3.6.47 \(\int \frac {\sqrt {d+e x}}{(a-c x^2)^2} \, dx\)
Optimal. Leaf size=194 \[ -\frac {\left (\frac {2 \sqrt {c} d}{\sqrt {a}}-e\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{4 a c^{3/4} \sqrt {\sqrt {c} d-\sqrt {a} e}}+\frac {\left (\frac {2 \sqrt {c} d}{\sqrt {a}}+e\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{4 a c^{3/4} \sqrt {\sqrt {a} e+\sqrt {c} d}}+\frac {x \sqrt {d+e x}}{2 a \left (a-c x^2\right )} \]
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Rubi [A] time = 0.19, antiderivative size = 194, normalized size of antiderivative = 1.00,
number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used =
{737, 827, 1166, 208} \begin {gather*} -\frac {\left (\frac {2 \sqrt {c} d}{\sqrt {a}}-e\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{4 a c^{3/4} \sqrt {\sqrt {c} d-\sqrt {a} e}}+\frac {\left (\frac {2 \sqrt {c} d}{\sqrt {a}}+e\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{4 a c^{3/4} \sqrt {\sqrt {a} e+\sqrt {c} d}}+\frac {x \sqrt {d+e x}}{2 a \left (a-c x^2\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[Sqrt[d + e*x]/(a - c*x^2)^2,x]
[Out]
(x*Sqrt[d + e*x])/(2*a*(a - c*x^2)) - (((2*Sqrt[c]*d)/Sqrt[a] - e)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c
]*d - Sqrt[a]*e]])/(4*a*c^(3/4)*Sqrt[Sqrt[c]*d - Sqrt[a]*e]) + (((2*Sqrt[c]*d)/Sqrt[a] + e)*ArcTanh[(c^(1/4)*S
qrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(4*a*c^(3/4)*Sqrt[Sqrt[c]*d + Sqrt[a]*e])
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 737
Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[(x*(d + e*x)^m*(a + c*x^2)^(p + 1)
)/(2*a*(p + 1)), x] + Dist[1/(2*a*(p + 1)), Int[(d + e*x)^(m - 1)*(d*(2*p + 3) + e*(m + 2*p + 3)*x)*(a + c*x^2
)^(p + 1), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 0] && (LtQ[m, 1]
|| (ILtQ[m + 2*p + 3, 0] && NeQ[m, 2])) && IntQuadraticQ[a, 0, c, d, e, m, p, x]
Rule 827
Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2, Subst[Int[(e*f
- d*g + g*x^2)/(c*d^2 + a*e^2 - 2*c*d*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e, f, g}, x]
&& NeQ[c*d^2 + a*e^2, 0]
Rule 1166
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]
Rubi steps
\begin {align*} \int \frac {\sqrt {d+e x}}{\left (a-c x^2\right )^2} \, dx &=\frac {x \sqrt {d+e x}}{2 a \left (a-c x^2\right )}-\frac {\int \frac {-d-\frac {e x}{2}}{\sqrt {d+e x} \left (a-c x^2\right )} \, dx}{2 a}\\ &=\frac {x \sqrt {d+e x}}{2 a \left (a-c x^2\right )}-\frac {\operatorname {Subst}\left (\int \frac {-\frac {d e}{2}-\frac {e x^2}{2}}{-c d^2+a e^2+2 c d x^2-c x^4} \, dx,x,\sqrt {d+e