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

Optimal. Leaf size=286 \[ \frac{5 x \left (a+b x^3\right )^2}{18 a^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}+\frac{x \left (a+b x^3\right )}{6 a \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}+\frac{5 \left (a+b x^3\right )^3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{8/3} \sqrt [3]{b} \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}-\frac{5 \left (a+b x^3\right )^3 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{8/3} \sqrt [3]{b} \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}-\frac{5 \left (a+b x^3\right )^3 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{8/3} \sqrt [3]{b} \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}} \]

[Out]

(x*(a + b*x^3))/(6*a*(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2)) + (5*x*(a + b*x^3)^2)/(18*a^2*(a^2 + 2*a*b*x^3 + b^2*x
^6)^(3/2)) - (5*(a + b*x^3)^3*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(8/3)*b^(1/3)*(a
^2 + 2*a*b*x^3 + b^2*x^6)^(3/2)) + (5*(a + b*x^3)^3*Log[a^(1/3) + b^(1/3)*x])/(27*a^(8/3)*b^(1/3)*(a^2 + 2*a*b
*x^3 + b^2*x^6)^(3/2)) - (5*(a + b*x^3)^3*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(8/3)*b^(1/3)*
(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2))

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Rubi [A]  time = 0.146766, antiderivative size = 286, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {1343, 199, 200, 31, 634, 617, 204, 628} \[ \frac{5 x \left (a+b x^3\right )^2}{18 a^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}+\frac{x \left (a+b x^3\right )}{6 a \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}+\frac{5 \left (a+b x^3\right )^3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{8/3} \sqrt [3]{b} \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}-\frac{5 \left (a+b x^3\right )^3 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{8/3} \sqrt [3]{b} \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}-\frac{5 \left (a+b x^3\right )^3 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{8/3} \sqrt [3]{b} \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(a + b*x^3))/(6*a*(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2)) + (5*x*(a + b*x^3)^2)/(18*a^2*(a^2 + 2*a*b*x^3 + b^2*x
^6)^(3/2)) - (5*(a + b*x^3)^3*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(8/3)*b^(1/3)*(a
^2 + 2*a*b*x^3 + b^2*x^6)^(3/2)) + (5*(a + b*x^3)^3*Log[a^(1/3) + b^(1/3)*x])/(27*a^(8/3)*b^(1/3)*(a^2 + 2*a*b
*x^3 + b^2*x^6)^(3/2)) - (5*(a + b*x^3)^3*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(8/3)*b^(1/3)*
(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2))

Rule 1343

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

Rule 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

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 31

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

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 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 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 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]

Rubi steps

\begin{align*} \int \frac{1}{\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}} \, dx &=\frac{\left (2 a b+2 b^2 x^3\right )^3 \int \frac{1}{\left (2 a b+2 b^2 x^3\right )^3} \, dx}{\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}\\ &=\frac{x \left (a+b x^3\right )}{6 a \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}+\frac{\left (5 \left (2 a b+2 b^2 x^3\right )^3\right ) \int \frac{1}{\left (2 a b+2 b^2 x^3\right )^2} \, dx}{12 a b \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}\\ &=\frac{x \left (a+b x^3\right )}{6 a \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}+\frac{5 x \left (a+b x^3\right )^2}{18 a^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}+\frac{\left (5 \left (2 a b+2 b^2 x^3\right )^3\right ) \int \frac{1}{2 a b+2 b^2 x^3} \, dx}{36 a^2 b^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}\\ &=\frac{x \left (a+b x^3\right )}{6 a \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}+\frac{5 x \left (a+b x^3\right )^2}{18 a^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}+\frac{\left (5 \left (2 a b+2 b^2 x^3\right )^3\right ) \int \frac{1}{\sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{b}+\sqrt [3]{2} b^{2/3} x} \, dx}{108\ 2^{2/3} a^{8/3} b^{8/3} \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}+\frac{\left (5 \left (2 a b+2 b^2 x^3\right )^3\right ) \int \frac{2 \sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{b}-\sqrt [3]{2} b^{2/3} x}{2^{2/3} a^{2/3} b^{2/3}-2^{2/3} \sqrt [3]{a} b x+2^{2/3} b^{4/3} x^2} \, dx}{108\ 2^{2/3} a^{8/3} b^{8/3} \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}\\ &=\frac{x \left (a+b x^3\right )}{6 a \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}+\frac{5 x \left (a+b x^3\right )^2}{18 a^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}+\frac{5 \left (a+b x^3\right )^3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{8/3} \sqrt [3]{b} \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}-\frac{\left (5 \left (2 a b+2 b^2 x^3\right )^3\right ) \int \frac{-2^{2/3} \sqrt [3]{a} b+2\ 2^{2/3} b^{4/3} x}{2^{2/3} a^{2/3} b^{2/3}-2^{2/3} \sqrt [3]{a} b x+2^{2/3} b^{4/3} x^2} \, dx}{432 a^{8/3} b^{10/3} \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}+\frac{\left (5 \left (2 a b+2 b^2 x^3\right )^3\right ) \int \frac{1}{2^{2/3} a^{2/3} b^{2/3}-2^{2/3} \sqrt [3]{a} b x+2^{2/3} b^{4/3} x^2} \, dx}{72 \sqrt [3]{2} a^{7/3} b^{7/3} \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}\\ &=\frac{x \left (a+b x^3\right )}{6 a \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}+\frac{5 x \left (a+b x^3\right )^2}{18 a^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}+\frac{5 \left (a+b x^3\right )^3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{8/3} \sqrt [3]{b} \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}-\frac{5 \left (a+b x^3\right )^3 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{8/3} \sqrt [3]{b} \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}+\frac{\left (5 \left (2 a b+2 b^2 x^3\right )^3\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{72 a^{8/3} b^{10/3} \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}\\ &=\frac{x \left (a+b x^3\right )}{6 a \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}+\frac{5 x \left (a+b x^3\right )^2}{18 a^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}-\frac{5 \left (a+b x^3\right )^3 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{8/3} \sqrt [3]{b} \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}+\frac{5 \left (a+b x^3\right )^3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{8/3} \sqrt [3]{b} \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}-\frac{5 \left (a+b x^3\right )^3 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{8/3} \sqrt [3]{b} \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0693797, size = 235, normalized size = 0.82 \[ \frac{15 a^{2/3} b^{4/3} x^4-5 b^2 x^6 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-10 a b x^3 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-5 a^2 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+24 a^{5/3} \sqrt [3]{b} x+10 \left (a+b x^3\right )^2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-10 \sqrt{3} \left (a+b x^3\right )^2 \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{54 a^{8/3} \sqrt [3]{b} \left (a+b x^3\right ) \sqrt{\left (a+b x^3\right )^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(24*a^(5/3)*b^(1/3)*x + 15*a^(2/3)*b^(4/3)*x^4 - 10*Sqrt[3]*(a + b*x^3)^2*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/S
qrt[3]] + 10*(a + b*x^3)^2*Log[a^(1/3) + b^(1/3)*x] - 5*a^2*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2] - 1
0*a*b*x^3*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2] - 5*b^2*x^6*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)
*x^2])/(54*a^(8/3)*b^(1/3)*(a + b*x^3)*Sqrt[(a + b*x^3)^2])

