3.1.35 \(\int \frac {A+B x}{\sqrt {d+e x+f x^2} (a e+b e x+b f x^2)^2} \, dx\)
Optimal. Leaf size=249 \[ \frac {(B e-2 A f) \left (8 a e f-b \left (4 d f+e^2\right )\right ) \tanh ^{-1}\left (\frac {(e+2 f x) \sqrt {b d-a e}}{\sqrt {e} \sqrt {b e-4 a f} \sqrt {d+e x+f x^2}}\right )}{2 e^{3/2} f (b d-a e)^{3/2} (b e-4 a f)^{3/2}}-\frac {\sqrt {d+e x+f x^2} (e (A b-2 a B)-b x (B e-2 A f))}{e (b d-a e) (b e-4 a f) \left (a e+b e x+b f x^2\right )}+\frac {B \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x+f x^2}}{\sqrt {b d-a e}}\right )}{2 \sqrt {b} f (b d-a e)^{3/2}} \]
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Rubi [A] time = 0.91, antiderivative size = 249, normalized size of antiderivative = 1.00,
number of steps used = 6, number of rules used = 5, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.139, Rules used =
{1016, 1025, 982, 208, 1024} \begin {gather*} \frac {(B e-2 A f) \left (8 a e f-b \left (4 d f+e^2\right )\right ) \tanh ^{-1}\left (\frac {(e+2 f x) \sqrt {b d-a e}}{\sqrt {e} \sqrt {b e-4 a f} \sqrt {d+e x+f x^2}}\right )}{2 e^{3/2} f (b d-a e)^{3/2} (b e-4 a f)^{3/2}}-\frac {\sqrt {d+e x+f x^2} (e (A b-2 a B)-b x (B e-2 A f))}{e (b d-a e) (b e-4 a f) \left (a e+b e x+b f x^2\right )}+\frac {B \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x+f x^2}}{\sqrt {b d-a e}}\right )}{2 \sqrt {b} f (b d-a e)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(A + B*x)/(Sqrt[d + e*x + f*x^2]*(a*e + b*e*x + b*f*x^2)^2),x]
[Out]
-((((A*b - 2*a*B)*e - b*(B*e - 2*A*f)*x)*Sqrt[d + e*x + f*x^2])/(e*(b*d - a*e)*(b*e - 4*a*f)*(a*e + b*e*x + b*
f*x^2))) + ((B*e - 2*A*f)*(8*a*e*f - b*(e^2 + 4*d*f))*ArcTanh[(Sqrt[b*d - a*e]*(e + 2*f*x))/(Sqrt[e]*Sqrt[b*e
- 4*a*f]*Sqrt[d + e*x + f*x^2])])/(2*e^(3/2)*(b*d - a*e)^(3/2)*f*(b*e - 4*a*f)^(3/2)) + (B*ArcTanh[(Sqrt[b]*Sq
rt[d + e*x + f*x^2])/Sqrt[b*d - a*e]])/(2*Sqrt[b]*(b*d - a*e)^(3/2)*f)
Rule 208
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]
Rule 982
Int[1/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[-2*e, Su
bst[Int[1/(e*(b*e - 4*a*f) - (b*d - a*e)*x^2), x], x, (e + 2*f*x)/Sqrt[d + e*x + f*x^2]], x] /; FreeQ[{a, b, c
, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && EqQ[c*e - b*f, 0]
Rule 1016
Int[((g_.) + (h_.)*(x_))*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_) + (e_.)*(x_) + (f_.)*(x_)^2)^(q_), x_Sy
mbol] :> Simp[((a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^(q + 1)*(g*c*(2*a*c*e - b*(c*d + a*f)) + (g*b - a*h
)*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)) + c*(g*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)) - h*(b*c*d - 2*a*c*e + a*b*f)
)*x))/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1)), x] + Dist[1/((b^2 - 4*a*c)*((c*d - a*
f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1)), Int[(a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^q*Simp[(b*h - 2*g*c)
*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1) + (b^2*(g*f) - b*(h*c*d + g*c*e + a*h*f) + 2*(g*c*(c*d - a*
