3.1.79 \(\int \frac {x \sqrt {a+b x+c x^2}}{d-f x^2} \, dx\) [79]

Optimal. Leaf size=282 \[ -\frac {\sqrt {a+b x+c x^2}}{f}-\frac {b \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {c} f}-\frac {\sqrt {c d-b \sqrt {d} \sqrt {f}+a f} \tanh ^{-1}\left (\frac {b \sqrt {d}-2 a \sqrt {f}+\left (2 c \sqrt {d}-b \sqrt {f}\right ) x}{2 \sqrt {c d-b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 f^{3/2}}+\frac {\sqrt {c d+b \sqrt {d} \sqrt {f}+a f} \tanh ^{-1}\left (\frac {b \sqrt {d}+2 a \sqrt {f}+\left (2 c \sqrt {d}+b \sqrt {f}\right ) x}{2 \sqrt {c d+b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 f^{3/2}} \]

[Out]

-1/2*b*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/f/c^(1/2)-(c*x^2+b*x+a)^(1/2)/f-1/2*arctanh(1/2*(b*d
^(1/2)-2*a*f^(1/2)+x*(2*c*d^(1/2)-b*f^(1/2)))/(c*x^2+b*x+a)^(1/2)/(c*d+a*f-b*d^(1/2)*f^(1/2))^(1/2))*(c*d+a*f-
b*d^(1/2)*f^(1/2))^(1/2)/f^(3/2)+1/2*arctanh(1/2*(b*d^(1/2)+2*a*f^(1/2)+x*(2*c*d^(1/2)+b*f^(1/2)))/(c*x^2+b*x+
a)^(1/2)/(c*d+a*f+b*d^(1/2)*f^(1/2))^(1/2))*(c*d+a*f+b*d^(1/2)*f^(1/2))^(1/2)/f^(3/2)

________________________________________________________________________________________

Rubi [A]
time = 0.19, antiderivative size = 282, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {1035, 1092, 635, 212, 1047, 738} \begin {gather*} -\frac {\sqrt {a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d} \tanh ^{-1}\left (\frac {-2 a \sqrt {f}+x \left (2 c \sqrt {d}-b \sqrt {f}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d}}\right )}{2 f^{3/2}}+\frac {\sqrt {a f+b \sqrt {d} \sqrt {f}+c d} \tanh ^{-1}\left (\frac {2 a \sqrt {f}+x \left (b \sqrt {f}+2 c \sqrt {d}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \sqrt {d} \sqrt {f}+c d}}\right )}{2 f^{3/2}}-\frac {\sqrt {a+b x+c x^2}}{f}-\frac {b \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {c} f} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x*Sqrt[a + b*x + c*x^2])/(d - f*x^2),x]

[Out]

-(Sqrt[a + b*x + c*x^2]/f) - (b*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(2*Sqrt[c]*f) - (Sqrt[
c*d - b*Sqrt[d]*Sqrt[f] + a*f]*ArcTanh[(b*Sqrt[d] - 2*a*Sqrt[f] + (2*c*Sqrt[d] - b*Sqrt[f])*x)/(2*Sqrt[c*d - b
*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*f^(3/2)) + (Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*ArcTanh[(b
*Sqrt[d] + 2*a*Sqrt[f] + (2*c*Sqrt[d] + b*Sqrt[f])*x)/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*
x^2])])/(2*f^(3/2))

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 635

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

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 1035

Int[((g_.) + (h_.)*(x_))*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_) + (f_.)*(x_)^2)^(q_), x_Symbol] :> Simp
[h*(a + b*x + c*x^2)^p*((d + f*x^2)^(q + 1)/(2*f*(p + q + 1))), x] - Dist[1/(2*f*(p + q + 1)), Int[(a + b*x +
c*x^2)^(p - 1)*(d + f*x^2)^q*Simp[h*p*(b*d) + a*(-2*g*f)*(p + q + 1) + (2*h*p*(c*d - a*f) + b*(-2*g*f)*(p + q
+ 1))*x + (h*p*((-b)*f) + c*(-2*g*f)*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, f, g, h, q}, x] && NeQ
[b^2 - 4*a*c, 0] && GtQ[p, 0] && NeQ[p + q + 1, 0]

