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

Optimal. Leaf size=416 \[ \frac {\left (-B \sqrt {b^2-4 a c}-2 A c+b B\right ) \tanh ^{-1}\left (\frac {2 x \left (c e-f \left (b-\sqrt {b^2-4 a c}\right )\right )-e \left (b-\sqrt {b^2-4 a c}\right )+4 c d}{2 \sqrt {2} \sqrt {d+e x+f x^2} \sqrt {\sqrt {b^2-4 a c} (c e-b f)-2 a c f+b^2 f-b c e+2 c^2 d}}\right )}{\sqrt {2} \sqrt {b^2-4 a c} \sqrt {\sqrt {b^2-4 a c} (c e-b f)-2 a c f+b^2 f-b c e+2 c^2 d}}+\frac {\left (2 A c-B \left (\sqrt {b^2-4 a c}+b\right )\right ) \tanh ^{-1}\left (\frac {2 x \left (c e-f \left (\sqrt {b^2-4 a c}+b\right )\right )-e \left (\sqrt {b^2-4 a c}+b\right )+4 c d}{2 \sqrt {2} \sqrt {d+e x+f x^2} \sqrt {-\sqrt {b^2-4 a c} (c e-b f)-2 a c f+b^2 f-b c e+2 c^2 d}}\right )}{\sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\sqrt {b^2-4 a c} (c e-b f)-2 a c f+b^2 f-b c e+2 c^2 d}} \]

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Rubi [A]  time = 2.70, antiderivative size = 416, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {1032, 724, 206} \begin {gather*} \frac {\left (-B \sqrt {b^2-4 a c}-2 A c+b B\right ) \tanh ^{-1}\left (\frac {2 x \left (c e-f \left (b-\sqrt {b^2-4 a c}\right )\right )-e \left (b-\sqrt {b^2-4 a c}\right )+4 c d}{2 \sqrt {2} \sqrt {d+e x+f x^2} \sqrt {\sqrt {b^2-4 a c} (c e-b f)-2 a c f+b^2 f-b c e+2 c^2 d}}\right )}{\sqrt {2} \sqrt {b^2-4 a c} \sqrt {\sqrt {b^2-4 a c} (c e-b f)-2 a c f+b^2 f-b c e+2 c^2 d}}+\frac {\left (2 A c-B \left (\sqrt {b^2-4 a c}+b\right )\right ) \tanh ^{-1}\left (\frac {2 x \left (c e-f \left (\sqrt {b^2-4 a c}+b\right )\right )-e \left (\sqrt {b^2-4 a c}+b\right )+4 c d}{2 \sqrt {2} \sqrt {d+e x+f x^2} \sqrt {-\sqrt {b^2-4 a c} (c e-b f)-2 a c f+b^2 f-b c e+2 c^2 d}}\right )}{\sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\sqrt {b^2-4 a c} (c e-b f)-2 a c f+b^2 f-b c e+2 c^2 d}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((b*B - 2*A*c - B*Sqrt[b^2 - 4*a*c])*ArcTanh[(4*c*d - (b - Sqrt[b^2 - 4*a*c])*e + 2*(c*e - (b - Sqrt[b^2 - 4*a
*c])*f)*x)/(2*Sqrt[2]*Sqrt[2*c^2*d - b*c*e + b^2*f - 2*a*c*f + Sqrt[b^2 - 4*a*c]*(c*e - b*f)]*Sqrt[d + e*x + f
*x^2])])/(Sqrt[2]*Sqrt[b^2 - 4*a*c]*Sqrt[2*c^2*d - b*c*e + b^2*f - 2*a*c*f + Sqrt[b^2 - 4*a*c]*(c*e - b*f)]) +
((2*A*c - B*(b + Sqrt[b^2 - 4*a*c]))*ArcTanh[(4*c*d - (b + Sqrt[b^2 - 4*a*c])*e + 2*(c*e - (b + Sqrt[b^2 - 4*
a*c])*f)*x)/(2*Sqrt[2]*Sqrt[2*c^2*d - b*c*e + b^2*f - 2*a*c*f - Sqrt[b^2 - 4*a*c]*(c*e - b*f)]*Sqrt[d + e*x +
f*x^2])])/(Sqrt[2]*Sqrt[b^2 - 4*a*c]*Sqrt[2*c^2*d - b*c*e + b^2*f - 2*a*c*f - Sqrt[b^2 - 4*a*c]*(c*e - b*f)])