f) - a*(-(h*c*e))))*(a*f*(p + 1) - c*d*(p + 2)) - e*((g*c)*(2*a*c*e - b*(c*d + a*f)) + (g*b - a*h)*(2*c^2*d +
b^2*f - c*(b*e + 2*a*f)))*(p + q + 2) - (2*f*((g*c)*(2*a*c*e - b*(c*d + a*f)) + (g*b - a*h)*(2*c^2*d + b^2*f -
c*(b*e + 2*a*f)))*(p + q + 2) - (b^2*g*f - b*(h*c*d + g*c*e + a*h*f) + 2*(g*c*(c*d - a*f) - a*(-(h*c*e))))*(b
*f*(p + 1) - c*e*(2*p + q + 4)))*x - c*f*(b^2*(g*f) - b*(h*c*d + g*c*e + a*h*f) + 2*(g*c*(c*d - a*f) + a*h*c*e
))*(2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, q}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2
- 4*d*f, 0] && LtQ[p, -1] && NeQ[(c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f), 0] && !( !IntegerQ[p] && ILtQ[q, -1
])
Rule 1024
Int[((g_) + (h_.)*(x_))/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol
] :> Dist[-2*g, Subst[Int[1/(b*d - a*e - b*x^2), x], x, Sqrt[d + e*x + f*x^2]], 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] && EqQ[c*e - b*f, 0] && EqQ[h*e - 2*g*f, 0]
Rule 1025
Int[((g_.) + (h_.)*(x_))/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbo
l] :> -Dist[(h*e - 2*g*f)/(2*f), Int[1/((a + b*x + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x] + Dist[h/(2*f), Int[(
e + 2*f*x)/((a + b*x + c*x^2)*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] && EqQ[c*e - b*f, 0] && NeQ[h*e - 2*g*f, 0]
Rubi steps
\begin {align*} \int \frac {A+B x}{\sqrt {d+e x+f x^2} \left (a e+b e x+b f x^2\right )^2} \, dx &=-\frac {((A b-2 a B) e-b (B e-2 A f) x) \sqrt {d+e x+f x^2}}{e (b d-a e) (b e-4 a f) \left (a e+b e x+b f x^2\right )}-\frac {\int \frac {-\frac {1}{2} b (b d-a e) f^2 \left (2 b B d e-2 a e (B e-4 A f)-A b \left (e^2+4 d f\right )\right )+\frac {1}{2} b B e (b d-a e) f^2 (b e-4 a f) x}{\sqrt {d+e x+f x^2} \left (a e+b e x+b f x^2\right )} \, dx}{b e (b d-a e)^2 f^2 (b e-4 a f)}\\ &=-\frac {((A b-2 a B) e-b (B e-2 A f) x) \sqrt {d+e x+f x^2}}{e (b d-a e) (b e-4 a f) \left (a e+b e x+b f x^2\right )}-\frac {B \int \frac {e+2 f x}{\sqrt {d+e x+f x^2} \left (a e+b e x+b f x^2\right )} \, dx}{4 (b d-a e) f}-\frac {\left ((B e-2 A f) \left (8 a e f-b \left (e^2+4 d f\right )\right )\right ) \int \frac {1}{\sqrt {d+e x+f x^2} \left (a e+b e x+b f x^2\right )} \, dx}{4 e (b d-a e) f (b e-4 a f)}\\ &=-\frac {((A b-2 a B) e-b (B e-2 A f) x) \sqrt {d+e x+f x^2}}{e (b d-a e) (b e-4 a f) \left (a e+b e x+b f x^2\right )}+\frac {(B e) \operatorname {Subst}\left (\int \frac {1}{b d e-a e^2-b e x^2} \, dx,x,\sqrt {d+e x+f x^2}\right )}{2 (b d-a e) f}+\frac {\left ((B e-2 A f) \left (8 a e f-b \left (e^2+4 d f\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{e \left (b e^2-4 a e f\right )-\left (b d e-a e^2\right ) x^2} \, dx,x,\frac {e+2 f x}{\sqrt {d+e x+f x^2}}\right )}{2 (b d-a e) f (b e-4 a f)}\\ &=-\frac {((A b-2 a B) e-b (B e-2 A f) x) \sqrt {d+e x+f x^2}}{e (b d-a e) (b e-4 a f) \left (a e+b e x+b f x^2\right )}+\frac {(B e-2 A f) \left (8 a e f-b \left (e^2+4 d f\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {b d-a e} (e+2 f x)}{\sqrt {e} \sqrt {b e-4 a f} \sqrt {d+e x+f x^2}}\right )}{2 e^{3/2} (b d-a e)^{3/2} f (b e-4 a f)^{3/2}}+\frac {B \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x+f x^2}}{\sqrt {b d-a e}}\right )}{2 \sqrt {b} (b d-a e)^{3/2} f}\\ \end {align*}
