3.349 \(\int \frac{1}{x^2 (d+e x)^2 \sqrt{a+c x^2}} \, dx\)

Optimal. Leaf size=212 \[ \frac{2 e \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{\sqrt{a} d^3}-\frac{c e^2 \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{d \left (a e^2+c d^2\right )^{3/2}}-\frac{e^3 \sqrt{a+c x^2}}{d^2 (d+e x) \left (a e^2+c d^2\right )}-\frac{\sqrt{a+c x^2}}{a d^2 x}-\frac{2 e^2 \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{d^3 \sqrt{a e^2+c d^2}} \]

[Out]

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

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Rubi [A]  time = 0.412162, antiderivative size = 212, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364 \[ \frac{2 e \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{\sqrt{a} d^3}-\frac{c e^2 \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{d \left (a e^2+c d^2\right )^{3/2}}-\frac{e^3 \sqrt{a+c x^2}}{d^2 (d+e x) \left (a e^2+c d^2\right )}-\frac{\sqrt{a+c x^2}}{a d^2 x}-\frac{2 e^2 \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{d^3 \sqrt{a e^2+c d^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 41.1157, size = 187, normalized size = 0.88 \[ - \frac{c e^{2} \operatorname{atanh}{\left (\frac{a e - c d x}{\sqrt{a + c x^{2}} \sqrt{a e^{2} + c d^{2}}} \right )}}{d \left (a e^{2} + c d^{2}\right )^{\frac{3}{2}}} - \frac{e^{3} \sqrt{a + c x^{2}}}{d^{2} \left (d + e x\right ) \left (a e^{2} + c d^{2}\right )} - \frac{2 e^{2} \operatorname{atanh}{\left (\frac{a e - c d x}{\sqrt{a + c x^{2}} \sqrt{a e^{2} + c d^{2}}} \right )}}{d^{3} \sqrt{a e^{2} + c d^{2}}} - \frac{\sqrt{a + c x^{2}}}{a d^{2} x} + \frac{2 e \operatorname{atanh}{\left (\frac{\sqrt{a + c x^{2}}}{\sqrt{a}} \right )}}{\sqrt{a} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-c*e**2*atanh((a*e - c*d*x)/(sqrt(a + c*x**2)*sqrt(a*e**2 + c*d**2)))/(d*(a*e**2
+ c*d**2)**(3/2)) - e**3*sqrt(a + c*x**2)/(d**2*(d + e*x)*(a*e**2 + c*d**2)) -
2*e**2*atanh((a*e - c*d*x)/(sqrt(a + c*x**2)*sqrt(a*e**2 + c*d**2)))/(d**3*sqrt(
a*e**2 + c*d**2)) - sqrt(a + c*x**2)/(a*d**2*x) + 2*e*atanh(sqrt(a + c*x**2)/sqr
t(a))/(sqrt(a)*d**3)

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Mathematica [A]  time = 0.563207, size = 197, normalized size = 0.93 \[ \frac{-\frac{e^2 \left (2 a e^2+3 c d^2\right ) \log \left (\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right )}{\left (a e^2+c d^2\right )^{3/2}}+\frac{e^2 \left (2 a e^2+3 c d^2\right ) \log (d+e x)}{\left (a e^2+c d^2\right )^{3/2}}-d \sqrt{a+c x^2} \left (\frac{e^3}{(d+e x) \left (a e^2+c d^2\right )}+\frac{1}{a x}\right )+\frac{2 e \log \left (\sqrt{a} \sqrt{a+c x^2}+a\right )}{\sqrt{a}}-\frac{2 e \log (x)}{\sqrt{a}}}{d^3} \]

Antiderivative was successfully verified.

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

[Out]

(-(d*Sqrt[a + c*x^2]*(1/(a*x) + e^3/((c*d^2 + a*e^2)*(d + e*x)))) - (2*e*Log[x])
/Sqrt[a] + (e^2*(3*c*d^2 + 2*a*e^2)*Log[d + e*x])/(c*d^2 + a*e^2)^(3/2) + (2*e*L
og[a + Sqrt[a]*Sqrt[a + c*x^2]])/Sqrt[a] - (e^2*(3*c*d^2 + 2*a*e^2)*Log[a*e - c*
d*x + Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]])/(c*d^2 + a*e^2)^(3/2))/d^3

