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

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 = 0.085703, 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, Rules used = {2155, 31} \[ \frac{\log \left (c \sqrt{a+b x^2}+d\right )}{b c} \]

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)

Rule 2155

Int[(x_)^(m_.)/((c_) + (d_.)*(x_)^(n_) + (e_.)*Sqrt[(a_) + (b_.)*(x_)^(n_)]), x_Symbol] :> Dist[1/n, Subst[Int
[x^((m + 1)/n - 1)/(c + d*x + e*Sqrt[a + b*x]), x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && EqQ[b*c
- a*d, 0] && IntegerQ[(m + 1)/n]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{x}{a c+b c x^2+d \sqrt{a+b x^2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{a c+b c x+d \sqrt{a+b x}} \, dx,x,x^2\right )\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{d+c x} \, dx,x,\sqrt{a+b x^2}\right )}{b}\\ &=\frac{\log \left (d+c \sqrt{a+b x^2}\right )}{b c}\\ \end{align*}

Mathematica [A]  time = 0.0222406, 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 = 0.015, size = 1931, normalized size = 84. \begin{align*} \text{result too large to display} \end{align*}

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+(-a
*b)^(1/2)/b)^2*b-2*(-a*b)^(1/2)*(x+(-a*b)^(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))*(-a*b)^(1/2)*ln((b*(x+(-a*b)^(1/2)/b)-(-a*b)^(1/2))/b^(1/2)
+((x+(-a*b)^(1/2)/b)^2*b-2*(-a*b)^(1/2)*(x+(-a*b)^(1/2)/b))^(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-(-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)+1/c^2*d^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+b*(x-(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/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)+1/c^2*d^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/(1/c^2*d^2)^(1/
2)*ln((2/c^2*d^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*(1/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)+1
/c^2*d^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)+1/c^2*d^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+b*(x+(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/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)+1/c^2*d^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/(1/c^2*d^2)^(1/2)*ln((2/c^2*d^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*(1/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)+1/c^2*d^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-(-a*b)^(1/2)/b)^2*b+2*(-a*b)^(1/2)*(x-(-a*b)^(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))*(-a*b)^(1/2)*ln((b*(x-(
-a*b)^(1/2)/b)+(-a*b)^(1/2))/b^(1/2)+((x-(-a*b)^(1/2)/b)^2*b+2*(-a*b)^(1/2)*(x-(-a*b)^(1/2)/b))^(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 = 1.57803, size = 28, normalized size = 1.22 \begin{align*} \frac{\log \left (\sqrt{b x^{2} + a} c + d\right )}{b c} \end{align*}

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, algorithm="maxima")

[Out]

log(sqrt(b*x^2 + a)*c + d)/(b*c)

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Fricas [B]  time = 1.2414, size = 227, normalized size = 9.87 \begin{align*} \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} \end{align*}

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, 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 [A]  time = 2.38927, size = 29, normalized size = 1.26 \begin{align*} \frac{\begin{cases} \frac{\sqrt{a + b x^{2}}}{d} & \text{for}\: c = 0 \\\frac{\log{\left (c \sqrt{a + b x^{2}} + d \right )}}{c} & \text{otherwise} \end{cases}}{b} \end{align*}

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]

Piecewise((sqrt(a + b*x**2)/d, Eq(c, 0)), (log(c*sqrt(a + b*x**2) + d)/c, True))/b

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Giac [A]  time = 1.1544, size = 30, normalized size = 1.3 \begin{align*} \frac{\log \left ({\left | \sqrt{b x^{2} + a} c + d \right |}\right )}{b c} \end{align*}

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, algorithm="giac")

[Out]

log(abs(sqrt(b*x^2 + a)*c + d))/(b*c)