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

Optimal. Leaf size=425 \[ -\frac {\log \left (\sqrt [3]{a^3 x^3-b^2 x^2}-a x\right )}{a}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} a x}{2 \sqrt [3]{a^3 x^3-b^2 x^2}+a x}\right )}{a}-\frac {i \left (\sqrt {3}-i\right ) \log \left (\sqrt [3]{-1} \sqrt [3]{a^3 x^3-b^2 x^2}+\sqrt [3]{a} x \sqrt [3]{a^2+b}\right )}{\sqrt [3]{a} \sqrt [3]{a^2+b}}+\frac {\log \left (a x \sqrt [3]{a^3 x^3-b^2 x^2}+\left (a^3 x^3-b^2 x^2\right )^{2/3}+a^2 x^2\right )}{2 a}+\frac {\sqrt {6 \left (-1+i \sqrt {3}\right )} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{a} x \sqrt [3]{a^2+b}}{\sqrt [3]{a} x \sqrt [3]{a^2+b}-2 \sqrt [3]{-1} \sqrt [3]{a^3 x^3-b^2 x^2}}\right )}{\sqrt [3]{a} \sqrt [3]{a^2+b}}+\frac {\left (1+i \sqrt {3}\right ) \log \left ((-1)^{2/3} \left (a^3 x^3-b^2 x^2\right )^{2/3}+a^{2/3} x^2 \left (a^2+b\right )^{2/3}-\sqrt [3]{-1} \sqrt [3]{a} x \sqrt [3]{a^2+b} \sqrt [3]{a^3 x^3-b^2 x^2}\right )}{2 \sqrt [3]{a} \sqrt [3]{a^2+b}} \]

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Rubi [A]  time = 0.23, antiderivative size = 477, normalized size of antiderivative = 1.12, number of steps used = 4, number of rules used = 4, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {2056, 157, 59, 91} \begin {gather*} -\frac {x^{2/3} \log (x) \sqrt [3]{a^3 x-b^2}}{2 a \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}}{a \sqrt [3]{x}}-1\right )}{2 a \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} a \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{a \sqrt [3]{a^3 x^3-b^2 x^2}}-\frac {x^{2/3} \sqrt [3]{a^3 x-b^2} \log (a x+b)}{\sqrt [3]{a} \sqrt [3]{a^2+b} \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} \sqrt [3]{a^2+b}}-\sqrt [3]{x}\right )}{\sqrt [3]{a} \sqrt [3]{a^2+b} \sqrt [3]{a^3 x^3-b^2 x^2}}+\frac {2 \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]{a} \sqrt [3]{x} \sqrt [3]{a^2+b}}+\frac {1}{\sqrt {3}}\right )}{\sqrt [3]{a} \sqrt [3]{a^2+b} \sqrt [3]{a^3 x^3-b^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 59

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

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 157

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/((a_.) + (b_.)*(x_)), x_Symbol]
:> Dist[h/b, Int[(c + d*x)^n*(e + f*x)^p, x], x] + Dist[(b*g - a*h)/b, Int[((c + d*x)^n*(e + f*x)^p)/(a + b*x
), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]

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

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Mathematica [C]  time = 0.06, size = 89, normalized size = 0.21 \begin {gather*} \frac {3 x \sqrt [3]{1-\frac {a^3 x}{b^2}} \, _2F_1\left (\frac {1}{3},\frac {1}{3};\frac {4}{3};\frac {a^3 x}{b^2}\right )-6 x \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {a \left (a^2+b\right ) x}{a^3 x-b^2}\right )}{\sqrt [3]{x^2 \left (a^3 x-b^2\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*x*(1 - (a^3*x)/b^2)^(1/3)*Hypergeometric2F1[1/3, 1/3, 4/3, (a^3*x)/b^2] - 6*x*Hypergeometric2F1[1/3, 1, 4/3
, (a*(a^2 + b)*x)/(-b^2 + a^3*x)])/(x^2*(-b^2 + a^3*x))^(1/3)