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Maple [B]  time = 0.016, size = 395, normalized size = 1.9 \[ -{\frac{1}{a{d}^{2}x}\sqrt{c{x}^{2}+a}}-{\frac{{e}^{2}}{{d}^{2} \left ( a{e}^{2}+c{d}^{2} \right ) }\sqrt{ \left ( x+{\frac{d}{e}} \right ) ^{2}c-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \left ( x+{\frac{d}{e}} \right ) ^{-1}}-{\frac{ce}{d \left ( a{e}^{2}+c{d}^{2} \right ) }\ln \left ({1 \left ( 2\,{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+2\,\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}\sqrt{ \left ( x+{\frac{d}{e}} \right ) ^{2}c-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \right ) \left ( x+{\frac{d}{e}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}}+2\,{\frac{e}{{d}^{3}\sqrt{a}}\ln \left ({\frac{2\,a+2\,\sqrt{a}\sqrt{c{x}^{2}+a}}{x}} \right ) }-2\,{\frac{e}{{d}^{3}}\ln \left ({1 \left ( 2\,{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+2\,\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}\sqrt{ \left ( x+{\frac{d}{e}} \right ) ^{2}c-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \right ) \left ( x+{\frac{d}{e}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(c*x^2+a)^(1/2)/a/d^2/x-1/d^2/(a*e^2+c*d^2)*e^2/(x+d/e)*((x+d/e)^2*c-2*c*d/e*(x
+d/e)+(a*e^2+c*d^2)/e^2)^(1/2)-1/d*c*e/(a*e^2+c*d^2)/((a*e^2+c*d^2)/e^2)^(1/2)*l
n((2*(a*e^2+c*d^2)/e^2-2*c*d/e*(x+d/e)+2*((a*e^2+c*d^2)/e^2)^(1/2)*((x+d/e)^2*c-
2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2))/(x+d/e))+2/d^3*e/a^(1/2)*ln((2*a+2*a^(
1/2)*(c*x^2+a)^(1/2))/x)-2/d^3*e/((a*e^2+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2+c*d^2)/e
^2-2*c*d/e*(x+d/e)+2*((a*e^2+c*d^2)/e^2)^(1/2)*((x+d/e)^2*c-2*c*d/e*(x+d/e)+(a*e
^2+c*d^2)/e^2)^(1/2))/(x+d/e))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.590383, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/2*(2*(c*d^4 + a*d^2*e^2 + (c*d^3*e + 2*a*d*e^3)*x)*sqrt(c*d^2 + a*e^2)*sqrt(
c*x^2 + a)*sqrt(a) - ((3*a*c*d^2*e^3 + 2*a^2*e^5)*x^2 + (3*a*c*d^3*e^2 + 2*a^2*d
*e^4)*x)*sqrt(a)*log(((2*a*c*d*e*x - a*c*d^2 - 2*a^2*e^2 - (2*c^2*d^2 + a*c*e^2)
*x^2)*sqrt(c*d^2 + a*e^2) + 2*(a*c*d^2*e + a^2*e^3 - (c^2*d^3 + a*c*d*e^2)*x)*sq
rt(c*x^2 + a))/(e^2*x^2 + 2*d*e*x + d^2)) - 2*sqrt(c*d^2 + a*e^2)*((a*c*d^2*e^2
+ a^2*e^4)*x^2 + (a*c*d^3*e + a^2*d*e^3)*x)*log(-((c*x^2 + 2*a)*sqrt(a) + 2*sqrt
(c*x^2 + a)*a)/x^2))/(sqrt(c*d^2 + a*e^2)*((a*c*d^5*e + a^2*d^3*e^3)*x^2 + (a*c*
d^6 + a^2*d^4*e^2)*x)*sqrt(a)), -((c*d^4 + a*d^2*e^2 + (c*d^3*e + 2*a*d*e^3)*x)*
sqrt(-c*d^2 - a*e^2)*sqrt(c*x^2 + a)*sqrt(a) - ((3*a*c*d^2*e^3 + 2*a^2*e^5)*x^2
+ (3*a*c*d^3*e^2 + 2*a^2*d*e^4)*x)*sqrt(a)*arctan(sqrt(-c*d^2 - a*e^2)*(c*d*x -
a*e)/((c*d^2 + a*e^2)*sqrt(c*x^2 + a))) - sqrt(-c*d^2 - a*e^2)*((a*c*d^2*e^2 + a
^2*e^4)*x^2 + (a*c*d^3*e + a^2*d*e^3)*x)*log(-((c*x^2 + 2*a)*sqrt(a) + 2*sqrt(c*
x^2 + a)*a)/x^2))/(sqrt(-c*d^2 - a*e^2)*((a*c*d^5*e + a^2*d^3*e^3)*x^2 + (a*c*d^
6 + a^2*d^4*e^2)*x)*sqrt(a)), -1/2*(2*(c*d^4 + a*d^2*e^2 + (c*d^3*e + 2*a*d*e^3)
*x)*sqrt(c*d^2 + a*e^2)*sqrt(c*x^2 + a)*sqrt(-a) - 4*sqrt(c*d^2 + a*e^2)*((a*c*d
^2*e^2 + a^2*e^4)*x^2 + (a*c*d^3*e + a^2*d*e^3)*x)*arctan(sqrt(-a)/sqrt(c*x^2 +
a)) - ((3*a*c*d^2*e^3 + 2*a^2*e^5)*x^2 + (3*a*c*d^3*e^2 + 2*a^2*d*e^4)*x)*sqrt(-
a)*log(((2*a*c*d*e*x - a*c*d^2 - 2*a^2*e^2 - (2*c^2*d^2 + a*c*e^2)*x^2)*sqrt(c*d
^2 + a*e^2) + 2*(a*c*d^2*e + a^2*e^3 - (c^2*d^3 + a*c*d*e^2)*x)*sqrt(c*x^2 + a))
/(e^2*x^2 + 2*d*e*x + d^2)))/(sqrt(c*d^2 + a*e^2)*((a*c*d^5*e + a^2*d^3*e^3)*x^2
+ (a*c*d^6 + a^2*d^4*e^2)*x)*sqrt(-a)), -((c*d^4 + a*d^2*e^2 + (c*d^3*e + 2*a*d
*e^3)*x)*sqrt(-c*d^2 - a*e^2)*sqrt(c*x^2 + a)*sqrt(-a) - ((3*a*c*d^2*e^3 + 2*a^2
*e^5)*x^2 + (3*a*c*d^3*e^2 + 2*a^2*d*e^4)*x)*sqrt(-a)*arctan(sqrt(-c*d^2 - a*e^2
)*(c*d*x - a*e)/((c*d^2 + a*e^2)*sqrt(c*x^2 + a))) - 2*sqrt(-c*d^2 - a*e^2)*((a*
c*d^2*e^2 + a^2*e^4)*x^2 + (a*c*d^3*e + a^2*d*e^3)*x)*arctan(sqrt(-a)/sqrt(c*x^2
+ a)))/(sqrt(-c*d^2 - a*e^2)*((a*c*d^5*e + a^2*d^3*e^3)*x^2 + (a*c*d^6 + a^2*d^
4*e^2)*x)*sqrt(-a))]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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