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

Optimal. Leaf size=88 \[ -\frac{c \log \left (c \sqrt{a+b x^2}+d\right )}{a c^2-d^2}+\frac{d \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{\sqrt{a} \left (a c^2-d^2\right )}+\frac{c \log (x)}{a c^2-d^2} \]

[Out]

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

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Rubi [A]  time = 0.415814, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207 \[ -\frac{c \log \left (c \sqrt{a+b x^2}+d\right )}{a c^2-d^2}+\frac{d \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{\sqrt{a} \left (a c^2-d^2\right )}+\frac{c \log (x)}{a c^2-d^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [C]  time = 0.91718, size = 282, normalized size = 3.2 \[ -\frac{c \log \left (a c^2+b c^2 x^2-d^2\right )+c \log \left (\frac{\left (a c^2-d^2\right ) \left (-i \sqrt{b} x \sqrt{a c^2-d^2}+d \sqrt{a+b x^2}+a c\right )}{\sqrt{b} c d^2 \left (\sqrt{b} c x+i \sqrt{a c^2-d^2}\right )}\right )+c \log \left (\frac{\left (a c^2-d^2\right ) \left (i \sqrt{b} x \sqrt{a c^2-d^2}+d \sqrt{a+b x^2}+a c\right )}{\sqrt{b} c d^2 \left (\sqrt{b} c x-i \sqrt{a c^2-d^2}\right )}\right )-\frac{2 d \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )}{\sqrt{a}}+\log (x) \left (\frac{2 d}{\sqrt{a}}-2 c\right )+c \log (4)}{2 a c^2-2 d^2} \]

Antiderivative was successfully verified.

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

[Out]

-((c*Log[4] + (-2*c + (2*d)/Sqrt[a])*Log[x] + c*Log[a*c^2 - d^2 + b*c^2*x^2] - (
2*d*Log[a + Sqrt[a]*Sqrt[a + b*x^2]])/Sqrt[a] + c*Log[((a*c^2 - d^2)*(a*c - I*Sq
rt[b]*Sqrt[a*c^2 - d^2]*x + d*Sqrt[a + b*x^2]))/(Sqrt[b]*c*d^2*(I*Sqrt[a*c^2 - d
^2] + Sqrt[b]*c*x))] + c*Log[((a*c^2 - d^2)*(a*c + I*Sqrt[b]*Sqrt[a*c^2 - d^2]*x
+ d*Sqrt[a + b*x^2]))/(Sqrt[b]*c*d^2*((-I)*Sqrt[a*c^2 - d^2] + Sqrt[b]*c*x))])/
(2*a*c^2 - 2*d^2))

