3.5.13 \(\int \frac {\sqrt {d+e x}}{\sqrt {f+g x} (a+c x^2)} \, dx\)
Optimal. Leaf size=240 \[ \frac {\sqrt {\sqrt {c} d-\sqrt {-a} e} \tanh ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {\sqrt {c} f-\sqrt {-a} g}}{\sqrt {f+g x} \sqrt {\sqrt {c} d-\sqrt {-a} e}}\right )}{\sqrt {-a} \sqrt {c} \sqrt {\sqrt {c} f-\sqrt {-a} g}}-\frac {\sqrt {\sqrt {-a} e+\sqrt {c} d} \tanh ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {\sqrt {-a} g+\sqrt {c} f}}{\sqrt {f+g x} \sqrt {\sqrt {-a} e+\sqrt {c} d}}\right )}{\sqrt {-a} \sqrt {c} \sqrt {\sqrt {-a} g+\sqrt {c} f}} \]
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Rubi [A] time = 0.33, antiderivative size = 240, normalized size of antiderivative = 1.00,
number of steps used = 6, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used =
{910, 93, 208} \begin {gather*} \frac {\sqrt {\sqrt {c} d-\sqrt {-a} e} \tanh ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {\sqrt {c} f-\sqrt {-a} g}}{\sqrt {f+g x} \sqrt {\sqrt {c} d-\sqrt {-a} e}}\right )}{\sqrt {-a} \sqrt {c} \sqrt {\sqrt {c} f-\sqrt {-a} g}}-\frac {\sqrt {\sqrt {-a} e+\sqrt {c} d} \tanh ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {\sqrt {-a} g+\sqrt {c} f}}{\sqrt {f+g x} \sqrt {\sqrt {-a} e+\sqrt {c} d}}\right )}{\sqrt {-a} \sqrt {c} \sqrt {\sqrt {-a} g+\sqrt {c} f}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[Sqrt[d + e*x]/(Sqrt[f + g*x]*(a + c*x^2)),x]
[Out]
(Sqrt[Sqrt[c]*d - Sqrt[-a]*e]*ArcTanh[(Sqrt[Sqrt[c]*f - Sqrt[-a]*g]*Sqrt[d + e*x])/(Sqrt[Sqrt[c]*d - Sqrt[-a]*
e]*Sqrt[f + g*x])])/(Sqrt[-a]*Sqrt[c]*Sqrt[Sqrt[c]*f - Sqrt[-a]*g]) - (Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*ArcTanh[(S
qrt[Sqrt[c]*f + Sqrt[-a]*g]*Sqrt[d + e*x])/(Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*Sqrt[f + g*x])])/(Sqrt[-a]*Sqrt[c]*Sq
rt[Sqrt[c]*f + Sqrt[-a]*g])
Rule 93
Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]
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 910
Int[((d_.) + (e_.)*(x_))^(m_)/(Sqrt[(f_.) + (g_.)*(x_)]*((a_.) + (c_.)*(x_)^2)), x_Symbol] :> Int[ExpandIntegr
and[1/(Sqrt[d + e*x]*Sqrt[f + g*x]), (d + e*x)^(m + 1/2)/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g}, x] &
& NeQ[c*d^2 + a*e^2, 0] && IGtQ[m + 1/2, 0]
Rubi steps
