∖int 1___________________ (d+ex)3∕2∖left(a−cx2∖right)∖,dx

3.605 \(\int \frac{1}{(d+e x)^{3/2} \left (a-c x^2\right )} \, dx\)

Optimal. Leaf size=160 \[ \frac{2 e}{\sqrt{d+e x} \left (c d^2-a e^2\right )}-\frac{\sqrt [4]{c} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{\sqrt{a} \left (\sqrt{c} d-\sqrt{a} e\right )^{3/2}}+\frac{\sqrt [4]{c} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{\sqrt{a} \left (\sqrt{a} e+\sqrt{c} d\right )^{3/2}} \]

[Out]

(2*e)/((c*d^2 - a*e^2)*Sqrt[d + e*x]) - (c^(1/4)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])

/Sqrt[Sqrt[c]*d - Sqrt[a]*e]])/(Sqrt[a]*(Sqrt[c]*d - Sqrt[a]*e)^(3/2)) + (c^(1/4

)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(Sqrt[a]*(Sqrt[c

]*d + Sqrt[a]*e)^(3/2))

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Rubi [A] time = 0.593389, antiderivative size = 160, 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 \[ \frac{2 e}{\sqrt{d+e x} \left (c d^2-a e^2\right )}-\frac{\sqrt [4]{c} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{\sqrt{a} \left (\sqrt{c} d-\sqrt{a} e\right )^{3/2}}+\frac{\sqrt [4]{c} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{\sqrt{a} \left (\sqrt{a} e+\sqrt{c} d\right )^{3/2}} \]

Antiderivative was successfully veriﬁed.

[In] Int[1/((d + e*x)^(3/2)*(a - c*x^2)),x]

[Out]

(2*e)/((c*d^2 - a*e^2)*Sqrt[d + e*x]) - (c^(1/4)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])

/Sqrt[Sqrt[c]*d - Sqrt[a]*e]])/(Sqrt[a]*(Sqrt[c]*d - Sqrt[a]*e)^(3/2)) + (c^(1/4

)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(Sqrt[a]*(Sqrt[c

]*d + Sqrt[a]*e)^(3/2))

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Rubi in Sympy [A] time = 85.4661, size = 187, normalized size = 1.17 \[ - \frac{2 e}{\sqrt{d + e x} \left (a e^{2} - c d^{2}\right )} + \frac{\sqrt [4]{c} \left (\sqrt{a} e - \sqrt{c} d\right ) \operatorname{atanh}{\left (\frac{\sqrt [4]{c} \sqrt{d + e x}}{\sqrt{\sqrt{a} e + \sqrt{c} d}} \right )}}{\sqrt{a} \sqrt{\sqrt{a} e + \sqrt{c} d} \left (a e^{2} - c d^{2}\right )} - \frac{\sqrt [4]{c} \left (\sqrt{a} e + \sqrt{c} d\right ) \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{d + e x}}{\sqrt{\sqrt{a} e - \sqrt{c} d}} \right )}}{\sqrt{a} \sqrt{\sqrt{a} e - \sqrt{c} d} \left (a e^{2} - c d^{2}\right )} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] rubi_integrate(1/(e*x+d)**(3/2)/(-c*x**2+a),x)

[Out]

-2*e/(sqrt(d + e*x)*(a*e**2 - c*d**2)) + c**(1/4)*(sqrt(a)*e - sqrt(c)*d)*atanh(

c**(1/4)*sqrt(d + e*x)/sqrt(sqrt(a)*e + sqrt(c)*d))/(sqrt(a)*sqrt(sqrt(a)*e + sq

rt(c)*d)*(a*e**2 - c*d**2)) - c**(1/4)*(sqrt(a)*e + sqrt(c)*d)*atan(c**(1/4)*sqr

t(d + e*x)/sqrt(sqrt(a)*e - sqrt(c)*d))/(sqrt(a)*sqrt(sqrt(a)*e - sqrt(c)*d)*(a*

e**2 - c*d**2))

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Mathematica [A] time = 0.653084, size = 189, normalized size = 1.18 \[ \frac{2 e}{\sqrt{d+e x} \left (c d^2-a e^2\right )}-\frac{\sqrt{c d-\sqrt{a} \sqrt{c} e} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-\sqrt{a} \sqrt{c} e}}\right )}{\sqrt{a} \left (\sqrt{c} d-\sqrt{a} e\right )^2}+\frac{\sqrt{\sqrt{a} \sqrt{c} e+c d} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} \sqrt{c} e+c d}}\right )}{\sqrt{a} \left (\sqrt{a} e+\sqrt{c} d\right )^2} \]

Antiderivative was successfully veriﬁed.

