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

Optimal. Leaf size=119 \[ \frac{\sqrt{a+b x^n+c x^{2 n}}}{n}-\frac{\sqrt{a} \tanh ^{-1}\left (\frac{2 a+b x^n}{2 \sqrt{a} \sqrt{a+b x^n+c x^{2 n}}}\right )}{n}+\frac{b \tanh ^{-1}\left (\frac{b+2 c x^n}{2 \sqrt{c} \sqrt{a+b x^n+c x^{2 n}}}\right )}{2 \sqrt{c} n} \]

[Out]

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

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Rubi [A]  time = 0.0952366, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {1357, 734, 843, 621, 206, 724} \[ \frac{\sqrt{a+b x^n+c x^{2 n}}}{n}-\frac{\sqrt{a} \tanh ^{-1}\left (\frac{2 a+b x^n}{2 \sqrt{a} \sqrt{a+b x^n+c x^{2 n}}}\right )}{n}+\frac{b \tanh ^{-1}\left (\frac{b+2 c x^n}{2 \sqrt{c} \sqrt{a+b x^n+c x^{2 n}}}\right )}{2 \sqrt{c} n} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b*x^n + c*x^(2*n)]/x,x]

[Out]

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

Rule 1357

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*x + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[n2, 2*n] && NeQ[
b^2 - 4*a*c, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 734

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(
a + b*x + c*x^2)^p)/(e*(m + 2*p + 1)), x] - Dist[p/(e*(m + 2*p + 1)), Int[(d + e*x)^m*Simp[b*d - 2*a*e + (2*c*
d - b*e)*x, x]*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ
[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && GtQ[p, 0] && NeQ[m + 2*p + 1, 0] && ( !RationalQ[m] || Lt
Q[m, 1]) &&  !ILtQ[m + 2*p, 0] && IntQuadraticQ[a, b, c, d, e, m, p, x]

Rule 843

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + b*x + c*x^
2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
&&  !IGtQ[m, 0]

Rule 621

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 724

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rubi steps

\begin{align*} \int \frac{\sqrt{a+b x^n+c x^{2 n}}}{x} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\sqrt{a+b x+c x^2}}{x} \, dx,x,x^n\right )}{n}\\ &=\frac{\sqrt{a+b x^n+c x^{2 n}}}{n}-\frac{\operatorname{Subst}\left (\int \frac{-2 a-b x}{x \sqrt{a+b x+c x^2}} \, dx,x,x^n\right )}{2 n}\\ &=\frac{\sqrt{a+b x^n+c x^{2 n}}}{n}+\frac{a \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x+c x^2}} \, dx,x,x^n\right )}{n}+\frac{b \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x+c x^2}} \, dx,x,x^n\right )}{2 n}\\ &=\frac{\sqrt{a+b x^n+c x^{2 n}}}{n}-\frac{(2 a) \operatorname{Subst}\left (\int \frac{1}{4 a-x^2} \, dx,x,\frac{2 a+b x^n}{\sqrt{a+b x^n+c x^{2 n}}}\right )}{n}+\frac{b \operatorname{Subst}\left (\int \frac{1}{4 c-x^2} \, dx,x,\frac{b+2 c x^n}{\sqrt{a+b x^n+c x^{2 n}}}\right )}{n}\\ &=\frac{\sqrt{a+b x^n+c x^{2 n}}}{n}-\frac{\sqrt{a} \tanh ^{-1}\left (\frac{2 a+b x^n}{2 \sqrt{a} \sqrt{a+b x^n+c x^{2 n}}}\right )}{n}+\frac{b \tanh ^{-1}\left (\frac{b+2 c x^n}{2 \sqrt{c} \sqrt{a+b x^n+c x^{2 n}}}\right )}{2 \sqrt{c} n}\\ \end{align*}

