\(\int \frac {\sqrt {a^2+2 a b x^3+b^2 x^6}}{x^4} \, dx\) [15]

Optimal result

Integrand size = 26, antiderivative size = 75 \[ \int \frac {\sqrt {a^2+2 a b x^3+b^2 x^6}}{x^4} \, dx=-\frac {a \sqrt {a^2+2 a b x^3+b^2 x^6}}{3 x^3 \left (a+b x^3\right )}+\frac {b \sqrt {a^2+2 a b x^3+b^2 x^6} \log (x)}{a+b x^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1369, 14} \[ \int \frac {\sqrt {a^2+2 a b x^3+b^2 x^6}}{x^4} \, dx=\frac {b \log (x) \sqrt {a^2+2 a b x^3+b^2 x^6}}{a+b x^3}-\frac {a \sqrt {a^2+2 a b x^3+b^2 x^6}}{3 x^3 \left (a+b x^3\right )} \]

[In]

Int[Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]/x^4,x]

[Out]

-1/3*(a*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(x^3*(a + b*x^3)) + (b*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 1369

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*} \text {integral}& = \frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int \frac {a b+b^2 x^3}{x^4} \, 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^4}+\frac {b^2}{x}\right ) \, dx}{a b+b^2 x^3} \\ & = -\frac {a \sqrt {a^2+2 a b x^3+b^2 x^6}}{3 x^3 \left (a+b x^3\right )}+\frac {b \sqrt {a^2+2 a b x^3+b^2 x^6} \log (x)}{a+b x^3} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(178\) vs. \(2(75)=150\).

Time = 0.22 (sec) , antiderivative size = 178, normalized size of antiderivative = 2.37 \[ \int \frac {\sqrt {a^2+2 a b x^3+b^2 x^6}}{x^4} \, dx=\frac {a \sqrt {a^2}-a \sqrt {\left (a+b x^3\right )^2}-2 a b x^3 \text {arctanh}\left (\frac {b x^3}{\sqrt {a^2}-\sqrt {\left (a+b x^3\right )^2}}\right )-2 \sqrt {a^2} b x^3 \log \left (x^3\right )+\sqrt {a^2} b x^3 \log \left (a \left (\sqrt {a^2}-b x^3-\sqrt {\left (a+b x^3\right )^2}\right )\right )+\sqrt {a^2} b x^3 \log \left (a \left (\sqrt {a^2}+b x^3-\sqrt {\left (a+b x^3\right )^2}\right )\right )}{6 a x^3} \]

[In]

Integrate[Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]/x^4,x]

[Out]

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

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.07 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.37

method	result	size
pseudoelliptic	\(-\frac {\operatorname {csgn}\left (b \,x^{3}+a \right ) \left (-\ln \left (b \,x^{3}\right ) b \,x^{3}+a \right )}{3 x^{3}}\)	\(28\)
default	\(\frac {\sqrt {\left (b \,x^{3}+a \right )^{2}}\, \left (3 b \ln \left (x \right ) x^{3}-a \right )}{3 x^{3} \left (b \,x^{3}+a \right )}\)	\(38\)
risch	\(-\frac {a \sqrt {\left (b \,x^{3}+a \right )^{2}}}{3 x^{3} \left (b \,x^{3}+a \right )}+\frac {b \ln \left (x \right ) \sqrt {\left (b \,x^{3}+a \right )^{2}}}{b \,x^{3}+a}\)	\(52\)

[In]

int(((b*x^3+a)^2)^(1/2)/x^4,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.23 \[ \int \frac {\sqrt {a^2+2 a b x^3+b^2 x^6}}{x^4} \, dx=\frac {3 \, b x^{3} \log \left (x\right ) - a}{3 \, x^{3}} \]

[In]

integrate(((b*x^3+a)^2)^(1/2)/x^4,x, algorithm="fricas")

[Out]

1/3*(3*b*x^3*log(x) - a)/x^3

Sympy [F(-1)]

Timed out. \[ \int \frac {\sqrt {a^2+2 a b x^3+b^2 x^6}}{x^4} \, dx=\text {Timed out} \]

[In]

integrate(((b*x**3+a)**2)**(1/2)/x**4,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.32 \[ \int \frac {\sqrt {a^2+2 a b x^3+b^2 x^6}}{x^4} \, dx=\frac {1}{3} \, \left (-1\right )^{2 \, b^{2} x^{3} + 2 \, a b} b \log \left (2 \, b^{2} x^{3} + 2 \, a b\right ) - \frac {1}{3} \, \left (-1\right )^{2 \, a b x^{3} + 2 \, a^{2}} b \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{x^{2} {\left | x \right |}}\right ) - \frac {\sqrt {b^{2} x^{6} + 2 \, a b x^{3} + a^{2}}}{3 \, x^{3}} \]

[In]

integrate(((b*x^3+a)^2)^(1/2)/x^4,x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.57 \[ \int \frac {\sqrt {a^2+2 a b x^3+b^2 x^6}}{x^4} \, dx=b \log \left ({\left | x \right |}\right ) \mathrm {sgn}\left (b x^{3} + a\right ) - \frac {b x^{3} \mathrm {sgn}\left (b x^{3} + a\right ) + a \mathrm {sgn}\left (b x^{3} + a\right )}{3 \, x^{3}} \]

[In]

integrate(((b*x^3+a)^2)^(1/2)/x^4,x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 8.36 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.49 \[ \int \frac {\sqrt {a^2+2 a b x^3+b^2 x^6}}{x^4} \, dx=\frac {\ln \left (a\,b+\sqrt {{\left (b\,x^3+a\right )}^2}\,\sqrt {b^2}+b^2\,x^3\right )\,\sqrt {b^2}}{3}-\frac {\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{3\,x^3}-\frac {a\,b\,\ln \left (a\,b+\frac {a^2}{x^3}+\frac {\sqrt {a^2}\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{x^3}\right )}{3\,\sqrt {a^2}} \]

[In]

int(((a + b*x^3)^2)^(1/2)/x^4,x)

[Out]

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