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Maple [B]  time = 0.042, size = 2175, normalized size = 24.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c*ln(x)/(a*c^2-d^2)-1/2*a*c^3/(a*c^2-d^2)/d^2*ln(b*c^2*x^2+a*c^2-d^2)+1/2*c/d^2*
ln(b*c^2*x^2+a*c^2-d^2)+d/a^(1/2)/(a*c^2-d^2)*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))
/x)-d/a/(a*c^2-d^2)*(b*x^2+a)^(1/2)+1/2*d*b*c^2/a/((-a*b)^(1/2)*c^2+(-c^2*b*(a*c
^2-d^2))^(1/2))/((-a*b)^(1/2)*c^2-(-c^2*b*(a*c^2-d^2))^(1/2))*((x-1/b*(-a*b)^(1/
2))^2*b+2*(-a*b)^(1/2)*(x-1/b*(-a*b)^(1/2)))^(1/2)+1/2*d*b^(1/2)*c^2/a/((-a*b)^(
1/2)*c^2+(-c^2*b*(a*c^2-d^2))^(1/2))/((-a*b)^(1/2)*c^2-(-c^2*b*(a*c^2-d^2))^(1/2
))*(-a*b)^(1/2)*ln(((x-1/b*(-a*b)^(1/2))*b+(-a*b)^(1/2))/b^(1/2)+((x-1/b*(-a*b)^
(1/2))^2*b+2*(-a*b)^(1/2)*(x-1/b*(-a*b)^(1/2)))^(1/2))+1/2*d*b*c^2/a/((-a*b)^(1/
2)*c^2+(-c^2*b*(a*c^2-d^2))^(1/2))/((-a*b)^(1/2)*c^2-(-c^2*b*(a*c^2-d^2))^(1/2))
*((x+1/b*(-a*b)^(1/2))^2*b-2*(-a*b)^(1/2)*(x+1/b*(-a*b)^(1/2)))^(1/2)-1/2*d*b^(1
/2)*c^2/a/((-a*b)^(1/2)*c^2+(-c^2*b*(a*c^2-d^2))^(1/2))/((-a*b)^(1/2)*c^2-(-c^2*
b*(a*c^2-d^2))^(1/2))*(-a*b)^(1/2)*ln(((x+1/b*(-a*b)^(1/2))*b-(-a*b)^(1/2))/b^(1
/2)+((x+1/b*(-a*b)^(1/2))^2*b-2*(-a*b)^(1/2)*(x+1/b*(-a*b)^(1/2)))^(1/2))-1/2*d*
b*c^4/(a*c^2-d^2)/((-a*b)^(1/2)*c^2+(-c^2*b*(a*c^2-d^2))^(1/2))/((-a*b)^(1/2)*c^
2-(-c^2*b*(a*c^2-d^2))^(1/2))*((x-(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)^2*b+2*(-c^2*
b*(a*c^2-d^2))^(1/2)/c^2*(x-(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)+d^2/c^2)^(1/2)-1/2
*d*b^(1/2)*c^2/(a*c^2-d^2)/((-a*b)^(1/2)*c^2+(-c^2*b*(a*c^2-d^2))^(1/2))/((-a*b)
^(1/2)*c^2-(-c^2*b*(a*c^2-d^2))^(1/2))*(-c^2*b*(a*c^2-d^2))^(1/2)*ln(((-c^2*b*(a
*c^2-d^2))^(1/2)/c^2+(x-(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)*b)/b^(1/2)+((x-(-c^2*b
*(a*c^2-d^2))^(1/2)/c^2/b)^2*b+2*(-c^2*b*(a*c^2-d^2))^(1/2)/c^2*(x-(-c^2*b*(a*c^
2-d^2))^(1/2)/c^2/b)+d^2/c^2)^(1/2))+1/2*b*c^2/(a*c^2-d^2)/((-a*b)^(1/2)*c^2+(-c
^2*b*(a*c^2-d^2))^(1/2))/((-a*b)^(1/2)*c^2-(-c^2*b*(a*c^2-d^2))^(1/2))*d^3/(d^2/
c^2)^(1/2)*ln((2*d^2/c^2+2*(-c^2*b*(a*c^2-d^2))^(1/2)/c^2*(x-(-c^2*b*(a*c^2-d^2)
)^(1/2)/c^2/b)+2*(d^2/c^2)^(1/2)*((x-(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)^2*b+2*(-c
^2*b*(a*c^2-d^2))^(1/2)/c^2*(x-(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)+d^2/c^2)^(1/2))
/(x-(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b))-1/2*d*b*c^4/(a*c^2-d^2)/((-a*b)^(1/2)*c^2
+(-c^2*b*(a*c^2-d^2))^(1/2))/((-a*b)^(1/2)*c^2-(-c^2*b*(a*c^2-d^2))^(1/2))*((x+(
-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)^2*b-2*(-c^2*b*(a*c^2-d^2))^(1/2)/c^2*(x+(-c^2*b
*(a*c^2-d^2))^(1/2)/c^2/b)+d^2/c^2)^(1/2)+1/2*d*b^(1/2)*c^2/(a*c^2-d^2)/((-a*b)^
(1/2)*c^2+(-c^2*b*(a*c^2-d^2))^(1/2))/((-a*b)^(1/2)*c^2-(-c^2*b*(a*c^2-d^2))^(1/
2))*(-c^2*b*(a*c^2-d^2))^(1/2)*ln((-(-c^2*b*(a*c^2-d^2))^(1/2)/c^2+(x+(-c^2*b*(a
