3.1.24 \(\int \frac {A+B x}{(a+c x^2) \sqrt {d+f x^2}} \, dx\) [24]

Optimal. Leaf size=101 \[ \frac {A \tan ^{-1}\left (\frac {\sqrt {c d-a f} x}{\sqrt {a} \sqrt {d+f x^2}}\right )}{\sqrt {a} \sqrt {c d-a f}}-\frac {B \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+f x^2}}{\sqrt {c d-a f}}\right )}{\sqrt {c} \sqrt {c d-a f}} \]

[Out]

A*arctan(x*(-a*f+c*d)^(1/2)/a^(1/2)/(f*x^2+d)^(1/2))/a^(1/2)/(-a*f+c*d)^(1/2)-B*arctanh(c^(1/2)*(f*x^2+d)^(1/2
)/(-a*f+c*d)^(1/2))/c^(1/2)/(-a*f+c*d)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.08, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {1024, 385, 211, 455, 65, 214} \begin {gather*} \frac {A \text {ArcTan}\left (\frac {x \sqrt {c d-a f}}{\sqrt {a} \sqrt {d+f x^2}}\right )}{\sqrt {a} \sqrt {c d-a f}}-\frac {B \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+f x^2}}{\sqrt {c d-a f}}\right )}{\sqrt {c} \sqrt {c d-a f}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 211

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

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 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 455

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m
- n + 1, 0]

Rule 1024

Int[((g_) + (h_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_)*((d_) + (f_.)*(x_)^2)^(q_), x_Symbol] :> Dist[g, Int[(a + c
*x^2)^p*(d + f*x^2)^q, x], x] + Dist[h, Int[x*(a + c*x^2)^p*(d + f*x^2)^q, x], x] /; FreeQ[{a, c, d, f, g, h,
p, q}, x]

Rubi steps

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

________________________________________________________________________________________

Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(349\) vs. \(2(101)=202\).
time = 1.77, size = 349, normalized size = 3.46 \begin {gather*} \frac {\sqrt {a} B \left (\left (\sqrt {a} \sqrt {f}+\sqrt {-c d+a f}\right ) \sqrt {-c d+2 a f-2 \sqrt {a} \sqrt {f} \sqrt {-c d+a f}} \tan ^{-1}\left (\frac {\sqrt {c} \left (\sqrt {f} x-\sqrt {d+f x^2}\right )}{\sqrt {-c d+2 a f-2 \sqrt {a} \sqrt {f} \sqrt {-c d+a f}}}\right )+\left (-\sqrt {a} \sqrt {f}+\sqrt {-c d+a f}\right ) \sqrt {-c d+2 a f+2 \sqrt {a} \sqrt {f} \sqrt {-c d+a f}} \tan ^{-1}\left (\frac {\sqrt {c} \left (\sqrt {f} x-\sqrt {d+f x^2}\right )}{\sqrt {-c d+2 a f+2 \sqrt {a} \sqrt {f} \sqrt {-c d+a f}}}\right )\right )+A c^{3/2} d \tanh ^{-1}\left (\frac {a \sqrt {f}+c x \left (\sqrt {f} x-\sqrt {d+f x^2}\right )}{\sqrt {a} \sqrt {-c d+a f}}\right )}{\sqrt {a} c^{3/2} d \sqrt {-c d+a f}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(336\) vs. \(2(81)=162\).
time = 0.08, size = 337, normalized size = 3.34