Rule 206

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

Rule 724

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 1032

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

Rubi steps

\begin {align*} \int \frac {A+B x}{\left (a+b x+c x^2\right ) \sqrt {d+e x+f x^2}} \, dx &=\frac {\left (2 A c-B \left (b-\sqrt {b^2-4 a c}\right )\right ) \int \frac {1}{\left (b-\sqrt {b^2-4 a c}+2 c x\right ) \sqrt {d+e x+f x^2}} \, dx}{\sqrt {b^2-4 a c}}-\frac {\left (2 A c-B \left (b+\sqrt {b^2-4 a c}\right )\right ) \int \frac {1}{\left (b+\sqrt {b^2-4 a c}+2 c x\right ) \sqrt {d+e x+f x^2}} \, dx}{\sqrt {b^2-4 a c}}\\ &=\frac {\left (2 \left (b B-2 A c-B \sqrt {b^2-4 a c}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{16 c^2 d-8 c \left (b-\sqrt {b^2-4 a c}\right ) e+4 \left (b-\sqrt {b^2-4 a c}\right )^2 f-x^2} \, dx,x,\frac {4 c d-\left (b-\sqrt {b^2-4 a c}\right ) e-\left (-2 c e+2 \left (b-\sqrt {b^2-4 a c}\right ) f\right ) x}{\sqrt {d+e x+f x^2}}\right )}{\sqrt {b^2-4 a c}}+\frac {\left (2 \left (2 A c-B \left (b+\sqrt {b^2-4 a c}\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{16 c^2 d-8 c \left (b+\sqrt {b^2-4 a c}\right ) e+4 \left (b+\sqrt {b^2-4 a c}\right )^2 f-x^2} \, dx,x,\frac {4 c d-\left (b+\sqrt {b^2-4 a c}\right ) e-\left (-2 c e+2 \left (b+\sqrt {b^2-4 a c}\right ) f\right ) x}{\sqrt {d+e x+f x^2}}\right )}{\sqrt {b^2-4 a c}}\\ &=\frac {\left (b B-2 A c-B \sqrt {b^2-4 a c}\right ) \tanh ^{-1}\left (\frac {4 c d-\left (b-\sqrt {b^2-4 a c}\right ) e+2 \left (c e-\left (b-\sqrt {b^2-4 a c}\right ) f\right ) x}{2 \sqrt {2} \sqrt {2 c^2 d-b c e+b^2 f-2 a c f+\sqrt {b^2-4 a c} (c e-b f)} \sqrt {d+e x+f x^2}}\right )}{\sqrt {2} \sqrt {b^2-4 a c} \sqrt {2 c^2 d-b c e+b^2 f-2 a c f+\sqrt {b^2-4 a c} (c e-b f)}}+\frac {\left (2 A c-B \left (b+\sqrt {b^2-4 a c}\right )\right ) \tanh ^{-1}\left (\frac {4 c d-\left (b+\sqrt {b^2-4 a c}\right ) e+2 \left (c e-\left (b+\sqrt {b^2-4 a c}\right ) f\right ) x}{2 \sqrt {2} \sqrt {2 c^2 d-b c e+b^2 f-2 a c f-\sqrt {b^2-4 a c} (c e-b f)} \sqrt {d+e x+f x^2}}\right )}{\sqrt {2} \sqrt {b^2-4 a c} \sqrt {2 c^2 d-b c e+b^2 f-2 a c f-\sqrt {b^2-4 a c} (c e-b f)}}\\ \end {align*}

