3.1.35 \(\int \frac {A+B x}{\sqrt {d+e x+f x^2} (a e+b e x+b f x^2)^2} \, dx\) [35]

Optimal. Leaf size=249 \[ -\frac {((A b-2 a B) e-b (B e-2 A f) x) \sqrt {d+e x+f x^2}}{e (b d-a e) (b e-4 a f) \left (a e+b e x+b f x^2\right )}+\frac {(B e-2 A f) \left (8 a e f-b \left (e^2+4 d f\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {b d-a e} (e+2 f x)}{\sqrt {e} \sqrt {b e-4 a f} \sqrt {d+e x+f x^2}}\right )}{2 e^{3/2} (b d-a e)^{3/2} f (b e-4 a f)^{3/2}}+\frac {B \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x+f x^2}}{\sqrt {b d-a e}}\right )}{2 \sqrt {b} (b d-a e)^{3/2} f} \]

[Out]

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

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Rubi [A]
time = 0.54, 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 = {1030, 1039, 996, 214, 1038} \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 214

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

Rule 996

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 1030

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)/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)
*(c*e - b*f))*(p + 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), 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 1038

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 1039

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) \text {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 ) \text {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 [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 1.89, size = 1014, normalized size = 4.07 \begin {gather*} \frac {\sqrt {d+x (e+f x)} (-B e (2 a+b x)+A b (e+2 f x))}{e (-b d+a e) (b e-4 a f) (a e+b x (e+f x))}-\frac {4 B \text {RootSum}\left [-b d e^2+a e^3+b d^2 f+2 b d e \sqrt {f} \text {$\#$1}-4 a e^2 \sqrt {f} \text {$\#$1}+b e^2 \text {$\#$1}^2-2 b d f \text {$\#$1}^2+4 a e f \text {$\#$1}^2-2 b e \sqrt {f} \text {$\#$1}^3+b f \text {$\#$1}^4\&,\frac {\log \left (-\sqrt {f} x+\sqrt {d+e x+f x^2}-\text {$\#$1}\right )}{-b d e \sqrt {f}+2 a e^2 \sqrt {f}-b e^2 \text {$\#$1}+2 b d f \text {$\#$1}-4 a e f \text {$\#$1}+3 b e \sqrt {f} \text {$\#$1}^2-2 b f \text {$\#$1}^3}\&\right ]}{b}-\frac {\text {RootSum}\left [-b d e^2+a e^3+b d^2 f+2 b d e \sqrt {f} \text {$\#$1}-4 a e^2 \sqrt {f} \text {$\#$1}+b e^2 \text {$\#$1}^2-2 b d f \text {$\#$1}^2+4 a e f \text {$\#$1}^2-2 b e \sqrt {f} \text {$\#$1}^3+b f \text {$\#$1}^4\&,\frac {-5 b^2 B d e^2 \log \left (-\sqrt {f} x+\sqrt {d+e x+f x^2}-\text {$\#$1}\right )-A b^2 e^3 \log \left (-\sqrt {f} x+\sqrt {d+e x+f x^2}-\text {$\#$1}\right )+6 a b B e^3 \log \left (-\sqrt {f} x+\sqrt {d+e x+f x^2}-\text {$\#$1}\right )-4 A b^2 d e f \log \left (-\sqrt {f} x+\sqrt {d+e x+f x^2}-\text {$\#$1}\right )+28 a b B d e f \log \left (-\sqrt {f} x+\sqrt {d+e x+f x^2}-\text {$\#$1}\right )+8 a A b e^2 f \log \left (-\sqrt {f} x+\sqrt {d+e x+f x^2}-\text {$\#$1}\right )-32 a^2 B e^2 f \log \left (-\sqrt {f} x+\sqrt {d+e x+f x^2}-\text {$\#$1}\right )-4 b^2 B d e \sqrt {f} \log \left (-\sqrt {f} x+\sqrt {d+e x+f x^2}-\text {$\#$1}\right ) \text {$\#$1}+2 A b^2 e^2 \sqrt {f} \log \left (-\sqrt {f} x+\sqrt {d+e x+f x^2}-\text {$\#$1}\right ) \text {$\#$1}+4 a b B e^2 \sqrt {f} \log \left (-\sqrt {f} x+\sqrt {d+e x+f x^2}-\text {$\#$1}\right ) \text {$\#$1}+8 A b^2 d f^{3/2} \log \left (-\sqrt {f} x+\sqrt {d+e x+f x^2}-\text {$\#$1}\right ) \text {$\#$1}-16 a A b e f^{3/2} \log \left (-\sqrt {f} x+\sqrt {d+e x+f x^2}-\text {$\#$1}\right ) \text {$\#$1}-b^2 B e^2 \log \left (-\sqrt {f} x+\sqrt {d+e x+f x^2}-\text {$\#$1}\right ) \text {$\#$1}^2+4 a b B e f \log \left (-\sqrt {f} x+\sqrt {d+e x+f x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-b d e \sqrt {f}+2 a e^2 \sqrt {f}-b e^2 \text {$\#$1}+2 b d f \text {$\#$1}-4 a e f \text {$\#$1}+3 b e \sqrt {f} \text {$\#$1}^2-2 b f \text {$\#$1}^3}\&\right ]}{2 b e (b d-a e) (b e-4 a f)} \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]