x}\right )}{a}\\ &=\frac {x \sqrt {d+e x}}{2 a \left (a-c x^2\right )}-\frac {\left (\frac {2 \sqrt {c} d}{\sqrt {a}}-e\right ) \operatorname {Subst}\left (\int \frac {1}{c d-\sqrt {a} \sqrt {c} e-c x^2} \, dx,x,\sqrt {d+e x}\right )}{4 a}+\frac {\left (2 \sqrt {c} d+\sqrt {a} e\right ) \operatorname {Subst}\left (\int \frac {1}{c d+\sqrt {a} \sqrt {c} e-c x^2} \, dx,x,\sqrt {d+e x}\right )}{4 a^{3/2}}\\ &=\frac {x \sqrt {d+e x}}{2 a \left (a-c x^2\right )}-\frac {\left (\frac {2 \sqrt {c} d}{\sqrt {a}}-e\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{4 a c^{3/4} \sqrt {\sqrt {c} d-\sqrt {a} e}}+\frac {\left (2 \sqrt {c} d+\sqrt {a} e\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {a} e}}\right )}{4 a^{3/2} c^{3/4} \sqrt {\sqrt {c} d+\sqrt {a} e}}\\ \end {align*}
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Mathematica [A] time = 0.29, size = 267, normalized size = 1.38 \begin {gather*} \frac {\left (\sqrt {c} d-\sqrt {a} e\right ) \left (\left (c x^2-a\right ) \sqrt {\sqrt {a} e+\sqrt {c} d} \left (\sqrt {a} e+2 \sqrt {c} d\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )-2 \sqrt {a} c^{3/4} x \sqrt {d+e x} \left (\sqrt {a} e+\sqrt {c} d\right )\right )-\left (c x^2-a\right ) \sqrt {\sqrt {c} d-\sqrt {a} e} \left (\sqrt {a} \sqrt {c} d e-a e^2+2 c d^2\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{4 a^{3/2} c^{3/4} \left (a-c x^2\right ) \left (a e^2-c d^2\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[Sqrt[d + e*x]/(a - c*x^2)^2,x]
[Out]
(-(Sqrt[Sqrt[c]*d - Sqrt[a]*e]*(2*c*d^2 + Sqrt[a]*Sqrt[c]*d*e - a*e^2)*(-a + c*x^2)*ArcTanh[(c^(1/4)*Sqrt[d +
e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]]) + (Sqrt[c]*d - Sqrt[a]*e)*(-2*Sqrt[a]*c^(3/4)*(Sqrt[c]*d + Sqrt[a]*e)*x*Sq
rt[d + e*x] + Sqrt[Sqrt[c]*d + Sqrt[a]*e]*(2*Sqrt[c]*d + Sqrt[a]*e)*(-a + c*x^2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x
])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]]))/(4*a^(3/2)*c^(3/4)*(-(c*d^2) + a*e^2)*(a - c*x^2))
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IntegrateAlgebraic [A] time = 0.55, size = 267, normalized size = 1.38 \begin {gather*} \frac {\left (\sqrt {a} e+2 \sqrt {c} d\right ) \tan ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {-\sqrt {a} \sqrt {c} e-c d}}{\sqrt {a} e+\sqrt {c} d}\right )}{4 a^{3/2} \sqrt {c} \sqrt {-\sqrt {c} \left (\sqrt {a} e+\sqrt {c} d\right )}}+\frac {\left (\sqrt {a} e-2 \sqrt {c} d\right ) \tan ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {\sqrt {a} \sqrt {c} e-c d}}{\sqrt {c} d-\sqrt {a} e}\right )}{4 a^{3/2} \sqrt {c} \sqrt {-\sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right )}}+\frac {e^2 x \sqrt {d+e x}}{2 a \left (a e^2-c d^2+2 c d (d+e x)-c (d+e x)^2\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[Sqrt[d + e*x]/(a - c*x^2)^2,x]
