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

Optimal. Leaf size=243 \[ \frac {-a-b x^3}{2 a x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {b^{2/3} \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{5/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {b^{2/3} \left (a+b x^3\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{5/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {b^{2/3} \left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{5/3} \sqrt {a^2+2 a b x^3+b^2 x^6}} \]

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Rubi [A]  time = 0.11, antiderivative size = 240, normalized size of antiderivative = 0.99, number of steps used = 8, number of rules used = 8, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {1355, 325, 200, 31, 634, 617, 204, 628} \begin {gather*} -\frac {a+b x^3}{2 a x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {b^{2/3} \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{5/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {b^{2/3} \left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{5/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {b^{2/3} \left (a+b x^3\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{5/3} \sqrt {a^2+2 a b x^3+b^2 x^6}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(a + b*x^3)/(2*a*x^2*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) + (b^(2/3)*(a + b*x^3)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(
Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(5/3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) - (b^(2/3)*(a + b*x^3)*Log[a^(1/3) + b^(1
/3)*x])/(3*a^(5/3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) + (b^(2/3)*(a + b*x^3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b
^(2/3)*x^2])/(6*a^(5/3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])

Rule 31

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

Rule 200

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Dist[1/(3*Rt[a, 3]^2), Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Di
st[1/(3*Rt[a, 3]^2), Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x]
/; FreeQ[{a, b}, x]

Rule 204

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

Rule 325

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 634

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

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^3 \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^3 \left (a b+b^2 x^3\right )} \, dx}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\\ &=-\frac {a+b x^3}{2 a x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {\left (b \left (a b+b^2 x^3\right )\right ) \int \frac {1}{a b+b^2 x^3} \, dx}{a \sqrt {a^2+2 a b x^3+b^2 x^6}}\\ &=-\frac {a+b x^3}{2 a x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {\left (\sqrt [3]{b} \left (a b+b^2 x^3\right )\right ) \int \frac {1}{\sqrt [3]{a} \sqrt [3]{b}+b^{2/3} x} \, dx}{3 a^{5/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {\left (\sqrt [3]{b} \left (a b+b^2 x^3\right )\right ) \int \frac {2 \sqrt [3]{a} \sqrt [3]{b}-b^{2/3} x}{a^{2/3} b^{2/3}-\sqrt [3]{a} b x+b^{4/3} x^2} \, dx}{3 a^{5/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}\\ &=-\frac {a+b x^3}{2 a x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {b^{2/3} \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{5/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {\left (a b+b^2 x^3\right ) \int \frac {-\sqrt [3]{a} b+2 b^{4/3} x}{a^{2/3} b^{2/3}-\sqrt [3]{a} b x+b^{4/3} x^2} \, dx}{6 a^{5/3} \sqrt [3]{b} \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {\left (b^{2/3} \left (a b+b^2 x^3\right )\right ) \int \frac {1}{a^{2/3} b^{2/3}-\sqrt [3]{a} b x+b^{4/3} x^2} \, dx}{2 a^{4/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}\\ &=-\frac {a+b x^3}{2 a x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {b^{2/3} \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{5/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {b^{2/3} \left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{5/3} \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 \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{a^{5/3} \sqrt [3]{b} \sqrt {a^2+2 a b x^3+b^2 x^6}}\\ &=-\frac {a+b x^3}{2 a x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {b^{2/3} \left (a+b x^3\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{5/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {b^{2/3} \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{5/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {b^{2/3} \left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{5/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 140, normalized size = 0.58 \begin {gather*} -\frac {\left (a+b x^3\right ) \left (-b^{2/3} x^2 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+3 a^{2/3}+2 b^{2/3} x^2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-2 \sqrt {3} b^{2/3} x^2 \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )\right )}{6 a^{5/3} x^2 \sqrt {\left (a+b x^3\right )^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/6*((a + b*x^3)*(3*a^(2/3) - 2*Sqrt[3]*b^(2/3)*x^2*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 2*b^(2/3)*x
^2*Log[a^(1/3) + b^(1/3)*x] - b^(2/3)*x^2*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2]))/(a^(5/3)*x^2*Sqrt[(
a + b*x^3)^2])

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IntegrateAlgebraic [A]  time = 17.85, size = 144, normalized size = 0.59 \begin {gather*} \frac {\left (a+b x^3\right ) \left (\frac {b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{5/3}}-\frac {b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{5/3}}+\frac {b^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{5/3}}-\frac {1}{2 a x^2}\right )}{\sqrt {\left (a+b x^3\right )^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b*x^3)*(-1/2*1/(a*x^2) + (b^(2/3)*ArcTan[1/Sqrt[3] - (2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(5/3))
- (b^(2/3)*Log[a^(1/3) + b^(1/3)*x])/(3*a^(5/3)) + (b^(2/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(
6*a^(5/3))))/Sqrt[(a + b*x^3)^2]

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fricas [A]  time = 1.30, size = 143, normalized size = 0.59 \begin {gather*} \frac {2 \, \sqrt {3} x^{2} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} a x \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}} - \sqrt {3} b}{3 \, b}\right ) - x^{2} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b^{2} x^{2} + a b x \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} + a^{2} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}}\right ) + 2 \, x^{2} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b x - a \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}}\right ) - 3}{6 \, a x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.33, size = 125, normalized size = 0.51 \begin {gather*} \frac {1}{6} \, {\left (\frac {2 \, b \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{a^{2}} - \frac {2 \, \sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{a^{2}} - \frac {\left (-a b^{2}\right )^{\frac {1}{3}} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{a^{2}} - \frac {3}{a x^{2}}\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^3/((b*x^3+a)^2)^(1/2),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.01, size = 117, normalized size = 0.48 \begin {gather*} \frac {\left (b \,x^{3}+a \right ) \left (2 \sqrt {3}\, x^{2} \arctan \left (\frac {\sqrt {3}\, \left (-2 x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )-2 x^{2} \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )+x^{2} \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )-3 \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \sqrt {\left (b \,x^{3}+a \right )^{2}}\, \left (\frac {a}{b}\right )^{\frac {2}{3}} a \,x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 2.21, size = 106, normalized size = 0.44 \begin {gather*} -\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, a \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {\log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, a \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {\log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, a \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {1}{2 \, a x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{x^3\,\sqrt {{\left (b\,x^3+a\right )}^2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^3*((a + b*x^3)^2)^(1/2)),x)

[Out]

int(1/(x^3*((a + b*x^3)^2)^(1/2)), x)

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sympy [A]  time = 0.26, size = 32, normalized size = 0.13 \begin {gather*} \operatorname {RootSum} {\left (27 t^{3} a^{5} + b^{2}, \left (t \mapsto t \log {\left (- \frac {3 t a^{2}}{b} + x \right )} \right )\right )} - \frac {1}{2 a x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(27*_t**3*a**5 + b**2, Lambda(_t, _t*log(-3*_t*a**2/b + x))) - 1/(2*a*x**2)

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