3.1.12 \(\int \frac {\sqrt {a^2+2 a b x^3+b^2 x^6}}{x} \, dx\)

Optimal. Leaf size=75 \[ \frac {b x^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{3 \left (a+b x^3\right )}+\frac {a \log (x) \sqrt {a^2+2 a b x^3+b^2 x^6}}{a+b x^3} \]

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1355, 14} \begin {gather*} \frac {b x^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{3 \left (a+b x^3\right )}+\frac {a \log (x) \sqrt {a^2+2 a b x^3+b^2 x^6}}{a+b x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]/x,x]

[Out]

(b*x^3*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(3*(a + b*x^3)) + (a*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]*Log[x])/(a + b*x^
3)

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 1355

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.))^(p_), x_Symbol] :> Dist[(a + b*x^n + c*x^
(2*n))^FracPart[p]/(c^IntPart[p]*(b/2 + c*x^n)^(2*FracPart[p])), Int[(d*x)^m*(b/2 + c*x^n)^(2*p), x], x] /; Fr
eeQ[{a, b, c, d, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2]

Rubi steps

\begin {align*} \int \frac {\sqrt {a^2+2 a b x^3+b^2 x^6}}{x} \, dx &=\frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int \frac {a b+b^2 x^3}{x} \, dx}{a b+b^2 x^3}\\ &=\frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int \left (\frac {a b}{x}+b^2 x^2\right ) \, dx}{a b+b^2 x^3}\\ &=\frac {b x^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{3 \left (a+b x^3\right )}+\frac {a \sqrt {a^2+2 a b x^3+b^2 x^6} \log (x)}{a+b x^3}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 37, normalized size = 0.49 \begin {gather*} \frac {\sqrt {\left (a+b x^3\right )^2} \left (3 a \log (x)+b x^3\right )}{3 \left (a+b x^3\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]/x,x]

[Out]

(Sqrt[(a + b*x^3)^2]*(b*x^3 + 3*a*Log[x]))/(3*(a + b*x^3))

________________________________________________________________________________________

IntegrateAlgebraic [B]  time = 0.24, size = 197, normalized size = 2.63 \begin {gather*} \frac {1}{6} \sqrt {a^2+2 a b x^3+b^2 x^6}+\frac {1}{6} a \log \left (\sqrt {a^2+2 a b x^3+b^2 x^6}-a-\sqrt {b^2} x^3\right )-\frac {a \left (\sqrt {b^2}+b\right ) \log \left (\sqrt {a^2+2 a b x^3+b^2 x^6}+a-\sqrt {b^2} x^3\right )}{6 b}-\frac {a \sqrt {b^2} \log \left (b \sqrt {a^2+2 a b x^3+b^2 x^6}-a b-b \sqrt {b^2} x^3\right )}{6 b}-\frac {1}{6} \sqrt {b^2} x^3 \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]/x,x]

[Out]

-1/6*(Sqrt[b^2]*x^3) + Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]/6 + (a*Log[-a - Sqrt[b^2]*x^3 + Sqrt[a^2 + 2*a*b*x^3 +
b^2*x^6]])/6 - (a*(b + Sqrt[b^2])*Log[a - Sqrt[b^2]*x^3 + Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]])/(6*b) - (a*Sqrt[b^
2]*Log[-(a*b) - b*Sqrt[b^2]*x^3 + b*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]])/(6*b)

________________________________________________________________________________________

fricas [A]  time = 1.18, size = 11, normalized size = 0.15 \begin {gather*} \frac {1}{3} \, b x^{3} + a \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((b*x^3+a)^2)^(1/2)/x,x, algorithm="fricas")

[Out]

1/3*b*x^3 + a*log(x)

________________________________________________________________________________________

giac [A]  time = 0.29, size = 28, normalized size = 0.37 \begin {gather*} \frac {1}{3} \, b x^{3} \mathrm {sgn}\left (b x^{3} + a\right ) + a \log \left ({\left | x \right |}\right ) \mathrm {sgn}\left (b x^{3} + a\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((b*x^3+a)^2)^(1/2)/x,x, algorithm="giac")

[Out]

1/3*b*x^3*sgn(b*x^3 + a) + a*log(abs(x))*sgn(b*x^3 + a)

________________________________________________________________________________________

maple [A]  time = 0.01, size = 34, normalized size = 0.45 \begin {gather*} \frac {\sqrt {\left (b \,x^{3}+a \right )^{2}}\, \left (b \,x^{3}+3 a \ln \relax (x )\right )}{3 b \,x^{3}+3 a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((b*x^3+a)^2)^(1/2)/x,x)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.46, size = 96, normalized size = 1.28 \begin {gather*} \frac {1}{3} \, \left (-1\right )^{2 \, b^{2} x^{3} + 2 \, a b} a \log \left (2 \, b^{2} x^{3} + 2 \, a b\right ) - \frac {1}{3} \, \left (-1\right )^{2 \, a b x^{3} + 2 \, a^{2}} a \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{x^{2} {\left | x \right |}}\right ) + \frac {1}{3} \, \sqrt {b^{2} x^{6} + 2 \, a b x^{3} + a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((b*x^3+a)^2)^(1/2)/x,x, algorithm="maxima")

[Out]

1/3*(-1)^(2*b^2*x^3 + 2*a*b)*a*log(2*b^2*x^3 + 2*a*b) - 1/3*(-1)^(2*a*b*x^3 + 2*a^2)*a*log(2*a*b*x/abs(x) + 2*
a^2/(x^2*abs(x))) + 1/3*sqrt(b^2*x^6 + 2*a*b*x^3 + a^2)

________________________________________________________________________________________

mupad [B]  time = 1.38, size = 109, normalized size = 1.45 \begin {gather*} \frac {\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{3}-\frac {\ln \left (a\,b+\frac {a^2}{x^3}+\frac {\sqrt {a^2}\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{x^3}\right )\,\sqrt {a^2}}{3}+\frac {a\,b\,\ln \left (a\,b+\sqrt {{\left (b\,x^3+a\right )}^2}\,\sqrt {b^2}+b^2\,x^3\right )}{3\,\sqrt {b^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*x^3)^2)^(1/2)/x,x)

[Out]

(a^2 + b^2*x^6 + 2*a*b*x^3)^(1/2)/3 - (log(a*b + a^2/x^3 + ((a^2)^(1/2)*(a^2 + b^2*x^6 + 2*a*b*x^3)^(1/2))/x^3
)*(a^2)^(1/2))/3 + (a*b*log(a*b + ((a + b*x^3)^2)^(1/2)*(b^2)^(1/2) + b^2*x^3))/(3*(b^2)^(1/2))

________________________________________________________________________________________

sympy [A]  time = 0.13, size = 10, normalized size = 0.13 \begin {gather*} a \log {\relax (x )} + \frac {b x^{3}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((b*x**3+a)**2)**(1/2)/x,x)

[Out]

a*log(x) + b*x**3/3

________________________________________________________________________________________