(Sqrt[d + x*(e + f*x)]*(-(B*e*(2*a + b*x)) + A*b*(e + 2*f*x)))/(e*(-(b*d) + a*e)*(b*e - 4*a*f)*(a*e + b*x*(e +
f*x))) - (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) + Sqrt[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] - 32*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*Sqrt[f]*Lo
g[-(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*S
qrt[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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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1429\) vs. \(2(219)=438\).
time = 0.22, size = 1430, normalized size = 5.74

method	result	size
default	\(-\frac {\left (2 A b f -B b e -B \sqrt {-e b \left (4 f a -e b \right )}\right ) \left (\frac {b \sqrt {\left (x +\frac {e b +\sqrt {-e b \left (4 f a -e b \right )}}{2 b f}\right )^{2} f -\frac {\sqrt {-e b \left (4 f a -e b \right )}\, \left (x +\frac {e b +\sqrt {-e b \left (4 f a -e b \right )}}{2 b f}\right )}{b}-\frac {a e -b d}{b}}}{\left (a e -b d \right ) \left (x +\frac {e b +\sqrt {-e b \left (4 f a -e b \right )}}{2 b f}\right )}+\frac {\sqrt {-e b \left (4 f a -e b \right )}\, \ln \left (\frac {-\frac {2 \left (a e -b d \right )}{b}-\frac {\sqrt {-e b \left (4 f a -e b \right )}\, \left (x +\frac {e b +\sqrt {-e b \left (4 f a -e b \right )}}{2 b f}\right )}{b}+2 \sqrt {-\frac {a e -b d}{b}}\, \sqrt {\left (x +\frac {e b +\sqrt {-e b \left (4 f a -e b \right )}}{2 b f}\right )^{2} f -\frac {\sqrt {-e b \left (4 f a -e b \right )}\, \left (x +\frac {e b +\sqrt {-e b \left (4 f a -e b \right )}}{2 b f}\right )}{b}-\frac {a e -b d}{b}}}{x +\frac {e b +\sqrt {-e b \left (4 f a -e b \right )}}{2 b f}}\right )}{2 \left (a e -b d \right ) \sqrt {-\frac {a e -b d}{b}}}\right )}{2 f e \left (4 f a -e b \right ) b^{2}}+\frac {\left (2 A f -B e \right ) \ln \left (\frac {-\frac {2 \left (a e -b d \right )}{b}-\frac {\sqrt {-e b \left (4 f a -e b \right )}\, \left (x +\frac {e b +\sqrt {-e b \left (4 f a -e b \right )}}{2 b f}\right )}{b}+2 \sqrt {-\frac {a e -b d}{b}}\, \sqrt {\left (x +\frac {e b +\sqrt {-e b \left (4 f a -e b \right )}}{2 b f}\right )^{2} f -\frac {\sqrt {-e b \left (4 f a -e b \right )}\, \left (x +\frac {e b +\sqrt {-e b \left (4 f a -e b \right )}}{2 b f}\right )}{b}-\frac {a e -b d}{b}}}{x +\frac {e b +\sqrt {-e b \left (4 f a -e b \right )}}{2 b f}}\right )}{e \left (4 f a -e b \right ) \sqrt {-e b \left (4 f a -e b \right )}\, \sqrt {-\frac {a e -b d}{b}}}-\frac {\left (2 A f -B e \right ) \ln \left (\frac {-\frac {2 \left (a e -b d \right )}{b}+\frac {\sqrt {-e b \left (4 f a -e b \right )}\, \left (x -\frac {-e b +\sqrt {-e b \left (4 f a -e b \right )}}{2 b f}\right )}{b}+2 \sqrt {-\frac {a e -b d}{b}}\, \sqrt {\left (x -\frac {-e b +\sqrt {-e b \left (4 f a -e b \right )}}{2 b f}\right )^{2} f +\frac {\sqrt {-e b \left (4 f a -e b \right )}\, \left (x -\frac {-e b +\sqrt {-e b \left (4 f a -e b \right )}}{2 b f}\right )}{b}-\frac {a e -b d}{b}}}{x -\frac {-e b +\sqrt {-e b \left (4 f a -e b \right )}}{2 b f}}\right )}{e \left (4 f a -e b \right ) \sqrt {-e b \left (4 f a -e b \right )}\, \sqrt {-\frac {a e -b d}{b}}}-\frac {\left (2 A b f -B b e +B \sqrt {-e b \left (4 f a -e b \right )}\right ) \left (\frac {b \sqrt {\left (x -\frac {-e b +\sqrt {-e b \left (4 f a -e b \right )}}{2 b f}\right )^{2} f +\frac {\sqrt {-e b \left (4 f a -e b \right )}\, \left (x -\frac {-e b +\sqrt {-e b \left (4 f a -e b \right )}}{2 b f}\right )}{b}-\frac {a e -b d}{b}}}{\left (a e -b d \right ) \left (x -\frac {-e b +\sqrt {-e b \left (4 f a -e b \right )}}{2 b f}\right )}-\frac {\sqrt {-e b \left (4 f a -e b \right )}\, \ln \left (\frac {-\frac {2 \left (a e -b d \right )}{b}+\frac {\sqrt {-e b \left (4 f a -e b \right )}\, \left (x -\frac {-e b +\sqrt {-e b \left (4 f a -e b \right )}}{2 b f}\right )}{b}+2 \sqrt {-\frac {a e -b d}{b}}\, \sqrt {\left (x -\frac {-e b +\sqrt {-e b \left (4 f a -e b \right )}}{2 b f}\right )^{2} f +\frac {\sqrt {-e b \left (4 f a -e b \right )}\, \left (x -\frac {-e b +\sqrt {-e b \left (4 f a -e b \right )}}{2 b f}\right )}{b}-\frac {a e -b d}{b}}}{x -\frac {-e b +\sqrt {-e b \left (4 f a -e b \right )}}{2 b f}}\right )}{2 \left (a e -b d \right ) \sqrt {-\frac {a e -b d}{b}}}\right )}{2 f e \left (4 f a -e b \right ) b^{2}}\)	\(1430\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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*x*e + a*e)^2*sqrt(f*x^2 + x*e + d)), x)

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Fricas [F(-1)] Timed out
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

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Sympy [F(-1)] Timed out
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

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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)/(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,sageVARx):;OUTP
UT:Unable to divide, perhaps due to rounding error%%%{%%%{1,[2]%%%},[8,2,0,0,0]%%%}+%%%{%%{[%%%{-4,[1]%%%},0]:
[1,0,%%%{-1

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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)

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