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Mathematica [B] time = 1.49, size = 767, normalized size = 3.08 \begin {gather*} -\frac {-(a e+b x (e+f x)) \log \left (b (e+2 f x)-\sqrt {b} \sqrt {e} \sqrt {b e-4 a f}\right ) \left (-8 a b e f (B e-2 A f)-b^{3/2} B e^{5/2} \sqrt {b e-4 a f}+4 a \sqrt {b} B e^{3/2} f \sqrt {b e-4 a f}+b^2 \left (4 d f+e^2\right ) (B e-2 A f)\right )+(a e+b x (e+f x)) \log \left (\sqrt {b} \sqrt {e} \sqrt {b e-4 a f}+b (e+2 f x)\right ) \left (-8 a b e f (B e-2 A f)+b^{3/2} B e^{5/2} \sqrt {b e-4 a f}-4 a \sqrt {b} B e^{3/2} f \sqrt {b e-4 a f}+b^2 \left (4 d f+e^2\right ) (B e-2 A f)\right )-(a e+b x (e+f x)) \left (-8 a b e f (B e-2 A f)+b^{3/2} B e^{5/2} \sqrt {b e-4 a f}-4 a \sqrt {b} B e^{3/2} f \sqrt {b e-4 a f}+b^2 \left (4 d f+e^2\right ) (B e-2 A f)\right ) \log \left (\sqrt {b} \left (-4 f \sqrt {b d-a e} \sqrt {d+x (e+f x)}+e^{3/2} \sqrt {b e-4 a f}+2 \sqrt {e} f x \sqrt {b e-4 a f}+\sqrt {b} \left (e^2-4 d f\right )\right )\right )+(a e+b x (e+f x)) \left (-8 a b e f (B e-2 A f)-b^{3/2} B e^{5/2} \sqrt {b e-4 a f}+4 a \sqrt {b} B e^{3/2} f \sqrt {b e-4 a f}+b^2 \left (4 d f+e^2\right ) (B e-2 A f)\right ) \log \left (\sqrt {b} \left (4 f \sqrt {b d-a e} \sqrt {d+x (e+f x)}+e^{3/2} \sqrt {b e-4 a f}+2 \sqrt {e} f x \sqrt {b e-4 a f}-\sqrt {b} \left (e^2-4 d f\right )\right )\right )+4 b \sqrt {e} f \sqrt {b d-a e} \sqrt {b e-4 a f} \sqrt {d+x (e+f x)} (A b (e+2 f x)-B e (2 a+b x))}{4 b e^{3/2} f (b d-a e)^{3/2} (b e-4 a f)^{3/2} (a e+b x (e+f x))} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(A + B*x)/(Sqrt[d + e*x + f*x^2]*(a*e + b*e*x + b*f*x^2)^2),x]
[Out]
-1/4*(4*b*Sqrt[e]*Sqrt[b*d - a*e]*f*Sqrt[b*e - 4*a*f]*Sqrt[d + x*(e + f*x)]*(-(B*e*(2*a + b*x)) + A*b*(e + 2*f
*x)) - (-(b^(3/2)*B*e^(5/2)*Sqrt[b*e - 4*a*f]) + 4*a*Sqrt[b]*B*e^(3/2)*f*Sqrt[b*e - 4*a*f] - 8*a*b*e*f*(B*e -
2*A*f) + b^2*(B*e - 2*A*f)*(e^2 + 4*d*f))*(a*e + b*x*(e + f*x))*Log[-(Sqrt[b]*Sqrt[e]*Sqrt[b*e - 4*a*f]) + b*(
e + 2*f*x)] + (b^(3/2)*B*e^(5/2)*Sqrt[b*e - 4*a*f] - 4*a*Sqrt[b]*B*e^(3/2)*f*Sqrt[b*e - 4*a*f] - 8*a*b*e*f*(B*
e - 2*A*f) + b^2*(B*e - 2*A*f)*(e^2 + 4*d*f))*(a*e + b*x*(e + f*x))*Log[Sqrt[b]*Sqrt[e]*Sqrt[b*e - 4*a*f] + b*
(e + 2*f*x)] - (b^(3/2)*B*e^(5/2)*Sqrt[b*e - 4*a*f] - 4*a*Sqrt[b]*B*e^(3/2)*f*Sqrt[b*e - 4*a*f] - 8*a*b*e*f*(B
*e - 2*A*f) + b^2*(B*e - 2*A*f)*(e^2 + 4*d*f))*(a*e + b*x*(e + f*x))*Log[Sqrt[b]*(e^(3/2)*Sqrt[b*e - 4*a*f] +
Sqrt[b]*(e^2 - 4*d*f) + 2*Sqrt[e]*f*Sqrt[b*e - 4*a*f]*x - 4*Sqrt[b*d - a*e]*f*Sqrt[d + x*(e + f*x)])] + (-(b^(
3/2)*B*e^(5/2)*Sqrt[b*e - 4*a*f]) + 4*a*Sqrt[b]*B*e^(3/2)*f*Sqrt[b*e - 4*a*f] - 8*a*b*e*f*(B*e - 2*A*f) + b^2*
(B*e - 2*A*f)*(e^2 + 4*d*f))*(a*e + b*x*(e + f*x))*Log[Sqrt[b]*(e^(3/2)*Sqrt[b*e - 4*a*f] - Sqrt[b]*(e^2 - 4*d