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Mathematica [A]  time = 4.18, size = 393, normalized size = 0.94 \begin {gather*} \frac {-\frac {\left (B \sqrt {b^2-4 a c}+2 A c-b B\right ) \tanh ^{-1}\left (\frac {\left (\sqrt {b^2-4 a c}-b\right ) (e+2 f x)+2 c (2 d+e x)}{2 \sqrt {2} \sqrt {d+x (e+f x)} \sqrt {c \left (e \sqrt {b^2-4 a c}-2 a f-b e\right )+b f \left (b-\sqrt {b^2-4 a c}\right )+2 c^2 d}}\right )}{\sqrt {c \left (e \sqrt {b^2-4 a c}-2 a f-b e\right )+b f \left (b-\sqrt {b^2-4 a c}\right )+2 c^2 d}}-\frac {\left (B \sqrt {b^2-4 a c}-2 A c+b B\right ) \tanh ^{-1}\left (\frac {2 c (2 d+e x)-\left (\sqrt {b^2-4 a c}+b\right ) (e+2 f x)}{2 \sqrt {d+x (e+f x)} \sqrt {-2 c \left (e \sqrt {b^2-4 a c}+2 a f+b e\right )+2 b f \left (\sqrt {b^2-4 a c}+b\right )+4 c^2 d}}\right )}{\sqrt {-c \left (e \sqrt {b^2-4 a c}+2 a f+b e\right )+b f \left (\sqrt {b^2-4 a c}+b\right )+2 c^2 d}}}{\sqrt {2} \sqrt {b^2-4 a c}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-(((-(b*B) + 2*A*c + B*Sqrt[b^2 - 4*a*c])*ArcTanh[(2*c*(2*d + e*x) + (-b + Sqrt[b^2 - 4*a*c])*(e + 2*f*x))/(2
*Sqrt[2]*Sqrt[2*c^2*d + b*(b - Sqrt[b^2 - 4*a*c])*f + c*(-(b*e) + Sqrt[b^2 - 4*a*c]*e - 2*a*f)]*Sqrt[d + x*(e
+ f*x)])])/Sqrt[2*c^2*d + b*(b - Sqrt[b^2 - 4*a*c])*f + c*(-(b*e) + Sqrt[b^2 - 4*a*c]*e - 2*a*f)]) - ((b*B - 2
*A*c + B*Sqrt[b^2 - 4*a*c])*ArcTanh[(2*c*(2*d + e*x) - (b + Sqrt[b^2 - 4*a*c])*(e + 2*f*x))/(2*Sqrt[4*c^2*d +
2*b*(b + Sqrt[b^2 - 4*a*c])*f - 2*c*(b*e + Sqrt[b^2 - 4*a*c]*e + 2*a*f)]*Sqrt[d + x*(e + f*x)])])/Sqrt[2*c^2*d
+ b*(b + Sqrt[b^2 - 4*a*c])*f - c*(b*e + Sqrt[b^2 - 4*a*c]*e + 2*a*f)])/(Sqrt[2]*Sqrt[b^2 - 4*a*c])