Rule 1047

Int[((g_.) + (h_.)*(x_))/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> With[{q
= Rt[(-a)*c, 2]}, Dist[h/2 + c*(g/(2*q)), Int[1/((-q + c*x)*Sqrt[d + e*x + f*x^2]), x], x] + Dist[h/2 - c*(g/
(2*q)), Int[1/((q + c*x)*Sqrt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, c, d, e, f, g, h}, x] && NeQ[e^2 - 4*d*f
, 0] && PosQ[(-a)*c]

Rule 1092

Int[((A_.) + (B_.)*(x_) + (C_.)*(x_)^2)/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Sym
bol] :> Dist[C/c, Int[1/Sqrt[d + e*x + f*x^2], x], x] + Dist[1/c, Int[(A*c - a*C + B*c*x)/((a + c*x^2)*Sqrt[d
+ e*x + f*x^2]), x], x] /; FreeQ[{a, c, d, e, f, A, B, C}, x] && NeQ[e^2 - 4*d*f, 0]

Rubi steps

\begin {align*} \int \frac {x \sqrt {a+b x+c x^2}}{d-f x^2} \, dx &=-\frac {\sqrt {a+b x+c x^2}}{f}+\frac {\int \frac {\frac {b d}{2}+(c d+a f) x+\frac {1}{2} b f x^2}{\sqrt {a+b x+c x^2} \left (d-f x^2\right )} \, dx}{f}\\ &=-\frac {\sqrt {a+b x+c x^2}}{f}-\frac {\int \frac {-b d f-f (c d+a f) x}{\sqrt {a+b x+c x^2} \left (d-f x^2\right )} \, dx}{f^2}-\frac {b \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{2 f}\\ &=-\frac {\sqrt {a+b x+c x^2}}{f}-\frac {b \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )}{f}+\frac {\left (c d-b \sqrt {d} \sqrt {f}+a f\right ) \int \frac {1}{\left (-\sqrt {d} \sqrt {f}-f x\right ) \sqrt {a+b x+c x^2}} \, dx}{2 f}+\frac {\left (c d+b \sqrt {d} \sqrt {f}+a f\right ) \int \frac {1}{\left (\sqrt {d} \sqrt {f}-f x\right ) \sqrt {a+b x+c x^2}} \, dx}{2 f}\\ &=-\frac {\sqrt {a+b x+c x^2}}{f}-\frac {b \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {c} f}-\frac {\left (c d-b \sqrt {d} \sqrt {f}+a f\right ) \text {Subst}\left (\int \frac {1}{4 c d f-4 b \sqrt {d} f^{3/2}+4 a f^2-x^2} \, dx,x,\frac {b \sqrt {d} \sqrt {f}-2 a f-\left (-2 c \sqrt {d} \sqrt {f}+b f\right ) x}{\sqrt {a+b x+c x^2}}\right )}{f}-\frac {\left (c d+b \sqrt {d} \sqrt {f}+a f\right ) \text {Subst}\left (\int \frac {1}{4 c d f+4 b \sqrt {d} f^{3/2}+4 a f^2-x^2} \, dx,x,\frac {-b \sqrt {d} \sqrt {f}-2 a f-\left (2 c \sqrt {d} \sqrt {f}+b f\right ) x}{\sqrt {a+b x+c x^2}}\right )}{f}\\ &=-\frac {\sqrt {a+b x+c x^2}}{f}-\frac {b \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {c} f}-\frac {\sqrt {c d-b \sqrt {d} \sqrt {f}+a f} \tanh ^{-1}\left (\frac {b \sqrt {d}-2 a \sqrt {f}+\left (2 c \sqrt {d}-b \sqrt {f}\right ) x}{2 \sqrt {c d-b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 f^{3/2}}+\frac {\sqrt {c d+b \sqrt {d} \sqrt {f}+a f} \tanh ^{-1}\left (\frac {b \sqrt {d}+2 a \sqrt {f}+\left (2 c \sqrt {d}+b \sqrt {f}\right ) x}{2 \sqrt {c d+b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 f^{3/2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 0.41, size = 349, normalized size = 1.24 \begin {gather*} -\frac {2 \sqrt {a+x (b+c x)}-\frac {b \log \left (f \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )\right )}{\sqrt {c}}+\text {RootSum}\left [b^2 d-a^2 f-4 b \sqrt {c} d \text {$\#$1}+4 c d \text {$\#$1}^2+2 a f \text {$\#$1}^2-f \text {$\#$1}^4\&,\frac {b^2 d \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )-a c d \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )-a^2 f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )-2 b \sqrt {c} d \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}+c d \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2+a f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{b \sqrt {c} d-2 c d \text {$\#$1}-a f \text {$\#$1}+f \text {$\#$1}^3}\&\right ]}{2 f} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x*Sqrt[a + b*x + c*x^2])/(d - f*x^2),x]