*f) + 2*Sqrt[e]*f*Sqrt[b*e - 4*a*f]*x + 4*Sqrt[b*d - a*e]*f*Sqrt[d + x*(e + f*x)])])/(b*e^(3/2)*(b*d - a*e)^(3
/2)*f*(b*e - 4*a*f)^(3/2)*(a*e + b*x*(e + f*x)))
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IntegrateAlgebraic [C] time = 2.43, size = 1019, normalized size = 4.09 \begin {gather*} -\frac {\sqrt {f x^2+e x+d} (-A b e+2 a B e+b B x e-2 A b f x)}{e (a e-b d) (b e-4 a f) \left (b f x^2+b e x+a e\right )}-\frac {4 B \text {RootSum}\left [b f \text {$\#$1}^4-2 b e \sqrt {f} \text {$\#$1}^3+b e^2 \text {$\#$1}^2-2 b d f \text {$\#$1}^2+4 a e f \text {$\#$1}^2-4 a e^2 \sqrt {f} \text {$\#$1}+2 b d e \sqrt {f} \text {$\#$1}+a e^3-b d e^2+b d^2 f\&,\frac {\log \left (-\sqrt {f} x-\text {$\#$1}+\sqrt {f x^2+e x+d}\right )}{-2 b f \text {$\#$1}^3+3 b e \sqrt {f} \text {$\#$1}^2-b e^2 \text {$\#$1}+2 b d f \text {$\#$1}-4 a e f \text {$\#$1}+2 a e^2 \sqrt {f}-b d e \sqrt {f}}\&\right ]}{b}-\frac {\text {RootSum}\left [b f \text {$\#$1}^4-2 b e \sqrt {f} \text {$\#$1}^3+b e^2 \text {$\#$1}^2-2 b d f \text {$\#$1}^2+4 a e f \text {$\#$1}^2-4 a e^2 \sqrt {f} \text {$\#$1}+2 b d e \sqrt {f} \text {$\#$1}+a e^3-b d e^2+b d^2 f\&,\frac {-A b^2 \log \left (-\sqrt {f} x-\text {$\#$1}+\sqrt {f x^2+e x+d}\right ) e^3+6 a b B \log \left (-\sqrt {f} x-\text {$\#$1}+\sqrt {f x^2+e x+d}\right ) e^3-b^2 B \log \left (-\sqrt {f} x-\text {$\#$1}+\sqrt {f x^2+e x+d}\right ) \text {$\#$1}^2 e^2-5 b^2 B d \log \left (-\sqrt {f} x-\text {$\#$1}+\sqrt {f x^2+e x+d}\right ) e^2+8 a A b f \log \left (-\sqrt {f} x-\text {$\#$1}+\sqrt {f x^2+e x+d}\right ) e^2-32 a^2 B f \log \left (-\sqrt {f} x-\text {$\#$1}+\sqrt {f x^2+e x+d}\right ) e^2+2 A b^2 \sqrt {f} \log \left (-\sqrt {f} x-\text {$\#$1}+\sqrt {f x^2+e x+d}\right ) \text {$\#$1} e^2+4 a b B \sqrt {f} \log \left (-\sqrt {f} x-\text {$\#$1}+\sqrt {f x^2+e x+d}\right ) \text {$\#$1} e^2+4 a b B f \log \left (-\sqrt {f} x-\text {$\#$1}+\sqrt {f x^2+e x+d}\right ) \text {$\#$1}^2 e-4 A b^2 d f \log \left (-\sqrt {f} x-\text {$\#$1}+\sqrt {f x^2+e x+d}\right ) e+28 a b B d f \log \left (-\sqrt {f} x-\text {$\#$1}+\sqrt {f x^2+e x+d}\right ) e-16 a A b f^{3/2} \log \left (-\sqrt {f} x-\text {$\#$1}+\sqrt {f x^2+e x+d}\right ) \text {$\#$1} e-4 b^2 B d \sqrt {f} \log \left (-\sqrt {f} x-\text {$\#$1}+\sqrt {f x^2+e x+d}\right ) \text {$\#$1} e+8 A b^2 d f^{3/2} \log \left (-\sqrt {f} x-\text {$\#$1}+\sqrt {f x^2+e x+d}\right ) \text {$\#$1}}{-2 b f \text {$\#$1}^3+3 b e \sqrt {f} \text {$\#$1}^2-b e^2 \text {$\#$1}+2 b d f \text {$\#$1}-4 a e f \text {$\#$1}+2 a e^2 \sqrt {f}-b d e \sqrt {f}}\&\right ]}{2 b e (b d-a e) (b e-4 a f)} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[(A + B*x)/(Sqrt[d + e*x + f*x^2]*(a*e + b*e*x + b*f*x^2)^2),x]
[Out]
-(((-(A*b*e) + 2*a*B*e + b*B*e*x - 2*A*b*f*x)*Sqrt[d + e*x + f*x^2])/(e*(-(b*d) + a*e)*(b*e - 4*a*f)*(a*e + b*
e*x + b*f*x^2))) - (4*B*RootSum[-(b*d*e^2) + a*e^3 + b*d^2*f + 2*b*d*e*Sqrt[f]*#1 - 4*a*e^2*Sqrt[f]*#1 + b*e^2