*c^2-d^2))^(1/2)/c^2/b)*b)/b^(1/2)+((x+(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)^2*b-2*(
-c^2*b*(a*c^2-d^2))^(1/2)/c^2*(x+(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)+d^2/c^2)^(1/2
))+1/2*b*c^2/(a*c^2-d^2)/((-a*b)^(1/2)*c^2+(-c^2*b*(a*c^2-d^2))^(1/2))/((-a*b)^(
1/2)*c^2-(-c^2*b*(a*c^2-d^2))^(1/2))*d^3/(d^2/c^2)^(1/2)*ln((2*d^2/c^2-2*(-c^2*b
*(a*c^2-d^2))^(1/2)/c^2*(x+(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)+2*(d^2/c^2)^(1/2)*(
(x+(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)^2*b-2*(-c^2*b*(a*c^2-d^2))^(1/2)/c^2*(x+(-c
^2*b*(a*c^2-d^2))^(1/2)/c^2/b)+d^2/c^2)^(1/2))/(x+(-c^2*b*(a*c^2-d^2))^(1/2)/c^2
/b))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.374207, size = 1, normalized size = 0.01 \[ \left [-\frac{\sqrt{a} c \log \left (-\frac{b c^{2} x^{2} + a c^{2} + 2 \, \sqrt{b x^{2} + a} c d + d^{2}}{x^{2}}\right ) - \sqrt{a} c \log \left (-\frac{b c^{2} x^{2} + a c^{2} - 2 \, \sqrt{b x^{2} + a} c d + d^{2}}{x^{2}}\right ) + 2 \, d \log \left (-\frac{{\left (b x^{2} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x^{2} + a} a}{x^{2}}\right ) + 2 \,{\left (c \log \left (b c^{2} x^{2} + a c^{2} - d^{2}\right ) - 2 \, c \log \left (x\right )\right )} \sqrt{a}}{4 \,{\left (a c^{2} - d^{2}\right )} \sqrt{a}}, -\frac{\sqrt{-a} c \log \left (-\frac{b c^{2} x^{2} + a c^{2} + 2 \, \sqrt{b x^{2} + a} c d + d^{2}}{x^{2}}\right ) - \sqrt{-a} c \log \left (-\frac{b c^{2} x^{2} + a c^{2} - 2 \, \sqrt{b x^{2} + a} c d + d^{2}}{x^{2}}\right ) - 4 \, d \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right ) + 2 \,{\left (c \log \left (b c^{2} x^{2} + a c^{2} - d^{2}\right ) - 2 \, c \log \left (x\right )\right )} \sqrt{-a}}{4 \,{\left (a c^{2} - d^{2}\right )} \sqrt{-a}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/4*(sqrt(a)*c*log(-(b*c^2*x^2 + a*c^2 + 2*sqrt(b*x^2 + a)*c*d + d^2)/x^2) - s
qrt(a)*c*log(-(b*c^2*x^2 + a*c^2 - 2*sqrt(b*x^2 + a)*c*d + d^2)/x^2) + 2*d*log(-
((b*x^2 + 2*a)*sqrt(a) - 2*sqrt(b*x^2 + a)*a)/x^2) + 2*(c*log(b*c^2*x^2 + a*c^2
- d^2) - 2*c*log(x))*sqrt(a))/((a*c^2 - d^2)*sqrt(a)), -1/4*(sqrt(-a)*c*log(-(b*
c^2*x^2 + a*c^2 + 2*sqrt(b*x^2 + a)*c*d + d^2)/x^2) - sqrt(-a)*c*log(-(b*c^2*x^2
+ a*c^2 - 2*sqrt(b*x^2 + a)*c*d + d^2)/x^2) - 4*d*arctan(sqrt(-a)/sqrt(b*x^2 +
a)) + 2*(c*log(b*c^2*x^2 + a*c^2 - d^2) - 2*c*log(x))*sqrt(-a))/((a*c^2 - d^2)*s
qrt(-a))]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.281115, size = 127, normalized size = 1.44 \[ -\frac{c^{2}{\rm ln}\left ({\left | \sqrt{b x^{2} + a} c + d \right |}\right )}{a c^{3} - c d^{2}} + \frac{c{\rm ln}\left (b x^{2}\right )}{2 \,{\left (a c^{2} - d^{2}\right )}} - \frac{d \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{{\left (a c^{2} - d^{2}\right )} \sqrt{-a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-c^2*ln(abs(sqrt(b*x^2 + a)*c + d))/(a*c^3 - c*d^2) + 1/2*c*ln(b*x^2)/(a*c^2 - d
^2) - d*arctan(sqrt(b*x^2 + a)/sqrt(-a))/((a*c^2 - d^2)*sqrt(-a))