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

Optimal. Leaf size=41 \[ \frac{2 B \tan ^{-1}\left (\frac{\sqrt [3]{a}+2 \sqrt [3]{-b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} \sqrt [3]{a}} \]

[Out]

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

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Rubi [A]  time = 0.0431953, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {1586, 617, 204} \[ \frac{2 B \tan ^{-1}\left (\frac{\sqrt [3]{a}+2 \sqrt [3]{-b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} \sqrt [3]{a}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 1586

Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px, Qx, x]^p*Qx^(p + q), x] /; FreeQ[
q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]

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])

Rubi steps

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

Mathematica [B]  time = 0.0490526, size = 129, normalized size = 3.15 \[ \frac{\sqrt [3]{-b} B \left (\left (\sqrt [3]{-b}+\sqrt [3]{b}\right ) \left (2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )\right )+2 \sqrt{3} \left (\sqrt [3]{-b}-\sqrt [3]{b}\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )\right )}{6 \sqrt [3]{a} b^{2/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.01, size = 228, normalized size = 5.6 \begin{align*}{\frac{B\sqrt [3]{-1}}{3}\sqrt [3]{a}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){b}^{-{\frac{2}{3}}} \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{B\sqrt [3]{-1}}{6}\sqrt [3]{a}\ln \left ({x}^{2}-\sqrt [3]{{\frac{a}{b}}}x+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){b}^{-{\frac{2}{3}}} \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{B\sqrt [3]{-1}\sqrt{3}}{3}\sqrt [3]{a}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){b}^{-{\frac{2}{3}}} \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{B\sqrt [3]{-1}}{3}\sqrt [3]{-b}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){b}^{-{\frac{2}{3}}}{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{B\sqrt [3]{-1}}{6}\sqrt [3]{-b}\ln \left ({x}^{2}-\sqrt [3]{{\frac{a}{b}}}x+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){b}^{-{\frac{2}{3}}}{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{B\sqrt [3]{-1}\sqrt{3}}{3}\sqrt [3]{-b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){b}^{-{\frac{2}{3}}}{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.08293, size = 339, normalized size = 8.27 \begin{align*} \left [\sqrt{\frac{1}{3}} B \sqrt{-\frac{1}{a^{\frac{2}{3}}}} \log \left (\frac{2 \, b x^{3} + 3 \, a^{\frac{2}{3}} \left (-b\right )^{\frac{1}{3}} x - 3 \, \sqrt{\frac{1}{3}}{\left (2 \, a^{\frac{2}{3}} \left (-b\right )^{\frac{2}{3}} x^{2} - a \left (-b\right )^{\frac{1}{3}} x - a^{\frac{4}{3}}\right )} \sqrt{-\frac{1}{a^{\frac{2}{3}}}} - a}{b x^{3} + a}\right ), \frac{2 \, \sqrt{\frac{1}{3}} B \arctan \left (\frac{\sqrt{\frac{1}{3}}{\left (2 \, \left (-b\right )^{\frac{1}{3}} x + a^{\frac{1}{3}}\right )}}{a^{\frac{1}{3}}}\right )}{a^{\frac{1}{3}}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[sqrt(1/3)*B*sqrt(-1/a^(2/3))*log((2*b*x^3 + 3*a^(2/3)*(-b)^(1/3)*x - 3*sqrt(1/3)*(2*a^(2/3)*(-b)^(2/3)*x^2 -
a*(-b)^(1/3)*x - a^(4/3))*sqrt(-1/a^(2/3)) - a)/(b*x^3 + a)), 2*sqrt(1/3)*B*arctan(sqrt(1/3)*(2*(-b)^(1/3)*x +
a^(1/3))/a^(1/3))/a^(1/3)]

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Sympy [C]  time = 0.47654, size = 105, normalized size = 2.56 \begin{align*} - \frac{B \left (- \frac{\sqrt{3} i \log{\left (- \frac{\sqrt [3]{a} \left (- b\right )^{\frac{2}{3}}}{2 b} - \frac{\sqrt{3} i \sqrt [3]{a} \left (- b\right )^{\frac{2}{3}}}{2 b} + x \right )}}{3} + \frac{\sqrt{3} i \log{\left (- \frac{\sqrt [3]{a} \left (- b\right )^{\frac{2}{3}}}{2 b} + \frac{\sqrt{3} i \sqrt [3]{a} \left (- b\right )^{\frac{2}{3}}}{2 b} + x \right )}}{3}\right )}{\sqrt [3]{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.09758, size = 78, normalized size = 1.9 \begin{align*} \frac{2 \, \sqrt{3} B b \arctan \left (-\frac{\sqrt{3}{\left (2 \, \left (-b\right )^{\frac{2}{3}} x + a^{\frac{1}{3}} \left (-b\right )^{\frac{1}{3}}\right )}}{3 \, \sqrt{a^{\frac{2}{3}} \left (-b\right )^{\frac{2}{3}}}}\right )}{3 \, \sqrt{a^{\frac{2}{3}} \left (-b\right )^{\frac{2}{3}}} \left (-b\right )^{\frac{2}{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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