*#1^2 - 2*b*d*f*#1^2 + 4*a*e*f*#1^2 - 2*b*e*Sqrt[f]*#1^3 + b*f*#1^4 & , Log[-(Sqrt[f]*x) + Sqrt[d + e*x + f*x^
2] - #1]/(-(b*d*e*Sqrt[f]) + 2*a*e^2*Sqrt[f] - b*e^2*#1 + 2*b*d*f*#1 - 4*a*e*f*#1 + 3*b*e*Sqrt[f]*#1^2 - 2*b*f
*#1^3) & ])/b - RootSum[-(b*d*e^2) + a*e^3 + b*d^2*f + 2*b*d*e*Sqrt[f]*#1 - 4*a*e^2*Sqrt[f]*#1 + b*e^2*#1^2 -
2*b*d*f*#1^2 + 4*a*e*f*#1^2 - 2*b*e*Sqrt[f]*#1^3 + b*f*#1^4 & , (-5*b^2*B*d*e^2*Log[-(Sqrt[f]*x) + Sqrt[d + e*
x + f*x^2] - #1] - A*b^2*e^3*Log[-(Sqrt[f]*x) + Sqrt[d + e*x + f*x^2] - #1] + 6*a*b*B*e^3*Log[-(Sqrt[f]*x) + S
qrt[d + e*x + f*x^2] - #1] - 4*A*b^2*d*e*f*Log[-(Sqrt[f]*x) + Sqrt[d + e*x + f*x^2] - #1] + 28*a*b*B*d*e*f*Log
[-(Sqrt[f]*x) + Sqrt[d + e*x + f*x^2] - #1] + 8*a*A*b*e^2*f*Log[-(Sqrt[f]*x) + Sqrt[d + e*x + f*x^2] - #1] - 3
2*a^2*B*e^2*f*Log[-(Sqrt[f]*x) + Sqrt[d + e*x + f*x^2] - #1] - 4*b^2*B*d*e*Sqrt[f]*Log[-(Sqrt[f]*x) + Sqrt[d +
e*x + f*x^2] - #1]*#1 + 2*A*b^2*e^2*Sqrt[f]*Log[-(Sqrt[f]*x) + Sqrt[d + e*x + f*x^2] - #1]*#1 + 4*a*b*B*e^2*S
qrt[f]*Log[-(Sqrt[f]*x) + Sqrt[d + e*x + f*x^2] - #1]*#1 + 8*A*b^2*d*f^(3/2)*Log[-(Sqrt[f]*x) + Sqrt[d + e*x +
f*x^2] - #1]*#1 - 16*a*A*b*e*f^(3/2)*Log[-(Sqrt[f]*x) + Sqrt[d + e*x + f*x^2] - #1]*#1 - b^2*B*e^2*Log[-(Sqrt
[f]*x) + Sqrt[d + e*x + f*x^2] - #1]*#1^2 + 4*a*b*B*e*f*Log[-(Sqrt[f]*x) + Sqrt[d + e*x + f*x^2] - #1]*#1^2)/(
-(b*d*e*Sqrt[f]) + 2*a*e^2*Sqrt[f] - b*e^2*#1 + 2*b*d*f*#1 - 4*a*e*f*#1 + 3*b*e*Sqrt[f]*#1^2 - 2*b*f*#1^3) & ]
/(2*b*e*(b*d - a*e)*(b*e - 4*a*f))
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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)/(b*f*x^2+b*e*x+a*e)^2/(f*x^2+e*x+d)^(1/2),x, algorithm="fricas")
[Out]
Timed out
________________________________________________________________________________________
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)/(b*f*x^2+b*e*x+a*e)^2/(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%%%{%%%{1,[2]%%%},[8,2,0,0,0]%%%}+%%%{%%{[%%%{-4,[1]%%%},0]:[1,0,%%
%{-1,[1]%%%}]%%},[7,2,1,0,0]%%%}+%%%{%%%{6,[1]%%%},[6,2,2,0,0]%%%}+%%%{%%%{-4,[2]%%%},[6,2,0,0,1]%%%}+%%%{%%%{
8,[2]%%%},[6,1,1,1,0]%%%}+%%%{%%{[-4,0]:[1,0,%%%{-1,[1]%%%}]%%},[5,2,3,0,0]%%%}+%%%{%%{[%%%{12,[1]%%%},0]:[1,0
,%%%{-1,[1]%%%}]%%},[5,2,1,0,1]%%%}+%%%{%%{[%%%{-24,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,1,2,1,0]%%%}+%%%{1,[
4,2,4,0,0]%%%}+%%%{%%%{-14,[1]%%%},[4,2,2,0,1]%%%}+%%%{%%%{6,[2]%%%},[4,2,0,0,2]%%%}+%%%{%%%{26,[1]%%%},[4,1,3
,1,0]%%%}+%%%{%%%{-16,[2]%%%},[4,1,1,1,1]%%%}+%%%{%%%{16,[2]%%%},[4,0,2,2,0]%%%}+%%%{%%{[8,0]:[1,0,%%%{-1,[1]%
%%}]%%},[3,2,3,0,1]%%%}+%%%{%%{[%%%{-12,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,2,1,0,2]%%%}+%%%{%%{[-12,0]:[1,0
,%%%{-1,[1]%%%}]%%},[3,1,4,1,0]%%%}+%%%{%%{[%%%{32,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,1,2,1,1]%%%}+%%%{%%{[
%%%{-32,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,0,3,2,0]%%%}+%%%{-2,[2,2,4,0,1]%%%}+%%%{%%%{10,[1]%%%},[2,2,2,0,
2]%%%}+%%%{%%%{-4,[2]%%%},[2,2,0,0,3]%%%}+%%%{2,[2,1,5,1,0]%%%}+%%%{%%%{-28,[1]%%%},[2,1,3,1,1]%%%}+%%%{%%%{8,