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IntegrateAlgebraic [C]  time = 0.53, size = 278, normalized size = 0.67 \begin {gather*} -\text {RootSum}\left [\text {$\#$1}^4 c-2 \text {$\#$1}^3 b \sqrt {f}+4 \text {$\#$1}^2 a f+\text {$\#$1}^2 b e-2 \text {$\#$1}^2 c d-4 \text {$\#$1} a e \sqrt {f}+2 \text {$\#$1} b d \sqrt {f}+a e^2-b d e+c d^2\&,\frac {\text {$\#$1}^2 (-B) \log \left (-\text {$\#$1}+\sqrt {d+e x+f x^2}-\sqrt {f} x\right )+2 \text {$\#$1} A \sqrt {f} \log \left (-\text {$\#$1}+\sqrt {d+e x+f x^2}-\sqrt {f} x\right )-A e \log \left (-\text {$\#$1}+\sqrt {d+e x+f x^2}-\sqrt {f} x\right )+B d \log \left (-\text {$\#$1}+\sqrt {d+e x+f x^2}-\sqrt {f} x\right )}{2 \text {$\#$1}^3 c-3 \text {$\#$1}^2 b \sqrt {f}+4 \text {$\#$1} a f+\text {$\#$1} b e-2 \text {$\#$1} c d-2 a e \sqrt {f}+b d \sqrt {f}}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Unab
le to divide, perhaps due to rounding error%%%{%%{poly1[%%%{-4,[3,2,0]%%%}+%%%{16,[1,3,1]%%%},%%%{4,[4,2,0]%%%
}+%%%{-24,[2,3,1]%%%}+%%%{32,[0,4,2]%%%}]:[1,0,%%%{-1,[2,0,0]%%%}+%%%{4,[0,1,1]%%%}]%%},[2,1,0,0,0]%%%}+%%%{%%
{[%%%{-4,[5,0,0]%%%}+%%%{24,[3,1,1]%%%}+%%%{-32,[1,2,2]%%%},%%%{4,[6,0,0]%%%}+%%%{-32,[4,1,1]%%%}+%%%{72,[2,2,
2]%%%}+%%%{-32,[0,3,3]%%%}]:[1,0,%%%{-1,[2,0,0]%%%}+%%%{4,[0,1,1]%%%}]%%},[2,0,0,1,0]%%%}+%%%{%%{[%%%{4,[4,1,0
]%%%}+%%%{-20,[2,2,1]%%%}+%%%{16,[0,3,2]%%%},%%%{-4,[5,1,0]%%%}+%%%{28,[3,2,1]%%%}+%%%{-48,[1,3,2]%%%}]:[1,0,%
%%{-1,[2,0,0]%%%}+%%%{4,[0,1,1]%%%}]%%},[2,0,0,0,1]%%%}+%%%{%%{poly1[%%%{8,[2,3,0]%%%}+%%%{-32,[0,4,1]%%%},%%%
{-8,[3,3,0]%%%}+%%%{32,[1,4,1]%%%}]:[1,0,%%%{-1,[2,0,0]%%%}+%%%{4,[0,1,1]%%%}]%%},[1,1,1,0,0]%%%}+%%%{%%{[%%%{
