3.7.28 \(\int \frac {1}{\sqrt {d+e x} (a-c x^2)^2} \, dx\) [628]
Optimal. Leaf size=222 \[ -\frac {(a e-c d x) \sqrt {d+e x}}{2 a \left (c d^2-a e^2\right ) \left (a-c x^2\right )}-\frac {\left (2 \sqrt {c} d-3 \sqrt {a} e\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{4 a^{3/2} \sqrt [4]{c} \left (\sqrt {c} d-\sqrt {a} e\right )^{3/2}}+\frac {\left (2 \sqrt {c} d+3 \sqrt {a} e\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {a} e}}\right )}{4 a^{3/2} \sqrt [4]{c} \left (\sqrt {c} d+\sqrt {a} e\right )^{3/2}} \]
[Out]
-1/4*arctanh(c^(1/4)*(e*x+d)^(1/2)/(-e*a^(1/2)+d*c^(1/2))^(1/2))*(-3*e*a^(1/2)+2*d*c^(1/2))/a^(3/2)/c^(1/4)/(-
e*a^(1/2)+d*c^(1/2))^(3/2)+1/4*arctanh(c^(1/4)*(e*x+d)^(1/2)/(e*a^(1/2)+d*c^(1/2))^(1/2))*(3*e*a^(1/2)+2*d*c^(
1/2))/a^(3/2)/c^(1/4)/(e*a^(1/2)+d*c^(1/2))^(3/2)-1/2*(-c*d*x+a*e)*(e*x+d)^(1/2)/a/(-a*e^2+c*d^2)/(-c*x^2+a)
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Rubi [A]
time = 0.23, antiderivative size = 222, 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 = {755, 841, 1180,
214} \begin {gather*} -\frac {\left (2 \sqrt {c} d-3 \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} \sqrt [4]{c} \left (\sqrt {c} d-\sqrt {a} e\right )^{3/2}}+\frac {\left (3 \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 )}{4 a^{3/2} \sqrt [4]{c} \left (\sqrt {a} e+\sqrt {c} d\right )^{3/2}}-\frac {\sqrt {d+e x} (a e-c d x)}{2 a \left (a-c x^2\right ) \left (c d^2-a e^2\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[1/(Sqrt[d + e*x]*(a - c*x^2)^2),x]
[Out]
-1/2*((a*e - c*d*x)*Sqrt[d + e*x])/(a*(c*d^2 - a*e^2)*(a - c*x^2)) - ((2*Sqrt[c]*d - 3*Sqrt[a]*e)*ArcTanh[(c^(
1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]])/(4*a^(3/2)*c^(1/4)*(Sqrt[c]*d - Sqrt[a]*e)^(3/2)) + ((2*Sqrt
[c]*d + 3*Sqrt[a]*e)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(4*a^(3/2)*c^(1/4)*(Sqrt[c]
*d + Sqrt[a]*e)^(3/2))
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 755
Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*(a*e + c*d*x)*
((a + c*x^2)^(p + 1)/(2*a*(p + 1)*(c*d^2 + a*e^2))), x] + Dist[1/(2*a*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^
m*Simp[c*d^2*(2*p + 3) + a*e^2*(m + 2*p + 3) + c*e*d*(m + 2*p + 4)*x, x]*(a + c*x^2)^(p + 1), x], x] /; FreeQ[
{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && IntQuadraticQ[a, 0, c, d, e, m, p, x]
Rule 841
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 1180
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 {1}{\sqrt {d+e x} \left (a-c x^2\right )^2} \, dx &=-\frac {(a e-c d x) \sqrt {d+e x}}{2 a \left (c d^2-a e^2\right ) \left (a-c x^2\right )}+\frac {\int \frac {\frac {1}{2} \left (2 c d^2-3 a e^2\right )+\frac {1}{2} c d e x}{\sqrt {d+e x} \left (a-c x^2\right )} \, dx}{2 a \left (c d^2-a e^2\right )}\\ &=-\frac {(a e-c d x) \sqrt {d+e x}}{2 a \left (c d^2-a e^2\right ) \left (a-c x^2\right )}+\frac {\text {Subst}\left (\int \frac {-\frac {1}{2} c d^2 e+\frac {1}{2} e \left (2 c d^2-3 a e^2\right )+\frac {1}{2} c d e x^2}{-c d^2+a e^2+2 c d x^2-c x^4} \, dx,x,\sqrt {d+e x}\right )}{a \left (c d^2-a e^2\right )}\\ &=-\frac {(a e-c d x) \sqrt {d+e x}}{2 a \left (c d^2-a e^2\right ) \left (a-c x^2\right )}-\frac {\left (\sqrt {c} \left (2 \sqrt {c} d-3 \sqrt {a} e\right )\right ) \text {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} \left (\sqrt {c} d-\sqrt {a} e\right )}+\frac {\left (\sqrt {c} \left (2 \sqrt {c} d+3 \sqrt {a} e\right )\right ) \text {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} \left (\sqrt {c} d+\sqrt {a} e\right )}\\ &=-\frac {(a e-c d x) \sqrt {d+e x}}{2 a \left (c d^2-a e^2\right ) \left (a-c x^2\right )}-\frac {\left (2 \sqrt {c} d-3 \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} \sqrt [4]{c} \left (\sqrt {c} d-\sqrt {a} e\right )^{3/2}}+\frac {\left (2 \sqrt {c} d+3 \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} \sqrt [4]{c} \left (\sqrt {c} d+\sqrt {a} e\right )^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.68, size = 