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Maple [A]  time = 0.007, size = 299, normalized size = 1.1 \begin{align*}{\frac{b{x}^{3}+a}{54\,b{a}^{2}} \left ( -10\,\sqrt{3}\arctan \left ( 1/3\,{\sqrt{3} \left ( -2\,x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \right ){x}^{6}{b}^{2}+10\,\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){x}^{6}{b}^{2}-5\,\ln \left ({x}^{2}-\sqrt [3]{{\frac{a}{b}}}x+ \left ({\frac{a}{b}} \right ) ^{2/3} \right ){x}^{6}{b}^{2}+15\, \left ({\frac{a}{b}} \right ) ^{2/3}{x}^{4}{b}^{2}-20\,\sqrt{3}\arctan \left ( 1/3\,{\sqrt{3} \left ( -2\,x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \right ){x}^{3}ab+20\,\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){x}^{3}ab-10\,\ln \left ({x}^{2}-\sqrt [3]{{\frac{a}{b}}}x+ \left ({\frac{a}{b}} \right ) ^{2/3} \right ){x}^{3}ab+24\, \left ({\frac{a}{b}} \right ) ^{2/3}xab-10\,\sqrt{3}\arctan \left ( 1/3\,{\sqrt{3} \left ( -2\,x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \right ){a}^{2}+10\,\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){a}^{2}-5\,\ln \left ({x}^{2}-\sqrt [3]{{\frac{a}{b}}}x+ \left ({\frac{a}{b}} \right ) ^{2/3} \right ){a}^{2} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}} \left ( \left ( b{x}^{3}+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b^2*x^6+2*a*b*x^3+a^2)^(3/2),x)

[Out]