[In] Integrate[1/((d + e*x)^(3/2)*(a - c*x^2)),x]

[Out]

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

Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - Sqrt[a]*Sqrt[c]*e]])/(Sqrt[a]*(Sqrt[c]*d - Sqr

t[a]*e)^2) + (Sqrt[c*d + Sqrt[a]*Sqrt[c]*e]*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt

[c*d + Sqrt[a]*Sqrt[c]*e]])/(Sqrt[a]*(Sqrt[c]*d + Sqrt[a]*e)^2)

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Maple [B] time = 0.03, size = 291, normalized size = 1.8 \[ -{\frac{e{c}^{2}d}{a{e}^{2}-c{d}^{2}}{\it Artanh} \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( cd+\sqrt{ac{e}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{ac{e}^{2}}}}{\frac{1}{\sqrt{ \left ( cd+\sqrt{ac{e}^{2}} \right ) c}}}}+{\frac{ce}{a{e}^{2}-c{d}^{2}}{\it Artanh} \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( cd+\sqrt{ac{e}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{ \left ( cd+\sqrt{ac{e}^{2}} \right ) c}}}}-{\frac{e{c}^{2}d}{a{e}^{2}-c{d}^{2}}\arctan \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( -cd+\sqrt{ac{e}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{ac{e}^{2}}}}{\frac{1}{\sqrt{ \left ( -cd+\sqrt{ac{e}^{2}} \right ) c}}}}-{\frac{ce}{a{e}^{2}-c{d}^{2}}\arctan \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( -cd+\sqrt{ac{e}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{ \left ( -cd+\sqrt{ac{e}^{2}} \right ) c}}}}-2\,{\frac{e}{ \left ( a{e}^{2}-c{d}^{2} \right ) \sqrt{ex+d}}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] int(1/(e*x+d)^(3/2)/(-c*x^2+a),x)

[Out]

-e*c^2/(a*e^2-c*d^2)/(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+e*c/(a*e^2-c*d^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))-e*c^2

/(a*e^2-c*d^2)/(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-e*c/(a*e^2-c*d^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))-2*e/(a*e^

2-c*d^2)/(e*x+d)^(1/2)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] integrate(-1/((c*x^2 - a)*(e*x + d)^(3/2)),x, algorithm="maxima")

[Out]

-integrate(1/((c*x^2 - a)*(e*x + d)^(3/2)), x)

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Fricas [A] time = 0.257452, size = 3792, normalized size = 23.7 \[ \text{result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] integrate(-1/((c*x^2 - a)*(e*x + d)^(3/2)),x, algorithm="fricas")

[Out]

1/2*((c*d^2 - a*e^2)*sqrt(e*x + d)*sqrt((c^2*d^3 + 3*a*c*d*e^2 + (a*c^3*d^6 - 3*

a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 - a^4*e^6)*sqrt((9*c^3*d^4*e^2 + 6*a*c^2*d^2*e

^4 + a^2*c*e^6)/(a*c^6*d^12 - 6*a^2*c^5*d^10*e^2 + 15*a^3*c^4*d^8*e^4 - 20*a^4*c

^3*d^6*e^6 + 15*a^5*c^2*d^4*e^8 - 6*a^6*c*d^2*e^10 + a^7*e^12)))/(a*c^3*d^6 - 3*

a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 - a^4*e^6))*log((3*c^2*d^2*e + a*c*e^3)*sqrt(e

