3.9.97 \(\int \frac {1}{(b+a x^2) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx\)

Optimal. Leaf size=68 \[ -\frac {\text {RootSum}\left [\text {$\#$1}^6-2 \text {$\#$1}^3 a^3+a^6+a b^3\& ,\frac {\log \left (\sqrt [3]{a^3 x^3+b^2 x^2}-\text {$\#$1} x\right )-\log (x)}{\text {$\#$1}}\& \right ]}{2 b} \]

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Rubi [B]  time = 0.34, antiderivative size = 633, normalized size of antiderivative = 9.31, number of steps used = 5, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {2056, 912, 91} \begin {gather*} \frac {x^{2/3} \sqrt [3]{a^3 x+b^2} \log \left (\sqrt {b}-\sqrt {-a} x\right )}{4 b \sqrt [3]{a^3+\sqrt {-a} b^{3/2}} \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {x^{2/3} \sqrt [3]{a^3 x+b^2} \log \left (\sqrt {-a} x+\sqrt {b}\right )}{4 b \sqrt [3]{a^3-\sqrt {-a} b^{3/2}} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \log \left (\frac {\sqrt [3]{a^3 x+b^2}}{\sqrt [3]{a^3-\sqrt {-a} b^{3/2}}}-\sqrt [3]{x}\right )}{4 b \sqrt [3]{a^3-\sqrt {-a} b^{3/2}} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \log \left (\frac {\sqrt [3]{a^3 x+b^2}}{\sqrt [3]{a^3+\sqrt {-a} b^{3/2}}}-\sqrt [3]{x}\right )}{4 b \sqrt [3]{a^3+\sqrt {-a} b^{3/2}} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {\sqrt {3} x^{2/3} \sqrt [3]{a^3 x+b^2} \tan ^{-1}\left (\frac {2 \sqrt [3]{a^3 x+b^2}}{\sqrt {3} \sqrt [3]{x} \sqrt [3]{a^3-\sqrt {-a} b^{3/2}}}+\frac {1}{\sqrt {3}}\right )}{2 b \sqrt [3]{a^3-\sqrt {-a} b^{3/2}} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {\sqrt {3} x^{2/3} \sqrt [3]{a^3 x+b^2} \tan ^{-1}\left (\frac {2 \sqrt [3]{a^3 x+b^2}}{\sqrt {3} \sqrt [3]{x} \sqrt [3]{a^3+\sqrt {-a} b^{3/2}}}+\frac {1}{\sqrt {3}}\right )}{2 b \sqrt [3]{a^3+\sqrt {-a} b^{3/2}} \sqrt [3]{a^3 x^3+b^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(Sqrt[3]*x^(2/3)*(b^2 + a^3*x)^(1/3)*ArcTan[1/Sqrt[3] + (2*(b^2 + a^3*x)^(1/3))/(Sqrt[3]*(a^3 - Sqrt[-a]*
b^(3/2))^(1/3)*x^(1/3))])/(b*(a^3 - Sqrt[-a]*b^(3/2))^(1/3)*(b^2*x^2 + a^3*x^3)^(1/3)) - (Sqrt[3]*x^(2/3)*(b^2
+ a^3*x)^(1/3)*ArcTan[1/Sqrt[3] + (2*(b^2 + a^3*x)^(1/3))/(Sqrt[3]*(a^3 + Sqrt[-a]*b^(3/2))^(1/3)*x^(1/3))])/
(2*b*(a^3 + Sqrt[-a]*b^(3/2))^(1/3)*(b^2*x^2 + a^3*x^3)^(1/3)) + (x^(2/3)*(b^2 + a^3*x)^(1/3)*Log[Sqrt[b] - Sq
rt[-a]*x])/(4*b*(a^3 + Sqrt[-a]*b^(3/2))^(1/3)*(b^2*x^2 + a^3*x^3)^(1/3)) + (x^(2/3)*(b^2 + a^3*x)^(1/3)*Log[S
qrt[b] + Sqrt[-a]*x])/(4*b*(a^3 - Sqrt[-a]*b^(3/2))^(1/3)*(b^2*x^2 + a^3*x^3)^(1/3)) - (3*x^(2/3)*(b^2 + a^3*x
)^(1/3)*Log[-x^(1/3) + (b^2 + a^3*x)^(1/3)/(a^3 - Sqrt[-a]*b^(3/2))^(1/3)])/(4*b*(a^3 - Sqrt[-a]*b^(3/2))^(1/3
)*(b^2*x^2 + a^3*x^3)^(1/3)) - (3*x^(2/3)*(b^2 + a^3*x)^(1/3)*Log[-x^(1/3) + (b^2 + a^3*x)^(1/3)/(a^3 + Sqrt[-
a]*b^(3/2))^(1/3)])/(4*b*(a^3 + Sqrt[-a]*b^(3/2))^(1/3)*(b^2*x^2 + a^3*x^3)^(1/3))

