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

Optimal. Leaf size=198 \[ \frac{a \tan ^{-1}\left (\frac{x}{\sqrt [3]{2} \sqrt [3]{x^2+1}+1}\right )}{2\ 2^{2/3}}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{2} \sqrt [3]{x^2+1}\right )}{x}\right )}{2\ 2^{2/3} \sqrt{3}}-\frac{a \tan ^{-1}(x)}{6\ 2^{2/3}}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{3}}{x}\right )}{2\ 2^{2/3} \sqrt{3}}+\frac{b \log \left (3-x^2\right )}{4\ 2^{2/3}}-\frac{3 b \log \left (2^{2/3}-\sqrt [3]{x^2+1}\right )}{4\ 2^{2/3}}-\frac{\sqrt{3} b \tan ^{-1}\left (\frac{\sqrt [3]{2} \sqrt [3]{x^2+1}+1}{\sqrt{3}}\right )}{2\ 2^{2/3}} \]

[Out]

-(a*ArcTan[x])/(6*2^(2/3)) + (a*ArcTan[x/(1 + 2^(1/3)*(1 + x^2)^(1/3))])/(2*2^(2/3)) - (Sqrt[3]*b*ArcTan[(1 +
2^(1/3)*(1 + x^2)^(1/3))/Sqrt[3]])/(2*2^(2/3)) - (a*ArcTanh[Sqrt[3]/x])/(2*2^(2/3)*Sqrt[3]) - (a*ArcTanh[(Sqrt
[3]*(1 - 2^(1/3)*(1 + x^2)^(1/3)))/x])/(2*2^(2/3)*Sqrt[3]) + (b*Log[3 - x^2])/(4*2^(2/3)) - (3*b*Log[2^(2/3) -
(1 + x^2)^(1/3)])/(4*2^(2/3))

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Rubi [A]  time = 0.0905163, antiderivative size = 198, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {1010, 392, 444, 55, 617, 204, 31} \[ \frac{a \tan ^{-1}\left (\frac{x}{\sqrt [3]{2} \sqrt [3]{x^2+1}+1}\right )}{2\ 2^{2/3}}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{2} \sqrt [3]{x^2+1}\right )}{x}\right )}{2\ 2^{2/3} \sqrt{3}}-\frac{a \tan ^{-1}(x)}{6\ 2^{2/3}}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{3}}{x}\right )}{2\ 2^{2/3} \sqrt{3}}+\frac{b \log \left (3-x^2\right )}{4\ 2^{2/3}}-\frac{3 b \log \left (2^{2/3}-\sqrt [3]{x^2+1}\right )}{4\ 2^{2/3}}-\frac{\sqrt{3} b \tan ^{-1}\left (\frac{\sqrt [3]{2} \sqrt [3]{x^2+1}+1}{\sqrt{3}}\right )}{2\ 2^{2/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a*ArcTan[x])/(6*2^(2/3)) + (a*ArcTan[x/(1 + 2^(1/3)*(1 + x^2)^(1/3))])/(2*2^(2/3)) - (Sqrt[3]*b*ArcTan[(1 +
2^(1/3)*(1 + x^2)^(1/3))/Sqrt[3]])/(2*2^(2/3)) - (a*ArcTanh[Sqrt[3]/x])/(2*2^(2/3)*Sqrt[3]) - (a*ArcTanh[(Sqrt
[3]*(1 - 2^(1/3)*(1 + x^2)^(1/3)))/x])/(2*2^(2/3)*Sqrt[3]) + (b*Log[3 - x^2])/(4*2^(2/3)) - (3*b*Log[2^(2/3) -
(1 + x^2)^(1/3)])/(4*2^(2/3))

Rule 1010

Int[((g_) + (h_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_)*((d_) + (f_.)*(x_)^2)^(q_), x_Symbol] :> Dist[g, Int[(a + c
*x^2)^p*(d + f*x^2)^q, x], x] + Dist[h, Int[x*(a + c*x^2)^p*(d + f*x^2)^q, x], x] /; FreeQ[{a, c, d, f, g, h,
p, q}, x]

Rule 392

Int[1/(((a_) + (b_.)*(x_)^2)^(1/3)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> With[{q = Rt[b/a, 2]}, Simp[(q*ArcTanh
[Sqrt[3]/(q*x)])/(2*2^(2/3)*Sqrt[3]*a^(1/3)*d), x] + (-Simp[(q*ArcTan[(a^(1/3)*q*x)/(a^(1/3) + 2^(1/3)*(a + b*
x^2)^(1/3))])/(2*2^(2/3)*a^(1/3)*d), x] + Simp[(q*ArcTan[q*x])/(6*2^(2/3)*a^(1/3)*d), x] + Simp[(q*ArcTanh[(Sq
rt[3]*(a^(1/3) - 2^(1/3)*(a + b*x^2)^(1/3)))/(a^(1/3)*q*x)])/(2*2^(2/3)*Sqrt[3]*a^(1/3)*d), x])] /; FreeQ[{a,
b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[b*c + 3*a*d, 0] && PosQ[b/a]

Rule 444

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m
- n + 1, 0]

Rule 55

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/b, 3]}, -Simp[L
og[RemoveContent[a + b*x, x]]/(2*b*q), x] + (Dist[3/(2*b), Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1/
3)], x] - Dist[3/(2*b*q), Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x] && PosQ
[(b*c - a*d)/b]

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 31

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

Rubi steps

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

Mathematica [C]  time = 0.244593, size = 153, normalized size = 0.77 \[ \frac{1}{6} b x^2 F_1\left (1;\frac{1}{3},1;2;-x^2,\frac{x^2}{3}\right )-\frac{9 a x F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};-x^2,\frac{x^2}{3}\right )}{\left (x^2-3\right ) \sqrt [3]{x^2+1} \left (2 x^2 \left (F_1\left (\frac{3}{2};\frac{1}{3},2;\frac{5}{2};-x^2,\frac{x^2}{3}\right )-F_1\left (\frac{3}{2};\frac{4}{3},1;\frac{5}{2};-x^2,\frac{x^2}{3}\right )\right )+9 F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};-x^2,\frac{x^2}{3}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(b*x^2*AppellF1[1, 1/3, 1, 2, -x^2, x^2/3])/6 - (9*a*x*AppellF1[1/2, 1/3, 1, 3/2, -x^2, x^2/3])/((-3 + x^2)*(1
+ x^2)^(1/3)*(9*AppellF1[1/2, 1/3, 1, 3/2, -x^2, x^2/3] + 2*x^2*(AppellF1[3/2, 1/3, 2, 5/2, -x^2, x^2/3] - Ap
pellF1[3/2, 4/3, 1, 5/2, -x^2, x^2/3])))

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Maple [F]  time = 0.046, size = 0, normalized size = 0. \begin{align*} \int{\frac{bx+a}{-{x}^{2}+3}{\frac{1}{\sqrt [3]{{x}^{2}+1}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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