[2]%%%},[2,1,1,1,2]%%%}+%%%{%%%{24,[1]%%%},[2,0,4,2,0]%%%}+%%%{%%{[-4,0]:[1,0,%%%{-1,[1]%%%}]%%},[1,2,3,0,2]%%
%}+%%%{%%{[%%%{4,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,2,1,0,3]%%%}+%%%{%%{[12,0]:[1,0,%%%{-1,[1]%%%}]%%},[1,1
,4,1,1]%%%}+%%%{%%{[%%%{-8,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,1,2,1,2]%%%}+%%%{%%{[-8,0]:[1,0,%%%{-1,[1]%%%
}]%%},[1,0,5,2,0]%%%}+%%%{1,[0,2,4,0,2]%%%}+%%%{%%%{-2,[1]%%%},[0,2,2,0,3]%%%}+%%%{%%%{1,[2]%%%},[0,2,0,0,4]%%
%}+%%%{-2,[0,1,5,1,1]%%%}+%%%{%%%{2,[1]%%%},[0,1,3,1,2]%%%}+%%%{1,[0,0,6,2,0]%%%} / %%%{%%%{1,[3]%%%},[8,2,0,0
,0]%%%}+%%%{%%{poly1[%%%{-4,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[7,2,1,0,0]%%%}+%%%{%%%{6,[2]%%%},[6,2,2,0,0]%%
%}+%%%{%%%{-4,[3]%%%},[6,2,0,0,1]%%%}+%%%{%%%{8,[3]%%%},[6,1,1,1,0]%%%}+%%%{%%{poly1[%%%{-4,[1]%%%},0]:[1,0,%%
%{-1,[1]%%%}]%%},[5,2,3,0,0]%%%}+%%%{%%{poly1[%%%{12,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,2,1,0,1]%%%}+%%%{%%
{poly1[%%%{-24,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[5,1,2,1,0]%%%}+%%%{%%%{1,[1]%%%},[4,2,4,0,0]%%%}+%%%{%%%{-1
4,[2]%%%},[4,2,2,0,1]%%%}+%%%{%%%{6,[3]%%%},[4,2,0,0,2]%%%}+%%%{%%%{26,[2]%%%},[4,1,3,1,0]%%%}+%%%{%%%{-16,[3]
%%%},[4,1,1,1,1]%%%}+%%%{%%%{16,[3]%%%},[4,0,2,2,0]%%%}+%%%{%%{poly1[%%%{8,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},
[3,2,3,0,1]%%%}+%%%{%%{poly1[%%%{-12,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,2,1,0,2]%%%}+%%%{%%{poly1[%%%{-12,[
1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,1,4,1,0]%%%}+%%%{%%{poly1[%%%{32,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,1,
2,1,1]%%%}+%%%{%%{poly1[%%%{-32,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[3,0,3,2,0]%%%}+%%%{%%%{-2,[1]%%%},[2,2,4,0
,1]%%%}+%%%{%%%{10,[2]%%%},[2,2,2,0,2]%%%}+%%%{%%%{-4,[3]%%%},[2,2,0,0,3]%%%}+%%%{%%%{2,[1]%%%},[2,1,5,1,0]%%%
}+%%%{%%%{-28,[2]%%%},[2,1,3,1,1]%%%}+%%%{%%%{8,[3]%%%},[2,1,1,1,2]%%%}+%%%{%%%{24,[2]%%%},[2,0,4,2,0]%%%}+%%%
{%%{poly1[%%%{-4,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,2,3,0,2]%%%}+%%%{%%{poly1[%%%{4,[2]%%%},0]:[1,0,%%%{-1,
[1]%%%}]%%},[1,2,1,0,3]%%%}+%%%{%%{poly1[%%%{12,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,1,4,1,1]%%%}+%%%{%%{poly
1[%%%{-8,[2]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[1,1,2,1,2]%%%}+%%%{%%{poly1[%%%{-8,[1]%%%},0]:[1,0,%%%{-1,[1]%%%}
]%%},[1,0,5,2,0]%%%}+%%%{%%%{1,[1]%%%},[0,2,4,0,2]%%%}+%%%{%%%{-2,[2]%%%},[0,2,2,0,3]%%%}+%%%{%%%{1,[3]%%%},[0
,2,0,0,4]%%%}+%%%{%%%{-2,[1]%%%},[0,1,5,1,1]%%%}+%%%{%%%{2,[2]%%%},[0,1,3,1,2]%%%}+%%%{%%%{1,[1]%%%},[0,0,6,2,
0]%%%} Error: Bad Argument Value
________________________________________________________________________________________
maple [B] time = 0.05, size = 3606, normalized size = 14.48 \begin {gather*} \text {output too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((B*x+A)/(b*f*x^2+b*e*x+a*e)^2/(f*x^2+e*x+d)^(1/2),x)
[Out]
