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

Optimal. Leaf size=330 \[ \frac{f \left (\sqrt{e^2-4 d f}+e\right ) \tanh ^{-1}\left (\frac{2 a f-c x \left (e-\sqrt{e^2-4 d f}\right )}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left (-e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt{2} d \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (-e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}-\frac{f \left (e-\sqrt{e^2-4 d f}\right ) \tanh ^{-1}\left (\frac{2 a f-c x \left (\sqrt{e^2-4 d f}+e\right )}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt{2} d \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{\sqrt{a} d} \]

[Out]

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

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Rubi [A]  time = 1.85615, antiderivative size = 330, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259 \[ \frac{f \left (\sqrt{e^2-4 d f}+e\right ) \tanh ^{-1}\left (\frac{2 a f-c x \left (e-\sqrt{e^2-4 d f}\right )}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left (-e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt{2} d \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (-e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}-\frac{f \left (e-\sqrt{e^2-4 d f}\right ) \tanh ^{-1}\left (\frac{2 a f-c x \left (\sqrt{e^2-4 d f}+e\right )}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt{2} d \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{\sqrt{a} d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 131.601, size = 325, normalized size = 0.98 \[ - \frac{\sqrt{2} f \left (e - \sqrt{- 4 d f + e^{2}}\right ) \operatorname{atanh}{\left (\frac{\sqrt{2} \left (2 a f - c x \left (e + \sqrt{- 4 d f + e^{2}}\right )\right )}{2 \sqrt{a + c x^{2}} \sqrt{2 a f^{2} - 2 c d f + c e^{2} + c e \sqrt{- 4 d f + e^{2}}}} \right )}}{2 d \sqrt{- 4 d f + e^{2}} \sqrt{2 a f^{2} - 2 c d f + c e^{2} + c e \sqrt{- 4 d f + e^{2}}}} + \frac{\sqrt{2} f \left (e + \sqrt{- 4 d f + e^{2}}\right ) \operatorname{atanh}{\left (\frac{\sqrt{2} \left (2 a f - c x \left (e - \sqrt{- 4 d f + e^{2}}\right )\right )}{2 \sqrt{a + c x^{2}} \sqrt{2 a f^{2} - 2 c d f + c e^{2} - c e \sqrt{- 4 d f + e^{2}}}} \right )}}{2 d \sqrt{- 4 d f + e^{2}} \sqrt{2 a f^{2} - 2 c d f + c e^{2} - c e \sqrt{- 4 d f + e^{2}}}} - \frac{\operatorname{atanh}{\left (\frac{\sqrt{a + c x^{2}}}{\sqrt{a}} \right )}}{\sqrt{a} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(2)*f*(e - sqrt(-4*d*f + e**2))*atanh(sqrt(2)*(2*a*f - c*x*(e + sqrt(-4*d*f
+ e**2)))/(2*sqrt(a + c*x**2)*sqrt(2*a*f**2 - 2*c*d*f + c*e**2 + c*e*sqrt(-4*d*
f + e**2))))/(2*d*sqrt(-4*d*f + e**2)*sqrt(2*a*f**2 - 2*c*d*f + c*e**2 + c*e*sqr
t(-4*d*f + e**2))) + sqrt(2)*f*(e + sqrt(-4*d*f + e**2))*atanh(sqrt(2)*(2*a*f -
c*x*(e - sqrt(-4*d*f + e**2)))/(2*sqrt(a + c*x**2)*sqrt(2*a*f**2 - 2*c*d*f + c*e
**2 - c*e*sqrt(-4*d*f + e**2))))/(2*d*sqrt(-4*d*f + e**2)*sqrt(2*a*f**2 - 2*c*d*
f + c*e**2 - c*e*sqrt(-4*d*f + e**2))) - atanh(sqrt(a + c*x**2)/sqrt(a))/(sqrt(a
)*d)