8,[4,1,0]%%%}+%%%{-40,[2,2,1]%%%}+%%%{32,[0,3,2]%%%},%%%{-8,[5,1,0]%%%}+%%%{56,[3,2,1]%%%}+%%%{-96,[1,3,2]%%%}
]:[1,0,%%%{-1,[2,0,0]%%%}+%%%{4,[0,1,1]%%%}]%%},[1,0,1,1,0]%%%}+%%%{%%{[%%%{-8,[3,2,0]%%%}+%%%{32,[1,3,1]%%%},
%%%{8,[4,2,0]%%%}+%%%{-48,[2,3,1]%%%}+%%%{64,[0,4,2]%%%}]:[1,0,%%%{-1,[2,0,0]%%%}+%%%{4,[0,1,1]%%%}]%%},[1,0,1
,0,1]%%%}+%%%{%%%{8,[2,4,0]%%%}+%%%{-32,[0,5,1]%%%},[0,1,2,0,0]%%%}+%%%{%%{poly1[%%%{-4,[3,2,0]%%%}+%%%{16,[1,
3,1]%%%},%%%{4,[4,2,0]%%%}+%%%{-24,[2,3,1]%%%}+%%%{32,[0,4,2]%%%}]:[1,0,%%%{-1,[2,0,0]%%%}+%%%{4,[0,1,1]%%%}]%
%},[0,0,2,1,0]%%%}+%%%{%%{poly1[%%%{4,[2,3,0]%%%}+%%%{-16,[0,4,1]%%%},%%%{-4,[3,3,0]%%%}+%%%{16,[1,4,1]%%%}]:[
1,0,%%%{-1,[2,0,0]%%%}+%%%{4,[0,1,1]%%%}]%%},[0,0,2,0,1]%%%} / %%%{%%{[%%%{4,[3,2,0]%%%}+%%%{-16,[1,3,1]%%%},%
%%{-4,[4,2,0]%%%}+%%%{24,[2,3,1]%%%}+%%%{-32,[0,4,2]%%%}]:[1,0,%%%{-1,[2,0,0]%%%}+%%%{4,[0,1,1]%%%}]%%},[0,1,0
,0,0]%%%}+%%%{%%{[%%%{4,[5,0,0]%%%}+%%%{-24,[3,1,1]%%%}+%%%{32,[1,2,2]%%%},%%%{-4,[6,0,0]%%%}+%%%{32,[4,1,1]%%
%}+%%%{-72,[2,2,2]%%%}+%%%{32,[0,3,3]%%%}]:[1,0,%%%{-1,[2,0,0]%%%}+%%%{4,[0,1,1]%%%}]%%},[0,0,0,1,0]%%%}+%%%{%
%{[%%%{-4,[4,1,0]%%%}+%%%{20,[2,2,1]%%%}+%%%{-16,[0,3,2]%%%},%%%{4,[5,1,0]%%%}+%%%{-28,[3,2,1]%%%}+%%%{48,[1,3
,2]%%%}]:[1,0,%%%{-1,[2,0,0]%%%}+%%%{4,[0,1,1]%%%}]%%},[0,0,0,0,1]%%%} Error: Bad Argument ValueEvaluation tim
e: 0.92Unable to divide, perhaps due to rounding error%%%{%%{[%%%{1,[3,0,0]%%%}+%%%{-4,[1,1,1]%%%},%%%{1,[4,0,
0]%%%}+%%%{-6,[2,1,1]%%%}+%%%{8,[0,2,2]%%%}]:[1,0,%%%{-1,[2,0,0]%%%}+%%%{4,[0,1,1]%%%}]%%}/%%%{4,[0,0,2]%%%},[
2,1,0,0,0]%%%}+%%%{%%%{1,[2,0,0]%%%}+%%%{-4,[0,1,1]%%%}/2,[2,0,0,1,0]%%%}+%%%{%%{[%%%{-1,[2,0,0]%%%}+%%%{4,[0,
