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

Optimal. Leaf size=215 \[ \frac {\left (a^3 x^3+b^2 x^2\right )^{2/3} \left (3 a^3 x-4 b^2\right )}{6 a^5 x}+\frac {\left (9 a^5 b-2 b^4\right ) \log \left (\sqrt [3]{a^3 x^3+b^2 x^2}-a x\right )}{9 a^6}-\frac {\left (9 a^5 b-2 b^4\right ) \tan ^{-1}\left (\frac {\sqrt {3} a x}{2 \sqrt [3]{a^3 x^3+b^2 x^2}+a x}\right )}{3 \sqrt {3} a^6}+\frac {\left (2 b^4-9 a^5 b\right ) \log \left (a x \sqrt [3]{a^3 x^3+b^2 x^2}+\left (a^3 x^3+b^2 x^2\right )^{2/3}+a^2 x^2\right )}{18 a^6} \]

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Rubi [B]  time = 0.34, antiderivative size = 466, normalized size of antiderivative = 2.17, number of steps used = 8, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {2053, 2011, 59, 2024} \begin {gather*} \frac {b 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 b 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} b 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 {b^4 x^{2/3} \log (x) \sqrt [3]{a^3 x+b^2}}{9 a^6 \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {b^4 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 )}{3 a^6 \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {2 b^4 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 )}{3 \sqrt {3} a^6 \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {2 b^2 \left (a^3 x^3+b^2 x^2\right )^{2/3}}{3 a^5 x}+\frac {\left (a^3 x^3+b^2 x^2\right )^{2/3}}{2 a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b^2*x^2 + a^3*x^3)^(2/3)/(2*a^2) - (2*b^2*(b^2*x^2 + a^3*x^3)^(2/3))/(3*a^5*x) + (Sqrt[3]*b*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
*b^4*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))])/(3*Sqrt[3]*a
^6*(b^2*x^2 + a^3*x^3)^(1/3)) + (b*x^(2/3)*(b^2 + a^3*x)^(1/3)*Log[x])/(2*a*(b^2*x^2 + a^3*x^3)^(1/3)) - (b^4*
x^(2/3)*(b^2 + a^3*x)^(1/3)*Log[x])/(9*a^6*(b^2*x^2 + a^3*x^3)^(1/3)) + (3*b*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)) - (b^4*x^(2/3)*(b^2 + a^3*x)^(1/3)*Log[-
1 + (b^2 + a^3*x)^(1/3)/(a*x^(1/3))])/(3*a^6*(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 2011

Int[((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Dist[(a*x^j + b*x^n)^FracPart[p]/(x^(j*FracPart[p
])*(a + b*x^(n - j))^FracPart[p]), Int[x^(j*p)*(a + b*x^(n - j))^p, x], x] /; FreeQ[{a, b, j, n, p}, x] &&  !I
ntegerQ[p] && NeQ[n, j] && PosQ[n - j]

Rule 2024

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n +
1)*(a*x^j + b*x^n)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^(n - j)*(m + j*p - n + j + 1))/(b*(m + n*p + 1)
), Int[(c*x)^(m - (n - j))*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IntegerQ[p] && LtQ[0, j
, n] && (IntegersQ[j, n] || GtQ[c, 0]) && GtQ[m + j*p + 1 - n + j, 0] && NeQ[m + n*p + 1, 0]

Rule 2053

Int[(Pq_)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[Pq*(a*x^j + b*x^n)^p, x]
, x] /; FreeQ[{a, b, j, n, p}, x] && (PolyQ[Pq, x] || PolyQ[Pq, x^n]) &&  !IntegerQ[p] && NeQ[n, j]