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Mathematica [A]  time = 3.4225, size = 513, normalized size = 1.55 \[ \frac{\frac{\sqrt{2} f \left (\sqrt{e^2-4 d f}+e\right ) \log \left (\sqrt{a+c x^2} \sqrt{4 a f^2-2 c e \sqrt{e^2-4 d f}-4 c d f+2 c e^2}+2 a f+c x \left (\sqrt{e^2-4 d f}-e\right )\right )}{\sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (-e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}+\frac{\sqrt{2} f \left (\sqrt{e^2-4 d f}-e\right ) \log \left (-\sqrt{a+c x^2} \sqrt{4 a f^2+2 c \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )}-2 a f+c x \sqrt{e^2-4 d f}+c e x\right )}{\sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}-\frac{\sqrt{2} f \left (\sqrt{e^2-4 d f}+e\right ) \log \left (\sqrt{e^2-4 d f}-e-2 f x\right )}{\sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (-e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}-\frac{\sqrt{2} f \left (\sqrt{e^2-4 d f}-e\right ) \log \left (\sqrt{e^2-4 d f}+e+2 f x\right )}{\sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}-\frac{2 \log \left (\sqrt{a} \sqrt{a+c x^2}+a\right )}{\sqrt{a}}+\frac{2 \log (x)}{\sqrt{a}}}{2 d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.021, size = 681, normalized size = 2.1 \[ 4\,{\frac{f}{ \left ( -e+\sqrt{-4\,df+{e}^{2}} \right ) \left ( e+\sqrt{-4\,df+{e}^{2}} \right ) \sqrt{a}}\ln \left ({\frac{2\,a+2\,\sqrt{a}\sqrt{c{x}^{2}+a}}{x}} \right ) }-2\,{\frac{f\sqrt{2}}{ \left ( -e+\sqrt{-4\,df+{e}^{2}} \right ) \sqrt{-4\,df+{e}^{2}}}\ln \left ({1 \left ({\frac{-\sqrt{-4\,df+{e}^{2}}ce+2\,a{f}^{2}-2\,cdf+{e}^{2}c}{{f}^{2}}}-{\frac{c \left ( e-\sqrt{-4\,df+{e}^{2}} \right ) }{f} \left ( x-1/2\,{\frac{-e+\sqrt{-4\,df+{e}^{2}}}{f}} \right ) }+1/2\,\sqrt{2}\sqrt{{\frac{-\sqrt{-4\,df+{e}^{2}}ce+2\,a{f}^{2}-2\,cdf+{e}^{2}c}{{f}^{2}}}}\sqrt{4\, \left ( x-1/2\,{\frac{-e+\sqrt{-4\,df+{e}^{2}}}{f}} \right ) ^{2}c-4\,{\frac{c \left ( e-\sqrt{-4\,df+{e}^{2}} \right ) }{f} \left ( x-1/2\,{\frac{-e+\sqrt{-4\,df+{e}^{2}}}{f}} \right ) }+2\,{\frac{-\sqrt{-4\,df+{e}^{2}}ce+2\,a{f}^{2}-2\,cdf+{e}^{2}c}{{f}^{2}}}} \right ) \left ( x-1/2\,{\frac{-e+\sqrt{-4\,df+{e}^{2}}}{f}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{-\sqrt{-4\,df+{e}^{2}}ce+2\,a{f}^{2}-2\,cdf+{e}^{2}c}{{f}^{2}}}}}}}-2\,{\frac{f\sqrt{2}}{ \left ( e+\sqrt{-4\,df+{e}^{2}} \right ) \sqrt{-4\,df+{e}^{2}}}\ln \left ({1 \left ({\frac{\sqrt{-4\,df+{e}^{2}}ce+2\,a{f}^{2}-2\,cdf+{e}^{2}c}{{f}^{2}}}-{\frac{c \left ( e+\sqrt{-4\,df+{e}^{2}} \right ) }{f} \left ( x+1/2\,{\frac{e+\sqrt{-4\,df+{e}^{2}}}{f}} \right ) }+1/2\,\sqrt{2}\sqrt{{\frac{\sqrt{-4\,df+{e}^{2}}ce+2\,a{f}^{2}-2\,cdf+{e}^{2}c}{{f}^{2}}}}\sqrt{4\, \left ( x+1/2\,{\frac{e+\sqrt{-4\,df+{e}^{2}}}{f}} \right ) ^{2}c-4\,{\frac{c \left ( e+\sqrt{-4\,df+{e}^{2}} \right ) }{f} \left ( x+1/2\,{\frac{e+\sqrt{-4\,df+{e}^{2}}}{f}} \right ) }+2\,{\frac{\sqrt{-4\,df+{e}^{2}}ce+2\,a{f}^{2}-2\,cdf+{e}^{2}c}{{f}^{2}}}} \right ) \left ( x+1/2\,{\frac{e+\sqrt{-4\,df+{e}^{2}}}{f}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{\sqrt{-4\,df+{e}^{2}}ce+2\,a{f}^{2}-2\,cdf+{e}^{2}c}{{f}^{2}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*f/(-e+(-4*d*f+e^2)^(1/2))/(e+(-4*d*f+e^2)^(1/2))/a^(1/2)*ln((2*a+2*a^(1/2)*(c*
x^2+a)^(1/2))/x)-2*f/(-e+(-4*d*f+e^2)^(1/2))/(-4*d*f+e^2)^(1/2)*2^(1/2)/((-(-4*d
*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+e^2*c)/f^2)^(1/2)*ln(((-(-4*d*f+e^2)^(1/2)*c*e
+2*a*f^2-2*c*d*f+e^2*c)/f^2-c*(e-(-4*d*f+e^2)^(1/2))/f*(x-1/2*(-e+(-4*d*f+e^2)^(
1/2))/f)+1/2*2^(1/2)*((-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+e^2*c)/f^2)^(1/2)
*(4*(x-1/2*(-e+(-4*d*f+e^2)^(1/2))/f)^2*c-4*c*(e-(-4*d*f+e^2)^(1/2))/f*(x-1/2*(-
e+(-4*d*f+e^2)^(1/2))/f)+2*(-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+e^2*c)/f^2)^
(1/2))/(x-1/2*(-e+(-4*d*f+e^2)^(1/2))/f))-2*f/(e+(-4*d*f+e^2)^(1/2))/(-4*d*f+e^2
)^(1/2)*2^(1/2)/(((-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+e^2*c)/f^2)^(1/2)*ln(((
(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+e^2*c)/f^2-c*(e+(-4*d*f+e^2)^(1/2))/f*(x+
1/2*(e+(-4*d*f+e^2)^(1/2))/f)+1/2*2^(1/2)*(((-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d
*f+e^2*c)/f^2)^(1/2)*(4*(x+1/2*(e+(-4*d*f+e^2)^(1/2))/f)^2*c-4*c*(e+(-4*d*f+e^2)
^(1/2))/f*(x+1/2*(e+(-4*d*f+e^2)^(1/2))/f)+2*((-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c
*d*f+e^2*c)/f^2)^(1/2))/(x+1/2*(e+(-4*d*f+e^2)^(1/2))/f))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [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/(sqrt(c*x^2 + a)*(f*x^2 + e*x + d)*x),x, algorithm="fricas")

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x*sqrt(a + c*x**2)*(d + e*x + f*x**2)), 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/(sqrt(c*x^2 + a)*(f*x^2 + e*x + d)*x),x, algorithm="giac")

[Out]

Timed out