Mathematica [A]  time = 0.149285, size = 110, normalized size = 0.92 \[ \frac{\sqrt{a+x^n \left (b+c x^n\right )}-\sqrt{a} \tanh ^{-1}\left (\frac{2 a+b x^n}{2 \sqrt{a} \sqrt{a+x^n \left (b+c x^n\right )}}\right )+\frac{b \tanh ^{-1}\left (\frac{b+2 c x^n}{2 \sqrt{c} \sqrt{a+x^n \left (b+c x^n\right )}}\right )}{2 \sqrt{c}}}{n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.108, size = 125, normalized size = 1.1 \begin{align*}{\frac{1}{n}\sqrt{a+b{{\rm e}^{n\ln \left ( x \right ) }}+c \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}}+{\frac{b}{2\,n}\ln \left ({ \left ({\frac{b}{2}}+c{{\rm e}^{n\ln \left ( x \right ) }} \right ){\frac{1}{\sqrt{c}}}}+\sqrt{a+b{{\rm e}^{n\ln \left ( x \right ) }}+c \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}} \right ){\frac{1}{\sqrt{c}}}}-{\frac{1}{n}\sqrt{a}\ln \left ({\frac{1}{{{\rm e}^{n\ln \left ( x \right ) }}} \left ( 2\,a+b{{\rm e}^{n\ln \left ( x \right ) }}+2\,\sqrt{a}\sqrt{a+b{{\rm e}^{n\ln \left ( x \right ) }}+c \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/n*(a+b*exp(n*ln(x))+c*exp(n*ln(x))^2)^(1/2)+1/2/n*b*ln((1/2*b+c*exp(n*ln(x)))/c^(1/2)+(a+b*exp(n*ln(x))+c*ex
p(n*ln(x))^2)^(1/2))/c^(1/2)-1/n*a^(1/2)*ln((2*a+b*exp(n*ln(x))+2*a^(1/2)*(a+b*exp(n*ln(x))+c*exp(n*ln(x))^2)^
(1/2))/exp(n*ln(x)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c x^{2 \, n} + b x^{n} + a}}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^n+c*x^(2*n))^(1/2)/x,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.00975, size = 1554, normalized size = 13.06 \begin{align*} \left [\frac{b \sqrt{c} \log \left (-8 \, c^{2} x^{2 \, n} - 8 \, b c x^{n} - b^{2} - 4 \, a c - 4 \,{\left (2 \, c^{\frac{3}{2}} x^{n} + b \sqrt{c}\right )} \sqrt{c x^{2 \, n} + b x^{n} + a}\right ) + 2 \, \sqrt{a} c \log \left (-\frac{8 \, a b x^{n} + 8 \, a^{2} +{\left (b^{2} + 4 \, a c\right )} x^{2 \, n} - 4 \,{\left (\sqrt{a} b x^{n} + 2 \, a^{\frac{3}{2}}\right )} \sqrt{c x^{2 \, n} + b x^{n} + a}}{x^{2 \, n}}\right ) + 4 \, \sqrt{c x^{2 \, n} + b x^{n} + a} c}{4 \, c n}, -\frac{b \sqrt{-c} \arctan \left (\frac{{\left (2 \, \sqrt{-c} c x^{n} + b \sqrt{-c}\right )} \sqrt{c x^{2 \, n} + b x^{n} + a}}{2 \,{\left (c^{2} x^{2 \, n} + b c x^{n} + a c\right )}}\right ) - \sqrt{a} c \log \left (-\frac{8 \, a b x^{n} + 8 \, a^{2} +{\left (b^{2} + 4 \, a c\right )} x^{2 \, n} - 4 \,{\left (\sqrt{a} b x^{n} + 2 \, a^{\frac{3}{2}}\right )} \sqrt{c x^{2 \, n} + b x^{n} + a}}{x^{2 \, n}}\right ) - 2 \, \sqrt{c x^{2 \, n} + b x^{n} + a} c}{2 \, c n}, \frac{4 \, \sqrt{-a} c \arctan \left (\frac{{\left (\sqrt{-a} b x^{n} + 2 \, \sqrt{-a} a\right )} \sqrt{c x^{2 \, n} + b x^{n} + a}}{2 \,{\left (a c x^{2 \, n} + a b x^{n} + a^{2}\right )}}\right ) + b \sqrt{c} \log \left (-8 \, c^{2} x^{2 \, n} - 8 \, b c x^{n} - b^{2} - 4 \, a c - 4 \,{\left (2 \, c^{\frac{3}{2}} x^{n} + b \sqrt{c}\right )} \sqrt{c x^{2 \, n} + b x^{n} + a}\right ) + 4 \, \sqrt{c x^{2 \, n} + b x^{n} + a} c}{4 \, c n}, \frac{2 \, \sqrt{-a} c \arctan \left (\frac{{\left (\sqrt{-a} b x^{n} + 2 \, \sqrt{-a} a\right )} \sqrt{c x^{2 \, n} + b x^{n} + a}}{2 \,{\left (a c x^{2 \, n} + a b x^{n} + a^{2}\right )}}\right ) - b \sqrt{-c} \arctan \left (\frac{{\left (2 \, \sqrt{-c} c x^{n} + b \sqrt{-c}\right )} \sqrt{c x^{2 \, n} + b x^{n} + a}}{2 \,{\left (c^{2} x^{2 \, n} + b c x^{n} + a c\right )}}\right ) + 2 \, \sqrt{c x^{2 \, n} + b x^{n} + a} c}{2 \, c n}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^n+c*x^(2*n))^(1/2)/x,x, algorithm="fricas")