1,1]%%%},%%%{-1,[3,0,0]%%%}+%%%{4,[1,1,1]%%%}]:[1,0,%%%{-1,[2,0,0]%%%}+%%%{4,[0,1,1]%%%}]%%}/%%%{4,[0,0,1]%%%}
,[2,0,0,0,1]%%%}+%%%{%%{[%%%{-1,[4,0,0]%%%}+%%%{5,[2,1,1]%%%}+%%%{-4,[0,2,2]%%%},%%%{-1,[5,0,0]%%%}+%%%{7,[3,1
,1]%%%}+%%%{-12,[1,2,2]%%%}]:[1,0,%%%{-1,[2,0,0]%%%}+%%%{4,[0,1,1]%%%}]%%}/%%%{2,[0,0,3]%%%},[1,1,1,0,0]%%%}+%
%%{%%{[%%%{-1,[2,0,0]%%%}+%%%{4,[0,1,1]%%%},%%%{-1,[3,0,0]%%%}+%%%{4,[1,1,1]%%%}]:[1,0,%%%{-1,[2,0,0]%%%}+%%%{
4,[0,1,1]%%%}]%%}/%%%{2,[0,0,1]%%%},[1,0,1,1,0]%%%}+%%%{%%{[%%%{1,[3,0,0]%%%}+%%%{-4,[1,1,1]%%%},%%%{1,[4,0,0]
%%%}+%%%{-6,[2,1,1]%%%}+%%%{8,[0,2,2]%%%}]:[1,0,%%%{-1,[2,0,0]%%%}+%%%{4,[0,1,1]%%%}]%%}/%%%{2,[0,0,2]%%%},[1,
0,1,0,1]%%%}+%%%{%%{[%%%{1,[5,0,0]%%%}+%%%{-6,[3,1,1]%%%}+%%%{8,[1,2,2]%%%},%%%{1,[6,0,0]%%%}+%%%{-8,[4,1,1]%%
%}+%%%{18,[2,2,2]%%%}+%%%{-8,[0,3,3]%%%}]:[1,0,%%%{-1,[2,0,0]%%%}+%%%{4,[0,1,1]%%%}]%%}/%%%{4,[0,0,4]%%%},[0,1
,2,0,0]%%%}+%%%{%%{[%%%{1,[3,0,0]%%%}+%%%{-4,[1,1,1]%%%},%%%{1,[4,0,0]%%%}+%%%{-6,[2,1,1]%%%}+%%%{8,[0,2,2]%%%
}]:[1,0,%%%{-1,[2,0,0]%%%}+%%%{4,[0,1,1]%%%}]%%}/%%%{4,[0,0,2]%%%},[0,0,2,1,0]%%%}+%%%{%%{[%%%{-1,[4,0,0]%%%}+
%%%{5,[2,1,1]%%%}+%%%{-4,[0,2,2]%%%},%%%{-1,[5,0,0]%%%}+%%%{7,[3,1,1]%%%}+%%%{-12,[1,2,2]%%%}]:[1,0,%%%{-1,[2,
0,0]%%%}+%%%{4,[0,1,1]%%%}]%%}/%%%{4,[0,0,3]%%%},[0,0,2,0,1]%%%} / %%%{%%{[%%%{-1,[3,0,0]%%%}+%%%{4,[1,1,1]%%%
},%%%{-1,[4,0,0]%%%}+%%%{6,[2,1,1]%%%}+%%%{-8,[0,2,2]%%%}]:[1,0,%%%{-1,[2,0,0]%%%}+%%%{4,[0,1,1]%%%}]%%}/%%%{4
,[0,0,4]%%%},[0,1,0,0,0]%%%}+%%%{%%%{-1,[2,0,0]%%%}+%%%{4,[0,1,1]%%%}/%%%{2,[0,0,2]%%%},[0,0,0,1,0]%%%}+%%%{%%
{[%%%{1,[2,0,0]%%%}+%%%{-4,[0,1,1]%%%},%%%{1,[3,0,0]%%%}+%%%{-4,[1,1,1]%%%}]:[1,0,%%%{-1,[2,0,0]%%%}+%%%{4,[0,
1,1]%%%}]%%}/%%%{4,[0,0,3]%%%},[0,0,0,0,1]%%%} Error: Bad Argument Value