*x + d) + (6*a*c^2*d^3*e^2 + 2*a^2*c*d*e^4 - (a*c^4*d^8 - 2*a^2*c^3*d^6*e^2 + 2*

a^4*c*d^2*e^6 - a^5*e^8)*sqrt((9*c^3*d^4*e^2 + 6*a*c^2*d^2*e^4 + a^2*c*e^6)/(a*c

^6*d^12 - 6*a^2*c^5*d^10*e^2 + 15*a^3*c^4*d^8*e^4 - 20*a^4*c^3*d^6*e^6 + 15*a^5*

c^2*d^4*e^8 - 6*a^6*c*d^2*e^10 + a^7*e^12)))*sqrt((c^2*d^3 + 3*a*c*d*e^2 + (a*c^

3*d^6 - 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 - a^4*e^6)*sqrt((9*c^3*d^4*e^2 + 6*a

*c^2*d^2*e^4 + a^2*c*e^6)/(a*c^6*d^12 - 6*a^2*c^5*d^10*e^2 + 15*a^3*c^4*d^8*e^4

- 20*a^4*c^3*d^6*e^6 + 15*a^5*c^2*d^4*e^8 - 6*a^6*c*d^2*e^10 + a^7*e^12)))/(a*c^

3*d^6 - 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 - a^4*e^6))) - (c*d^2 - a*e^2)*sqrt(

e*x + d)*sqrt((c^2*d^3 + 3*a*c*d*e^2 + (a*c^3*d^6 - 3*a^2*c^2*d^4*e^2 + 3*a^3*c*

d^2*e^4 - a^4*e^6)*sqrt((9*c^3*d^4*e^2 + 6*a*c^2*d^2*e^4 + a^2*c*e^6)/(a*c^6*d^1

2 - 6*a^2*c^5*d^10*e^2 + 15*a^3*c^4*d^8*e^4 - 20*a^4*c^3*d^6*e^6 + 15*a^5*c^2*d^

4*e^8 - 6*a^6*c*d^2*e^10 + a^7*e^12)))/(a*c^3*d^6 - 3*a^2*c^2*d^4*e^2 + 3*a^3*c*

d^2*e^4 - a^4*e^6))*log((3*c^2*d^2*e + a*c*e^3)*sqrt(e*x + d) - (6*a*c^2*d^3*e^2

+ 2*a^2*c*d*e^4 - (a*c^4*d^8 - 2*a^2*c^3*d^6*e^2 + 2*a^4*c*d^2*e^6 - a^5*e^8)*s

qrt((9*c^3*d^4*e^2 + 6*a*c^2*d^2*e^4 + a^2*c*e^6)/(a*c^6*d^12 - 6*a^2*c^5*d^10*e

^2 + 15*a^3*c^4*d^8*e^4 - 20*a^4*c^3*d^6*e^6 + 15*a^5*c^2*d^4*e^8 - 6*a^6*c*d^2*

e^10 + a^7*e^12)))*sqrt((c^2*d^3 + 3*a*c*d*e^2 + (a*c^3*d^6 - 3*a^2*c^2*d^4*e^2

+ 3*a^3*c*d^2*e^4 - a^4*e^6)*sqrt((9*c^3*d^4*e^2 + 6*a*c^2*d^2*e^4 + a^2*c*e^6)/

(a*c^6*d^12 - 6*a^2*c^5*d^10*e^2 + 15*a^3*c^4*d^8*e^4 - 20*a^4*c^3*d^6*e^6 + 15*

a^5*c^2*d^4*e^8 - 6*a^6*c*d^2*e^10 + a^7*e^12)))/(a*c^3*d^6 - 3*a^2*c^2*d^4*e^2

+ 3*a^3*c*d^2*e^4 - a^4*e^6))) + (c*d^2 - a*e^2)*sqrt(e*x + d)*sqrt((c^2*d^3 + 3

*a*c*d*e^2 - (a*c^3*d^6 - 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 - a^4*e^6)*sqrt((9

*c^3*d^4*e^2 + 6*a*c^2*d^2*e^4 + a^2*c*e^6)/(a*c^6*d^12 - 6*a^2*c^5*d^10*e^2 + 1

5*a^3*c^4*d^8*e^4 - 20*a^4*c^3*d^6*e^6 + 15*a^5*c^2*d^4*e^8 - 6*a^6*c*d^2*e^10 +

a^7*e^12)))/(a*c^3*d^6 - 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 - a^4*e^6))*log((3