274, normalized size = 1.23 \begin {gather*} \frac {-\frac {2 \sqrt {a} (-a e+c d x) \sqrt {d+e x}}{\left (-c d^2+a e^2\right ) \left (a-c x^2\right )}+\frac {\left (2 \sqrt {c} d+3 \sqrt {a} e\right ) \tan ^{-1}\left (\frac {\sqrt {-c d-\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+\sqrt {a} e}\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) \sqrt {-c d-\sqrt {a} \sqrt {c} e}}-\frac {\left (2 \sqrt {c} d-3 \sqrt {a} e\right ) \tan ^{-1}\left (\frac {\sqrt {-c d+\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-\sqrt {a} e}\right )}{\left (\sqrt {c} d-\sqrt {a} e\right ) \sqrt {-c d+\sqrt {a} \sqrt {c} e}}}{4 a^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[1/(Sqrt[d + e*x]*(a - c*x^2)^2),x]
[Out]
((-2*Sqrt[a]*(-(a*e) + c*d*x)*Sqrt[d + e*x])/((-(c*d^2) + a*e^2)*(a - c*x^2)) + ((2*Sqrt[c]*d + 3*Sqrt[a]*e)*A
rcTan[(Sqrt[-(c*d) - Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d + Sqrt[a]*e)])/((Sqrt[c]*d + Sqrt[a]*e)*Sqrt
[-(c*d) - Sqrt[a]*Sqrt[c]*e]) - ((2*Sqrt[c]*d - 3*Sqrt[a]*e)*ArcTan[(Sqrt[-(c*d) + Sqrt[a]*Sqrt[c]*e]*Sqrt[d +
e*x])/(Sqrt[c]*d - Sqrt[a]*e)])/((Sqrt[c]*d - Sqrt[a]*e)*Sqrt[-(c*d) + Sqrt[a]*Sqrt[c]*e]))/(4*a^(3/2))
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Maple [A]
time = 0.45, size = 308, normalized size = 1.39
| | |
| method |
result |
size |
| | |
| derivativedivides |
\(2 e^{3} c^{2} \left (\frac {\frac {\sqrt {a c \,e^{2}}\, \sqrt {e x +d}}{2 c \left (c d -\sqrt {a c \,e^{2}}\right ) \left (-e x -\frac {\sqrt {a c \,e^{2}}}{c}\right )}+\frac {\left (-2 c d +3 \sqrt {a c \,e^{2}}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \left (-c d +\sqrt {a c \,e^{2}}\right ) \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}}{4 \sqrt {a c \,e^{2}}\, a \,e^{2} c}+\frac {\frac {\sqrt {a c \,e^{2}}\, \sqrt {e x +d}}{2 c \left (c d +\sqrt {a c \,e^{2}}\right ) \left (-e x +\frac {\sqrt {a c \,e^{2}}}{c}\right )}+\frac {\left (2 c d +3 \sqrt {a c \,e^{2}}\right ) \arctanh \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \left (c d +\sqrt {a c \,e^{2}}\right ) \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}}{4 \sqrt {a c \,e^{2}}\, a \,e^{2} c}\right )\) |
\(308\) |
| default |
\(2 e^{3} c^{2} \left (\frac {\frac {\sqrt {a c \,e^{2}}\, \sqrt {e x +d}}{2 c \left (c d -\sqrt {a c \,e^{2}}\right ) \left (-e x -\frac {\sqrt {a c \,e^{2}}}{c}\right )}+\frac {\left (-2 c d +3 \sqrt {a c \,e^{2}}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \left (-c d +\sqrt {a c \,e^{2}}\right ) \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}}{4 \sqrt {a c \,e^{2}}\, a \,e^{2} c}+\frac {\frac {\sqrt {a c \,e^{2}}\, \sqrt {e x +d}}{2 c \left (c d +\sqrt {a c \,e^{2}}\right ) \left (-e x +\frac {\sqrt {a c \,e^{2}}}{c}\right )}+\frac {\left (2 c d +3 \sqrt {a c \,e^{2}}\right ) \arctanh \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \left (c d +\sqrt {a c \,e^{2}}\right ) \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}}{4 \sqrt {a c \,e^{2}}\, a \,e^{2} c}\right )\) |
\(308\) |
| | |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(-c*x^2+a)^2/(e*x+d)^(1/2),x,method=_RETURNVERBOSE)
[Out]
2*e^3*c^2*(1/4/(a*c*e^2)^(1/2)/a/e^2/c*(1/2/c/(c*d-(a*c*e^2)^(1/2))*(a*c*e^2)^(1/2)*(e*x+d)^(1/2)/(-e*x-(a*c*e
^2)^(1/2)/c)+1/2*(-2*c*d+3*(a*c*e^2)^(1/2))/(-c*d+(a*c*e^2)^(1/2))/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctan(c*(
e*x+d)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)))+1/4/(a*c*e^2)^(1/2)/a/e^2/c*(1/2/c/(c*d+(a*c*e^2)^(1/2))*(a*c*
e^2)^(1/2)*(e*x+d)^(1/2)/(-e*x+(a*c*e^2)^(1/2)/c)+1/2*(2*c*d+3*(a*c*e^2)^(1/2))/(c*d+(a*c*e^2)^(1/2))/((c*d+(a
*c*e^2)^(1/2))*c)^(1/2)*arctanh(c*(e*x+d)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2))))
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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(1/(-c*x^2+a)^2/(e*x+d)^(1/2),x, algorithm="maxima")
[Out]
integrate(1/((c*x^2 - a)^2*sqrt(x*e + d)), x)
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 3068 vs.