\begin {align*} \int \frac {\sqrt {d+e x}}{\sqrt {f+g x} \left (a+c x^2\right )} \, dx &=\int \left (\frac {\sqrt {-a} d-\frac {a e}{\sqrt {c}}}{2 a \left (\sqrt {-a}-\sqrt {c} x\right ) \sqrt {d+e x} \sqrt {f+g x}}+\frac {\sqrt {-a} d+\frac {a e}{\sqrt {c}}}{2 a \left (\sqrt {-a}+\sqrt {c} x\right ) \sqrt {d+e x} \sqrt {f+g x}}\right ) \, dx\\ &=\frac {1}{2} \left (\frac {a d}{(-a)^{3/2}}-\frac {e}{\sqrt {c}}\right ) \int \frac {1}{\left (\sqrt {-a}-\sqrt {c} x\right ) \sqrt {d+e x} \sqrt {f+g x}} \, dx+\frac {1}{2} \left (\frac {a d}{(-a)^{3/2}}+\frac {e}{\sqrt {c}}\right ) \int \frac {1}{\left (\sqrt {-a}+\sqrt {c} x\right ) \sqrt {d+e x} \sqrt {f+g x}} \, dx\\ &=\left (\frac {a d}{(-a)^{3/2}}-\frac {e}{\sqrt {c}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c} d+\sqrt {-a} e-\left (\sqrt {c} f+\sqrt {-a} g\right ) x^2} \, dx,x,\frac {\sqrt {d+e x}}{\sqrt {f+g x}}\right )+\left (\frac {a d}{(-a)^{3/2}}+\frac {e}{\sqrt {c}}\right ) \operatorname {Subst}\left (\int \frac {1}{-\sqrt {c} d+\sqrt {-a} e-\left (-\sqrt {c} f+\sqrt {-a} g\right ) x^2} \, dx,x,\frac {\sqrt {d+e x}}{\sqrt {f+g x}}\right )\\ &=\frac {\sqrt {\sqrt {c} d-\sqrt {-a} e} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {c} f-\sqrt {-a} g} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {-a} e} \sqrt {f+g x}}\right )}{\sqrt {-a} \sqrt {c} \sqrt {\sqrt {c} f-\sqrt {-a} g}}-\frac {\sqrt {\sqrt {c} d+\sqrt {-a} e} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {c} f+\sqrt {-a} g} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {-a} e} \sqrt {f+g x}}\right )}{\sqrt {-a} \sqrt {c} \sqrt {\sqrt {c} f+\sqrt {-a} g}}\\ \end {align*}
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Mathematica [A] time = 0.33, size = 229, normalized size = 0.95 \begin {gather*} \frac {\frac {\sqrt {\sqrt {-a} e-\sqrt {c} d} \tanh ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {\sqrt {-a} g-\sqrt {c} f}}{\sqrt {f+g x} \sqrt {\sqrt {-a} e-\sqrt {c} d}}\right )}{\sqrt {\sqrt {-a} g-\sqrt {c} f}}-\frac {\sqrt {\sqrt {-a} e+\sqrt {c} d} \tanh ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {\sqrt {-a} g+\sqrt {c} f}}{\sqrt {f+g x} \sqrt {\sqrt {-a} e+\sqrt {c} d}}\right )}{\sqrt {\sqrt {-a} g+\sqrt {c} f}}}{\sqrt {-a} \sqrt {c}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[Sqrt[d + e*x]/(Sqrt[f + g*x]*(a + c*x^2)),x]
[Out]
((Sqrt[-(Sqrt[c]*d) + Sqrt[-a]*e]*ArcTanh[(Sqrt[-(Sqrt[c]*f) + Sqrt[-a]*g]*Sqrt[d + e*x])/(Sqrt[-(Sqrt[c]*d) +
Sqrt[-a]*e]*Sqrt[f + g*x])])/Sqrt[-(Sqrt[c]*f) + Sqrt[-a]*g] - (Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*ArcTanh[(Sqrt[Sq