-1/e/(4*a*f-b*e)/(a*e-b*d)/(x+1/2/f*e-1/2/b/f*(-b*e*(4*a*f-b*e))^(1/2))*((x-1/2*(-b*e+(-b*e*(4*a*f-b*e))^(1/2)
)/b/f)^2*f+(-b*e*(4*a*f-b*e))^(1/2)/b*(x-1/2*(-b*e+(-b*e*(4*a*f-b*e))^(1/2))/b/f)-1/b*(a*e-b*d))^(1/2)*A+1/2/f
/(4*a*f-b*e)/(a*e-b*d)/(x+1/2/f*e-1/2/b/f*(-b*e*(4*a*f-b*e))^(1/2))*((x-1/2*(-b*e+(-b*e*(4*a*f-b*e))^(1/2))/b/
f)^2*f+(-b*e*(4*a*f-b*e))^(1/2)/b*(x-1/2*(-b*e+(-b*e*(4*a*f-b*e))^(1/2))/b/f)-1/b*(a*e-b*d))^(1/2)*B-1/2/f/e/(
4*a*f-b*e)/b/(a*e-b*d)/(x+1/2/f*e-1/2/b/f*(-b*e*(4*a*f-b*e))^(1/2))*((x-1/2*(-b*e+(-b*e*(4*a*f-b*e))^(1/2))/b/
f)^2*f+(-b*e*(4*a*f-b*e))^(1/2)/b*(x-1/2*(-b*e+(-b*e*(4*a*f-b*e))^(1/2))/b/f)-1/b*(a*e-b*d))^(1/2)*B*(-b*e*(4*
a*f-b*e))^(1/2)+1/2/e/(4*a*f-b*e)/b*(-b*e*(4*a*f-b*e))^(1/2)/(a*e-b*d)/(-1/b*(a*e-b*d))^(1/2)*ln((-2/b*(a*e-b*
d)+(-b*e*(4*a*f-b*e))^(1/2)/b*(x-1/2*(-b*e+(-b*e*(4*a*f-b*e))^(1/2))/b/f)+2*(-1/b*(a*e-b*d))^(1/2)*((x-1/2*(-b
*e+(-b*e*(4*a*f-b*e))^(1/2))/b/f)^2*f+(-b*e*(4*a*f-b*e))^(1/2)/b*(x-1/2*(-b*e+(-b*e*(4*a*f-b*e))^(1/2))/b/f)-1
/b*(a*e-b*d))^(1/2))/(x-1/2*(-b*e+(-b*e*(4*a*f-b*e))^(1/2))/b/f))*A-1/4/f/(4*a*f-b*e)/b*(-b*e*(4*a*f-b*e))^(1/
2)/(a*e-b*d)/(-1/b*(a*e-b*d))^(1/2)*ln((-2/b*(a*e-b*d)+(-b*e*(4*a*f-b*e))^(1/2)/b*(x-1/2*(-b*e+(-b*e*(4*a*f-b*
e))^(1/2))/b/f)+2*(-1/b*(a*e-b*d))^(1/2)*((x-1/2*(-b*e+(-b*e*(4*a*f-b*e))^(1/2))/b/f)^2*f+(-b*e*(4*a*f-b*e))^(
1/2)/b*(x-1/2*(-b*e+(-b*e*(4*a*f-b*e))^(1/2))/b/f)-1/b*(a*e-b*d))^(1/2))/(x-1/2*(-b*e+(-b*e*(4*a*f-b*e))^(1/2)
)/b/f))*B-1/4/f/b/(a*e-b*d)/(-1/b*(a*e-b*d))^(1/2)*ln((-2/b*(a*e-b*d)+(-b*e*(4*a*f-b*e))^(1/2)/b*(x-1/2*(-b*e+
(-b*e*(4*a*f-b*e))^(1/2))/b/f)+2*(-1/b*(a*e-b*d))^(1/2)*((x-1/2*(-b*e+(-b*e*(4*a*f-b*e))^(1/2))/b/f)^2*f+(-b*e
*(4*a*f-b*e))^(1/2)/b*(x-1/2*(-b*e+(-b*e*(4*a*f-b*e))^(1/2))/b/f)-1/b*(a*e-b*d))^(1/2))/(x-1/2*(-b*e+(-b*e*(4*
a*f-b*e))^(1/2))/b/f))*B-2/e/(4*a*f-b*e)/(-b*e*(4*a*f-b*e))^(1/2)/(-1/b*(a*e-b*d))^(1/2)*ln((-2/b*(a*e-b*d)+(-
b*e*(4*a*f-b*e))^(1/2)/b*(x-1/2*(-b*e+(-b*e*(4*a*f-b*e))^(1/2))/b/f)+2*(-1/b*(a*e-b*d))^(1/2)*((x-1/2*(-b*e+(-
b*e*(4*a*f-b*e))^(1/2))/b/f)^2*f+(-b*e*(4*a*f-b*e))^(1/2)/b*(x-1/2*(-b*e+(-b*e*(4*a*f-b*e))^(1/2))/b/f)-1/b*(a
*e-b*d))^(1/2))/(x-1/2*(-b*e+(-b*e*(4*a*f-b*e))^(1/2))/b/f))*A*f+1/(4*a*f-b*e)/(-b*e*(4*a*f-b*e))^(1/2)/(-1/b*
(a*e-b*d))^(1/2)*ln((-2/b*(a*e-b*d)+(-b*e*(4*a*f-b*e))^(1/2)/b*(x-1/2*(-b*e+(-b*e*(4*a*f-b*e))^(1/2))/b/f)+2*(
-1/b*(a*e-b*d))^(1/2)*((x-1/2*(-b*e+(-b*e*(4*a*f-b*e))^(1/2))/b/f)^2*f+(-b*e*(4*a*f-b*e))^(1/2)/b*(x-1/2*(-b*e
+(-b*e*(4*a*f-b*e))^(1/2))/b/f)-1/b*(a*e-b*d))^(1/2))/(x-1/2*(-b*e+(-b*e*(4*a*f-b*e))^(1/2))/b/f))*B+2/e/(4*a*
f-b*e)/(-b*e*(4*a*f-b*e))^(1/2)/(-1/b*(a*e-b*d))^(1/2)*ln((-2/b*(a*e-b*d)-(-b*e*(4*a*f-b*e))^(1/2)/b*(x+1/2*(b
*e+(-b*e*(4*a*f-b*e))^(1/2))/b/f)+2*(-1/b*(a*e-b*d))^(1/2)*((x+1/2*(b*e+(-b*e*(4*a*f-b*e))^(1/2))/b/f)^2*f-(-b
*e*(4*a*f-b*e))^(1/2)/b*(x+1/2*(b*e+(-b*e*(4*a*f-b*e))^(1/2))/b/f)-1/b*(a*e-b*d))^(1/2))/(x+1/2*(b*e+(-b*e*(4*
a*f-b*e))^(1/2))/b/f))*A*f-1/(4*a*f-b*e)/(-b*e*(4*a*f-b*e))^(1/2)/(-1/b*(a*e-b*d))^(1/2)*ln((-2/b*(a*e-b*d)-(-