\(2 (176) = 352\).
time = 7.38, size = 3068, normalized size = 13.82 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(-c*x^2+a)^2/(e*x+d)^(1/2),x, algorithm="fricas")
[Out]
1/8*((a*c^2*d^2*x^2 - a^2*c*d^2 - (a^2*c*x^2 - a^3)*e^2)*sqrt((4*c^2*d^5 - 15*a*c*d^3*e^2 + 15*a^2*d*e^4 + (a^
3*c^3*d^6 - 3*a^4*c^2*d^4*e^2 + 3*a^5*c*d^2*e^4 - a^6*e^6)*sqrt((25*c^2*d^4*e^6 - 90*a*c*d^2*e^8 + 81*a^2*e^10
)/(a^3*c^7*d^12 - 6*a^4*c^6*d^10*e^2 + 15*a^5*c^5*d^8*e^4 - 20*a^6*c^4*d^6*e^6 + 15*a^7*c^3*d^4*e^8 - 6*a^8*c^
2*d^2*e^10 + a^9*c*e^12)))/(a^3*c^3*d^6 - 3*a^4*c^2*d^4*e^2 + 3*a^5*c*d^2*e^4 - a^6*e^6))*log((20*c^2*d^4*e^3
- 81*a*c*d^2*e^5 + 81*a^2*e^7)*sqrt(x*e + d) + (5*a^2*c^2*d^4*e^4 - 24*a^3*c*d^2*e^6 + 27*a^4*e^8 + 2*(a^3*c^5
*d^9 - 5*a^4*c^4*d^7*e^2 + 9*a^5*c^3*d^5*e^4 - 7*a^6*c^2*d^3*e^6 + 2*a^7*c*d*e^8)*sqrt((25*c^2*d^4*e^6 - 90*a*
c*d^2*e^8 + 81*a^2*e^10)/(a^3*c^7*d^12 - 6*a^4*c^6*d^10*e^2 + 15*a^5*c^5*d^8*e^4 - 20*a^6*c^4*d^6*e^6 + 15*a^7
*c^3*d^4*e^8 - 6*a^8*c^2*d^2*e^10 + a^9*c*e^12)))*sqrt((4*c^2*d^5 - 15*a*c*d^3*e^2 + 15*a^2*d*e^4 + (a^3*c^3*d
^6 - 3*a^4*c^2*d^4*e^2 + 3*a^5*c*d^2*e^4 - a^6*e^6)*sqrt((25*c^2*d^4*e^6 - 90*a*c*d^2*e^8 + 81*a^2*e^10)/(a^3*
c^7*d^12 - 6*a^4*c^6*d^10*e^2 + 15*a^5*c^5*d^8*e^4 - 20*a^6*c^4*d^6*e^6 + 15*a^7*c^3*d^4*e^8 - 6*a^8*c^2*d^2*e
^10 + a^9*c*e^12)))/(a^3*c^3*d^6 - 3*a^4*c^2*d^4*e^2 + 3*a^5*c*d^2*e^4 - a^6*e^6))) - (a*c^2*d^2*x^2 - a^2*c*d
^2 - (a^2*c*x^2 - a^3)*e^2)*sqrt((4*c^2*d^5 - 15*a*c*d^3*e^2 + 15*a^2*d*e^4 + (a^3*c^3*d^6 - 3*a^4*c^2*d^4*e^2
+ 3*a^5*c*d^2*e^4 - a^6*e^6)*sqrt((25*c^2*d^4*e^6 - 90*a*c*d^2*e^8 + 81*a^2*e^10)/(a^3*c^7*d^12 - 6*a^4*c^6*d
^10*e^2 + 15*a^5*c^5*d^8*e^4 - 20*a^6*c^4*d^6*e^6 + 15*a^7*c^3*d^4*e^8 - 6*a^8*c^2*d^2*e^10 + a^9*c*e^12)))/(a
^3*c^3*d^6 - 3*a^4*c^2*d^4*e^2 + 3*a^5*c*d^2*e^4 - a^6*e^6))*log((20*c^2*d^4*e^3 - 81*a*c*d^2*e^5 + 81*a^2*e^7
)*sqrt(x*e + d) - (5*a^2*c^2*d^4*e^4 - 24*a^3*c*d^2*e^6 + 27*a^4*e^8 + 2*(a^3*c^5*d^9 - 5*a^4*c^4*d^7*e^2 + 9*
a^5*c^3*d^5*e^4 - 7*a^6*c^2*d^3*e^6 + 2*a^7*c*d*e^8)*sqrt((25*c^2*d^4*e^6 - 90*a*c*d^2*e^8 + 81*a^2*e^10)/(a^3
*c^7*d^12 - 6*a^4*c^6*d^10*e^2 + 15*a^5*c^5*d^8*e^4 - 20*a^6*c^4*d^6*e^6 + 15*a^7*c^3*d^4*e^8 - 6*a^8*c^2*d^2*
e^10 + a^9*c*e^12)))*sqrt((4*c^2*d^5 - 15*a*c*d^3*e^2 + 15*a^2*d*e^4 + (a^3*c^3*d^6 - 3*a^4*c^2*d^4*e^2 + 3*a^
5*c*d^2*e^4 - a^6*e^6)*sqrt((25*c^2*d^4*e^6 - 90*a*c*d^2*e^8 + 81*a^2*e^10)/(a^3*c^7*d^12 - 6*a^4*c^6*d^10*e^2
