3.385 \(\int \frac{x}{a c+b c x^2+d \sqrt{a+b x^2}} \, dx\)

Optimal. Leaf size=23 \[ \frac{\log \left (c \sqrt{a+b x^2}+d\right )}{b c} \]

[Out]

Log[d + c*Sqrt[a + b*x^2]]/(b*c)

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Rubi [A]  time = 0.136047, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074 \[ \frac{\log \left (c \sqrt{a+b x^2}+d\right )}{b c} \]

Antiderivative was successfully verified.

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

[Out]

Log[d + c*Sqrt[a + b*x^2]]/(b*c)

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Rubi in Sympy [A]  time = 7.2676, size = 17, normalized size = 0.74 \[ \frac{\log{\left (c \sqrt{a + b x^{2}} + d \right )}}{b c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(c*sqrt(a + b*x**2) + d)/(b*c)

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Mathematica [A]  time = 0.0165623, size = 23, normalized size = 1. \[ \frac{\log \left (c \sqrt{a+b x^2}+d\right )}{b c} \]

Antiderivative was successfully verified.

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

[Out]

Log[d + c*Sqrt[a + b*x^2]]/(b*c)

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Maple [B]  time = 0.026, size = 1941, normalized size = 84.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.697275, size = 28, normalized size = 1.22 \[ \frac{\log \left (\sqrt{b x^{2} + a} c + d\right )}{b c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(sqrt(b*x^2 + a)*c + d)/(b*c)

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Fricas [A]  time = 0.306467, size = 142, normalized size = 6.17 \[ \frac{2 \, \log \left (b c^{2} x^{2} + a c^{2} - d^{2}\right ) + \log \left (-\frac{b c^{2} x^{2} + a c^{2} + 2 \, \sqrt{b x^{2} + a} c d + d^{2}}{x^{2}}\right ) - \log \left (-\frac{b c^{2} x^{2} + a c^{2} - 2 \, \sqrt{b x^{2} + a} c d + d^{2}}{x^{2}}\right )}{4 \, b c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(2*log(b*c^2*x^2 + a*c^2 - d^2) + log(-(b*c^2*x^2 + a*c^2 + 2*sqrt(b*x^2 + a
)*c*d + d^2)/x^2) - log(-(b*c^2*x^2 + a*c^2 - 2*sqrt(b*x^2 + a)*c*d + d^2)/x^2))
/(b*c)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.273608, size = 30, normalized size = 1.3 \[ \frac{{\rm ln}\left ({\left | \sqrt{b x^{2} + a} c + d \right |}\right )}{b c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

ln(abs(sqrt(b*x^2 + a)*c + d))/(b*c)