method	result	size
default	\(-\frac {\left (A c +B \sqrt {-a c}\right ) \ln \left (\frac {-\frac {2 \left (f a -c d \right )}{c}+\frac {2 f \sqrt {-a c}\, \left (x -\frac {\sqrt {-a c}}{c}\right )}{c}+2 \sqrt {-\frac {f a -c d}{c}}\, \sqrt {f \left (x -\frac {\sqrt {-a c}}{c}\right )^{2}+\frac {2 f \sqrt {-a c}\, \left (x -\frac {\sqrt {-a c}}{c}\right )}{c}-\frac {f a -c d}{c}}}{x -\frac {\sqrt {-a c}}{c}}\right )}{2 \sqrt {-a c}\, c \sqrt {-\frac {f a -c d}{c}}}-\frac {\left (-A c +B \sqrt {-a c}\right ) \ln \left (\frac {-\frac {2 \left (f a -c d \right )}{c}-\frac {2 f \sqrt {-a c}\, \left (x +\frac {\sqrt {-a c}}{c}\right )}{c}+2 \sqrt {-\frac {f a -c d}{c}}\, \sqrt {f \left (x +\frac {\sqrt {-a c}}{c}\right )^{2}-\frac {2 f \sqrt {-a c}\, \left (x +\frac {\sqrt {-a c}}{c}\right )}{c}-\frac {f a -c d}{c}}}{x +\frac {\sqrt {-a c}}{c}}\right )}{2 \sqrt {-a c}\, c \sqrt {-\frac {f a -c d}{c}}}\)	\(337\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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)/(c*x^2+a)/(f*x^2+d)^(1/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1515 vs. \(2 (81) = 162\).
time = 0.42, size = 1515, normalized size = 15.00 \begin {gather*} -\frac {1}{4} \, \sqrt {\frac {B^{2} a - A^{2} c + 2 \, {\left (a c^{2} d - a^{2} c f\right )} \sqrt {-\frac {A^{2} B^{2}}{a c^{3} d^{2} - 2 \, a^{2} c^{2} d f + a^{3} c f^{2}}}}{a c^{2} d - a^{2} c f}} \log \left (\frac {{\left (A B^{3} a + A^{3} B c\right )} f x + {\left (A^{2} B c^{2} d - A^{2} B a c f + {\left (B a c^{3} d^{2} - 2 \, B a^{2} c^{2} d f + B a^{3} c f^{2}\right )} \sqrt {-\frac {A^{2} B^{2}}{a c^{3} d^{2} - 2 \, a^{2} c^{2} d f + a^{3} c f^{2}}}\right )} \sqrt {f x^{2} + d} \sqrt {\frac {B^{2} a - A^{2} c + 2 \, {\left (a c^{2} d - a^{2} c f\right )} \sqrt {-\frac {A^{2} B^{2}}{a c^{3} d^{2} - 2 \, a^{2} c^{2} d f + a^{3} c f^{2}}}}{a c^{2} d - a^{2} c f}} + \sqrt {-\frac {A^{2} B^{2}}{a c^{3} d^{2} - 2 \, a^{2} c^{2} d f + a^{3} c f^{2}}} {\left ({\left (B^{2} a c^{2} + A^{2} c^{3}\right )} d^{2} - {\left (B^{2} a^{2} c + A^{2} a c^{2}\right )} d f\right )}}{x}\right ) + \frac {1}{4} \, \sqrt {\frac {B^{2} a - A^{2} c + 2 \, {\left (a c^{2} d - a^{2} c f\right )} \sqrt {-\frac {A^{2} B^{2}}{a c^{3} d^{2} - 2 \, a^{2} c^{2} d f + a^{3} c f^{2}}}}{a c^{2} d - a^{2} c f}} \log \left (\frac {{\left (A B^{3} a + A^{3} B c\right )} f x - {\left (A^{2} B c^{2} d - A^{2} B a c f + {\left (B a c^{3} d^{2} - 2 \, B a^{2} c^{2} d f + B a^{3} c f^{2}\right )} \sqrt {-\frac {A^{2} B^{2}}{a c^{3} d^{2} - 2 \, a^{2} c^{2} d f + a^{3} c f^{2}}}\right )} \sqrt {f x^{2} + d} \sqrt {\frac {B^{2} a - A^{2} c + 2 \, {\left (a c^{2} d - a^{2} c f\right )} \sqrt {-\frac {A^{2} B^{2}}{a c^{3} d^{2} - 2 \, a^{2} c^{2} d f + a^{3} c f^{2}}}}{a c^{2} d - a^{2} c f}} + \sqrt {-\frac {A^{2} B^{2}}{a c^{3} d^{2} - 2 \, a^{2} c^{2} d f + a^{3} c f^{2}}} {\left ({\left (B^{2} a c^{2} + A^{2} c^{3}\right )} d^{2} - {\left (B^{2} a^{2} c + A^{2} a c^{2}\right )} d f\right )}}{x}\right ) - \frac {1}{4} \, \sqrt {\frac {B^{2} a - A^{2} c - 2 \, {\left (a c^{2} d - a^{2} c f\right )} \sqrt {-\frac {A^{2} B^{2}}{a c^{3} d^{2} - 2 \, a^{2} c^{2} d f + a^{3} c f^{2}}}}{a c^{2} d - a^{2} c f}} \log \left (\frac {{\left (A B^{3} a + A^{3} B c\right )} f x + {\left (A^{2} B c^{2} d - A^{2} B a c f - {\left (B a c^{3} d^{2} - 2 \, B a^{2} c^{2} d f + B a^{3} c f^{2}\right )} \sqrt {-\frac {A^{2} B^{2}}{a c^{3} d^{2} - 2 \, a^{2} c^{2} d f + a^{3} c f^{2}}}\right )} \sqrt {f x^{2} + d} \sqrt {\frac {B^{2} a - A^{2} c - 2 \, {\left (a c^{2} d - a^{2} c f\right )} \sqrt {-\frac {A^{2} B^{2}}{a c^{3} d^{2} - 2 \, a^{2} c^{2} d f + a^{3} c f^{2}}}}{a c^{2} d - a^{2} c f}} - \sqrt {-\frac {A^{2} B^{2}}{a c^{3} d^{2} - 2 \, a^{2} c^{2} d f + a^{3} c f^{2}}} {\left ({\left (B^{2} a c^{2} + A^{2} c^{3}\right )} d^{2} - {\left (B^{2} a^{2} c + A^{2} a c^{2}\right )} d f\right )}}{x}\right ) + \frac {1}{4} \, \sqrt {\frac {B^{2} a - A^{2} c - 2 \, {\left (a c^{2} d - a^{2} c f\right )} \sqrt {-\frac {A^{2} B^{2}}{a c^{3} d^{2} - 2 \, a^{2} c^{2} d f + a^{3} c f^{2}}}}{a c^{2} d - a^{2} c f}} \log \left (\frac {{\left (A B^{3} a + A^{3} B c\right )} f x - {\left (A^{2} B c^{2} d - A^{2} B a c f - {\left (B a c^{3} d^{2} - 2 \, B a^{2} c^{2} d f + B a^{3} c f^{2}\right )} \sqrt {-\frac {A^{2} B^{2}}{a c^{3} d^{2} - 2 \, a^{2} c^{2} d f + a^{3} c f^{2}}}\right )} \sqrt {f x^{2} + d} \sqrt {\frac {B^{2} a - A^{2} c - 2 \, {\left (a c^{2} d - a^{2} c f\right )} \sqrt {-\frac {A^{2} B^{2}}{a c^{3} d^{2} - 2 \, a^{2} c^{2} d f + a^{3} c f^{2}}}}{a c^{2} d - a^{2} c f}} - \sqrt {-\frac {A^{2} B^{2}}{a c^{3} d^{2} - 2 \, a^{2} c^{2} d f + a^{3} c f^{2}}} {\left ({\left (B^{2} a c^{2} + A^{2} c^{3}\right )} d^{2} - {\left (B^{2} a^{2} c + A^{2} a c^{2}\right )} d f\right )}}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*sqrt((B^2*a - A^2*c + 2*(a*c^2*d - a^2*c*f)*sqrt(-A^2*B^2/(a*c^3*d^2 - 2*a^2*c^2*d*f + a^3*c*f^2)))/(a*c^
2*d - a^2*c*f))*log(((A*B^3*a + A^3*B*c)*f*x + (A^2*B*c^2*d - A^2*B*a*c*f + (B*a*c^3*d^2 - 2*B*a^2*c^2*d*f + B
*a^3*c*f^2)*sqrt(-A^2*B^2/(a*c^3*d^2 - 2*a^2*c^2*d*f + a^3*c*f^2)))*sqrt(f*x^2 + d)*sqrt((B^2*a - A^2*c + 2*(a
*c^2*d - a^2*c*f)*sqrt(-A^2*B^2/(a*c^3*d^2 - 2*a^2*c^2*d*f + a^3*c*f^2)))/(a*c^2*d - a^2*c*f)) + sqrt(-A^2*B^2
/(a*c^3*d^2 - 2*a^2*c^2*d*f + a^3*c*f^2))*((B^2*a*c^2 + A^2*c^3)*d^2 - (B^2*a^2*c + A^2*a*c^2)*d*f))/x) + 1/4*
sqrt((B^2*a - A^2*c + 2*(a*c^2*d - a^2*c*f)*sqrt(-A^2*B^2/(a*c^3*d^2 - 2*a^2*c^2*d*f + a^3*c*f^2)))/(a*c^2*d -
a^2*c*f))*log(((A*B^3*a + A^3*B*c)*f*x - (A^2*B*c^2*d - A^2*B*a*c*f + (B*a*c^3*d^2 - 2*B*a^2*c^2*d*f + B*a^3*
c*f^2)*sqrt(-A^2*B^2/(a*c^3*d^2 - 2*a^2*c^2*d*f + a^3*c*f^2)))*sqrt(f*x^2 + d)*sqrt((B^2*a - A^2*c + 2*(a*c^2*
d - a^2*c*f)*sqrt(-A^2*B^2/(a*c^3*d^2 - 2*a^2*c^2*d*f + a^3*c*f^2)))/(a*c^2*d - a^2*c*f)) + sqrt(-A^2*B^2/(a*c
^3*d^2 - 2*a^2*c^2*d*f + a^3*c*f^2))*((B^2*a*c^2 + A^2*c^3)*d^2 - (B^2*a^2*c + A^2*a*c^2)*d*f))/x) - 1/4*sqrt(
(B^2*a - A^2*c - 2*(a*c^2*d - a^2*c*f)*sqrt(-A^2*B^2/(a*c^3*d^2 - 2*a^2*c^2*d*f + a^3*c*f^2)))/(a*c^2*d - a^2*
c*f))*log(((A*B^3*a + A^3*B*c)*f*x + (A^2*B*c^2*d - A^2*B*a*c*f - (B*a*c^3*d^2 - 2*B*a^2*c^2*d*f + B*a^3*c*f^2
)*sqrt(-A^2*B^2/(a*c^3*d^2 - 2*a^2*c^2*d*f + a^3*c*f^2)))*sqrt(f*x^2 + d)*sqrt((B^2*a - A^2*c - 2*(a*c^2*d - a
^2*c*f)*sqrt(-A^2*B^2/(a*c^3*d^2 - 2*a^2*c^2*d*f + a^3*c*f^2)))/(a*c^2*d - a^2*c*f)) - sqrt(-A^2*B^2/(a*c^3*d^
2 - 2*a^2*c^2*d*f + a^3*c*f^2))*((B^2*a*c^2 + A^2*c^3)*d^2 - (B^2*a^2*c + A^2*a*c^2)*d*f))/x) + 1/4*sqrt((B^2*
a - A^2*c - 2*(a*c^2*d - a^2*c*f)*sqrt(-A^2*B^2/(a*c^3*d^2 - 2*a^2*c^2*d*f + a^3*c*f^2)))/(a*c^2*d - a^2*c*f))
*log(((A*B^3*a + A^3*B*c)*f*x - (A^2*B*c^2*d - A^2*B*a*c*f - (B*a*c^3*d^2 - 2*B*a^2*c^2*d*f + B*a^3*c*f^2)*sqr
t(-A^2*B^2/(a*c^3*d^2 - 2*a^2*c^2*d*f + a^3*c*f^2)))*sqrt(f*x^2 + d)*sqrt((B^2*a - A^2*c - 2*(a*c^2*d - a^2*c*
f)*sqrt(-A^2*B^2/(a*c^3*d^2 - 2*a^2*c^2*d*f + a^3*c*f^2)))/(a*c^2*d - a^2*c*f)) - sqrt(-A^2*B^2/(a*c^3*d^2 - 2
*a^2*c^2*d*f + a^3*c*f^2))*((B^2*a*c^2 + A^2*c^3)*d^2 - (B^2*a^2*c + A^2*a*c^2)*d*f))/x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [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)/(c*x^2+a)/(f*x^2+d)^(1/2),x, algorithm="giac")