*c^2*d^2*e + a*c*e^3)*sqrt(e*x + d) + (6*a*c^2*d^3*e^2 + 2*a^2*c*d*e^4 + (a*c^4*

d^8 - 2*a^2*c^3*d^6*e^2 + 2*a^4*c*d^2*e^6 - a^5*e^8)*sqrt((9*c^3*d^4*e^2 + 6*a*c

^2*d^2*e^4 + a^2*c*e^6)/(a*c^6*d^12 - 6*a^2*c^5*d^10*e^2 + 15*a^3*c^4*d^8*e^4 -

20*a^4*c^3*d^6*e^6 + 15*a^5*c^2*d^4*e^8 - 6*a^6*c*d^2*e^10 + a^7*e^12)))*sqrt((c

^2*d^3 + 3*a*c*d*e^2 - (a*c^3*d^6 - 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 - a^4*e^

6)*sqrt((9*c^3*d^4*e^2 + 6*a*c^2*d^2*e^4 + a^2*c*e^6)/(a*c^6*d^12 - 6*a^2*c^5*d^

10*e^2 + 15*a^3*c^4*d^8*e^4 - 20*a^4*c^3*d^6*e^6 + 15*a^5*c^2*d^4*e^8 - 6*a^6*c*

d^2*e^10 + a^7*e^12)))/(a*c^3*d^6 - 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 - a^4*e^

6))) - (c*d^2 - a*e^2)*sqrt(e*x + d)*sqrt((c^2*d^3 + 3*a*c*d*e^2 - (a*c^3*d^6 -

3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 - a^4*e^6)*sqrt((9*c^3*d^4*e^2 + 6*a*c^2*d^2

*e^4 + a^2*c*e^6)/(a*c^6*d^12 - 6*a^2*c^5*d^10*e^2 + 15*a^3*c^4*d^8*e^4 - 20*a^4

*c^3*d^6*e^6 + 15*a^5*c^2*d^4*e^8 - 6*a^6*c*d^2*e^10 + a^7*e^12)))/(a*c^3*d^6 -

3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 - a^4*e^6))*log((3*c^2*d^2*e + a*c*e^3)*sqrt

(e*x + d) - (6*a*c^2*d^3*e^2 + 2*a^2*c*d*e^4 + (a*c^4*d^8 - 2*a^2*c^3*d^6*e^2 +

2*a^4*c*d^2*e^6 - a^5*e^8)*sqrt((9*c^3*d^4*e^2 + 6*a*c^2*d^2*e^4 + a^2*c*e^6)/(a

*c^6*d^12 - 6*a^2*c^5*d^10*e^2 + 15*a^3*c^4*d^8*e^4 - 20*a^4*c^3*d^6*e^6 + 15*a^

5*c^2*d^4*e^8 - 6*a^6*c*d^2*e^10 + a^7*e^12)))*sqrt((c^2*d^3 + 3*a*c*d*e^2 - (a*

c^3*d^6 - 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 - a^4*e^6)*sqrt((9*c^3*d^4*e^2 + 6

*a*c^2*d^2*e^4 + a^2*c*e^6)/(a*c^6*d^12 - 6*a^2*c^5*d^10*e^2 + 15*a^3*c^4*d^8*e^

4 - 20*a^4*c^3*d^6*e^6 + 15*a^5*c^2*d^4*e^8 - 6*a^6*c*d^2*e^10 + a^7*e^12)))/(a*

c^3*d^6 - 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 - a^4*e^6))) + 4*e)/((c*d^2 - a*e^

2)*sqrt(e*x + d))

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Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \int \frac{1}{- a d \sqrt{d + e x} - a e x \sqrt{d + e x} + c d x^{2} \sqrt{d + e x} + c e x^{3} \sqrt{d + e x}}\, dx \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] integrate(1/(e*x+d)**(3/2)/(-c*x**2+a),x)

[Out]

-Integral(1/(-a*d*sqrt(d + e*x) - a*e*x*sqrt(d + e*x) + c*d*x**2*sqrt(d + e*x) +

c*e*x**3*sqrt(d + e*x)), x)

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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)*(e*x + d)^(3/2)),x, algorithm="giac")

[Out]

Timed out

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