[Out]
(e^2*x*Sqrt[d + e*x])/(2*a*(-(c*d^2) + a*e^2 + 2*c*d*(d + e*x) - c*(d + e*x)^2)) + ((2*Sqrt[c]*d + Sqrt[a]*e)*
ArcTan[(Sqrt[-(c*d) - Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d + Sqrt[a]*e)])/(4*a^(3/2)*Sqrt[c]*Sqrt[-(Sq
rt[c]*(Sqrt[c]*d + Sqrt[a]*e))]) + ((-2*Sqrt[c]*d + Sqrt[a]*e)*ArcTan[(Sqrt[-(c*d) + Sqrt[a]*Sqrt[c]*e]*Sqrt[d
+ e*x])/(Sqrt[c]*d - Sqrt[a]*e)])/(4*a^(3/2)*Sqrt[c]*Sqrt[-(Sqrt[c]*(Sqrt[c]*d - Sqrt[a]*e))])
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fricas [B] time = 0.44, size = 1385, normalized size = 7.14
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)^2,x, algorithm="fricas")
[Out]
-1/8*((a*c*x^2 - a^2)*sqrt((4*c*d^3 - 3*a*d*e^2 + (a^3*c^2*d^2 - a^4*c*e^2)*sqrt(e^6/(a^3*c^5*d^4 - 2*a^4*c^4*
d^2*e^2 + a^5*c^3*e^4)))/(a^3*c^2*d^2 - a^4*c*e^2))*log(-(4*c*d^2*e^3 - a*e^5)*sqrt(e*x + d) + (a^2*c*d*e^4 -
(2*a^3*c^4*d^4 - 3*a^4*c^3*d^2*e^2 + a^5*c^2*e^4)*sqrt(e^6/(a^3*c^5*d^4 - 2*a^4*c^4*d^2*e^2 + a^5*c^3*e^4)))*s
qrt((4*c*d^3 - 3*a*d*e^2 + (a^3*c^2*d^2 - a^4*c*e^2)*sqrt(e^6/(a^3*c^5*d^4 - 2*a^4*c^4*d^2*e^2 + a^5*c^3*e^4))
)/(a^3*c^2*d^2 - a^4*c*e^2))) - (a*c*x^2 - a^2)*sqrt((4*c*d^3 - 3*a*d*e^2 + (a^3*c^2*d^2 - a^4*c*e^2)*sqrt(e^6
/(a^3*c^5*d^4 - 2*a^4*c^4*d^2*e^2 + a^5*c^3*e^4)))/(a^3*c^2*d^2 - a^4*c*e^2))*log(-(4*c*d^2*e^3 - a*e^5)*sqrt(
e*x + d) - (a^2*c*d*e^4 - (2*a^3*c^4*d^4 - 3*a^4*c^3*d^2*e^2 + a^5*c^2*e^4)*sqrt(e^6/(a^3*c^5*d^4 - 2*a^4*c^4*
d^2*e^2 + a^5*c^3*e^4)))*sqrt((4*c*d^3 - 3*a*d*e^2 + (a^3*c^2*d^2 - a^4*c*e^2)*sqrt(e^6/(a^3*c^5*d^4 - 2*a^4*c
^4*d^2*e^2 + a^5*c^3*e^4)))/(a^3*c^2*d^2 - a^4*c*e^2))) + (a*c*x^2 - a^2)*sqrt((4*c*d^3 - 3*a*d*e^2 - (a^3*c^2
*d^2 - a^4*c*e^2)*sqrt(e^6/(a^3*c^5*d^4 - 2*a^4*c^4*d^2*e^2 + a^5*c^3*e^4)))/(a^3*c^2*d^2 - a^4*c*e^2))*log(-(
4*c*d^2*e^3 - a*e^5)*sqrt(e*x + d) + (a^2*c*d*e^4 + (2*a^3*c^4*d^4 - 3*a^4*c^3*d^2*e^2 + a^5*c^2*e^4)*sqrt(e^6
/(a^3*c^5*d^4 - 2*a^4*c^4*d^2*e^2 + a^5*c^3*e^4)))*sqrt((4*c*d^3 - 3*a*d*e^2 - (a^3*c^2*d^2 - a^4*c*e^2)*sqrt(
e^6/(a^3*c^5*d^4 - 2*a^4*c^4*d^2*e^2 + a^5*c^3*e^4)))/(a^3*c^2*d^2 - a^4*c*e^2))) - (a*c*x^2 - a^2)*sqrt((4*c*
d^3 - 3*a*d*e^2 - (a^3*c^2*d^2 - a^4*c*e^2)*sqrt(e^6/(a^3*c^5*d^4 - 2*a^4*c^4*d^2*e^2 + a^5*c^3*e^4)))/(a^3*c^
2*d^2 - a^4*c*e^2))*log(-(4*c*d^2*e^3 - a*e^5)*sqrt(e*x + d) - (a^2*c*d*e^4 + (2*a^3*c^4*d^4 - 3*a^4*c^3*d^2*e