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IntegrateAlgebraic [A]  time = 3.09, size = 475, normalized size = 1.12 \begin {gather*} \frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} a x}{a x+2 \sqrt [3]{-b^2 x^2+a^3 x^3}}\right )}{a}+\frac {\sqrt {6 \left (-1+i \sqrt {3}\right )} \tan ^{-1}\left (\frac {3 \sqrt [3]{a} \sqrt [3]{a^2+b} x}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{a^2+b} x-3 i \sqrt [3]{-b^2 x^2+a^3 x^3}-\sqrt {3} \sqrt [3]{-b^2 x^2+a^3 x^3}}\right )}{\sqrt [3]{a} \sqrt [3]{a^2+b}}-\frac {i \left (-i+\sqrt {3}\right ) \log \left (2 \sqrt [3]{a} \sqrt [3]{a^2+b} x+\left (1+i \sqrt {3}\right ) \sqrt [3]{-b^2 x^2+a^3 x^3}\right )}{\sqrt [3]{a} \sqrt [3]{a^2+b}}-\frac {\log \left (a^2 x-a \sqrt [3]{-b^2 x^2+a^3 x^3}\right )}{a}+\frac {\log \left (a^2 x^2+a x \sqrt [3]{-b^2 x^2+a^3 x^3}+\left (-b^2 x^2+a^3 x^3\right )^{2/3}\right )}{2 a}+\frac {\left (1+i \sqrt {3}\right ) \log \left (-2 i a^{2/3} \left (a^2+b\right )^{2/3} x^2+\sqrt [3]{a} \sqrt [3]{a^2+b} \left (i x-\sqrt {3} x\right ) \sqrt [3]{-b^2 x^2+a^3 x^3}+\left (i+\sqrt {3}\right ) \left (-b^2 x^2+a^3 x^3\right )^{2/3}\right )}{2 \sqrt [3]{a} \sqrt [3]{a^2+b}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.56, size = 806, normalized size = 1.90 \begin {gather*} \left [\frac {2 \, \sqrt {3} {\left (a^{3} + a b\right )} \sqrt {-\frac {1}{{\left (a^{3} + a b\right )}^{\frac {2}{3}}}} \log \left (\frac {2 \, b^{2} x - {\left (3 \, a^{3} + a b\right )} x^{2} + 3 \, {\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {1}{3}} {\left (a^{3} + a b\right )}^{\frac {2}{3}} x + \sqrt {3} {\left ({\left (a^{3} + a b\right )}^{\frac {4}{3}} x^{2} + {\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {1}{3}} {\left (a^{3} + a b\right )} x - 2 \, {\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {2}{3}} {\left (a^{3} + a b\right )}^{\frac {2}{3}}\right )} \sqrt {-\frac {1}{{\left (a^{3} + a b\right )}^{\frac {2}{3}}}}}{a x^{2} + b x}\right ) - 2 \, \sqrt {3} {\left (a^{2} + b\right )} \arctan \left (\frac {\sqrt {3} a x + 2 \, \sqrt {3} {\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {1}{3}}}{3 \, a x}\right ) - 2 \, {\left (a^{2} + b\right )} \log \left (-\frac {a x - {\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + {\left (a^{2} + b\right )} \log \left (\frac {a^{2} x^{2} + {\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {1}{3}} a x + {\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {2}{3}}}{x^{2}}\right ) + 4 \, {\left (a^{3} + a b\right )}^{\frac {2}{3}} \log \left (-\frac {{\left (a^{3} + a b\right )}^{\frac {1}{3}} x - {\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - 2 \, {\left (a^{3} + a b\right )}^{\frac {2}{3}} \log \left (\frac {{\left (a^{3} + a b\right )}^{\frac {2}{3}} x^{2} + {\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {1}{3}} {\left (a^{3} + a b\right )}^{\frac {1}{3}} x + {\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {2}{3}}}{x^{2}}\right )}{2 \, {\left (a^{3} + a b\right )}}, -\frac {2 \, \sqrt {3} {\left (a^{2} + b\right )} \arctan \left (\frac {\sqrt {3} a x + 2 \, \sqrt {3} {\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {1}{3}}}{3 \, a x}\right ) - 4 \, \sqrt {3} {\left (a^{3} + a b\right )}^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left ({\left (a^{3} + a b\right )}^{\frac {1}{3}} x + 2 \, {\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {1}{3}}\right )}}{3 \, {\left (a^{3} + a b\right )}^{\frac {1}{3}} x}\right ) + 2 \, {\left (a^{2} + b\right )} \log \left (-\frac {a x - {\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - {\left (a^{2} + b\right )} \log \left (\frac {a^{2} x^{2} + {\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {1}{3}} a x + {\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {2}{3}}}{x^{2}}\right ) - 4 \, {\left (a^{3} + a b\right )}^{\frac {2}{3}} \log \left (-\frac {{\left (a^{3} + a b\right )}^{\frac {1}{3}} x - {\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + 2 \, {\left (a^{3} + a b\right )}^{\frac {2}{3}} \log \left (\frac {{\left (a^{3} + a b\right )}^{\frac {2}{3}} x^{2} + {\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {1}{3}} {\left (a^{3} + a b\right )}^{\frac {1}{3}} x + {\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {2}{3}}}{x^{2}}\right )}{2 \, {\left (a^{3} + a b\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 143.52, size = 255, normalized size = 0.60 \begin {gather*} \frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left ({\left (a^{3} + a b\right )}^{\frac {1}{3}} + 2 \, {\left (a^{3} - \frac {b^{2}}{x}\right )}^{\frac {1}{3}}\right )}}{3 \, {\left (a^{3} + a b\right )}^{\frac {1}{3}}}\right )}{{\left (a^{3} + a b\right )}^{\frac {1}{3}}} - \frac {\log \left ({\left (a^{3} + a b\right )}^{\frac {2}{3}} + {\left (a^{3} + a b\right )}^{\frac {1}{3}} {\left (a^{3} - \frac {b^{2}}{x}\right )}^{\frac {1}{3}} + {\left (a^{3} - \frac {b^{2}}{x}\right )}^{\frac {2}{3}}\right )}{{\left (a^{3} + a b\right )}^{\frac {1}{3}}} + \frac {2 \, \log \left ({\left | -{\left (a^{3} + a b\right )}^{\frac {1}{3}} + {\left (a^{3} - \frac {b^{2}}{x}\right )}^{\frac {1}{3}} \right |}\right )}{{\left (a^{3} + a b\right )}^{\frac {1}{3}}} - \frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (a + 2 \, {\left (a^{3} - \frac {b^{2}}{x}\right )}^{\frac {1}{3}}\right )}}{3 \, a}\right )}{a} + \frac {\log \left (a^{2} + {\left (a^{3} - \frac {b^{2}}{x}\right )}^{\frac {1}{3}} a + {\left (a^{3} - \frac {b^{2}}{x}\right )}^{\frac {2}{3}}\right )}{2 \, a} - \frac {\log \left ({\left | -a + {\left (a^{3} - \frac {b^{2}}{x}\right )}^{\frac {1}{3}} \right |}\right )}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.07, size = 0, normalized size = 0.00 \[\int \frac {a x -b}{\left (a x +b \right ) \left (a^{3} x^{3}-b^{2} x^{2}\right )^{\frac {1}{3}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((a*x-b)/(a*x+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 {a x - b}{{\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {1}{3}} {\left (a x + b\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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