rt[c]*f + Sqrt[-a]*g]*Sqrt[d + e*x])/(Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*Sqrt[f + g*x])])/Sqrt[Sqrt[c]*f + Sqrt[-a]*
g])/(Sqrt[-a]*Sqrt[c])
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IntegrateAlgebraic [C] time = 116.51, size = 1253, normalized size = 5.22 \begin {gather*} -e \sqrt {\frac {e}{g}} \text {RootSum}\left [c g^4 \text {$\#$1}^8-4 c d g^4 \text {$\#$1}^6-4 c e f g^3 \text {$\#$1}^6+6 c d^2 g^4 \text {$\#$1}^4+16 a e^2 g^4 \text {$\#$1}^4+4 c d e f g^3 \text {$\#$1}^4+6 c e^2 f^2 g^2 \text {$\#$1}^4-4 c d^3 g^4 \text {$\#$1}^2+4 c d^2 e f g^3 \text {$\#$1}^2+4 c d e^2 f^2 g^2 \text {$\#$1}^2-4 c e^3 f^3 g \text {$\#$1}^2+c e^4 f^4+c d^4 g^4-4 c d^3 e f g^3+6 c d^2 e^2 f^2 g^2-4 c d e^3 f^3 g\&,\frac {\log \left (-\text {$\#$1}-\sqrt {\frac {e}{g}} \sqrt {f+g x}+\sqrt {d+\frac {e (f+g x)}{g}-\frac {e f}{g}}\right ) \text {$\#$1}^4}{c g^3 \text {$\#$1}^6-3 c d g^3 \text {$\#$1}^4-3 c e f g^2 \text {$\#$1}^4+3 c d^2 g^3 \text {$\#$1}^2+8 a e^2 g^3 \text {$\#$1}^2+2 c d e f g^2 \text {$\#$1}^2+3 c e^2 f^2 g \text {$\#$1}^2-c e^3 f^3-c d^3 g^3+c d^2 e f g^2+c d e^2 f^2 g}\&\right ] g^3+\left (f^2 \sqrt {\frac {e}{g}} g e^3-2 d f \sqrt {\frac {e}{g}} g^2 e^2+d^2 \sqrt {\frac {e}{g}} g^3 e\right ) \text {RootSum}\left [c g^4 \text {$\#$1}^8-4 c d g^4 \text {$\#$1}^6-4 c e f g^3 \text {$\#$1}^6+6 c d^2 g^4 \text {$\#$1}^4+16 a e^2 g^4 \text {$\#$1}^4+4 c d e f g^3 \text {$\#$1}^4+6 c e^2 f^2 g^2 \text {$\#$1}^4-4 c d^3 g^4 \text {$\#$1}^2+4 c d^2 e f g^3 \text {$\#$1}^2+4 c d e^2 f^2 g^2 \text {$\#$1}^2-4 c e^3 f^3 g \text {$\#$1}^2+c e^4 f^4+c d^4 g^4-4 c d^3 e f g^3+6 c d^2 e^2 f^2 g^2-4 c d e^3 f^3 g\&,\frac {\log \left (-\text {$\#$1}-\sqrt {\frac {e}{g}} \sqrt {f+g x}+\sqrt {d+\frac {e (f+g x)}{g}-\frac {e f}{g}}\right )}{-c g^3 \text {$\#$1}^6+3 c d g^3 \text {$\#$1}^4+3 c e f g^2 \text {$\#$1}^4-3 c d^2 g^3 \text {$\#$1}^2-8 a e^2 g^3 \text {$\#$1}^2-2 c d e f g^2 \text {$\#$1}^2-3 c e^2 f^2 g \text {$\#$1}^2+c e^3 f^3+c d^3 g^3-c d^2 e f g^2-c d e^2 f^2 g}\&\right ]+2 \left (e^2 f \sqrt {\frac {e}{g}} g^2-d e \sqrt {\frac {e}{g}} g^3\right ) \text {RootSum}\left [c g^4 \text {$\#$1}^8-4 c d g^4 \text {$\#$1}^6-4 c e f g^3 \text {$\#$1}^6+6 c d^2 g^4 \text {$\#$1}^4+16 a e^2 g^4 \text {$\#$1}^4+4 c d e f g^3 \text {$\#$1}^4+6 c e^2 f^2 g^2 \text {$\#$1}^4-4 c d^3 g^4 \text {$\#$1}^2+4 c d^2 e f g^3 \text {$\#$1}^2+4 c d e^2 f^2 g^2 \text {$\#$1}^2-4 c e^3 f^3 g \text {$\#$1}^2+c e^4 f^4+c d^4 g^4-4 c d^3 e f g^3+6 c d^2 e^2 f^2 g^2-4 c d e^3 f^3 g\&,\frac {\log \left (-\text {$\#$1}-\sqrt {\frac {e}{g}} \sqrt {f+g x}+\sqrt {d+\frac {e (f+g x)}{g}-\frac {e f}{g}}\right ) \text {$\#$1}^2}{c g^3 \text {$\#$1}^6-3 c d g^3 \text {$\#$1}^4-3 c e f g^2 \text {$\#$1}^4+3 c d^2 g^3 \text {$\#$1}^2+8 a e^2 g^3 \text {$\#$1}^2+2 c d e f g^2 \text {$\#$1}^2+3 c e^2 f^2 g \text {$\#$1}^2-c e^3 f^3-c d^3 g^3+c d^2 e f g^2+c d e^2 f^2 g}\&\right ] \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[Sqrt[d + e*x]/(Sqrt[f + g*x]*(a + c*x^2)),x]
[Out]
(e^3*f^2*Sqrt[e/g]*g - 2*d*e^2*f*Sqrt[e/g]*g^2 + d^2*e*Sqrt[e/g]*g^3)*RootSum[c*e^4*f^4 - 4*c*d*e^3*f^3*g + 6*
c*d^2*e^2*f^2*g^2 - 4*c*d^3*e*f*g^3 + c*d^4*g^4 - 4*c*e^3*f^3*g*#1^2 + 4*c*d*e^2*f^2*g^2*#1^2 + 4*c*d^2*e*f*g^
3*#1^2 - 4*c*d^3*g^4*#1^2 + 6*c*e^2*f^2*g^2*#1^4 + 4*c*d*e*f*g^3*#1^4 + 6*c*d^2*g^4*#1^4 + 16*a*e^2*g^4*#1^4 -
4*c*e*f*g^3*#1^6 - 4*c*d*g^4*#1^6 + c*g^4*#1^8 & , Log[-(Sqrt[e/g]*Sqrt[f + g*x]) + Sqrt[d - (e*f)/g + (e*(f
+ g*x))/g] - #1]/(c*e^3*f^3 - c*d*e^2*f^2*g - c*d^2*e*f*g^2 + c*d^3*g^3 - 3*c*e^2*f^2*g*#1^2 - 2*c*d*e*f*g^2*#
1^2 - 3*c*d^2*g^3*#1^2 - 8*a*e^2*g^3*#1^2 + 3*c*e*f*g^2*#1^4 + 3*c*d*g^3*#1^4 - c*g^3*#1^6) & ] + 2*(e^2*f*Sqr
t[e/g]*g^2 - d*e*Sqrt[e/g]*g^3)*RootSum[c*e^4*f^4 - 4*c*d*e^3*f^3*g + 6*c*d^2*e^2*f^2*g^2 - 4*c*d^3*e*f*g^3 +
c*d^4*g^4 - 4*c*e^3*f^3*g*#1^2 + 4*c*d*e^2*f^2*g^2*#1^2 + 4*c*d^2*e*f*g^3*#1^2 - 4*c*d^3*g^4*#1^2 + 6*c*e^2*f^
2*g^2*#1^4 + 4*c*d*e*f*g^3*#1^4 + 6*c*d^2*g^4*#1^4 + 16*a*e^2*g^4*#1^4 - 4*c*e*f*g^3*#1^6 - 4*c*d*g^4*#1^6 + c
*g^4*#1^8 & , (Log[-(Sqrt[e/g]*Sqrt[f + g*x]) + Sqrt[d - (e*f)/g + (e*(f + g*x))/g] - #1]*#1^2)/(-(c*e^3*f^3)
+ c*d*e^2*f^2*g + c*d^2*e*f*g^2 - c*d^3*g^3 + 3*c*e^2*f^2*g*#1^2 + 2*c*d*e*f*g^2*#1^2 + 3*c*d^2*g^3*#1^2 + 8*a
*e^2*g^3*#1^2 - 3*c*e*f*g^2*#1^4 - 3*c*d*g^3*#1^4 + c*g^3*#1^6) & ] - e*Sqrt[e/g]*g^3*RootSum[c*e^4*f^4 - 4*c*
d*e^3*f^3*g + 6*c*d^2*e^2*f^2*g^2 - 4*c*d^3*e*f*g^3 + c*d^4*g^4 - 4*c*e^3*f^3*g*#1^2 + 4*c*d*e^2*f^2*g^2*#1^2
+ 4*c*d^2*e*f*g^3*#1^2 - 4*c*d^3*g^4*#1^2 + 6*c*e^2*f^2*g^2*#1^4 + 4*c*d*e*f*g^3*#1^4 + 6*c*d^2*g^4*#1^4 + 16*
a*e^2*g^4*#1^4 - 4*c*e*f*g^3*#1^6 - 4*c*d*g^4*#1^6 + c*g^4*#1^8 & , (Log[-(Sqrt[e/g]*Sqrt[f + g*x]) + Sqrt[d -
(e*f)/g + (e*(f + g*x))/g] - #1]*#1^4)/(-(c*e^3*f^3) + c*d*e^2*f^2*g + c*d^2*e*f*g^2 - c*d^3*g^3 + 3*c*e^2*f^