Rule 91

Int[1/(((a_.) + (b_.)*(x_))^(1/3)*((c_.) + (d_.)*(x_))^(2/3)*((e_.) + (f_.)*(x_))), x_Symbol] :> With[{q = Rt[
(d*e - c*f)/(b*e - a*f), 3]}, -Simp[(Sqrt[3]*q*ArcTan[1/Sqrt[3] + (2*q*(a + b*x)^(1/3))/(Sqrt[3]*(c + d*x)^(1/
3))])/(d*e - c*f), x] + (Simp[(q*Log[e + f*x])/(2*(d*e - c*f)), x] - Simp[(3*q*Log[q*(a + b*x)^(1/3) - (c + d*
x)^(1/3)])/(2*(d*e - c*f)), x])] /; FreeQ[{a, b, c, d, e, f}, x]

Rule 912

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegr
and[(d + e*x)^m*(f + g*x)^n, 1/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g, m, n}, x] && NeQ[c*d^2 + a*e^2,
0] &&  !IntegerQ[m] &&  !IntegerQ[n]

Rule 2056

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, x, 2]

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 103, normalized size = 1.51 \begin {gather*} \frac {3 x \left (\, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {\left (a^3-\sqrt {-a} b^{3/2}\right ) x}{x a^3+b^2}\right )+\, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {\left (a^3+\sqrt {-a} b^{3/2}\right ) x}{x a^3+b^2}\right )\right )}{2 b \sqrt [3]{x^2 \left (a^3 x+b^2\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*x*(Hypergeometric2F1[1/3, 1, 4/3, ((a^3 - Sqrt[-a]*b^(3/2))*x)/(b^2 + a^3*x)] + Hypergeometric2F1[1/3, 1, 4
/3, ((a^3 + Sqrt[-a]*b^(3/2))*x)/(b^2 + a^3*x)]))/(2*b*(x^2*(b^2 + a^3*x))^(1/3))

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IntegrateAlgebraic [A]  time = 0.00, size = 68, normalized size = 1.00 \begin {gather*} -\frac {\text {RootSum}\left [a^6+a b^3-2 a^3 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x)+\log \left (\sqrt [3]{b^2 x^2+a^3 x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{2 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*RootSum[a^6 + a*b^3 - 2*a^3*#1^3 + #1^6 & , (-Log[x] + Log[(b^2*x^2 + a^3*x^3)^(1/3) - x*#1])/#1 & ]/b

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fricas [B]  time = 0.53, size = 2062, normalized size = 30.32