+ 15*a^5*c^5*d^8*e^4 - 20*a^6*c^4*d^6*e^6 + 15*a^7*c^3*d^4*e^8 - 6*a^8*c^2*d^2*e^10 + a^9*c*e^12)))/(a^3*c^3*
d^6 - 3*a^4*c^2*d^4*e^2 + 3*a^5*c*d^2*e^4 - a^6*e^6))) + (a*c^2*d^2*x^2 - a^2*c*d^2 - (a^2*c*x^2 - a^3)*e^2)*s
qrt((4*c^2*d^5 - 15*a*c*d^3*e^2 + 15*a^2*d*e^4 - (a^3*c^3*d^6 - 3*a^4*c^2*d^4*e^2 + 3*a^5*c*d^2*e^4 - a^6*e^6)
*sqrt((25*c^2*d^4*e^6 - 90*a*c*d^2*e^8 + 81*a^2*e^10)/(a^3*c^7*d^12 - 6*a^4*c^6*d^10*e^2 + 15*a^5*c^5*d^8*e^4
- 20*a^6*c^4*d^6*e^6 + 15*a^7*c^3*d^4*e^8 - 6*a^8*c^2*d^2*e^10 + a^9*c*e^12)))/(a^3*c^3*d^6 - 3*a^4*c^2*d^4*e^
2 + 3*a^5*c*d^2*e^4 - a^6*e^6))*log((20*c^2*d^4*e^3 - 81*a*c*d^2*e^5 + 81*a^2*e^7)*sqrt(x*e + d) + (5*a^2*c^2*
d^4*e^4 - 24*a^3*c*d^2*e^6 + 27*a^4*e^8 - 2*(a^3*c^5*d^9 - 5*a^4*c^4*d^7*e^2 + 9*a^5*c^3*d^5*e^4 - 7*a^6*c^2*d
^3*e^6 + 2*a^7*c*d*e^8)*sqrt((25*c^2*d^4*e^6 - 90*a*c*d^2*e^8 + 81*a^2*e^10)/(a^3*c^7*d^12 - 6*a^4*c^6*d^10*e^
2 + 15*a^5*c^5*d^8*e^4 - 20*a^6*c^4*d^6*e^6 + 15*a^7*c^3*d^4*e^8 - 6*a^8*c^2*d^2*e^10 + a^9*c*e^12)))*sqrt((4*
c^2*d^5 - 15*a*c*d^3*e^2 + 15*a^2*d*e^4 - (a^3*c^3*d^6 - 3*a^4*c^2*d^4*e^2 + 3*a^5*c*d^2*e^4 - a^6*e^6)*sqrt((
25*c^2*d^4*e^6 - 90*a*c*d^2*e^8 + 81*a^2*e^10)/(a^3*c^7*d^12 - 6*a^4*c^6*d^10*e^2 + 15*a^5*c^5*d^8*e^4 - 20*a^
6*c^4*d^6*e^6 + 15*a^7*c^3*d^4*e^8 - 6*a^8*c^2*d^2*e^10 + a^9*c*e^12)))/(a^3*c^3*d^6 - 3*a^4*c^2*d^4*e^2 + 3*a
^5*c*d^2*e^4 - a^6*e^6))) - (a*c^2*d^2*x^2 - a^2*c*d^2 - (a^2*c*x^2 - a^3)*e^2)*sqrt((4*c^2*d^5 - 15*a*c*d^3*e
^2 + 15*a^2*d*e^4 - (a^3*c^3*d^6 - 3*a^4*c^2*d^4*e^2 + 3*a^5*c*d^2*e^4 - a^6*e^6)*sqrt((25*c^2*d^4*e^6 - 90*a*
c*d^2*e^8 + 81*a^2*e^10)/(a^3*c^7*d^12 - 6*a^4*c^6*d^10*e^2 + 15*a^5*c^5*d^8*e^4 - 20*a^6*c^4*d^6*e^6 + 15*a^7
*c^3*d^4*e^8 - 6*a^8*c^2*d^2*e^10 + a^9*c*e^12)))/(a^3*c^3*d^6 - 3*a^4*c^2*d^4*e^2 + 3*a^5*c*d^2*e^4 - a^6*e^6
))*log((20*c^2*d^4*e^3 - 81*a*c*d^2*e^5 + 81*a^2*e^7)*sqrt(x*e + d) - (5*a^2*c^2*d^4*e^4 - 24*a^3*c*d^2*e^6 +
27*a^4*e^8 - 2*(a^3*c^5*d^9 - 5*a^4*c^4*d^7*e^2 + 9*a^5*c^3*d^5*e^4 - 7*a^6*c^2*d^3*e^6 + 2*a^7*c*d*e^8)*sqrt(
(25*c^2*d^4*e^6 - 90*a*c*d^2*e^8 + 81*a^2*e^10)/(a^3*c^7*d^12 - 6*a^4*c^6*d^10*e^2 + 15*a^5*c^5*d^8*e^4 - 20*a
^6*c^4*d^6*e^6 + 15*a^7*c^3*d^4*e^8 - 6*a^8*c^2*d^2*e^10 + a^9*c*e^12)))*sqrt((4*c^2*d^5 - 15*a*c*d^3*e^2 + 15
*a^2*d*e^4 - (a^3*c^3*d^6 - 3*a^4*c^2*d^4*e^2 + 3*a^5*c*d^2*e^4 - a^6*e^6)*sqrt((25*c^2*d^4*e^6 - 90*a*c*d^2*e