[Out]

-1/2*(2*Sqrt[a + x*(b + c*x)] - (b*Log[f*(b + 2*c*x - 2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/Sqrt[c] + RootSum[b^2
*d - a^2*f - 4*b*Sqrt[c]*d*#1 + 4*c*d*#1^2 + 2*a*f*#1^2 - f*#1^4 & , (b^2*d*Log[-(Sqrt[c]*x) + Sqrt[a + b*x +
c*x^2] - #1] - a*c*d*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1] - a^2*f*Log[-(Sqrt[c]*x) + Sqrt[a + b*x +
c*x^2] - #1] - 2*b*Sqrt[c]*d*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1]*#1 + c*d*Log[-(Sqrt[c]*x) + Sqrt[a
+ b*x + c*x^2] - #1]*#1^2 + a*f*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1]*#1^2)/(b*Sqrt[c]*d - 2*c*d*#1
- a*f*#1 + f*#1^3) & ])/f

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(767\) vs. \(2(214)=428\).
time = 0.14, size = 768, normalized size = 2.72

method	result	size
default	\(-\frac {\sqrt {\left (x +\frac {\sqrt {d f}}{f}\right )^{2} c +\frac {\left (-2 c \sqrt {d f}+b f \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+\frac {-b \sqrt {d f}+f a +c d}{f}}+\frac {\left (-2 c \sqrt {d f}+b f \right ) \ln \left (\frac {\frac {-2 c \sqrt {d f}+b f}{2 f}+c \left (x +\frac {\sqrt {d f}}{f}\right )}{\sqrt {c}}+\sqrt {\left (x +\frac {\sqrt {d f}}{f}\right )^{2} c +\frac {\left (-2 c \sqrt {d f}+b f \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+\frac {-b \sqrt {d f}+f a +c d}{f}}\right )}{2 f \sqrt {c}}-\frac {\left (-b \sqrt {d f}+f a +c d \right ) \ln \left (\frac {\frac {-2 b \sqrt {d f}+2 f a +2 c d}{f}+\frac {\left (-2 c \sqrt {d f}+b f \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+2 \sqrt {\frac {-b \sqrt {d f}+f a +c d}{f}}\, \sqrt {\left (x +\frac {\sqrt {d f}}{f}\right )^{2} c +\frac {\left (-2 c \sqrt {d f}+b f \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+\frac {-b \sqrt {d f}+f a +c d}{f}}}{x +\frac {\sqrt {d f}}{f}}\right )}{f \sqrt {\frac {-b \sqrt {d f}+f a +c d}{f}}}}{2 f}-\frac {\sqrt {\left (x -\frac {\sqrt {d f}}{f}\right )^{2} c +\frac {\left (2 c \sqrt {d f}+b f \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+\frac {b \sqrt {d f}+f a +c d}{f}}+\frac {\left (2 c \sqrt {d f}+b f \right ) \ln \left (\frac {\frac {2 c \sqrt {d f}+b f}{2 f}+c \left (x -\frac {\sqrt {d f}}{f}\right )}{\sqrt {c}}+\sqrt {\left (x -\frac {\sqrt {d f}}{f}\right )^{2} c +\frac {\left (2 c \sqrt {d f}+b f \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+\frac {b \sqrt {d f}+f a +c d}{f}}\right )}{2 f \sqrt {c}}-\frac {\left (b \sqrt {d f}+f a +c d \right ) \ln \left (\frac {\frac {2 b \sqrt {d f}+2 f a +2 c d}{f}+\frac {\left (2 c \sqrt {d f}+b f \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+2 \sqrt {\frac {b \sqrt {d f}+f a +c d}{f}}\, \sqrt {\left (x -\frac {\sqrt {d f}}{f}\right )^{2} c +\frac {\left (2 c \sqrt {d f}+b f \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+\frac {b \sqrt {d f}+f a +c d}{f}}}{x -\frac {\sqrt {d f}}{f}}\right )}{f \sqrt {\frac {b \sqrt {d f}+f a +c d}{f}}}}{2 f}\)	\(768\)