^2 + a^5*c^2*e^4)*sqrt(e^6/(a^3*c^5*d^4 - 2*a^4*c^4*d^2*e^2 + a^5*c^3*e^4)))*sqrt((4*c*d^3 - 3*a*d*e^2 - (a^3*
c^2*d^2 - a^4*c*e^2)*sqrt(e^6/(a^3*c^5*d^4 - 2*a^4*c^4*d^2*e^2 + a^5*c^3*e^4)))/(a^3*c^2*d^2 - a^4*c*e^2))) +
4*sqrt(e*x + d)*x)/(a*c*x^2 - a^2)
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giac [B] time = 0.53, size = 342, normalized size = 1.76 \begin {gather*} \frac {{\left (2 \, a c d^{2} {\left | c \right |} - \sqrt {a c} d {\left | a \right |} {\left | c \right |} e - a^{2} {\left | c \right |} e^{2}\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {a c d + \sqrt {a^{2} c^{2} d^{2} - {\left (a c d^{2} - a^{2} e^{2}\right )} a c}}{a c}}}\right )}{4 \, {\left (a^{2} c e - \sqrt {a c} a c d\right )} \sqrt {-c^{2} d - \sqrt {a c} c e} {\left | a \right |}} + \frac {{\left (a c d {\left | a \right |} {\left | c \right |} e + 2 \, \sqrt {a c} a c d^{2} {\left | c \right |} - \sqrt {a c} a^{2} {\left | c \right |} e^{2}\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {a c d - \sqrt {a^{2} c^{2} d^{2} - {\left (a c d^{2} - a^{2} e^{2}\right )} a c}}{a c}}}\right )}{4 \, {\left (a^{2} c^{2} d + \sqrt {a c} a^{2} c e\right )} \sqrt {-c^{2} d + \sqrt {a c} c e} {\left | a \right |}} - \frac {{\left (x e + d\right )}^{\frac {3}{2}} e - \sqrt {x e + d} d e}{2 \, {\left ({\left (x e + d\right )}^{2} c - 2 \, {\left (x e + d\right )} c d + c d^{2} - a e^{2}\right )} a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)^(1/2)/(-c*x^2+a)^2,x, algorithm="giac")
[Out]
1/4*(2*a*c*d^2*abs(c) - sqrt(a*c)*d*abs(a)*abs(c)*e - a^2*abs(c)*e^2)*arctan(sqrt(x*e + d)/sqrt(-(a*c*d + sqrt
(a^2*c^2*d^2 - (a*c*d^2 - a^2*e^2)*a*c))/(a*c)))/((a^2*c*e - sqrt(a*c)*a*c*d)*sqrt(-c^2*d - sqrt(a*c)*c*e)*abs
(a)) + 1/4*(a*c*d*abs(a)*abs(c)*e + 2*sqrt(a*c)*a*c*d^2*abs(c) - sqrt(a*c)*a^2*abs(c)*e^2)*arctan(sqrt(x*e + d
)/sqrt(-(a*c*d - sqrt(a^2*c^2*d^2 - (a*c*d^2 - a^2*e^2)*a*c))/(a*c)))/((a^2*c^2*d + sqrt(a*c)*a^2*c*e)*sqrt(-c
^2*d + sqrt(a*c)*c*e)*abs(a)) - 1/2*((x*e + d)^(3/2)*e - sqrt(x*e + d)*d*e)/(((x*e + d)^2*c - 2*(x*e + d)*c*d
+ c*d^2 - a*e^2)*a)
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maple [B] time = 0.30, size = 287, normalized size = 1.48 \begin {gather*} \frac {c d e \arctanh \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, a}+\frac {c d e \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, a}+\frac {e \arctanh \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{4 \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, a}-\frac {e \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{4 \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, a}-\frac {\sqrt {e x +d}\, e}{4 \left (e x -\frac {\sqrt {a c \,e^{2}}}{c}\right ) a c}-\frac {\sqrt {e x +d}\, e}{4 \left (e x +\frac {\sqrt {a c \,e^{2}}}{c}\right ) a c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((e*x+d)^(1/2)/(-c*x^2+a)^2,x)