^8 + 81*a^2*e^10)/(a^3*c^7*d^12 - 6*a^4*c^6*d^10*e^2 + 15*a^5*c^5*d^8*e^4 - 20*a^6*c^4*d^6*e^6 + 15*a^7*c^3*d^
4*e^8 - 6*a^8*c^2*d^2*e^10 + a^9*c*e^12)))/(a^3*c^3*d^6 - 3*a^4*c^2*d^4*e^2 + 3*a^5*c*d^2*e^4 - a^6*e^6))) - 4
*(c*d*x - a*e)*sqrt(x*e + d))/(a*c^2*d^2*x^2 - a^2*c*d^2 - (a^2*c*x^2 - a^3)*e^2)
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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(1/(-c*x**2+a)**2/(e*x+d)**(1/2),x)
[Out]
Timed out
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 859 vs.
\(2 (176) = 352\).
time = 2.44, size = 859, normalized size = 3.87 \begin {gather*} -\frac {{\left ({\left (a c d^{2} e - a^{2} e^{3}\right )}^{2} c d {\left | c \right |} e + {\left (\sqrt {a c} c^{2} d^{4} e - 4 \, \sqrt {a c} a c d^{2} e^{3} + 3 \, \sqrt {a c} a^{2} e^{5}\right )} {\left | a c d^{2} e - a^{2} e^{3} \right |} {\left | c \right |} - {\left (2 \, a c^{4} d^{7} e - 7 \, a^{2} c^{3} d^{5} e^{3} + 8 \, a^{3} c^{2} d^{3} e^{5} - 3 \, a^{4} c d e^{7}\right )} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {a c^{2} d^{3} - a^{2} c d e^{2} + \sqrt {{\left (a c^{2} d^{3} - a^{2} c d e^{2}\right )}^{2} - {\left (a c^{2} d^{4} - 2 \, a^{2} c d^{2} e^{2} + a^{3} e^{4}\right )} {\left (a c^{2} d^{2} - a^{2} c e^{2}\right )}}}{a c^{2} d^{2} - a^{2} c e^{2}}}}\right )}{4 \, {\left (a^{2} c^{3} d^{4} e - \sqrt {a c} a c^{3} d^{5} + 2 \, \sqrt {a c} a^{2} c^{2} d^{3} e^{2} - 2 \, a^{3} c^{2} d^{2} e^{3} - \sqrt {a c} a^{3} c d e^{4} + a^{4} c e^{5}\right )} \sqrt {-c^{2} d - \sqrt {a c} c e} {\left | a c d^{2} e - a^{2} e^{3} \right |}} - \frac {{\left ({\left (a c d^{2} e - a^{2} e^{3}\right )}^{2} \sqrt {a c} d {\left | c \right |} e - {\left (a c^{2} d^{4} e - 4 \, a^{2} c d^{2} e^{3} + 3 \, a^{3} e^{5}\right )} {\left | a c d^{2} e - a^{2} e^{3} \right |} {\left | c \right |} - {\left (2 \, \sqrt {a c} a c^{3} d^{7} e - 7 \, \sqrt {a c} a^{2} c^{2} d^{5} e^{3} + 8 \, \sqrt {a c} a^{3} c d^{3} e^{5} - 3 \, \sqrt {a c} a^{4} d e^{7}\right )} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {a c^{2} d^{3} - a^{2} c d e^{2} - \sqrt {{\left (a c^{2} d^{3} - a^{2} c d e^{2}\right )}^{2} - {\left (a c^{2} d^{4} - 2 \, a^{2} c d^{2} e^{2} + a^{3} e^{4}\right )} {\left (a c^{2} d^{2} - a^{2} c e^{2}\right )}}}{a c^{2} d^{2} - a^{2} c e^{2}}}}\right )}{4 \, {\left (a^{2} c^{3} d^{5} + \sqrt {a c} a^{2} c^{2} d^{4} e - 2 \, a^{3} c^{2} d^{3} e^{2} - 2 \, \sqrt {a c} a^{3} c