b*e*(4*a*f-b*e))^(1/2)/b*(x+1/2*(b*e+(-b*e*(4*a*f-b*e))^(1/2))/b/f)+2*(-1/b*(a*e-b*d))^(1/2)*((x+1/2*(b*e+(-b*
e*(4*a*f-b*e))^(1/2))/b/f)^2*f-(-b*e*(4*a*f-b*e))^(1/2)/b*(x+1/2*(b*e+(-b*e*(4*a*f-b*e))^(1/2))/b/f)-1/b*(a*e-
b*d))^(1/2))/(x+1/2*(b*e+(-b*e*(4*a*f-b*e))^(1/2))/b/f))*B-1/e/(4*a*f-b*e)/(a*e-b*d)/(x+1/2/f*e+1/2/b/f*(-b*e*
(4*a*f-b*e))^(1/2))*((x+1/2*(b*e+(-b*e*(4*a*f-b*e))^(1/2))/b/f)^2*f-(-b*e*(4*a*f-b*e))^(1/2)/b*(x+1/2*(b*e+(-b
*e*(4*a*f-b*e))^(1/2))/b/f)-1/b*(a*e-b*d))^(1/2)*A+1/2/f/(4*a*f-b*e)/(a*e-b*d)/(x+1/2/f*e+1/2/b/f*(-b*e*(4*a*f
-b*e))^(1/2))*((x+1/2*(b*e+(-b*e*(4*a*f-b*e))^(1/2))/b/f)^2*f-(-b*e*(4*a*f-b*e))^(1/2)/b*(x+1/2*(b*e+(-b*e*(4*
a*f-b*e))^(1/2))/b/f)-1/b*(a*e-b*d))^(1/2)*B+1/2/f/e/(4*a*f-b*e)/b/(a*e-b*d)/(x+1/2/f*e+1/2/b/f*(-b*e*(4*a*f-b
*e))^(1/2))*((x+1/2*(b*e+(-b*e*(4*a*f-b*e))^(1/2))/b/f)^2*f-(-b*e*(4*a*f-b*e))^(1/2)/b*(x+1/2*(b*e+(-b*e*(4*a*
f-b*e))^(1/2))/b/f)-1/b*(a*e-b*d))^(1/2)*B*(-b*e*(4*a*f-b*e))^(1/2)-1/2/e/(4*a*f-b*e)/b*(-b*e*(4*a*f-b*e))^(1/
2)/(a*e-b*d)/(-1/b*(a*e-b*d))^(1/2)*ln((-2/b*(a*e-b*d)-(-b*e*(4*a*f-b*e))^(1/2)/b*(x+1/2*(b*e+(-b*e*(4*a*f-b*e
))^(1/2))/b/f)+2*(-1/b*(a*e-b*d))^(1/2)*((x+1/2*(b*e+(-b*e*(4*a*f-b*e))^(1/2))/b/f)^2*f-(-b*e*(4*a*f-b*e))^(1/
2)/b*(x+1/2*(b*e+(-b*e*(4*a*f-b*e))^(1/2))/b/f)-1/b*(a*e-b*d))^(1/2))/(x+1/2*(b*e+(-b*e*(4*a*f-b*e))^(1/2))/b/
f))*A+1/4/f/(4*a*f-b*e)/b*(-b*e*(4*a*f-b*e))^(1/2)/(a*e-b*d)/(-1/b*(a*e-b*d))^(1/2)*ln((-2/b*(a*e-b*d)-(-b*e*(
4*a*f-b*e))^(1/2)/b*(x+1/2*(b*e+(-b*e*(4*a*f-b*e))^(1/2))/b/f)+2*(-1/b*(a*e-b*d))^(1/2)*((x+1/2*(b*e+(-b*e*(4*
a*f-b*e))^(1/2))/b/f)^2*f-(-b*e*(4*a*f-b*e))^(1/2)/b*(x+1/2*(b*e+(-b*e*(4*a*f-b*e))^(1/2))/b/f)-1/b*(a*e-b*d))
^(1/2))/(x+1/2*(b*e+(-b*e*(4*a*f-b*e))^(1/2))/b/f))*B-1/4/f/b/(a*e-b*d)/(-1/b*(a*e-b*d))^(1/2)*ln((-2/b*(a*e-b
*d)-(-b*e*(4*a*f-b*e))^(1/2)/b*(x+1/2*(b*e+(-b*e*(4*a*f-b*e))^(1/2))/b/f)+2*(-1/b*(a*e-b*d))^(1/2)*((x+1/2*(b*
e+(-b*e*(4*a*f-b*e))^(1/2))/b/f)^2*f-(-b*e*(4*a*f-b*e))^(1/2)/b*(x+1/2*(b*e+(-b*e*(4*a*f-b*e))^(1/2))/b/f)-1/b
*(a*e-b*d))^(1/2))/(x+1/2*(b*e+(-b*e*(4*a*f-b*e))^(1/2))/b/f))*B
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {B x + A}{{\left (b f x^{2} + b e x + a e\right )}^{2} \sqrt {f x^{2} + e x + d}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)/(b*f*x^2+b*e*x+a*e)^2/(f*x^2+e*x+d)^(1/2),x, algorithm="maxima")
[Out]
integrate((B*x + A)/((b*f*x^2 + b*e*x + a*e)^2*sqrt(f*x^2 + e*x + d)), x)
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {A+B\,x}{{\left (b\,f\,x^2+b\,e\,x+a\,e\right )}^2\,\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*e + b*e*x + b*f*x^2)^2*(d + e*x + f*x^2)^(1/2)),x)
[Out]
int((A + B*x)/((a*e + b*e*x + b*f*x^2)^2*(d + e*x + f*x^2)^(1/2)), x)
________________________________________________________________________________________
sympy [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)/(b*f*x**2+b*e*x+a*e)**2/(f*x**2+e*x+d)**(1/2),x)
[Out]
Timed out
________________________________________________________________________________________