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

Optimal. Leaf size=194 \[ -\frac{\left (\frac{2 \sqrt{c} d}{\sqrt{a}}-e\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{4 a c^{3/4} \sqrt{\sqrt{c} d-\sqrt{a} e}}+\frac{\left (\frac{2 \sqrt{c} d}{\sqrt{a}}+e\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{4 a c^{3/4} \sqrt{\sqrt{a} e+\sqrt{c} d}}+\frac{x \sqrt{d+e x}}{2 a \left (a-c x^2\right )} \]

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

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Rubi [A]  time = 0.50035, antiderivative size = 194, 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 \[ -\frac{\left (\frac{2 \sqrt{c} d}{\sqrt{a}}-e\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{4 a c^{3/4} \sqrt{\sqrt{c} d-\sqrt{a} e}}+\frac{\left (\frac{2 \sqrt{c} d}{\sqrt{a}}+e\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{4 a c^{3/4} \sqrt{\sqrt{a} e+\sqrt{c} d}}+\frac{x \sqrt{d+e x}}{2 a \left (a-c x^2\right )} \]

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

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Rubi in Sympy [A]  time = 83.0564, size = 172, normalized size = 0.89 \[ \frac{x \sqrt{d + e x}}{2 a \left (a - c x^{2}\right )} - \frac{\left (\sqrt{a} e - 2 \sqrt{c} d\right ) \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{d + e x}}{\sqrt{\sqrt{a} e - \sqrt{c} d}} \right )}}{4 a^{\frac{3}{2}} c^{\frac{3}{4}} \sqrt{\sqrt{a} e - \sqrt{c} d}} + \frac{\left (\sqrt{a} e + 2 \sqrt{c} d\right ) \operatorname{atanh}{\left (\frac{\sqrt [4]{c} \sqrt{d + e x}}{\sqrt{\sqrt{a} e + \sqrt{c} d}} \right )}}{4 a^{\frac{3}{2}} c^{\frac{3}{4}} \sqrt{\sqrt{a} e + \sqrt{c} d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.498485, size = 203, normalized size = 1.05 \[ -\frac{\left (2 c d-\sqrt{a} \sqrt{c} e\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-\sqrt{a} \sqrt{c} e}}\right )}{4 a^{3/2} c \sqrt{c d-\sqrt{a} \sqrt{c} e}}-\frac{\left (-\sqrt{a} \sqrt{c} e-2 c d\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} \sqrt{c} e+c d}}\right )}{4 a^{3/2} c \sqrt{\sqrt{a} \sqrt{c} e+c d}}-\frac{x \sqrt{d+e x}}{2 a \left (c x^2-a\right )} \]

Antiderivative was successfully verified.

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

[Out]

-(x*Sqrt[d + e*x])/(2*a*(-a + c*x^2)) - ((2*c*d - Sqrt[a]*Sqrt[c]*e)*ArcTanh[(Sq
rt[c]*Sqrt[d + e*x])/Sqrt[c*d - Sqrt[a]*Sqrt[c]*e]])/(4*a^(3/2)*c*Sqrt[c*d - Sqr
t[a]*Sqrt[c]*e]) - ((-2*c*d - Sqrt[a]*Sqrt[c]*e)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])
/Sqrt[c*d + Sqrt[a]*Sqrt[c]*e]])/(4*a^(3/2)*c*Sqrt[c*d + Sqrt[a]*Sqrt[c]*e])

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Maple [B]  time = 0.115, size = 287, normalized size = 1.5 \[ -{\frac{e}{4\,ac}\sqrt{ex+d} \left ( ex+{\frac{1}{c}\sqrt{ac{e}^{2}}} \right ) ^{-1}}+{\frac{ced}{2\,a}\arctan \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( -cd+\sqrt{ac{e}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{ac{e}^{2}}}}{\frac{1}{\sqrt{ \left ( -cd+\sqrt{ac{e}^{2}} \right ) c}}}}-{\frac{e}{4\,a}\arctan \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( -cd+\sqrt{ac{e}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{ \left ( -cd+\sqrt{ac{e}^{2}} \right ) c}}}}-{\frac{e}{4\,ac}\sqrt{ex+d} \left ( ex-{\frac{1}{c}\sqrt{ac{e}^{2}}} \right ) ^{-1}}+{\frac{ced}{2\,a}{\it Artanh} \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( cd+\sqrt{ac{e}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{ac{e}^{2}}}}{\frac{1}{\sqrt{ \left ( cd+\sqrt{ac{e}^{2}} \right ) c}}}}+{\frac{e}{4\,a}{\it Artanh} \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( cd+\sqrt{ac{e}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{ \left ( cd+\sqrt{ac{e}^{2}} \right ) c}}}} \]

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)*arctan(c*(e*x+d)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1
/2))*d-1/4*e/a/((-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/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)*arctanh(c*(e*x+d)^(1/2)/((c*d
+(a*c*e^2)^(1/2))*c)^(1/2))*d+1/4*e/a/((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. \[ \int \frac{\sqrt{e x + d}}{{\left (c x^{2} - a\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.240775, size = 1870, normalized size = 9.64 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError