3.1.97 \(\int \frac {1}{x^4 \sqrt {a^2+2 a b x^3+b^2 x^6}} \, dx\)

Optimal. Leaf size=125 \[ \frac {-a-b x^3}{3 a x^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {b \log (x) \left (a+b x^3\right )}{a^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {b \left (a+b x^3\right ) \log \left (a+b x^3\right )}{3 a^2 \sqrt {a^2+2 a b x^3+b^2 x^6}} \]

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Rubi [A]  time = 0.05, antiderivative size = 122, normalized size of antiderivative = 0.98, number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1355, 266, 44} \begin {gather*} -\frac {a+b x^3}{3 a x^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {b \log (x) \left (a+b x^3\right )}{a^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {b \left (a+b x^3\right ) \log \left (a+b x^3\right )}{3 a^2 \sqrt {a^2+2 a b x^3+b^2 x^6}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^4*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]),x]

[Out]

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

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

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 {1}{x^4 \sqrt {a^2+2 a b x^3+b^2 x^6}} \, dx &=\frac {\left (a b+b^2 x^3\right ) \int \frac {1}{x^4 \left (a b+b^2 x^3\right )} \, dx}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\\ &=\frac {\left (a b+b^2 x^3\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \left (a b+b^2 x\right )} \, dx,x,x^3\right )}{3 \sqrt {a^2+2 a b x^3+b^2 x^6}}\\ &=\frac {\left (a b+b^2 x^3\right ) \operatorname {Subst}\left (\int \left (\frac {1}{a b x^2}-\frac {1}{a^2 x}+\frac {b}{a^2 (a+b x)}\right ) \, dx,x,x^3\right )}{3 \sqrt {a^2+2 a b x^3+b^2 x^6}}\\ &=-\frac {a+b x^3}{3 a x^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {b \left (a+b x^3\right ) \log (x)}{a^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {b \left (a+b x^3\right ) \log \left (a+b x^3\right )}{3 a^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 54, normalized size = 0.43 \begin {gather*} -\frac {\left (a+b x^3\right ) \left (-b x^3 \log \left (a+b x^3\right )+a+3 b x^3 \log (x)\right )}{3 a^2 x^3 \sqrt {\left (a+b x^3\right )^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]),x]

[Out]

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

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IntegrateAlgebraic [B]  time = 0.68, size = 383, normalized size = 3.06 \begin {gather*} \frac {\left (\sqrt {a^2+2 a b x^3+b^2 x^6}-\sqrt {b^2} x^3\right )^2 \left (\frac {b \log \left (\sqrt {a^2+2 a b x^3+b^2 x^6}+a-\sqrt {b^2} x^3\right )}{3 a^2}-\frac {b \log \left (a^3+a^2 \sqrt {b^2} x^3-a^2 \sqrt {a^2+2 a b x^3+b^2 x^6}\right )}{3 a^2}\right )}{-2 \sqrt {b^2} x^3 \sqrt {a^2+2 a b x^3+b^2 x^6}+a^2+2 a b x^3+2 b^2 x^6}+\frac {-2 \sqrt {a^2+2 a b x^3+b^2 x^6} \left (-a^2 b-4 a b^2 x^3-4 b^3 x^6\right )-2 \sqrt {b^2} \left (a^3+5 a^2 b x^3+8 a b^2 x^6+4 b^3 x^9\right )}{3 a \sqrt {b^2} \sqrt {a^2+2 a b x^3+b^2 x^6} \left (2 a^2 x^3+8 a b x^6+8 b^2 x^9\right )+3 a \left (-2 a^3 b x^3-10 a^2 b^2 x^6-16 a b^3 x^9-8 b^4 x^{12}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/(x^4*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]),x]

[Out]

(-2*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]*(-(a^2*b) - 4*a*b^2*x^3 - 4*b^3*x^6) - 2*Sqrt[b^2]*(a^3 + 5*a^2*b*x^3 + 8*
a*b^2*x^6 + 4*b^3*x^9))/(3*a*Sqrt[b^2]*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]*(2*a^2*x^3 + 8*a*b*x^6 + 8*b^2*x^9) + 3
*a*(-2*a^3*b*x^3 - 10*a^2*b^2*x^6 - 16*a*b^3*x^9 - 8*b^4*x^12)) + ((-(Sqrt[b^2]*x^3) + Sqrt[a^2 + 2*a*b*x^3 +
b^2*x^6])^2*((b*Log[a - Sqrt[b^2]*x^3 + Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]])/(3*a^2) - (b*Log[a^3 + a^2*Sqrt[b^2]
*x^3 - a^2*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]])/(3*a^2)))/(a^2 + 2*a*b*x^3 + 2*b^2*x^6 - 2*Sqrt[b^2]*x^3*Sqrt[a^2
+ 2*a*b*x^3 + b^2*x^6])

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fricas [A]  time = 1.12, size = 33, normalized size = 0.26 \begin {gather*} \frac {b x^{3} \log \left (b x^{3} + a\right ) - 3 \, b x^{3} \log \relax (x) - a}{3 \, a^{2} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.37, size = 50, normalized size = 0.40 \begin {gather*} \frac {1}{3} \, {\left (\frac {b \log \left ({\left | b x^{3} + a \right |}\right )}{a^{2}} - \frac {3 \, b \log \left ({\left | x \right |}\right )}{a^{2}} + \frac {b x^{3} - a}{a^{2} x^{3}}\right )} \mathrm {sgn}\left (b x^{3} + a\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.01, size = 51, normalized size = 0.41 \begin {gather*} -\frac {\left (b \,x^{3}+a \right ) \left (3 b \,x^{3} \ln \relax (x )-b \,x^{3} \ln \left (b \,x^{3}+a \right )+a \right )}{3 \sqrt {\left (b \,x^{3}+a \right )^{2}}\, a^{2} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.20, size = 73, normalized size = 0.58 \begin {gather*} \frac {\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 )}{3 \, a^{2}} - \frac {\sqrt {b^{2} x^{6} + 2 \, a b x^{3} + a^{2}}}{3 \, a^{2} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.40, size = 75, normalized size = 0.60 \begin {gather*} \frac {a\,b\,\mathrm {atanh}\left (\frac {a^2+b\,a\,x^3}{\sqrt {a^2}\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}\right )}{3\,{\left (a^2\right )}^{3/2}}-\frac {\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{3\,a^2\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^4*((a + b*x^3)^2)^(1/2)),x)

[Out]

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

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sympy [A]  time = 0.36, size = 31, normalized size = 0.25 \begin {gather*} - \frac {1}{3 a x^{3}} - \frac {b \log {\relax (x )}}{a^{2}} + \frac {b \log {\left (\frac {a}{b} + x^{3} \right )}}{3 a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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