d^{2} e^{3} + a^{4} c d e^{4} + \sqrt {a c} a^{4} e^{5}\right )} \sqrt {-c^{2} d + \sqrt {a c} c e} {\left | a c d^{2} e - a^{2} e^{3} \right |}} - \frac {{\left (x e + d\right )}^{\frac {3}{2}} c d e - \sqrt {x e + d} c d^{2} e - \sqrt {x e + d} a e^{3}}{2 \, {\left (a c d^{2} - a^{2} e^{2}\right )} {\left ({\left (x e + d\right )}^{2} c - 2 \, {\left (x e + d\right )} c d + c d^{2} - a e^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(-c*x^2+a)^2/(e*x+d)^(1/2),x, algorithm="giac")
[Out]
-1/4*((a*c*d^2*e - a^2*e^3)^2*c*d*abs(c)*e + (sqrt(a*c)*c^2*d^4*e - 4*sqrt(a*c)*a*c*d^2*e^3 + 3*sqrt(a*c)*a^2*
e^5)*abs(a*c*d^2*e - a^2*e^3)*abs(c) - (2*a*c^4*d^7*e - 7*a^2*c^3*d^5*e^3 + 8*a^3*c^2*d^3*e^5 - 3*a^4*c*d*e^7)
*abs(c))*arctan(sqrt(x*e + d)/sqrt(-(a*c^2*d^3 - a^2*c*d*e^2 + sqrt((a*c^2*d^3 - a^2*c*d*e^2)^2 - (a*c^2*d^4 -
2*a^2*c*d^2*e^2 + a^3*e^4)*(a*c^2*d^2 - a^2*c*e^2)))/(a*c^2*d^2 - a^2*c*e^2)))/((a^2*c^3*d^4*e - sqrt(a*c)*a*
c^3*d^5 + 2*sqrt(a*c)*a^2*c^2*d^3*e^2 - 2*a^3*c^2*d^2*e^3 - sqrt(a*c)*a^3*c*d*e^4 + a^4*c*e^5)*sqrt(-c^2*d - s
qrt(a*c)*c*e)*abs(a*c*d^2*e - a^2*e^3)) - 1/4*((a*c*d^2*e - a^2*e^3)^2*sqrt(a*c)*d*abs(c)*e - (a*c^2*d^4*e - 4
*a^2*c*d^2*e^3 + 3*a^3*e^5)*abs(a*c*d^2*e - a^2*e^3)*abs(c) - (2*sqrt(a*c)*a*c^3*d^7*e - 7*sqrt(a*c)*a^2*c^2*d
^5*e^3 + 8*sqrt(a*c)*a^3*c*d^3*e^5 - 3*sqrt(a*c)*a^4*d*e^7)*abs(c))*arctan(sqrt(x*e + d)/sqrt(-(a*c^2*d^3 - a^
2*c*d*e^2 - sqrt((a*c^2*d^3 - a^2*c*d*e^2)^2 - (a*c^2*d^4 - 2*a^2*c*d^2*e^2 + a^3*e^4)*(a*c^2*d^2 - a^2*c*e^2)
))/(a*c^2*d^2 - a^2*c*e^2)))/((a^2*c^3*d^5 + sqrt(a*c)*a^2*c^2*d^4*e - 2*a^3*c^2*d^3*e^2 - 2*sqrt(a*c)*a^3*c*d
^2*e^3 + a^4*c*d*e^4 + sqrt(a*c)*a^4*e^5)*sqrt(-c^2*d + sqrt(a*c)*c*e)*abs(a*c*d^2*e - a^2*e^3)) - 1/2*((x*e +
d)^(3/2)*c*d*e - sqrt(x*e + d)*c*d^2*e - sqrt(x*e + d)*a*e^3)/((a*c*d^2 - a^2*e^2)*((x*e + d)^2*c - 2*(x*e +
d)*c*d + c*d^2 - a*e^2))
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Mupad [B]
time = 2.53, size = 2500, normalized size = 11.26 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/((a - c*x^2)^2*(d + e*x)^(1/2)),x)
[Out]
atan(((((192*a^5*c^3*e^7 + 64*a^3*c^5*d^4*e^3 - 256*a^4*c^4*d^2*e^5)/(8*(a^5*e^4 + a^3*c^2*d^4 - 2*a^4*c*d^2*e
^2)) + ((d + e*x)^(1/2)*(64*a^5*c^4*d*e^6 + 64*a^3*c^6*d^5*e^2 - 128*a^4*c^5*d^3*e^4)*(-(4*a^3*c^3*d^5 - 9*a*e
^5*(a^9*c)^(1/2) + 5*c*d^2*e^3*(a^9*c)^(1/2) - 15*a^4*c^2*d^3*e^2 + 15*a^5*c*d*e^4)/(64*(a^9*c*e^6 - a^6*c^4*d