2*g*#1^2 + 2*c*d*e*f*g^2*#1^2 + 3*c*d^2*g^3*#1^2 + 8*a*e^2*g^3*#1^2 - 3*c*e*f*g^2*#1^4 - 3*c*d*g^3*#1^4 + c*g^
3*#1^6) & ]
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fricas [B] time = 9.63, size = 1913, normalized size = 7.97
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)^(1/2)/(c*x^2+a)/(g*x+f)^(1/2),x, algorithm="fricas")
[Out]
-1/4*sqrt(-(c*d*f + a*e*g + (a*c^2*f^2 + a^2*c*g^2)*sqrt(-(e^2*f^2 - 2*d*e*f*g + d^2*g^2)/(a*c^3*f^4 + 2*a^2*c
^2*f^2*g^2 + a^3*c*g^4)))/(a*c^2*f^2 + a^2*c*g^2))*log(-(e^2*f^2 - d^2*g^2 + 2*(c*e*f^2 - c*d*f*g + (a*c^2*f^2
*g + a^2*c*g^3)*sqrt(-(e^2*f^2 - 2*d*e*f*g + d^2*g^2)/(a*c^3*f^4 + 2*a^2*c^2*f^2*g^2 + a^3*c*g^4)))*sqrt(e*x +
d)*sqrt(g*x + f)*sqrt(-(c*d*f + a*e*g + (a*c^2*f^2 + a^2*c*g^2)*sqrt(-(e^2*f^2 - 2*d*e*f*g + d^2*g^2)/(a*c^3*
f^4 + 2*a^2*c^2*f^2*g^2 + a^3*c*g^4)))/(a*c^2*f^2 + a^2*c*g^2)) + 2*(e^2*f*g - d*e*g^2)*x - (2*c^2*d*f^3 + 2*a
*c*d*f*g^2 + (c^2*e*f^3 + c^2*d*f^2*g + a*c*e*f*g^2 + a*c*d*g^3)*x)*sqrt(-(e^2*f^2 - 2*d*e*f*g + d^2*g^2)/(a*c
^3*f^4 + 2*a^2*c^2*f^2*g^2 + a^3*c*g^4)))/x) + 1/4*sqrt(-(c*d*f + a*e*g + (a*c^2*f^2 + a^2*c*g^2)*sqrt(-(e^2*f
^2 - 2*d*e*f*g + d^2*g^2)/(a*c^3*f^4 + 2*a^2*c^2*f^2*g^2 + a^3*c*g^4)))/(a*c^2*f^2 + a^2*c*g^2))*log(-(e^2*f^2
- d^2*g^2 - 2*(c*e*f^2 - c*d*f*g + (a*c^2*f^2*g + a^2*c*g^3)*sqrt(-(e^2*f^2 - 2*d*e*f*g + d^2*g^2)/(a*c^3*f^4
+ 2*a^2*c^2*f^2*g^2 + a^3*c*g^4)))*sqrt(e*x + d)*sqrt(g*x + f)*sqrt(-(c*d*f + a*e*g + (a*c^2*f^2 + a^2*c*g^2)
*sqrt(-(e^2*f^2 - 2*d*e*f*g + d^2*g^2)/(a*c^3*f^4 + 2*a^2*c^2*f^2*g^2 + a^3*c*g^4)))/(a*c^2*f^2 + a^2*c*g^2))
+ 2*(e^2*f*g - d*e*g^2)*x - (2*c^2*d*f^3 + 2*a*c*d*f*g^2 + (c^2*e*f^3 + c^2*d*f^2*g + a*c*e*f*g^2 + a*c*d*g^3)
*x)*sqrt(-(e^2*f^2 - 2*d*e*f*g + d^2*g^2)/(a*c^3*f^4 + 2*a^2*c^2*f^2*g^2 + a^3*c*g^4)))/x) - 1/4*sqrt(-(c*d*f
+ a*e*g - (a*c^2*f^2 + a^2*c*g^2)*sqrt(-(e^2*f^2 - 2*d*e*f*g + d^2*g^2)/(a*c^3*f^4 + 2*a^2*c^2*f^2*g^2 + a^3*c
*g^4)))/(a*c^2*f^2 + a^2*c*g^2))*log(-(e^2*f^2 - d^2*g^2 + 2*(c*e*f^2 - c*d*f*g - (a*c^2*f^2*g + a^2*c*g^3)*sq
rt(-(e^2*f^2 - 2*d*e*f*g + d^2*g^2)/(a*c^3*f^4 + 2*a^2*c^2*f^2*g^2 + a^3*c*g^4)))*sqrt(e*x + d)*sqrt(g*x + f)*
sqrt(-(c*d*f + a*e*g - (a*c^2*f^2 + a^2*c*g^2)*sqrt(-(e^2*f^2 - 2*d*e*f*g + d^2*g^2)/(a*c^3*f^4 + 2*a^2*c^2*f^
2*g^2 + a^3*c*g^4)))/(a*c^2*f^2 + a^2*c*g^2)) + 2*(e^2*f*g - d*e*g^2)*x + (2*c^2*d*f^3 + 2*a*c*d*f*g^2 + (c^2*