risch	\(-\frac {\sqrt {c \,x^{2}+b x +a}}{f}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 f \sqrt {c}}+\frac {\ln \left (\frac {\frac {2 b \sqrt {d f}+2 f a +2 c d}{f}+\frac {\left (2 c \sqrt {d f}+b f \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+2 \sqrt {\frac {b \sqrt {d f}+f a +c d}{f}}\, \sqrt {\left (x -\frac {\sqrt {d f}}{f}\right )^{2} c +\frac {\left (2 c \sqrt {d f}+b f \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+\frac {b \sqrt {d f}+f a +c d}{f}}}{x -\frac {\sqrt {d f}}{f}}\right ) a}{2 f \sqrt {\frac {b \sqrt {d f}+f a +c d}{f}}}+\frac {\ln \left (\frac {\frac {2 b \sqrt {d f}+2 f a +2 c d}{f}+\frac {\left (2 c \sqrt {d f}+b f \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+2 \sqrt {\frac {b \sqrt {d f}+f a +c d}{f}}\, \sqrt {\left (x -\frac {\sqrt {d f}}{f}\right )^{2} c +\frac {\left (2 c \sqrt {d f}+b f \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+\frac {b \sqrt {d f}+f a +c d}{f}}}{x -\frac {\sqrt {d f}}{f}}\right ) c d}{2 f^{2} \sqrt {\frac {b \sqrt {d f}+f a +c d}{f}}}+\frac {\ln \left (\frac {\frac {2 b \sqrt {d f}+2 f a +2 c d}{f}+\frac {\left (2 c \sqrt {d f}+b f \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+2 \sqrt {\frac {b \sqrt {d f}+f a +c d}{f}}\, \sqrt {\left (x -\frac {\sqrt {d f}}{f}\right )^{2} c +\frac {\left (2 c \sqrt {d f}+b f \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+\frac {b \sqrt {d f}+f a +c d}{f}}}{x -\frac {\sqrt {d f}}{f}}\right ) b d}{2 f \sqrt {d f}\, \sqrt {\frac {b \sqrt {d f}+f a +c d}{f}}}+\frac {\ln \left (\frac {\frac {-2 b \sqrt {d f}+2 f a +2 c d}{f}+\frac {\left (-2 c \sqrt {d f}+b f \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+2 \sqrt {\frac {-b \sqrt {d f}+f a +c d}{f}}\, \sqrt {\left (x +\frac {\sqrt {d f}}{f}\right )^{2} c +\frac {\left (-2 c \sqrt {d f}+b f \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+\frac {-b \sqrt {d f}+f a +c d}{f}}}{x +\frac {\sqrt {d f}}{f}}\right ) a}{2 f \sqrt {\frac {-b \sqrt {d f}+f a +c d}{f}}}+\frac {\ln \left (\frac {\frac {-2 b \sqrt {d f}+2 f a +2 c d}{f}+\frac {\left (-2 c \sqrt {d f}+b f \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+2 \sqrt {\frac {-b \sqrt {d f}+f a +c d}{f}}\, \sqrt {\left (x +\frac {\sqrt {d f}}{f}\right )^{2} c +\frac {\left (-2 c \sqrt {d f}+b f \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+\frac {-b \sqrt {d f}+f a +c d}{f}}}{x +\frac {\sqrt {d f}}{f}}\right ) c d}{2 f^{2} \sqrt {\frac {-b \sqrt {d f}+f a +c d}{f}}}-\frac {\ln \left (\frac {\frac {-2 b \sqrt {d f}+2 f a +2 c d}{f}+\frac {\left (-2 c \sqrt {d f}+b f \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+2 \sqrt {\frac {-b \sqrt {d f}+f a +c d}{f}}\, \sqrt {\left (x +\frac {\sqrt {d f}}{f}\right )^{2} c +\frac {\left (-2 c \sqrt {d f}+b f \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+\frac {-b \sqrt {d f}+f a +c d}{f}}}{x +\frac {\sqrt {d f}}{f}}\right ) b d}{2 f \sqrt {d f}\, \sqrt {\frac {-b \sqrt {d f}+f a +c d}{f}}}\)	\(1129\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(c*x^2+b*x+a)^(1/2)/(-f*x^2+d),x,method=_RETURNVERBOSE)