Rubi steps

\begin {align*} \int \frac {-b+a x^2}{\sqrt [3]{b^2 x^2+a^3 x^3}} \, dx &=\int \left (-\frac {b}{\sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {a x^2}{\sqrt [3]{b^2 x^2+a^3 x^3}}\right ) \, dx\\ &=a \int \frac {x^2}{\sqrt [3]{b^2 x^2+a^3 x^3}} \, dx-b \int \frac {1}{\sqrt [3]{b^2 x^2+a^3 x^3}} \, dx\\ &=\frac {\left (b^2 x^2+a^3 x^3\right )^{2/3}}{2 a^2}-\frac {\left (2 b^2\right ) \int \frac {x}{\sqrt [3]{b^2 x^2+a^3 x^3}} \, dx}{3 a^2}-\frac {\left (b 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 (b^2 x^2+a^3 x^3\right )^{2/3}}{2 a^2}-\frac {2 b^2 \left (b^2 x^2+a^3 x^3\right )^{2/3}}{3 a^5 x}+\frac {\sqrt {3} b 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 {b 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 {3 b 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}}+\frac {\left (2 b^4\right ) \int \frac {1}{\sqrt [3]{b^2 x^2+a^3 x^3}} \, dx}{9 a^5}\\ &=\frac {\left (b^2 x^2+a^3 x^3\right )^{2/3}}{2 a^2}-\frac {2 b^2 \left (b^2 x^2+a^3 x^3\right )^{2/3}}{3 a^5 x}+\frac {\sqrt {3} b 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 {b 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 {3 b 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}}+\frac {\left (2 b^4 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}{9 a^5 \sqrt [3]{b^2 x^2+a^3 x^3}}\\ &=\frac {\left (b^2 x^2+a^3 x^3\right )^{2/3}}{2 a^2}-\frac {2 b^2 \left (b^2 x^2+a^3 x^3\right )^{2/3}}{3 a^5 x}+\frac {\sqrt {3} b 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 b^4 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 )}{3 \sqrt {3} a^6 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {b 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 {b^4 x^{2/3} \sqrt [3]{b^2+a^3 x} \log (x)}{9 a^6 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {3 b 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}}-\frac {b^4 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 )}{3 a^6 \sqrt [3]{b^2 x^2+a^3 x^3}}\\ \end {align*}

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Mathematica [C]  time = 0.12, size = 119, normalized size = 0.55 \begin {gather*} \frac {3 \left (x^2 \left (a^3 x+b^2\right )\right )^{2/3} \left (b^3 \, _2F_1\left (-\frac {5}{3},\frac {1}{3};\frac {4}{3};-\frac {a^3 x}{b^2}\right )-2 b^3 \, _2F_1\left (-\frac {2}{3},\frac {1}{3};\frac {4}{3};-\frac {a^3 x}{b^2}\right )+\left (b^3-a^5\right ) \, _2F_1\left (\frac {1}{3},\frac {1}{3};\frac {4}{3};-\frac {a^3 x}{b^2}\right )\right )}{a^5 b x \left (\frac {a^3 x}{b^2}+1\right )^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(x^2*(b^2 + a^3*x))^(2/3)*(b^3*Hypergeometric2F1[-5/3, 1/3, 4/3, -((a^3*x)/b^2)] - 2*b^3*Hypergeometric2F1[
-2/3, 1/3, 4/3, -((a^3*x)/b^2)] + (-a^5 + b^3)*Hypergeometric2F1[1/3, 1/3, 4/3, -((a^3*x)/b^2)]))/(a^5*b*x*(1
+ (a^3*x)/b^2)^(2/3))