[Out]

Timed out

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \left \{\begin {array}{cl} \frac {B\,\mathrm {atan}\left (\frac {c\,\sqrt {f\,x^2+d}}{\sqrt {a\,c\,f-c^2\,d}}\right )}{\sqrt {a\,c\,f-c^2\,d}}+\frac {A\,\mathrm {atan}\left (\frac {x\,\sqrt {c\,d-a\,f}}{\sqrt {a}\,\sqrt {f\,x^2+d}}\right )}{\sqrt {-a\,\left (a\,f-c\,d\right )}} & \text {\ if\ \ }0<c\,d-a\,f\\ \frac {A\,\ln \left (\frac {\sqrt {a\,\left (f\,x^2+d\right )}+x\,\sqrt {a\,f-c\,d}}{\sqrt {a\,\left (f\,x^2+d\right )}-x\,\sqrt {a\,f-c\,d}}\right )}{2\,\sqrt {a\,\left (a\,f-c\,d\right )}}+\frac {B\,\mathrm {atan}\left (\frac {c\,\sqrt {f\,x^2+d}}{\sqrt {a\,c\,f-c^2\,d}}\right )}{\sqrt {a\,c\,f-c^2\,d}} & \text {\ if\ \ }c\,d-a\,f<0\\ \int \frac {A+B\,x}{\left (c\,x^2+a\right )\,\sqrt {f\,x^2+d}} \,d x & \text {\ if\ \ }c\,d-a\,f\notin \mathbb {R}\vee a\,f=c\,d \end {array}\right . \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

piecewise(0 < - a*f + c*d, (B*atan((c*(d + f*x^2)^(1/2))/(- c^2*d + a*c*f)^(1/2)))/(- c^2*d + a*c*f)^(1/2) + (
A*atan((x*(- a*f + c*d)^(1/2))/(a^(1/2)*(d + f*x^2)^(1/2))))/(-a*(a*f - c*d))^(1/2), - a*f + c*d < 0, (A*log((
(a*(d + f*x^2))^(1/2) + x*(a*f - c*d)^(1/2))/((a*(d + f*x^2))^(1/2) - x*(a*f - c*d)^(1/2))))/(2*(a*(a*f - c*d)
)^(1/2)) + (B*atan((c*(d + f*x^2)^(1/2))/(- c^2*d + a*c*f)^(1/2)))/(- c^2*d + a*c*f)^(1/2), ~in(- a*f + c*d, '
real') | a*f == c*d, int((A + B*x)/((a + c*x^2)*(d + f*x^2)^(1/2)), x))

________________________________________________________________________________________