[Out]

-1/2/f*(((x+(d*f)^(1/2)/f)^2*c+1/f*(-2*c*(d*f)^(1/2)+b*f)*(x+(d*f)^(1/2)/f)+1/f*(-b*(d*f)^(1/2)+f*a+c*d))^(1/2
)+1/2/f*(-2*c*(d*f)^(1/2)+b*f)*ln((1/2/f*(-2*c*(d*f)^(1/2)+b*f)+c*(x+(d*f)^(1/2)/f))/c^(1/2)+((x+(d*f)^(1/2)/f
)^2*c+1/f*(-2*c*(d*f)^(1/2)+b*f)*(x+(d*f)^(1/2)/f)+1/f*(-b*(d*f)^(1/2)+f*a+c*d))^(1/2))/c^(1/2)-1/f*(-b*(d*f)^
(1/2)+f*a+c*d)/(1/f*(-b*(d*f)^(1/2)+f*a+c*d))^(1/2)*ln((2/f*(-b*(d*f)^(1/2)+f*a+c*d)+1/f*(-2*c*(d*f)^(1/2)+b*f
)*(x+(d*f)^(1/2)/f)+2*(1/f*(-b*(d*f)^(1/2)+f*a+c*d))^(1/2)*((x+(d*f)^(1/2)/f)^2*c+1/f*(-2*c*(d*f)^(1/2)+b*f)*(
x+(d*f)^(1/2)/f)+1/f*(-b*(d*f)^(1/2)+f*a+c*d))^(1/2))/(x+(d*f)^(1/2)/f)))-1/2/f*(((x-(d*f)^(1/2)/f)^2*c+(2*c*(
d*f)^(1/2)+b*f)/f*(x-(d*f)^(1/2)/f)+(b*(d*f)^(1/2)+f*a+c*d)/f)^(1/2)+1/2*(2*c*(d*f)^(1/2)+b*f)/f*ln((1/2*(2*c*
(d*f)^(1/2)+b*f)/f+c*(x-(d*f)^(1/2)/f))/c^(1/2)+((x-(d*f)^(1/2)/f)^2*c+(2*c*(d*f)^(1/2)+b*f)/f*(x-(d*f)^(1/2)/
f)+(b*(d*f)^(1/2)+f*a+c*d)/f)^(1/2))/c^(1/2)-(b*(d*f)^(1/2)+f*a+c*d)/f/((b*(d*f)^(1/2)+f*a+c*d)/f)^(1/2)*ln((2
*(b*(d*f)^(1/2)+f*a+c*d)/f+(2*c*(d*f)^(1/2)+b*f)/f*(x-(d*f)^(1/2)/f)+2*((b*(d*f)^(1/2)+f*a+c*d)/f)^(1/2)*((x-(
d*f)^(1/2)/f)^2*c+(2*c*(d*f)^(1/2)+b*f)/f*(x-(d*f)^(1/2)/f)+(b*(d*f)^(1/2)+f*a+c*d)/f)^(1/2))/(x-(d*f)^(1/2)/f
)))