^6 + 3*a^7*c^3*d^4*e^2 - 3*a^8*c^2*d^2*e^4)))^(1/2))/(a^4*e^4 + a^2*c^2*d^4 - 2*a^3*c*d^2*e^2))*(-(4*a^3*c^3*d
^5 - 9*a*e^5*(a^9*c)^(1/2) + 5*c*d^2*e^3*(a^9*c)^(1/2) - 15*a^4*c^2*d^3*e^2 + 15*a^5*c*d*e^4)/(64*(a^9*c*e^6 -
a^6*c^4*d^6 + 3*a^7*c^3*d^4*e^2 - 3*a^8*c^2*d^2*e^4)))^(1/2) - ((d + e*x)^(1/2)*(9*a^2*c^3*e^6 + 4*c^5*d^4*e^
2 - 11*a*c^4*d^2*e^4))/(a^4*e^4 + a^2*c^2*d^4 - 2*a^3*c*d^2*e^2))*(-(4*a^3*c^3*d^5 - 9*a*e^5*(a^9*c)^(1/2) + 5
*c*d^2*e^3*(a^9*c)^(1/2) - 15*a^4*c^2*d^3*e^2 + 15*a^5*c*d*e^4)/(64*(a^9*c*e^6 - a^6*c^4*d^6 + 3*a^7*c^3*d^4*e
^2 - 3*a^8*c^2*d^2*e^4)))^(1/2)*1i - (((192*a^5*c^3*e^7 + 64*a^3*c^5*d^4*e^3 - 256*a^4*c^4*d^2*e^5)/(8*(a^5*e^
4 + a^3*c^2*d^4 - 2*a^4*c*d^2*e^2)) - ((d + e*x)^(1/2)*(64*a^5*c^4*d*e^6 + 64*a^3*c^6*d^5*e^2 - 128*a^4*c^5*d^
3*e^4)*(-(4*a^3*c^3*d^5 - 9*a*e^5*(a^9*c)^(1/2) + 5*c*d^2*e^3*(a^9*c)^(1/2) - 15*a^4*c^2*d^3*e^2 + 15*a^5*c*d*
e^4)/(64*(a^9*c*e^6 - a^6*c^4*d^6 + 3*a^7*c^3*d^4*e^2 - 3*a^8*c^2*d^2*e^4)))^(1/2))/(a^4*e^4 + a^2*c^2*d^4 - 2
*a^3*c*d^2*e^2))*(-(4*a^3*c^3*d^5 - 9*a*e^5*(a^9*c)^(1/2) + 5*c*d^2*e^3*(a^9*c)^(1/2) - 15*a^4*c^2*d^3*e^2 + 1
5*a^5*c*d*e^4)/(64*(a^9*c*e^6 - a^6*c^4*d^6 + 3*a^7*c^3*d^4*e^2 - 3*a^8*c^2*d^2*e^4)))^(1/2) + ((d + e*x)^(1/2
)*(9*a^2*c^3*e^6 + 4*c^5*d^4*e^2 - 11*a*c^4*d^2*e^4))/(a^4*e^4 + a^2*c^2*d^4 - 2*a^3*c*d^2*e^2))*(-(4*a^3*c^3*
d^5 - 9*a*e^5*(a^9*c)^(1/2) + 5*c*d^2*e^3*(a^9*c)^(1/2) - 15*a^4*c^2*d^3*e^2 + 15*a^5*c*d*e^4)/(64*(a^9*c*e^6
- a^6*c^4*d^6 + 3*a^7*c^3*d^4*e^2 - 3*a^8*c^2*d^2*e^4)))^(1/2)*1i)/((((192*a^5*c^3*e^7 + 64*a^3*c^5*d^4*e^3 -
256*a^4*c^4*d^2*e^5)/(8*(a^5*e^4 + a^3*c^2*d^4 - 2*a^4*c*d^2*e^2)) + ((d + e*x)^(1/2)*(64*a^5*c^4*d*e^6 + 64*a
^3*c^6*d^5*e^2 - 128*a^4*c^5*d^3*e^4)*(-(4*a^3*c^3*d^5 - 9*a*e^5*(a^9*c)^(1/2) + 5*c*d^2*e^3*(a^9*c)^(1/2) - 1
5*a^4*c^2*d^3*e^2 + 15*a^5*c*d*e^4)/(64*(a^9*c*e^6 - a^6*c^4*d^6 + 3*a^7*c^3*d^4*e^2 - 3*a^8*c^2*d^2*e^4)))^(1
/2))/(a^4*e^4 + a^2*c^2*d^4 - 2*a^3*c*d^2*e^2))*(-(4*a^3*c^3*d^5 - 9*a*e^5*(a^9*c)^(1/2) + 5*c*d^2*e^3*(a^9*c)
^(1/2) - 15*a^4*c^2*d^3*e^2 + 15*a^5*c*d*e^4)/(64*(a^9*c*e^6 - a^6*c^4*d^6 + 3*a^7*c^3*d^4*e^2 - 3*a^8*c^2*d^2
*e^4)))^(1/2) - ((d + e*x)^(1/2)*(9*a^2*c^3*e^6 + 4*c^5*d^4*e^2 - 11*a*c^4*d^2*e^4))/(a^4*e^4 + a^2*c^2*d^4 -
2*a^3*c*d^2*e^2))*(-(4*a^3*c^3*d^5 - 9*a*e^5*(a^9*c)^(1/2) + 5*c*d^2*e^3*(a^9*c)^(1/2) - 15*a^4*c^2*d^3*e^2 +