[Out]
-1/4*e/c/a*(e*x+d)^(1/2)/(e*x-(a*c*e^2)^(1/2)/c)+1/2*e*c/a/(a*c*e^2)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arc
tanh((e*x+d)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)*d+1/4*e/a/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d
)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)-1/4*e/c/a*(e*x+d)^(1/2)/(e*x+(a*c*e^2)^(1/2)/c)+1/2*e*c/a/(a*c*e^2)
^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)*d-1/4*e/a/((-
c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)
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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 )}^{2}}\,{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)^2,x, algorithm="maxima")
[Out]
integrate(sqrt(e*x + d)/(c*x^2 - a)^2, x)
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mupad [B] time = 2.19, size = 2332, normalized size = 12.02
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((d + e*x)^(1/2)/(a - c*x^2)^2,x)
[Out]
- atan((((8*c^3*d*e^3 - 64*a*c^4*d*e^2*(d + e*x)^(1/2)*(-(e^3*(a^9*c^3)^(1/2) - 4*a^3*c^3*d^3 + 3*a^4*c^2*d*e^
2)/(64*(a^6*c^4*d^2 - a^7*c^3*e^2)))^(1/2))*(-(e^3*(a^9*c^3)^(1/2) - 4*a^3*c^3*d^3 + 3*a^4*c^2*d*e^2)/(64*(a^6
*c^4*d^2 - a^7*c^3*e^2)))^(1/2) + ((a*c^2*e^4 + 4*c^3*d^2*e^2)*(d + e*x)^(1/2))/a^2)*(-(e^3*(a^9*c^3)^(1/2) -
4*a^3*c^3*d^3 + 3*a^4*c^2*d*e^2)/(64*(a^6*c^4*d^2 - a^7*c^3*e^2)))^(1/2)*1i - ((8*c^3*d*e^3 + 64*a*c^4*d*e^2*(
d + e*x)^(1/2)*(-(e^3*(a^9*c^3)^(1/2) - 4*a^3*c^3*d^3 + 3*a^4*c^2*d*e^2)/(64*(a^6*c^4*d^2 - a^7*c^3*e^2)))^(1/
2))*(-(e^3*(a^9*c^3)^(1/2) - 4*a^3*c^3*d^3 + 3*a^4*c^2*d*e^2)/(64*(a^6*c^4*d^2 - a^7*c^3*e^2)))^(1/2) - ((a*c^
2*e^4 + 4*c^3*d^2*e^2)*(d + e*x)^(1/2))/a^2)*(-(e^3*(a^9*c^3)^(1/2) - 4*a^3*c^3*d^3 + 3*a^4*c^2*d*e^2)/(64*(a^
6*c^4*d^2 - a^7*c^3*e^2)))^(1/2)*1i)/(((8*c^3*d*e^3 - 64*a*c^4*d*e^2*(d + e*x)^(1/2)*(-(e^3*(a^9*c^3)^(1/2) -
4*a^3*c^3*d^3 + 3*a^4*c^2*d*e^2)/(64*(a^6*c^4*d^2 - a^7*c^3*e^2)))^(1/2))*(-(e^3*(a^9*c^3)^(1/2) - 4*a^3*c^3*d
^3 + 3*a^4*c^2*d*e^2)/(64*(a^6*c^4*d^2 - a^7*c^3*e^2)))^(1/2) + ((a*c^2*e^4 + 4*c^3*d^2*e^2)*(d + e*x)^(1/2))/
a^2)*(-(e^3*(a^9*c^3)^(1/2) - 4*a^3*c^3*d^3 + 3*a^4*c^2*d*e^2)/(64*(a^6*c^4*d^2 - a^7*c^3*e^2)))^(1/2) - (4*c^