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IntegrateAlgebraic [A]  time = 0.60, size = 215, normalized size = 1.00 \begin {gather*} \frac {\left (-4 b^2+3 a^3 x\right ) \left (b^2 x^2+a^3 x^3\right )^{2/3}}{6 a^5 x}-\frac {\left (9 a^5 b-2 b^4\right ) \tan ^{-1}\left (\frac {\sqrt {3} a x}{a x+2 \sqrt [3]{b^2 x^2+a^3 x^3}}\right )}{3 \sqrt {3} a^6}+\frac {\left (9 a^5 b-2 b^4\right ) \log \left (-a x+\sqrt [3]{b^2 x^2+a^3 x^3}\right )}{9 a^6}+\frac {\left (-9 a^5 b+2 b^4\right ) \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 )}{18 a^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-4*b^2 + 3*a^3*x)*(b^2*x^2 + a^3*x^3)^(2/3))/(6*a^5*x) - ((9*a^5*b - 2*b^4)*ArcTan[(Sqrt[3]*a*x)/(a*x + 2*(b
^2*x^2 + a^3*x^3)^(1/3))])/(3*Sqrt[3]*a^6) + ((9*a^5*b - 2*b^4)*Log[-(a*x) + (b^2*x^2 + a^3*x^3)^(1/3)])/(9*a^
6) + ((-9*a^5*b + 2*b^4)*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)])/(18*a^6)

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fricas [A]  time = 0.68, size = 206, normalized size = 0.96 \begin {gather*} \frac {2 \, \sqrt {3} {\left (9 \, a^{5} b - 2 \, b^{4}\right )} x \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 (9 \, a^{5} b - 2 \, b^{4}\right )} x \log \left (-\frac {a x - {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - {\left (9 \, a^{5} b - 2 \, b^{4}\right )} x \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 ) + 3 \, {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {2}{3}} {\left (3 \, a^{4} x - 4 \, a b^{2}\right )}}{18 \, a^{6} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/18*(2*sqrt(3)*(9*a^5*b - 2*b^4)*x*arctan(1/3*(sqrt(3)*a*x + 2*sqrt(3)*(a^3*x^3 + b^2*x^2)^(1/3))/(a*x)) + 2*
(9*a^5*b - 2*b^4)*x*log(-(a*x - (a^3*x^3 + b^2*x^2)^(1/3))/x) - (9*a^5*b - 2*b^4)*x*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) + 3*(a^3*x^3 + b^2*x^2)^(2/3)*(3*a^4*x - 4*a*b^2))/(a^6*x
)

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giac [A]  time = 0.31, size = 195, normalized size = 0.91 \begin {gather*} \frac {\frac {2 \, \sqrt {3} {\left (9 \, a^{5} b^{3} - 2 \, b^{6}\right )} \arctan \left (\frac {\sqrt {3} {\left (a + 2 \, {\left (a^{3} + \frac {b^{2}}{x}\right )}^{\frac {1}{3}}\right )}}{3 \, a}\right )}{a^{6}} - \frac {{\left (9 \, a^{5} b^{3} - 2 \, b^{6}\right )} \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 )}{a^{6}} + \frac {2 \, {\left (9 \, a^{5} b^{3} - 2 \, b^{6}\right )} \log \left ({\left | -a + {\left (a^{3} + \frac {b^{2}}{x}\right )}^{\frac {1}{3}} \right |}\right )}{a^{6}} + \frac {3 \, {\left (7 \, {\left (a^{3} + \frac {b^{2}}{x}\right )}^{\frac {2}{3}} a^{3} b^{6} - 4 \, {\left (a^{3} + \frac {b^{2}}{x}\right )}^{\frac {5}{3}} b^{6}\right )} x^{2}}{a^{5} b^{4}}}{18 \, b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/18*(2*sqrt(3)*(9*a^5*b^3 - 2*b^6)*arctan(1/3*sqrt(3)*(a + 2*(a^3 + b^2/x)^(1/3))/a)/a^6 - (9*a^5*b^3 - 2*b^6
)*log(a^2 + (a^3 + b^2/x)^(1/3)*a + (a^3 + b^2/x)^(2/3))/a^6 + 2*(9*a^5*b^3 - 2*b^6)*log(abs(-a + (a^3 + b^2/x
)^(1/3)))/a^6 + 3*(7*(a^3 + b^2/x)^(2/3)*a^3*b^6 - 4*(a^3 + b^2/x)^(5/3)*b^6)*x^2/(a^5*b^4))/b^2

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maple [F]  time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {a \,x^{2}-b}{\left (a^{3} x^{3}+b^{2} x^{2}\right )^{\frac {1}{3}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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