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maple [B]  time = 0.04, size = 2269, normalized size = 5.45

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/(-4*a*c+b^2)^(1/2)/(-2*(-(-4*a*c+b^2)^(1/2)*b*f+(-4*a*c+b^2)^(1/2)*c*e+2*a*c*f-b^2*f+b*c*e-2*c^2*d)/c^2)^(1/
2)*ln((-(-(-4*a*c+b^2)^(1/2)*b*f+(-4*a*c+b^2)^(1/2)*c*e+2*a*c*f-b^2*f+b*c*e-2*c^2*d)/c^2-(f*(-4*a*c+b^2)^(1/2)
+b*f-c*e)/c*(x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)+1/2*(-2*(-(-4*a*c+b^2)^(1/2)*b*f+(-4*a*c+b^2)^(1/2)*c*e+2*a*c*f-b
^2*f+b*c*e-2*c^2*d)/c^2)^(1/2)*(4*(x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)^2*f-4*(f*(-4*a*c+b^2)^(1/2)+b*f-c*e)/c*(x+1
/2*(b+(-4*a*c+b^2)^(1/2))/c)-2*(-(-4*a*c+b^2)^(1/2)*b*f+(-4*a*c+b^2)^(1/2)*c*e+2*a*c*f-b^2*f+b*c*e-2*c^2*d)/c^
2)^(1/2))/(x+1/2*(b+(-4*a*c+b^2)^(1/2))/c))*A-1/c/(-2*(-(-4*a*c+b^2)^(1/2)*b*f+(-4*a*c+b^2)^(1/2)*c*e+2*a*c*f-
b^2*f+b*c*e-2*c^2*d)/c^2)^(1/2)*ln((-(-(-4*a*c+b^2)^(1/2)*b*f+(-4*a*c+b^2)^(1/2)*c*e+2*a*c*f-b^2*f+b*c*e-2*c^2
*d)/c^2-(f*(-4*a*c+b^2)^(1/2)+b*f-c*e)/c*(x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)+1/2*(-2*(-(-4*a*c+b^2)^(1/2)*b*f+(-4
*a*c+b^2)^(1/2)*c*e+2*a*c*f-b^2*f+b*c*e-2*c^2*d)/c^2)^(1/2)*(4*(x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)^2*f-4*(f*(-4*a
*c+b^2)^(1/2)+b*f-c*e)/c*(x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)-2*(-(-4*a*c+b^2)^(1/2)*b*f+(-4*a*c+b^2)^(1/2)*c*e+2*
a*c*f-b^2*f+b*c*e-2*c^2*d)/c^2)^(1/2))/(x+1/2*(b+(-4*a*c+b^2)^(1/2))/c))*B-1/(-4*a*c+b^2)^(1/2)/c/(-2*(-(-4*a*
c+b^2)^(1/2)*b*f+(-4*a*c+b^2)^(1/2)*c*e+2*a*c*f-b^2*f+b*c*e-2*c^2*d)/c^2)^(1/2)*ln((-(-(-4*a*c+b^2)^(1/2)*b*f+
(-4*a*c+b^2)^(1/2)*c*e+2*a*c*f-b^2*f+b*c*e-2*c^2*d)/c^2-(f*(-4*a*c+b^2)^(1/2)+b*f-c*e)/c*(x+1/2*(b+(-4*a*c+b^2
)^(1/2))/c)+1/2*(-2*(-(-4*a*c+b^2)^(1/2)*b*f+(-4*a*c+b^2)^(1/2)*c*e+2*a*c*f-b^2*f+b*c*e-2*c^2*d)/c^2)^(1/2)*(4
*(x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)^2*f-4*(f*(-4*a*c+b^2)^(1/2)+b*f-c*e)/c*(x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)-2*(-
(-4*a*c+b^2)^(1/2)*b*f+(-4*a*c+b^2)^(1/2)*c*e+2*a*c*f-b^2*f+b*c*e-2*c^2*d)/c^2)^(1/2))/(x+1/2*(b+(-4*a*c+b^2)^