________________________________________________________________________________________

Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(c*x^2+b*x+a)^(1/2)/(-f*x^2+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(((c*sqrt(4*d*f))/(2*f^2)>0)',
see `assume?

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 595 vs. \(2 (214) = 428\).
time = 157.33, size = 1192, normalized size = 4.23 \begin {gather*} \left [\frac {c f \sqrt {\frac {f^{3} \sqrt {\frac {b^{2} d}{f^{5}}} + c d + a f}{f^{3}}} \log \left (\frac {2 \, \sqrt {c x^{2} + b x + a} f^{4} \sqrt {\frac {b^{2} d}{f^{5}}} \sqrt {\frac {f^{3} \sqrt {\frac {b^{2} d}{f^{5}}} + c d + a f}{f^{3}}} + 2 \, b c d x + b^{2} d + {\left (b f^{3} x + 2 \, a f^{3}\right )} \sqrt {\frac {b^{2} d}{f^{5}}}}{x}\right ) - c f \sqrt {\frac {f^{3} \sqrt {\frac {b^{2} d}{f^{5}}} + c d + a f}{f^{3}}} \log \left (-\frac {2 \, \sqrt {c x^{2} + b x + a} f^{4} \sqrt {\frac {b^{2} d}{f^{5}}} \sqrt {\frac {f^{3} \sqrt {\frac {b^{2} d}{f^{5}}} + c d + a f}{f^{3}}} - 2 \, b c d x - b^{2} d - {\left (b f^{3} x + 2 \, a f^{3}\right )} \sqrt {\frac {b^{2} d}{f^{5}}}}{x}\right ) - c f \sqrt {-\frac {f^{3} \sqrt {\frac {b^{2} d}{f^{5}}} - c d - a f}{f^{3}}} \log \left (\frac {2 \, \sqrt {c x^{2} + b x + a} f^{4} \sqrt {\frac {b^{2} d}{f^{5}}} \sqrt {-\frac {f^{3} \sqrt {\frac {b^{2} d}{f^{5}}} - c d - a f}{f^{3}}} + 2 \, b c d x + b^{2} d - {\left (b f^{3} x + 2 \, a f^{3}\right )} \sqrt {\frac {b^{2} d}{f^{5}}}}{x}\right ) + c f \sqrt {-\frac {f^{3} \sqrt {\frac {b^{2} d}{f^{5}}} - c d - a f}{f^{3}}} \log \left (-\frac {2 \, \sqrt {c x^{2} + b x + a} f^{4} \sqrt {\frac {b^{2} d}{f^{5}}} \sqrt {-\frac {f^{3} \sqrt {\frac {b^{2} d}{f^{5}}} - c d - a f}{f^{3}}} - 2 \, b c d x - b^{2} d + {\left (b f^{3} x + 2 \, a f^{3}\right )} \sqrt {\frac {b^{2} d}{f^{5}}}}{x}\right ) + b \sqrt {c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} + 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) - 4 \, \sqrt {c x^{2} + b x + a} c}{4 \, c f}, \frac {c f \sqrt {\frac {f^{3} \sqrt {\frac {b^{2} d}{f^{5}}} + c d + a f}{f^{3}}} \log \left (\frac {2 \, \sqrt {c x^{2} + b x + a} f^{4} \sqrt {\frac {b^{2} d}{f^{5}}} \sqrt {\frac {f^{3} \sqrt {\frac {b^{2} d}{f^{5}}} + c d + a f}{f^{3}}} + 2 \, b c d x + b^{2} d + {\left (b f^{3} x + 2 \, a f^{3}\right )} \sqrt {\frac {b^{2} d}{f^{5}}}}{x}\right ) - c f \sqrt {\frac {f^{3} \sqrt {\frac {b^{2} d}{f^{5}}} + c d + a f}{f^{3}}} \log \left (-\frac {2 \, \sqrt {c x^{2} + b x + a} f^{4} \sqrt {\frac {b^{2} d}{f^{5}}} \sqrt {\frac {f^{3} \sqrt {\frac {b^{2} d}{f^{5}}} + c d + a f}{f^{3}}} - 2 \, b c d x - b^{2} d - {\left (b f^{3} x + 2 \, a f^{3}\right )} \sqrt {\frac {b^{2} d}{f^{5}}}}{x}\right ) - c f \sqrt {-\frac {f^{3} \sqrt {\frac {b^{2} d}{f^{5}}} - c d - a f}{f^{3}}} \log \left (\frac {2 \, \sqrt {c x^{2} + b x + a} f^{4} \sqrt {\frac {b^{2} d}{f^{5}}} \sqrt {-\frac {f^{3} \sqrt {\frac {b^{2} d}{f^{5}}} - c d - a f}{f^{3}}} + 2 \, b c d x + b^{2} d - {\left (b f^{3} x + 2 \, a f^{3}\right )} \sqrt {\frac {b^{2} d}{f^{5}}}}{x}\right ) + c f \sqrt {-\frac {f^{3} \sqrt {\frac {b^{2} d}{f^{5}}} - c d - a f}{f^{3}}} \log \left (-\frac {2 \, \sqrt {c x^{2} + b x + a} f^{4} \sqrt {\frac {b^{2} d}{f^{5}}} \sqrt {-\frac {f^{3} \sqrt {\frac {b^{2} d}{f^{5}}} - c d - a f}{f^{3}}} - 2 \, b c d x - b^{2} d + {\left (b f^{3} x + 2 \, a f^{3}\right )} \sqrt {\frac {b^{2} d}{f^{5}}}}{x}\right ) + 2 \, b \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) - 4 \, \sqrt {c x^{2} + b x + a} c}{4 \, c f}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(c*x^2+b*x+a)^(1/2)/(-f*x^2+d),x, algorithm="fricas")

[Out]