1/54*(-10*3^(1/2)*arctan(1/3*3^(1/2)*(-2*x+(a/b)^(1/3))/(a/b)^(1/3))*x^6*b^2+10*ln(x+(a/b)^(1/3))*x^6*b^2-5*ln
(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))*x^6*b^2+15*(a/b)^(2/3)*x^4*b^2-20*3^(1/2)*arctan(1/3*3^(1/2)*(-2*x+(a/b)^(1/3)
)/(a/b)^(1/3))*x^3*a*b+20*ln(x+(a/b)^(1/3))*x^3*a*b-10*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))*x^3*a*b+24*(a/b)^(2/3
)*x*a*b-10*3^(1/2)*arctan(1/3*3^(1/2)*(-2*x+(a/b)^(1/3))/(a/b)^(1/3))*a^2+10*ln(x+(a/b)^(1/3))*a^2-5*ln(x^2-(a
/b)^(1/3)*x+(a/b)^(2/3))*a^2)*(b*x^3+a)/(a/b)^(2/3)/b/a^2/((b*x^3+a)^2)^(3/2)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b^2*x^6+2*a*b*x^3+a^2)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.87535, size = 1162, normalized size = 4.06 \begin{align*} \left [\frac{15 \, a^{2} b^{2} x^{4} + 24 \, a^{3} b x + 15 \, \sqrt{\frac{1}{3}}{\left (a b^{3} x^{6} + 2 \, a^{2} b^{2} x^{3} + a^{3} b\right )} \sqrt{-\frac{\left (a^{2} b\right )^{\frac{1}{3}}}{b}} \log \left (\frac{2 \, a b x^{3} - 3 \, \left (a^{2} b\right )^{\frac{1}{3}} a x - a^{2} + 3 \, \sqrt{\frac{1}{3}}{\left (2 \, a b x^{2} + \left (a^{2} b\right )^{\frac{2}{3}} x - \left (a^{2} b\right )^{\frac{1}{3}} a\right )} \sqrt{-\frac{\left (a^{2} b\right )^{\frac{1}{3}}}{b}}}{b x^{3} + a}\right ) - 5 \,{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \left (a^{2} b\right )^{\frac{2}{3}} \log \left (a b x^{2} - \left (a^{2} b\right )^{\frac{2}{3}} x + \left (a^{2} b\right )^{\frac{1}{3}} a\right ) + 10 \,{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \left (a^{2} b\right )^{\frac{2}{3}} \log \left (a b x + \left (a^{2} b\right )^{\frac{2}{3}}\right )}{54 \,{\left (a^{4} b^{3} x^{6} + 2 \, a^{5} b^{2} x^{3} + a^{6} b\right )}}, \frac{15 \, a^{2} b^{2} x^{4} + 24 \, a^{3} b x + 30 \, \sqrt{\frac{1}{3}}{\left (a b^{3} x^{6} + 2 \, a^{2} b^{2} x^{3} + a^{3} b\right )} \sqrt{\frac{\left (a^{2} b\right )^{\frac{1}{3}}}{b}} \arctan \left (\frac{\sqrt{\frac{1}{3}}{\left (2 \, \left (a^{2} b\right )^{\frac{2}{3}} x - \left (a^{2} b\right )^{\frac{1}{3}} a\right )} \sqrt{\frac{\left (a^{2} b\right )^{\frac{1}{3}}}{b}}}{a^{2}}\right ) - 5 \,{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \left (a^{2} b\right )^{\frac{2}{3}} \log \left (a b x^{2} - \left (a^{2} b\right )^{\frac{2}{3}} x + \left (a^{2} b\right )^{\frac{1}{3}} a\right ) + 10 \,{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \left (a^{2} b\right )^{\frac{2}{3}} \log \left (a b x + \left (a^{2} b\right )^{\frac{2}{3}}\right )}{54 \,{\left (a^{4} b^{3} x^{6} + 2 \, a^{5} b^{2} x^{3} + a^{6} b\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b^2*x^6+2*a*b*x^3+a^2)^(3/2),x, algorithm="fricas")

[Out]

[1/54*(15*a^2*b^2*x^4 + 24*a^3*b*x + 15*sqrt(1/3)*(a*b^3*x^6 + 2*a^2*b^2*x^3 + a^3*b)*sqrt(-(a^2*b)^(1/3)/b)*l
og((2*a*b*x^3 - 3*(a^2*b)^(1/3)*a*x - a^2 + 3*sqrt(1/3)*(2*a*b*x^2 + (a^2*b)^(2/3)*x - (a^2*b)^(1/3)*a)*sqrt(-
(a^2*b)^(1/3)/b))/(b*x^3 + a)) - 5*(b^2*x^6 + 2*a*b*x^3 + a^2)*(a^2*b)^(2/3)*log(a*b*x^2 - (a^2*b)^(2/3)*x + (
a^2*b)^(1/3)*a) + 10*(b^2*x^6 + 2*a*b*x^3 + a^2)*(a^2*b)^(2/3)*log(a*b*x + (a^2*b)^(2/3)))/(a^4*b^3*x^6 + 2*a^
5*b^2*x^3 + a^6*b), 1/54*(15*a^2*b^2*x^4 + 24*a^3*b*x + 30*sqrt(1/3)*(a*b^3*x^6 + 2*a^2*b^2*x^3 + a^3*b)*sqrt(
(a^2*b)^(1/3)/b)*arctan(sqrt(1/3)*(2*(a^2*b)^(2/3)*x - (a^2*b)^(1/3)*a)*sqrt((a^2*b)^(1/3)/b)/a^2) - 5*(b^2*x^
6 + 2*a*b*x^3 + a^2)*(a^2*b)^(2/3)*log(a*b*x^2 - (a^2*b)^(2/3)*x + (a^2*b)^(1/3)*a) + 10*(b^2*x^6 + 2*a*b*x^3
+ a^2)*(a^2*b)^(2/3)*log(a*b*x + (a^2*b)^(2/3)))/(a^4*b^3*x^6 + 2*a^5*b^2*x^3 + a^6*b)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b**2*x**6+2*a*b*x**3+a**2)**(3/2),x)

[Out]

Integral((a**2 + 2*a*b*x**3 + b**2*x**6)**(-3/2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b^2*x^6+2*a*b*x^3+a^2)^(3/2),x, algorithm="giac")

[Out]

sage0*x