(1/2))/c))*b*B-2/(-4*a*c+b^2)^(1/2)/(-2*((-4*a*c+b^2)^(1/2)*b*f-(-4*a*c+b^2)^(1/2)*c*e+2*a*c*f-b^2*f+b*c*e-2*c
^2*d)/c^2)^(1/2)*ln((-((-4*a*c+b^2)^(1/2)*b*f-(-4*a*c+b^2)^(1/2)*c*e+2*a*c*f-b^2*f+b*c*e-2*c^2*d)/c^2-(-f*(-4*
a*c+b^2)^(1/2)+b*f-c*e)/c*(x-1/2*(-b+(-4*a*c+b^2)^(1/2))/c)+1/2*(-2*((-4*a*c+b^2)^(1/2)*b*f-(-4*a*c+b^2)^(1/2)
*c*e+2*a*c*f-b^2*f+b*c*e-2*c^2*d)/c^2)^(1/2)*(4*(x-1/2*(-b+(-4*a*c+b^2)^(1/2))/c)^2*f-4*(-f*(-4*a*c+b^2)^(1/2)
+b*f-c*e)/c*(x-1/2*(-b+(-4*a*c+b^2)^(1/2))/c)-2*((-4*a*c+b^2)^(1/2)*b*f-(-4*a*c+b^2)^(1/2)*c*e+2*a*c*f-b^2*f+b
*c*e-2*c^2*d)/c^2)^(1/2))/(x-1/2*(-b+(-4*a*c+b^2)^(1/2))/c))*A-1/c/(-2*((-4*a*c+b^2)^(1/2)*b*f-(-4*a*c+b^2)^(1
/2)*c*e+2*a*c*f-b^2*f+b*c*e-2*c^2*d)/c^2)^(1/2)*ln((-((-4*a*c+b^2)^(1/2)*b*f-(-4*a*c+b^2)^(1/2)*c*e+2*a*c*f-b^
2*f+b*c*e-2*c^2*d)/c^2-(-f*(-4*a*c+b^2)^(1/2)+b*f-c*e)/c*(x-1/2*(-b+(-4*a*c+b^2)^(1/2))/c)+1/2*(-2*((-4*a*c+b^
2)^(1/2)*b*f-(-4*a*c+b^2)^(1/2)*c*e+2*a*c*f-b^2*f+b*c*e-2*c^2*d)/c^2)^(1/2)*(4*(x-1/2*(-b+(-4*a*c+b^2)^(1/2))/
c)^2*f-4*(-f*(-4*a*c+b^2)^(1/2)+b*f-c*e)/c*(x-1/2*(-b+(-4*a*c+b^2)^(1/2))/c)-2*((-4*a*c+b^2)^(1/2)*b*f-(-4*a*c
+b^2)^(1/2)*c*e+2*a*c*f-b^2*f+b*c*e-2*c^2*d)/c^2)^(1/2))/(x-1/2*(-b+(-4*a*c+b^2)^(1/2))/c))*B+1/(-4*a*c+b^2)^(
1/2)/c/(-2*((-4*a*c+b^2)^(1/2)*b*f-(-4*a*c+b^2)^(1/2)*c*e+2*a*c*f-b^2*f+b*c*e-2*c^2*d)/c^2)^(1/2)*ln((-((-4*a*
c+b^2)^(1/2)*b*f-(-4*a*c+b^2)^(1/2)*c*e+2*a*c*f-b^2*f+b*c*e-2*c^2*d)/c^2-(-f*(-4*a*c+b^2)^(1/2)+b*f-c*e)/c*(x-
1/2*(-b+(-4*a*c+b^2)^(1/2))/c)+1/2*(-2*((-4*a*c+b^2)^(1/2)*b*f-(-4*a*c+b^2)^(1/2)*c*e+2*a*c*f-b^2*f+b*c*e-2*c^
2*d)/c^2)^(1/2)*(4*(x-1/2*(-b+(-4*a*c+b^2)^(1/2))/c)^2*f-4*(-f*(-4*a*c+b^2)^(1/2)+b*f-c*e)/c*(x-1/2*(-b+(-4*a*
c+b^2)^(1/2))/c)-2*((-4*a*c+b^2)^(1/2)*b*f-(-4*a*c+b^2)^(1/2)*c*e+2*a*c*f-b^2*f+b*c*e-2*c^2*d)/c^2)^(1/2))/(x-
1/2*(-b+(-4*a*c+b^2)^(1/2))/c))*b*B

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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((B*x+A)/(c*x^2+b*x+a)/(f*x^2+e*x+d)^(1/2),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(4*a*c-b^2>0)', see `assume?` f
or more details)Is 4*a*c-b^2 positive, negative or zero?

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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