3.617 \(\int \frac{1}{\sqrt{d+e x} \left (a-c x^2\right )^2} \, dx\)

Optimal. Leaf size=222 \[ -\frac{\left (2 \sqrt{c} d-3 \sqrt{a} e\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{4 a^{3/2} \sqrt [4]{c} \left (\sqrt{c} d-\sqrt{a} e\right )^{3/2}}+\frac{\left (3 \sqrt{a} e+2 \sqrt{c} d\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{4 a^{3/2} \sqrt [4]{c} \left (\sqrt{a} e+\sqrt{c} d\right )^{3/2}}-\frac{\sqrt{d+e x} (a e-c d x)}{2 a \left (a-c x^2\right ) \left (c d^2-a e^2\right )} \]

[Out]

-((a*e - c*d*x)*Sqrt[d + e*x])/(2*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)*A
rcTanh[(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))

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Rubi [A]  time = 0.848175, antiderivative size = 222, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{\left (2 \sqrt{c} d-3 \sqrt{a} e\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{4 a^{3/2} \sqrt [4]{c} \left (\sqrt{c} d-\sqrt{a} e\right )^{3/2}}+\frac{\left (3 \sqrt{a} e+2 \sqrt{c} d\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{4 a^{3/2} \sqrt [4]{c} \left (\sqrt{a} e+\sqrt{c} d\right )^{3/2}}-\frac{\sqrt{d+e x} (a e-c d x)}{2 a \left (a-c x^2\right ) \left (c d^2-a e^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[d + e*x]*(a - c*x^2)^2),x]

[Out]

-((a*e - c*d*x)*Sqrt[d + e*x])/(2*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)*A
rcTanh[(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))

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Rubi in Sympy [A]  time = 162.888, size = 233, normalized size = 1.05 \[ \frac{\sqrt{d + e x} \left (a e - c d x\right )}{2 a \left (a - c x^{2}\right ) \left (a e^{2} - c d^{2}\right )} + \frac{\left (- \sqrt{a} \sqrt{c} d e + 3 a e^{2} - 2 c d^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt [4]{c} \sqrt{d + e x}}{\sqrt{\sqrt{a} e + \sqrt{c} d}} \right )}}{4 a^{\frac{3}{2}} \sqrt [4]{c} \sqrt{\sqrt{a} e + \sqrt{c} d} \left (a e^{2} - c d^{2}\right )} + \frac{\left (\sqrt{a} \sqrt{c} d e + 3 a e^{2} - 2 c d^{2}\right ) \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{d + e x}}{\sqrt{\sqrt{a} e - \sqrt{c} d}} \right )}}{4 a^{\frac{3}{2}} \sqrt [4]{c} \sqrt{\sqrt{a} e - \sqrt{c} d} \left (a e^{2} - c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(-c*x**2+a)**2/(e*x+d)**(1/2),x)

[Out]

sqrt(d + e*x)*(a*e - c*d*x)/(2*a*(a - c*x**2)*(a*e**2 - c*d**2)) + (-sqrt(a)*sqr
t(c)*d*e + 3*a*e**2 - 2*c*d**2)*atanh(c**(1/4)*sqrt(d + e*x)/sqrt(sqrt(a)*e + sq
rt(c)*d))/(4*a**(3/2)*c**(1/4)*sqrt(sqrt(a)*e + sqrt(c)*d)*(a*e**2 - c*d**2)) +
(sqrt(a)*sqrt(c)*d*e + 3*a*e**2 - 2*c*d**2)*atan(c**(1/4)*sqrt(d + e*x)/sqrt(sqr
t(a)*e - sqrt(c)*d))/(4*a**(3/2)*c**(1/4)*sqrt(sqrt(a)*e - sqrt(c)*d)*(a*e**2 -
c*d**2))

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Mathematica [A]  time = 0.449039, size = 230, normalized size = 1.04 \[ \frac{\frac{\left (-\sqrt{a} \sqrt{c} d e-3 a e^2+2 c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-\sqrt{a} \sqrt{c} e}}\right )}{\sqrt{c d-\sqrt{a} \sqrt{c} e}}-\frac{\left (\sqrt{a} \sqrt{c} d e-3 a e^2+2 c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} \sqrt{c} e+c d}}\right )}{\sqrt{\sqrt{a} \sqrt{c} e+c d}}+\frac{2 \sqrt{a} \sqrt{d+e x} (c d x-a e)}{c x^2-a}}{4 a^{3/2} \left (a e^2-c d^2\right )} \]