e*f^3 + c^2*d*f^2*g + a*c*e*f*g^2 + a*c*d*g^3)*x)*sqrt(-(e^2*f^2 - 2*d*e*f*g + d^2*g^2)/(a*c^3*f^4 + 2*a^2*c^2
*f^2*g^2 + a^3*c*g^4)))/x) + 1/4*sqrt(-(c*d*f + a*e*g - (a*c^2*f^2 + a^2*c*g^2)*sqrt(-(e^2*f^2 - 2*d*e*f*g + d
^2*g^2)/(a*c^3*f^4 + 2*a^2*c^2*f^2*g^2 + a^3*c*g^4)))/(a*c^2*f^2 + a^2*c*g^2))*log(-(e^2*f^2 - d^2*g^2 - 2*(c*
e*f^2 - c*d*f*g - (a*c^2*f^2*g + a^2*c*g^3)*sqrt(-(e^2*f^2 - 2*d*e*f*g + d^2*g^2)/(a*c^3*f^4 + 2*a^2*c^2*f^2*g
^2 + a^3*c*g^4)))*sqrt(e*x + d)*sqrt(g*x + f)*sqrt(-(c*d*f + a*e*g - (a*c^2*f^2 + a^2*c*g^2)*sqrt(-(e^2*f^2 -
2*d*e*f*g + d^2*g^2)/(a*c^3*f^4 + 2*a^2*c^2*f^2*g^2 + a^3*c*g^4)))/(a*c^2*f^2 + a^2*c*g^2)) + 2*(e^2*f*g - d*e
*g^2)*x + (2*c^2*d*f^3 + 2*a*c*d*f*g^2 + (c^2*e*f^3 + c^2*d*f^2*g + a*c*e*f*g^2 + a*c*d*g^3)*x)*sqrt(-(e^2*f^2
- 2*d*e*f*g + d^2*g^2)/(a*c^3*f^4 + 2*a^2*c^2*f^2*g^2 + a^3*c*g^4)))/x)
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giac [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((e*x+d)^(1/2)/(c*x^2+a)/(g*x+f)^(1/2),x, algorithm="giac")
[Out]
Timed out
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maple [B] time = 0.03, size = 1387, normalized size = 5.78
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((e*x+d)^(1/2)/(c*x^2+a)/(g*x+f)^(1/2),x)
[Out]
1/2*(e*x+d)^(1/2)*(g*x+f)^(1/2)*(ln((c*d*g*x+c*e*f*x+2*c*d*f-2*(-a*c)^(1/2)*e*g*x-(-a*c)^(1/2)*d*g-(-a*c)^(1/2
)*e*f+2*((e*x+d)*(g*x+f))^(1/2)*(-(a*e*g-c*d*f+(-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f)/c)^(1/2)*c)/(c*x+(-a*c)^(1/2
)))*a*c*d*g^2*((-a*e*g+c*d*f+(-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f)/c)^(1/2)-ln((c*d*g*x+c*e*f*x+2*c*d*f-2*(-a*c)^
(1/2)*e*g*x-(-a*c)^(1/2)*d*g-(-a*c)^(1/2)*e*f+2*((e*x+d)*(g*x+f))^(1/2)*(-(a*e*g-c*d*f+(-a*c)^(1/2)*d*g+(-a*c)
^(1/2)*e*f)/c)^(1/2)*c)/(c*x+(-a*c)^(1/2)))*a*e*g^2*((-a*e*g+c*d*f+(-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f)/c)^(1/2)
*(-a*c)^(1/2)+ln((c*d*g*x+c*e*f*x+2*c*d*f-2*(-a*c)^(1/2)*e*g*x-(-a*c)^(1/2)*d*g-(-a*c)^(1/2)*e*f+2*((e*x+d)*(g
*x+f))^(1/2)*(-(a*e*g-c*d*f+(-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f)/c)^(1/2)*c)/(c*x+(-a*c)^(1/2)))*c^2*d*f^2*((-a*
e*g+c*d*f+(-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f)/c)^(1/2)-ln((c*d*g*x+c*e*f*x+2*c*d*f-2*(-a*c)^(1/2)*e*g*x-(-a*c)^
(1/2)*d*g-(-a*c)^(1/2)*e*f+2*((e*x+d)*(g*x+f))^(1/2)*(-(a*e*g-c*d*f+(-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f)/c)^(1/2
)*c)/(c*x+(-a*c)^(1/2)))*c*e*f^2*((-a*e*g+c*d*f+(-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f)/c)^(1/2)*(-a*c)^(1/2)-ln((c