[1/4*(c*f*sqrt((f^3*sqrt(b^2*d/f^5) + c*d + a*f)/f^3)*log((2*sqrt(c*x^2 + b*x + a)*f^4*sqrt(b^2*d/f^5)*sqrt((f
^3*sqrt(b^2*d/f^5) + c*d + a*f)/f^3) + 2*b*c*d*x + b^2*d + (b*f^3*x + 2*a*f^3)*sqrt(b^2*d/f^5))/x) - c*f*sqrt(
(f^3*sqrt(b^2*d/f^5) + c*d + a*f)/f^3)*log(-(2*sqrt(c*x^2 + b*x + a)*f^4*sqrt(b^2*d/f^5)*sqrt((f^3*sqrt(b^2*d/
f^5) + c*d + a*f)/f^3) - 2*b*c*d*x - b^2*d - (b*f^3*x + 2*a*f^3)*sqrt(b^2*d/f^5))/x) - c*f*sqrt(-(f^3*sqrt(b^2
*d/f^5) - c*d - a*f)/f^3)*log((2*sqrt(c*x^2 + b*x + a)*f^4*sqrt(b^2*d/f^5)*sqrt(-(f^3*sqrt(b^2*d/f^5) - c*d -
a*f)/f^3) + 2*b*c*d*x + b^2*d - (b*f^3*x + 2*a*f^3)*sqrt(b^2*d/f^5))/x) + c*f*sqrt(-(f^3*sqrt(b^2*d/f^5) - c*d
- a*f)/f^3)*log(-(2*sqrt(c*x^2 + b*x + a)*f^4*sqrt(b^2*d/f^5)*sqrt(-(f^3*sqrt(b^2*d/f^5) - c*d - a*f)/f^3) -
2*b*c*d*x - b^2*d + (b*f^3*x + 2*a*f^3)*sqrt(b^2*d/f^5))/x) + b*sqrt(c)*log(-8*c^2*x^2 - 8*b*c*x - b^2 + 4*sqr
t(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(c) - 4*a*c) - 4*sqrt(c*x^2 + b*x + a)*c)/(c*f), 1/4*(c*f*sqrt((f^3*sqrt(b^
2*d/f^5) + c*d + a*f)/f^3)*log((2*sqrt(c*x^2 + b*x + a)*f^4*sqrt(b^2*d/f^5)*sqrt((f^3*sqrt(b^2*d/f^5) + c*d +
a*f)/f^3) + 2*b*c*d*x + b^2*d + (b*f^3*x + 2*a*f^3)*sqrt(b^2*d/f^5))/x) - c*f*sqrt((f^3*sqrt(b^2*d/f^5) + c*d
+ a*f)/f^3)*log(-(2*sqrt(c*x^2 + b*x + a)*f^4*sqrt(b^2*d/f^5)*sqrt((f^3*sqrt(b^2*d/f^5) + c*d + a*f)/f^3) - 2*
b*c*d*x - b^2*d - (b*f^3*x + 2*a*f^3)*sqrt(b^2*d/f^5))/x) - c*f*sqrt(-(f^3*sqrt(b^2*d/f^5) - c*d - a*f)/f^3)*l
og((2*sqrt(c*x^2 + b*x + a)*f^4*sqrt(b^2*d/f^5)*sqrt(-(f^3*sqrt(b^2*d/f^5) - c*d - a*f)/f^3) + 2*b*c*d*x + b^2
*d - (b*f^3*x + 2*a*f^3)*sqrt(b^2*d/f^5))/x) + c*f*sqrt(-(f^3*sqrt(b^2*d/f^5) - c*d - a*f)/f^3)*log(-(2*sqrt(c
*x^2 + b*x + a)*f^4*sqrt(b^2*d/f^5)*sqrt(-(f^3*sqrt(b^2*d/f^5) - c*d - a*f)/f^3) - 2*b*c*d*x - b^2*d + (b*f^3*
x + 2*a*f^3)*sqrt(b^2*d/f^5))/x) + 2*b*sqrt(-c)*arctan(1/2*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(-c)/(c^2*x^2
+ b*c*x + a*c)) - 4*sqrt(c*x^2 + b*x + a)*c)/(c*f)]

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {x \sqrt {a + b x + c x^{2}}}{- d + f x^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(c*x**2+b*x+a)**(1/2)/(-f*x**2+d),x)

[Out]

-Integral(x*sqrt(a + b*x + c*x**2)/(-d + f*x**2), x)

________________________________________________________________________________________

Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(c*x^2+b*x+a)^(1/2)/(-f*x^2+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]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x\,\sqrt {c\,x^2+b\,x+a}}{d-f\,x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*(a + b*x + c*x^2)^(1/2))/(d - f*x^2),x)

[Out]

int((x*(a + b*x + c*x^2)^(1/2))/(d - f*x^2), x)

________________________________________________________________________________________