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])/(-a + c*x^2) + ((2*c*d^2 - Sqrt[a]*S
qrt[c]*d*e - 3*a*e^2)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - Sqrt[a]*Sqrt[c]
*e]])/Sqrt[c*d - Sqrt[a]*Sqrt[c]*e] - ((2*c*d^2 + Sqrt[a]*Sqrt[c]*d*e - 3*a*e^2)
*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d + Sqrt[a]*Sqrt[c]*e]])/Sqrt[c*d + Sqrt
[a]*Sqrt[c]*e])/(4*a^(3/2)*(-(c*d^2) + a*e^2))

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Maple [B]  time = 0.076, size = 377, normalized size = 1.7 \[ -{\frac{e}{4\,a}\sqrt{ex+d} \left ( cd-\sqrt{ac{e}^{2}} \right ) ^{-1} \left ( ex+{\frac{1}{c}\sqrt{ac{e}^{2}}} \right ) ^{-1}}+{\frac{e{c}^{2}d}{2\,a}\arctan \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( -cd+\sqrt{ac{e}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{ac{e}^{2}}}} \left ( cd-\sqrt{ac{e}^{2}} \right ) ^{-1}{\frac{1}{\sqrt{ \left ( -cd+\sqrt{ac{e}^{2}} \right ) c}}}}-{\frac{3\,ce}{4\,a}\arctan \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( -cd+\sqrt{ac{e}^{2}} \right ) c}}}} \right ) \left ( cd-\sqrt{ac{e}^{2}} \right ) ^{-1}{\frac{1}{\sqrt{ \left ( -cd+\sqrt{ac{e}^{2}} \right ) c}}}}-{\frac{e}{4\,a}\sqrt{ex+d} \left ( cd+\sqrt{ac{e}^{2}} \right ) ^{-1} \left ( ex-{\frac{1}{c}\sqrt{ac{e}^{2}}} \right ) ^{-1}}+{\frac{e{c}^{2}d}{2\,a}{\it Artanh} \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( cd+\sqrt{ac{e}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{ac{e}^{2}}}} \left ( cd+\sqrt{ac{e}^{2}} \right ) ^{-1}{\frac{1}{\sqrt{ \left ( cd+\sqrt{ac{e}^{2}} \right ) c}}}}+{\frac{3\,ce}{4\,a}{\it Artanh} \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( cd+\sqrt{ac{e}^{2}} \right ) c}}}} \right ) \left ( cd+\sqrt{ac{e}^{2}} \right ) ^{-1}{\frac{1}{\sqrt{ \left ( cd+\sqrt{ac{e}^{2}} \right ) c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-c*x^2+a)^2/(e*x+d)^(1/2),x)

[Out]

-1/4*e/a/(c*d-(a*c*e^2)^(1/2))*(e*x+d)^(1/2)/(e*x+(a*c*e^2)^(1/2)/c)+1/2*e*c^2/a
/(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))*d-3/4*e*c/a/(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*e/a/(c*d+(a*c*e^2)^(1/2))*(e*x+d)^(1/2)/(e*x-(a*c*e^2)^(1/2)/c)
+1/2*e*c^2/a/(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))*d+3/4*e*c/a/(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., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{2} - a\right )}^{2} \sqrt{e x + d}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 - a)^2*sqrt(e*x + d)),x, algorithm="maxima")

[Out]

integrate(1/((c*x^2 - a)^2*sqrt(e*x + d)), x)

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Fricas [A]  time = 0.337186, size = 4427, normalized size = 19.94 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 - a)^2*sqrt(e*x + d)),x, algorithm="fricas")

[Out]

1/8*((a^2*c*d^2 - a^3*e^2 - (a*c^2*d^2 - a^2*c*e^2)*x^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(e*x
+ 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^2*c*d^2 - a
^3*e^2 - (a*c^2*d^2 - a^2*c*e^2)*x^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^1
0 + 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(e*x + 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^2*c*d^2 - a^3*e^2 - (a*c^2*d^2
- a^2*c*e^2)*x^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(e*x + 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^2*c*d^2 - a^3*e^2 - (a*c^2*d^2 - a^2*c*e^2)*x^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(e*x + 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^1
2 - 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(e*x + d))/(a^2*c*d^2 - a^3*e^2 - (a*c^2*d^2 - a^2*
c*e^2)*x^2)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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

_______________________________________________________________________________________

GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 - a)^2*sqrt(e*x + d)),x, algorithm="giac")

[Out]

Timed out