3.6.46 \(\int \frac {x}{a c+b c x^2+d \sqrt {a+b x^2}} \, dx\) [546]

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

[Out]

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

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Rubi [A]
time = 0.06, antiderivative size = 23, normalized size of antiderivative = 1.00, 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 = {2186, 31} \begin {gather*} \frac {\log \left (c \sqrt {a+b x^2}+d\right )}{b c} \end {gather*}

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 31

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

Rule 2186

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]

Rubi steps

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

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Mathematica [A]
time = 0.02, size = 26, normalized size = 1.13 \begin {gather*} \frac {\log \left (b d+b c \sqrt {a+b x^2}\right )}{b c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Log[b*d + b*c*Sqrt[a + b*x^2]]/(b*c)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1930\) vs. \(2(21)=42\).
time = 0.03, size = 1931, normalized size = 83.96

method	result	size
default	\(\text {Expression too large to display}\)	\(1931\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.28, size = 21, normalized size = 0.91 \begin {gather*} \frac {\log \left (\sqrt {b x^{2} + a} c + d\right )}{b c} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 105 vs. \(2 (21) = 42\).
time = 0.37, size = 105, normalized size = 4.57 \begin {gather*} \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 {gather*}

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 = 1.94, size = 29, normalized size = 1.26 \begin {gather*} \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 {gather*}

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 = 3.33, size = 22, normalized size = 0.96 \begin {gather*} \frac {\log \left ({\left | \sqrt {b x^{2} + a} c + d \right |}\right )}{b c} \end {gather*}

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)

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Mupad [B]
time = 3.47, size = 45, normalized size = 1.96 \begin {gather*} \frac {\mathrm {atanh}\left (\frac {c\,\sqrt {b\,x^2+a}}{d}\right )+\frac {\ln \left (b\,c^2\,x^2+a\,c^2-d^2\right )}{2}}{b\,c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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