*d*g*x+c*e*f*x+2*c*d*f+2*(-a*c)^(1/2)*e*g*x+(-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f+2*((e*x+d)*(g*x+f))^(1/2)*((-a*e
*g+c*d*f+(-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f)/c)^(1/2)*c)/(c*x-(-a*c)^(1/2)))*a*c*d*g^2*(-(a*e*g-c*d*f+(-a*c)^(1
/2)*d*g+(-a*c)^(1/2)*e*f)/c)^(1/2)-ln((c*d*g*x+c*e*f*x+2*c*d*f+2*(-a*c)^(1/2)*e*g*x+(-a*c)^(1/2)*d*g+(-a*c)^(1
/2)*e*f+2*((e*x+d)*(g*x+f))^(1/2)*((-a*e*g+c*d*f+(-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f)/c)^(1/2)*c)/(c*x-(-a*c)^(1
/2)))*a*e*g^2*(-(a*e*g-c*d*f+(-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f)/c)^(1/2)*(-a*c)^(1/2)-ln((c*d*g*x+c*e*f*x+2*c*
d*f+2*(-a*c)^(1/2)*e*g*x+(-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f+2*((e*x+d)*(g*x+f))^(1/2)*((-a*e*g+c*d*f+(-a*c)^(1/
2)*d*g+(-a*c)^(1/2)*e*f)/c)^(1/2)*c)/(c*x-(-a*c)^(1/2)))*c^2*d*f^2*(-(a*e*g-c*d*f+(-a*c)^(1/2)*d*g+(-a*c)^(1/2
)*e*f)/c)^(1/2)-ln((c*d*g*x+c*e*f*x+2*c*d*f+2*(-a*c)^(1/2)*e*g*x+(-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f+2*((e*x+d)*
(g*x+f))^(1/2)*((-a*e*g+c*d*f+(-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f)/c)^(1/2)*c)/(c*x-(-a*c)^(1/2)))*c*e*f^2*(-(a*
e*g-c*d*f+(-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f)/c)^(1/2)*(-a*c)^(1/2))/((e*x+d)*(g*x+f))^(1/2)/(c*f-(-a*c)^(1/2)*
g)/(-a*c)^(1/2)/(-(a*e*g-c*d*f+(-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f)/c)^(1/2)/(c*f+(-a*c)^(1/2)*g)/((-a*e*g+c*d*f
+(-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f)/c)^(1/2)
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {e x + d}}{{\left (c x^{2} + a\right )} \sqrt {g x + f}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)^(1/2)/(c*x^2+a)/(g*x+f)^(1/2),x, algorithm="maxima")
[Out]
integrate(sqrt(e*x + d)/((c*x^2 + a)*sqrt(g*x + f)), x)
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mupad [F(-1)] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \text {Hanged} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((d + e*x)^(1/2)/((f + g*x)^(1/2)*(a + c*x^2)),x)
[Out]
\text{Hanged}
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {d + e x}}{\left (a + c x^{2}\right ) \sqrt {f + g x}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)**(1/2)/(c*x**2+a)/(g*x+f)**(1/2),x)
[Out]
Integral(sqrt(d + e*x)/((a + c*x**2)*sqrt(f + g*x)), x)
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