^2)^(1/2)+e*b-2*c*d))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)+e*b-2*c*d)/(
e*(-4*a*c+b^2)^(1/2)-e*b+2*c*d))^(1/2))*(-4*a*c+b^2)^(1/2)*c*d^2*e-3*B*2^(1/2)*(-c*(e*x+d)/(e*(-4*a*c+b^2)^(1/
2)+e*b-2*c*d))^(1/2)*((-2*c*x+(-4*a*c+b^2)^(1/2)-b)*e/(e*(-4*a*c+b^2)^(1/2)-e*b+2*c*d))^(1/2)*((2*c*x+(-4*a*c+
b^2)^(1/2)+b)*e/(e*(-4*a*c+b^2)^(1/2)+e*b-2*c*d))^(1/2)*EllipticF(2^(1/2)*(-c*(e*x+d)/(e*(-4*a*c+b^2)^(1/2)+e*
b-2*c*d))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)+e*b-2*c*d)/(e*(-4*a*c+b^2)^(1/2)-e*b+2*c*d))^(1/2))*a*b*e^3+3*B*2^(1/2
)*(-c*(e*x+d)/(e*(-4*a*c+b^2)^(1/2)+e*b-2*c*d))^(1/2)*((-2*c*x+(-4*a*c+b^2)^(1/2)-b)*e/(e*(-4*a*c+b^2)^(1/2)-e
*b+2*c*d))^(1/2)*((2*c*x+(-4*a*c+b^2)^(1/2)+b)*e/(e*(-4*a*c+b^2)^(1/2)+e*b-2*c*d))^(1/2)*EllipticF(2^(1/2)*(-c
*(e*x+d)/(e*(-4*a*c+b^2)^(1/2)+e*b-2*c*d))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)+e*b-2*c*d)/(e*(-4*a*c+b^2)^(1/2)-e*b+
2*c*d))^(1/2))*b^2*d*e^2-3*B*2^(1/2)*(-c*(e*x+d)/(e*(-4*a*c+b^2)^(1/2)+e*b-2*c*d))^(1/2)*((-2*c*x+(-4*a*c+b^2)
^(1/2)-b)*e/(e*(-4*a*c+b^2)^(1/2)-e*b+2*c*d))^(1/2)*((2*c*x+(-4*a*c+b^2)^(1/2)+b)*e/(e*(-4*a*c+b^2)^(1/2)+e*b-
2*c*d))^(1/2)*EllipticF(2^(1/2)*(-c*(e*x+d)/(e*(-4*a*c+b^2)^(1/2)+e*b-2*c*d))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)+e*
b-2*c*d)/(e*(-4*a*c+b^2)^(1/2)-e*b+2*c*d))^(1/2))*b*c*d^2*e+4*B*2^(1/2)*(-c*(e*x+d)/(e*(-4*a*c+b^2)^(1/2)+e*b-
2*c*d))^(1/2)*((-2*c*x+(-4*a*c+b^2)^(1/2)-b)*e/(e*(-4*a*c+b^2)^(1/2)-e*b+2*c*d))^(1/2)*((2*c*x+(-4*a*c+b^2)^(1
/2)+b)*e/(e*(-4*a*c+b^2)^(1/2)+e*b-2*c*d))^(1/2)*EllipticE(2^(1/2)*(-c*(e*x+d)/(e*(-4*a*c+b^2)^(1/2)+e*b-2*c*d
))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)+e*b-2*c*d)/(e*(-4*a*c+b^2)^(1/2)-e*b+2*c*d))^(1/2))*a*b*e^3-2*B*2^(1/2)*(-c*(
e*x+d)/(e*(-4*a*c+b^2)^(1/2)+e*b-2*c*d))^(1/2)*((-2*c*x+(-4*a*c+b^2)^(1/2)-b)*e/(e*(-4*a*c+b^2)^(1/2)-e*b+2*c*
d))^(1/2)*((2*c*x+(-4*a*c+b^2)^(1/2)+b)*e/(e*(-4*a*c+b^2)^(1/2)+e*b-2*c*d))^(1/2)*EllipticE(2^(1/2)*(-c*(e*x+d
)/(e*(-4*a*c+b^2)^(1/2)+e*b-2*c*d))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)+e*b-2*c*d)/(e*(-4*a*c+b^2)^(1/2)-e*b+2*c*d))
^(1/2))*a*c*d*e^2-4*B*2^(1/2)*(-c*(e*x+d)/(e*(-4*a*c+b^2)^(1/2)+e*b-2*c*d))^(1/2)*((-2*c*x+(-4*a*c+b^2)^(1/2)-
b)*e/(e*(-4*a*c+b^2)^(1/2)-e*b+2*c*d))^(1/2)*((2*c*x+(-4*a*c+b^2)^(1/2)+b)*e/(e*(-4*a*c+b^2)^(1/2)+e*b-2*c*d))
^(1/2)*EllipticE(2^(1/2)*(-c*(e*x+d)/(e*(-4*a*c+b^2)^(1/2)+e*b-2*c*d))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)+e*b-2*c*d
)/(e*(-4*a*c+b^2)^(1/2)-e*b+2*c*d))^(1/2))*b^2*d*e^2+6*B*2^(1/2)*(-c*(e*x+d)/(e*(-4*a*c+b^2)^(1/2)+e*b-2*c*d))
^(1/2)*((-2*c*x+(-4*a*c+b^2)^(1/2)-b)*e/(e*(-4*a*c+b^2)^(1/2)-e*b+2*c*d))^(1/2)*((2*c*x+(-4*a*c+b^2)^(1/2)+b)*
e/(e*(-4*a*c+b^2)^(1/2)+e*b-2*c*d))^(1/2)*EllipticE(2^(1/2)*(-c*(e*x+d)/(e*(-4*a*c+b^2)^(1/2)+e*b-2*c*d))^(1/2
),(-(e*(-4*a*c+b^2)^(1/2)+e*b-2*c*d)/(e*(-4*a*c+b^2)^(1/2)-e*b+2*c*d))^(1/2))*b*c*d^2*e-2*B*2^(1/2)*(-c*(e*x+d
)/(e*(-4*a*c+b^2)^(1/2)+e*b-2*c*d))^(1/2)*((-2*c*x+(-4*a*c+b^2)^(1/2)-b)*e/(e*(-4*a*c+b^2)^(1/2)-e*b+2*c*d))^(
1/2)*((2*c*x+(-4*a*c+b^2)^(1/2)+b)*e/(e*(-4*a*c+b^2)^(1/2)+e*b-2*c*d))^(1/2)*EllipticE(2^(1/2)*(-c*(e*x+d)/(e*
(-4*a*c+b^2)^(1/2)+e*b-2*c*d))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)+e*b-2*c*d)/(e*(-4*a*c+b^2)^(1/2)-e*b+2*c*d))^(1/2
))*c^2*d^3+2*B*x^3*c^2*e^3+2*B*x^2*b*c*e^3+2*B*x^2*c^2*d*e^2+2*B*x*a*c*e^3+2*B*x*b*c*d*e^2+2*B*a*c*d*e^2)/(c*e
*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)/e^2

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (B x + A\right )} \sqrt {e x + d}}{\sqrt {c x^{2} + b x + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (A+B\,x\right )\,\sqrt {d+e\,x}}{\sqrt {c\,x^2+b\,x+a}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (A + B x\right ) \sqrt {d + e x}}{\sqrt {a + b x + c x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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