[Out]

[1/4*(b*sqrt(c)*log(-8*c^2*x^(2*n) - 8*b*c*x^n - b^2 - 4*a*c - 4*(2*c^(3/2)*x^n + b*sqrt(c))*sqrt(c*x^(2*n) +
b*x^n + a)) + 2*sqrt(a)*c*log(-(8*a*b*x^n + 8*a^2 + (b^2 + 4*a*c)*x^(2*n) - 4*(sqrt(a)*b*x^n + 2*a^(3/2))*sqrt
(c*x^(2*n) + b*x^n + a))/x^(2*n)) + 4*sqrt(c*x^(2*n) + b*x^n + a)*c)/(c*n), -1/2*(b*sqrt(-c)*arctan(1/2*(2*sqr
t(-c)*c*x^n + b*sqrt(-c))*sqrt(c*x^(2*n) + b*x^n + a)/(c^2*x^(2*n) + b*c*x^n + a*c)) - sqrt(a)*c*log(-(8*a*b*x
^n + 8*a^2 + (b^2 + 4*a*c)*x^(2*n) - 4*(sqrt(a)*b*x^n + 2*a^(3/2))*sqrt(c*x^(2*n) + b*x^n + a))/x^(2*n)) - 2*s
qrt(c*x^(2*n) + b*x^n + a)*c)/(c*n), 1/4*(4*sqrt(-a)*c*arctan(1/2*(sqrt(-a)*b*x^n + 2*sqrt(-a)*a)*sqrt(c*x^(2*
n) + b*x^n + a)/(a*c*x^(2*n) + a*b*x^n + a^2)) + b*sqrt(c)*log(-8*c^2*x^(2*n) - 8*b*c*x^n - b^2 - 4*a*c - 4*(2
*c^(3/2)*x^n + b*sqrt(c))*sqrt(c*x^(2*n) + b*x^n + a)) + 4*sqrt(c*x^(2*n) + b*x^n + a)*c)/(c*n), 1/2*(2*sqrt(-
a)*c*arctan(1/2*(sqrt(-a)*b*x^n + 2*sqrt(-a)*a)*sqrt(c*x^(2*n) + b*x^n + a)/(a*c*x^(2*n) + a*b*x^n + a^2)) - b
*sqrt(-c)*arctan(1/2*(2*sqrt(-c)*c*x^n + b*sqrt(-c))*sqrt(c*x^(2*n) + b*x^n + a)/(c^2*x^(2*n) + b*c*x^n + a*c)
) + 2*sqrt(c*x^(2*n) + b*x^n + a)*c)/(c*n)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x**n+c*x**(2*n))**(1/2)/x,x)

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c x^{2 \, n} + b x^{n} + a}}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^n+c*x^(2*n))^(1/2)/x,x, algorithm="giac")

[Out]

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