2*d^2*e^3 - a*c*e^5)/(4*a^3) + ((8*c^3*d*e^3 + 64*a*c^4*d*e^2*(d + e*x)^(1/2)*(-(e^3*(a^9*c^3)^(1/2) - 4*a^3*c
^3*d^3 + 3*a^4*c^2*d*e^2)/(64*(a^6*c^4*d^2 - a^7*c^3*e^2)))^(1/2))*(-(e^3*(a^9*c^3)^(1/2) - 4*a^3*c^3*d^3 + 3*
a^4*c^2*d*e^2)/(64*(a^6*c^4*d^2 - a^7*c^3*e^2)))^(1/2) - ((a*c^2*e^4 + 4*c^3*d^2*e^2)*(d + e*x)^(1/2))/a^2)*(-
(e^3*(a^9*c^3)^(1/2) - 4*a^3*c^3*d^3 + 3*a^4*c^2*d*e^2)/(64*(a^6*c^4*d^2 - a^7*c^3*e^2)))^(1/2)))*(-(e^3*(a^9*
c^3)^(1/2) - 4*a^3*c^3*d^3 + 3*a^4*c^2*d*e^2)/(64*(a^6*c^4*d^2 - a^7*c^3*e^2)))^(1/2)*2i - atan((((8*c^3*d*e^3
- 64*a*c^4*d*e^2*(d + e*x)^(1/2)*((e^3*(a^9*c^3)^(1/2) + 4*a^3*c^3*d^3 - 3*a^4*c^2*d*e^2)/(64*(a^6*c^4*d^2 -
a^7*c^3*e^2)))^(1/2))*((e^3*(a^9*c^3)^(1/2) + 4*a^3*c^3*d^3 - 3*a^4*c^2*d*e^2)/(64*(a^6*c^4*d^2 - a^7*c^3*e^2)
))^(1/2) + ((a*c^2*e^4 + 4*c^3*d^2*e^2)*(d + e*x)^(1/2))/a^2)*((e^3*(a^9*c^3)^(1/2) + 4*a^3*c^3*d^3 - 3*a^4*c^
2*d*e^2)/(64*(a^6*c^4*d^2 - a^7*c^3*e^2)))^(1/2)*1i - ((8*c^3*d*e^3 + 64*a*c^4*d*e^2*(d + e*x)^(1/2)*((e^3*(a^
9*c^3)^(1/2) + 4*a^3*c^3*d^3 - 3*a^4*c^2*d*e^2)/(64*(a^6*c^4*d^2 - a^7*c^3*e^2)))^(1/2))*((e^3*(a^9*c^3)^(1/2)
+ 4*a^3*c^3*d^3 - 3*a^4*c^2*d*e^2)/(64*(a^6*c^4*d^2 - a^7*c^3*e^2)))^(1/2) - ((a*c^2*e^4 + 4*c^3*d^2*e^2)*(d
+ e*x)^(1/2))/a^2)*((e^3*(a^9*c^3)^(1/2) + 4*a^3*c^3*d^3 - 3*a^4*c^2*d*e^2)/(64*(a^6*c^4*d^2 - a^7*c^3*e^2)))^
(1/2)*1i)/(((8*c^3*d*e^3 - 64*a*c^4*d*e^2*(d + e*x)^(1/2)*((e^3*(a^9*c^3)^(1/2) + 4*a^3*c^3*d^3 - 3*a^4*c^2*d*
e^2)/(64*(a^6*c^4*d^2 - a^7*c^3*e^2)))^(1/2))*((e^3*(a^9*c^3)^(1/2) + 4*a^3*c^3*d^3 - 3*a^4*c^2*d*e^2)/(64*(a^
6*c^4*d^2 - a^7*c^3*e^2)))^(1/2) + ((a*c^2*e^4 + 4*c^3*d^2*e^2)*(d + e*x)^(1/2))/a^2)*((e^3*(a^9*c^3)^(1/2) +
4*a^3*c^3*d^3 - 3*a^4*c^2*d*e^2)/(64*(a^6*c^4*d^2 - a^7*c^3*e^2)))^(1/2) - (4*c^2*d^2*e^3 - a*c*e^5)/(4*a^3) +
((8*c^3*d*e^3 + 64*a*c^4*d*e^2*(d + e*x)^(1/2)*((e^3*(a^9*c^3)^(1/2) + 4*a^3*c^3*d^3 - 3*a^4*c^2*d*e^2)/(64*(
a^6*c^4*d^2 - a^7*c^3*e^2)))^(1/2))*((e^3*(a^9*c^3)^(1/2) + 4*a^3*c^3*d^3 - 3*a^4*c^2*d*e^2)/(64*(a^6*c^4*d^2
- a^7*c^3*e^2)))^(1/2) - ((a*c^2*e^4 + 4*c^3*d^2*e^2)*(d + e*x)^(1/2))/a^2)*((e^3*(a^9*c^3)^(1/2) + 4*a^3*c^3*
d^3 - 3*a^4*c^2*d*e^2)/(64*(a^6*c^4*d^2 - a^7*c^3*e^2)))^(1/2)))*((e^3*(a^9*c^3)^(1/2) + 4*a^3*c^3*d^3 - 3*a^4
*c^2*d*e^2)/(64*(a^6*c^4*d^2 - a^7*c^3*e^2)))^(1/2)*2i - ((e*(d + e*x)^(3/2))/(2*a) - (d*e*(d + e*x)^(1/2))/(2
*a))/(c*(d + e*x)^2 - a*e^2 + c*d^2 - 2*c*d*(d + e*x))
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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((e*x+d)**(1/2)/(-c*x**2+a)**2,x)
[Out]
Timed out
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