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

Optimal. Leaf size=179 \[ \frac{e^2 \sqrt{a+c x^2}}{d (d+e x) \left (a e^2+c d^2\right )}+\frac{e \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{d^2 \sqrt{a e^2+c d^2}}+\frac{c e \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{\left (a e^2+c d^2\right )^{3/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{\sqrt{a} d^2} \]

[Out]

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

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Rubi [A]  time = 0.345129, antiderivative size = 179, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318 \[ \frac{e^2 \sqrt{a+c x^2}}{d (d+e x) \left (a e^2+c d^2\right )}+\frac{e \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{d^2 \sqrt{a e^2+c d^2}}+\frac{c e \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{\left (a e^2+c d^2\right )^{3/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{\sqrt{a} d^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 33.4604, size = 158, normalized size = 0.88 \[ \frac{c e \operatorname{atanh}{\left (\frac{a e - c d x}{\sqrt{a + c x^{2}} \sqrt{a e^{2} + c d^{2}}} \right )}}{\left (a e^{2} + c d^{2}\right )^{\frac{3}{2}}} + \frac{e^{2} \sqrt{a + c x^{2}}}{d \left (d + e x\right ) \left (a e^{2} + c d^{2}\right )} + \frac{e \operatorname{atanh}{\left (\frac{a e - c d x}{\sqrt{a + c x^{2}} \sqrt{a e^{2} + c d^{2}}} \right )}}{d^{2} \sqrt{a e^{2} + c d^{2}}} - \frac{\operatorname{atanh}{\left (\frac{\sqrt{a + c x^{2}}}{\sqrt{a}} \right )}}{\sqrt{a} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.399877, size = 178, normalized size = 0.99 \[ \frac{\frac{d e^2 \sqrt{a+c x^2}}{(d+e x) \left (a e^2+c d^2\right )}+\frac{e \left (a e^2+2 c d^2\right ) \log \left (\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right )}{\left (a e^2+c d^2\right )^{3/2}}-\frac{e \left (a e^2+2 c d^2\right ) \log (d+e x)}{\left (a e^2+c d^2\right )^{3/2}}-\frac{\log \left (\sqrt{a} \sqrt{a+c x^2}+a\right )}{\sqrt{a}}+\frac{\log (x)}{\sqrt{a}}}{d^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.016, size = 364, normalized size = 2. \[ -{\frac{1}{{d}^{2}}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{2}+a} \right ) } \right ){\frac{1}{\sqrt{a}}}}+{\frac{1}{{d}^{2}}\ln \left ({1 \left ( 2\,{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+2\,\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}\sqrt{ \left ( x+{\frac{d}{e}} \right ) ^{2}c-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \right ) \left ( x+{\frac{d}{e}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}}+{\frac{e}{d \left ( a{e}^{2}+c{d}^{2} \right ) }\sqrt{ \left ( x+{\frac{d}{e}} \right ) ^{2}c-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \left ( x+{\frac{d}{e}} \right ) ^{-1}}+{\frac{c}{a{e}^{2}+c{d}^{2}}\ln \left ({1 \left ( 2\,{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+2\,\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}\sqrt{ \left ( x+{\frac{d}{e}} \right ) ^{2}c-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \right ) \left ( x+{\frac{d}{e}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.62225, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2*(2*sqrt(c*d^2 + a*e^2)*sqrt(c*x^2 + a)*sqrt(a)*d*e^2 + (2*c*d^3*e + a*d*e^3
+ (2*c*d^2*e^2 + a*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)) + (c*d^3 + a*d*e^2
+ (c*d^2*e + a*e^3)*x)*sqrt(c*d^2 + a*e^2)*log(-((c*x^2 + 2*a)*sqrt(a) - 2*sqrt(
c*x^2 + a)*a)/x^2))/((c*d^5 + a*d^3*e^2 + (c*d^4*e + a*d^2*e^3)*x)*sqrt(c*d^2 +
a*e^2)*sqrt(a)), 1/2*(2*sqrt(-c*d^2 - a*e^2)*sqrt(c*x^2 + a)*sqrt(a)*d*e^2 - 2*(
2*c*d^3*e + a*d*e^3 + (2*c*d^2*e^2 + a*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))) + (c*d^3 + a*d*e^2 + (c*d^2*
e + a*e^3)*x)*sqrt(-c*d^2 - a*e^2)*log(-((c*x^2 + 2*a)*sqrt(a) - 2*sqrt(c*x^2 +
a)*a)/x^2))/((c*d^5 + a*d^3*e^2 + (c*d^4*e + a*d^2*e^3)*x)*sqrt(-c*d^2 - a*e^2)*
sqrt(a)), 1/2*(2*sqrt(c*d^2 + a*e^2)*sqrt(c*x^2 + a)*sqrt(-a)*d*e^2 - 2*(c*d^3 +
a*d*e^2 + (c*d^2*e + a*e^3)*x)*sqrt(c*d^2 + a*e^2)*arctan(sqrt(-a)/sqrt(c*x^2 +
a)) + (2*c*d^3*e + a*d*e^3 + (2*c*d^2*e^2 + a*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)))/((c*d^5 + a*d^3*e^2 + (c*d^4*e + a*d^2*e^3)*x)*sqrt(c*d^2 + a*e^2)*sqr
t(-a)), (sqrt(-c*d^2 - a*e^2)*sqrt(c*x^2 + a)*sqrt(-a)*d*e^2 - (2*c*d^3*e + a*d*
e^3 + (2*c*d^2*e^2 + a*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))) - (c*d^3 + a*d*e^2 + (c*d^2*e + a*e^3)*x)*s
qrt(-c*d^2 - a*e^2)*arctan(sqrt(-a)/sqrt(c*x^2 + a)))/((c*d^5 + a*d^3*e^2 + (c*d
^4*e + a*d^2*e^3)*x)*sqrt(-c*d^2 - a*e^2)*sqrt(-a))]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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