15*a^5*c*d*e^4)/(64*(a^9*c*e^6 - a^6*c^4*d^6 + 3*a^7*c^3*d^4*e^2 - 3*a^8*c^2*d^2*e^4)))^(1/2) - (4*c^4*d^3*e^3
- 9*a*c^3*d*e^5)/(4*(a^5*e^4 + a^3*c^2*d^4 - 2*a^4*c*d^2*e^2)) + (((192*a^5*c^3*e^7 + 64*a^3*c^5*d^4*e^3 - 25
6*a^4*c^4*d^2*e^5)/(8*(a^5*e^4 + a^3*c^2*d^4 - 2*a^4*c*d^2*e^2)) - ((d + e*x)^(1/2)*(64*a^5*c^4*d*e^6 + 64*a^3
*c^6*d^5*e^2 - 128*a^4*c^5*d^3*e^4)*(-(4*a^3*c^3*d^5 - 9*a*e^5*(a^9*c)^(1/2) + 5*c*d^2*e^3*(a^9*c)^(1/2) - 15*
a^4*c^2*d^3*e^2 + 15*a^5*c*d*e^4)/(64*(a^9*c*e^6 - a^6*c^4*d^6 + 3*a^7*c^3*d^4*e^2 - 3*a^8*c^2*d^2*e^4)))^(1/2
))/(a^4*e^4 + a^2*c^2*d^4 - 2*a^3*c*d^2*e^2))*(-(4*a^3*c^3*d^5 - 9*a*e^5*(a^9*c)^(1/2) + 5*c*d^2*e^3*(a^9*c)^(
1/2) - 15*a^4*c^2*d^3*e^2 + 15*a^5*c*d*e^4)/(64*(a^9*c*e^6 - a^6*c^4*d^6 + 3*a^7*c^3*d^4*e^2 - 3*a^8*c^2*d^2*e
^4)))^(1/2) + ((d + e*x)^(1/2)*(9*a^2*c^3*e^6 + 4*c^5*d^4*e^2 - 11*a*c^4*d^2*e^4))/(a^4*e^4 + a^2*c^2*d^4 - 2*
a^3*c*d^2*e^2))*(-(4*a^3*c^3*d^5 - 9*a*e^5*(a^9*c)^(1/2) + 5*c*d^2*e^3*(a^9*c)^(1/2) - 15*a^4*c^2*d^3*e^2 + 15
*a^5*c*d*e^4)/(64*(a^9*c*e^6 - a^6*c^4*d^6 + 3*a^7*c^3*d^4*e^2 - 3*a^8*c^2*d^2*e^4)))^(1/2)))*(-(4*a^3*c^3*d^5
- 9*a*e^5*(a^9*c)^(1/2) + 5*c*d^2*e^3*(a^9*c)^(1/2) - 15*a^4*c^2*d^3*e^2 + 15*a^5*c*d*e^4)/(64*(a^9*c*e^6 - a
^6*c^4*d^6 + 3*a^7*c^3*d^4*e^2 - 3*a^8*c^2*d^2*e^4)))^(1/2)*2i + atan(((((192*a^5*c^3*e^7 + 64*a^3*c^5*d^4*e^3
- 256*a^4*c^4*d^2*e^5)/(8*(a^5*e^4 + a^3*c^2*d^4 - 2*a^4*c*d^2*e^2)) + ((d + e*x)^(1/2)*(64*a^5*c^4*d*e^6 + 6
4*a^3*c^6*d^5*e^2 - 128*a^4*c^5*d^3*e^4)*(-(9*a*e^5*(a^9*c)^(1/2) + 4*a^3*c^3*d^5 - 5*c*d^2*e^3*(a^9*c)^(1/2)
- 15*a^4*c^2*d^3*e^2 + 15*a^5*c*d*e^4)/(64*(a^9*c*e^6 - a^6*c^4*d^6 + 3*a^7*c^3*d^4*e^2 - 3*a^8*c^2*d^2*e^4)))
^(1/2))/(a^4*e^4 + a^2*c^2*d^4 - 2*a^3*c*d^2*e^2))*(-(9*a*e^5*(a^9*c)^(1/2) + 4*a^3*c^3*d^5 - 5*c*d^2*e^3*(a^9
*c)^(1/2) - 15*a^4*c^2*d^3*e^2 + 15*a^5*c*d*e^4)/(64*(a^9*c*e^6 - a^6*c^4*d^6 + 3*a^7*c^3*d^4*e^2 - 3*a^8*c^2*
d^2*e^4)))^(1/2) - ((d + e*x)^(1/2)*(9*a^2*c^3*e^6 + 4*c^5*d^4*e^2 - 11*a*c^4*d^2*e^4))/(a^4*e^4 + a^2*c^2*d^4
- 2*a^3*c*d^2*e^2))*(-(9*a*e^5*(a^9*c)^(1/2) + 4*a^3*c^3*d^5 - 5*c*d^2*e^3*(a^9*c)^(1/2) - 15*a^4*c^2*d^3*e^2
+ 15*a^5*c*d*e^4)/(64*(a^9*c*e^6 - a^6*c^4*d^6 + 3*a^7*c^3*d^4*e^2 - 3*a^8*c^2*d^2*e^4)))^(1/2)*1i - (((192*a
^5*c^3*e^7 + 64*a^3*c^5*d^4*e^3 - 256*a^4*c^4*d^2*e^5)/(8*(a^5*e^4 + a^3*c^2*d^4 - 2*a^4*c*d^2*e^2)) - ((d + e
*x)^(1/2)*(64*a^5*c^4*d*e^6 + 64*a^3*c^6*d^5*e^...
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