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(3)*(-(a^2 + (a^5*b^3 + b^6)*sqrt(-1/(a^11*b^3 + 2*a^6*b^6 + a*b^9)))/(a^5*b^3 + b^6))^(1/3)*arctan(1/3*(2
*sqrt(3)*b*x*(-(a^2 + (a^5*b^3 + b^6)*sqrt(-1/(a^11*b^3 + 2*a^6*b^6 + a*b^9)))/(a^5*b^3 + b^6))^(1/3)*sqrt(((a
^3*b^2*x - (a^6*b^5 + a*b^8)*x*sqrt(-1/(a^11*b^3 + 2*a^6*b^6 + a*b^9)))*(a^3*x^3 + b^2*x^2)^(1/3)*(-(a^2 + (a^
5*b^3 + b^6)*sqrt(-1/(a^11*b^3 + 2*a^6*b^6 + a*b^9)))/(a^5*b^3 + b^6))^(2/3) - (a^3*b*x^2 - (a^6*b^4 + a*b^7)*
x^2*sqrt(-1/(a^11*b^3 + 2*a^6*b^6 + a*b^9)))*(-(a^2 + (a^5*b^3 + b^6)*sqrt(-1/(a^11*b^3 + 2*a^6*b^6 + a*b^9)))
/(a^5*b^3 + b^6))^(1/3) + (a^3*x^3 + b^2*x^2)^(2/3))/x^2) - 2*sqrt(3)*(a^3*x^3 + b^2*x^2)^(1/3)*b*(-(a^2 + (a^
5*b^3 + b^6)*sqrt(-1/(a^11*b^3 + 2*a^6*b^6 + a*b^9)))/(a^5*b^3 + b^6))^(1/3) + sqrt(3)*x)/x) + sqrt(3)*(-(a^2
- (a^5*b^3 + b^6)*sqrt(-1/(a^11*b^3 + 2*a^6*b^6 + a*b^9)))/(a^5*b^3 + b^6))^(1/3)*arctan(1/3*(2*sqrt(3)*b*x*(-
(a^2 - (a^5*b^3 + b^6)*sqrt(-1/(a^11*b^3 + 2*a^6*b^6 + a*b^9)))/(a^5*b^3 + b^6))^(1/3)*sqrt(((a^3*b^2*x + (a^6
*b^5 + a*b^8)*x*sqrt(-1/(a^11*b^3 + 2*a^6*b^6 + a*b^9)))*(a^3*x^3 + b^2*x^2)^(1/3)*(-(a^2 - (a^5*b^3 + b^6)*sq
rt(-1/(a^11*b^3 + 2*a^6*b^6 + a*b^9)))/(a^5*b^3 + b^6))^(2/3) - (a^3*b*x^2 + (a^6*b^4 + a*b^7)*x^2*sqrt(-1/(a^
11*b^3 + 2*a^6*b^6 + a*b^9)))*(-(a^2 - (a^5*b^3 + b^6)*sqrt(-1/(a^11*b^3 + 2*a^6*b^6 + a*b^9)))/(a^5*b^3 + b^6
))^(1/3) + (a^3*x^3 + b^2*x^2)^(2/3))/x^2) - 2*sqrt(3)*(a^3*x^3 + b^2*x^2)^(1/3)*b*(-(a^2 - (a^5*b^3 + b^6)*sq
rt(-1/(a^11*b^3 + 2*a^6*b^6 + a*b^9)))/(a^5*b^3 + b^6))^(1/3) + sqrt(3)*x)/x) + 1/2*(-(a^2 + (a^5*b^3 + b^6)*s
qrt(-1/(a^11*b^3 + 2*a^6*b^6 + a*b^9)))/(a^5*b^3 + b^6))^(1/3)*log(-((a^3*b^2*x - (a^6*b^5 + a*b^8)*x*sqrt(-1/
(a^11*b^3 + 2*a^6*b^6 + a*b^9)))*(-(a^2 + (a^5*b^3 + b^6)*sqrt(-1/(a^11*b^3 + 2*a^6*b^6 + a*b^9)))/(a^5*b^3 +
b^6))^(2/3) - (a^3*x^3 + b^2*x^2)^(1/3))/x) + 1/2*(-(a^2 - (a^5*b^3 + b^6)*sqrt(-1/(a^11*b^3 + 2*a^6*b^6 + a*b
^9)))/(a^5*b^3 + b^6))^(1/3)*log(-((a^3*b^2*x + (a^6*b^5 + a*b^8)*x*sqrt(-1/(a^11*b^3 + 2*a^6*b^6 + a*b^9)))*(
-(a^2 - (a^5*b^3 + b^6)*sqrt(-1/(a^11*b^3 + 2*a^6*b^6 + a*b^9)))/(a^5*b^3 + b^6))^(2/3) - (a^3*x^3 + b^2*x^2)^
(1/3))/x) - 1/4*(-(a^2 + (a^5*b^3 + b^6)*sqrt(-1/(a^11*b^3 + 2*a^6*b^6 + a*b^9)))/(a^5*b^3 + b^6))^(1/3)*log((
(a^3*b^2*x - (a^6*b^5 + a*b^8)*x*sqrt(-1/(a^11*b^3 + 2*a^6*b^6 + a*b^9)))*(a^3*x^3 + b^2*x^2)^(1/3)*(-(a^2 + (
a^5*b^3 + b^6)*sqrt(-1/(a^11*b^3 + 2*a^6*b^6 + a*b^9)))/(a^5*b^3 + b^6))^(2/3) - (a^3*b*x^2 - (a^6*b^4 + a*b^7
)*x^2*sqrt(-1/(a^11*b^3 + 2*a^6*b^6 + a*b^9)))*(-(a^2 + (a^5*b^3 + b^6)*sqrt(-1/(a^11*b^3 + 2*a^6*b^6 + a*b^9)
))/(a^5*b^3 + b^6))^(1/3) + (a^3*x^3 + b^2*x^2)^(2/3))/x^2) - 1/4*(-(a^2 - (a^5*b^3 + b^6)*sqrt(-1/(a^11*b^3 +
2*a^6*b^6 + a*b^9)))/(a^5*b^3 + b^6))^(1/3)*log(((a^3*b^2*x + (a^6*b^5 + a*b^8)*x*sqrt(-1/(a^11*b^3 + 2*a^6*b
^6 + a*b^9)))*(a^3*x^3 + b^2*x^2)^(1/3)*(-(a^2 - (a^5*b^3 + b^6)*sqrt(-1/(a^11*b^3 + 2*a^6*b^6 + a*b^9)))/(a^5
*b^3 + b^6))^(2/3) - (a^3*b*x^2 + (a^6*b^4 + a*b^7)*x^2*sqrt(-1/(a^11*b^3 + 2*a^6*b^6 + a*b^9)))*(-(a^2 - (a^5
*b^3 + b^6)*sqrt(-1/(a^11*b^3 + 2*a^6*b^6 + a*b^9)))/(a^5*b^3 + b^6))^(1/3) + (a^3*x^3 + b^2*x^2)^(2/3))/x^2)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}} {\left (a x^{2} + b\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a \,x^{2}+b \right ) \left (a^{3} x^{3}+b^{2} x^{2}\right )^{\frac {1}{3}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}} {\left (a x^{2} + b\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{{\left (a^3\,x^3+b^2\,x^2\right )}^{1/3}\,\left (a\,x^2+b\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt [3]{x^{2} \left (a^{3} x + b^{2}\right )} \left (a x^{2} + b\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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