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

Optimal. Leaf size=526 \[ \frac {\sqrt [3]{a^3 x^3+b^2 x^2}}{a}+\frac {\left (3 a^2 b-b^2\right ) \log \left (\sqrt [3]{a^3 x^3+b^2 x^2}-a x\right )}{3 a^3}+\frac {\left (b^2-3 a^2 b\right ) \log \left (a x \sqrt [3]{a^3 x^3+b^2 x^2}+\left (a^3 x^3+b^2 x^2\right )^{2/3}+a^2 x^2\right )}{6 a^3}+\frac {\left (3 a^2 b-b^2\right ) \tan ^{-1}\left (\frac {\sqrt {3} a x}{2 \sqrt [3]{a^3 x^3+b^2 x^2}+a x}\right )}{\sqrt {3} a^3}+\frac {\left (b \sqrt [3]{a^2-b}-i \sqrt {3} b \sqrt [3]{a^2-b}\right ) \log \left (\sqrt [3]{-1} \sqrt [3]{a^3 x^3+b^2 x^2}+\sqrt [3]{a} x \sqrt [3]{a^2-b}\right )}{2 a^{5/3}}+\frac {i \left (\sqrt {3} b \sqrt [3]{a^2-b}+i b \sqrt [3]{a^2-b}\right ) \log \left ((-1)^{2/3} \left (a^3 x^3+b^2 x^2\right )^{2/3}+a^{2/3} x^2 \left (a^2-b\right )^{2/3}-\sqrt [3]{-1} \sqrt [3]{a} x \sqrt [3]{a^2-b} \sqrt [3]{a^3 x^3+b^2 x^2}\right )}{4 a^{5/3}}+\frac {\sqrt {-3-3 i \sqrt {3}} b \sqrt [3]{a^2-b} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{a} x \sqrt [3]{a^2-b}}{\sqrt [3]{a} x \sqrt [3]{a^2-b}-2 \sqrt [3]{-1} \sqrt [3]{a^3 x^3+b^2 x^2}}\right )}{\sqrt {2} a^{5/3}} \]

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Rubi [A]  time = 0.27, antiderivative size = 524, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2056, 101, 157, 59, 91} \begin {gather*} \frac {\sqrt [3]{a^3 x^3+b^2 x^2}}{a}+\frac {b \left (3 a^2-b\right ) \sqrt [3]{a^3 x^3+b^2 x^2} \log \left (a^3 x+b^2\right )}{6 a^3 x^{2/3} \sqrt [3]{a^3 x+b^2}}+\frac {b \left (3 a^2-b\right ) \sqrt [3]{a^3 x^3+b^2 x^2} \log \left (\frac {a \sqrt [3]{x}}{\sqrt [3]{a^3 x+b^2}}-1\right )}{2 a^3 x^{2/3} \sqrt [3]{a^3 x+b^2}}+\frac {b \left (3 a^2-b\right ) \sqrt [3]{a^3 x^3+b^2 x^2} \tan ^{-1}\left (\frac {2 a \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a^3 x+b^2}}+\frac {1}{\sqrt {3}}\right )}{\sqrt {3} a^3 x^{2/3} \sqrt [3]{a^3 x+b^2}}+\frac {b \sqrt [3]{a^2-b} \sqrt [3]{a^3 x^3+b^2 x^2} \log (a x+b)}{2 a^{5/3} x^{2/3} \sqrt [3]{a^3 x+b^2}}-\frac {3 b \sqrt [3]{a^2-b} \sqrt [3]{a^3 x^3+b^2 x^2} \log \left (\sqrt [3]{a} \sqrt [3]{x} \sqrt [3]{a^2-b}-\sqrt [3]{a^3 x+b^2}\right )}{2 a^{5/3} x^{2/3} \sqrt [3]{a^3 x+b^2}}-\frac {\sqrt {3} b \sqrt [3]{a^2-b} \sqrt [3]{a^3 x^3+b^2 x^2} \tan ^{-1}\left (\frac {2 \sqrt [3]{a} \sqrt [3]{x} \sqrt [3]{a^2-b}}{\sqrt {3} \sqrt [3]{a^3 x+b^2}}+\frac {1}{\sqrt {3}}\right )}{a^{5/3} x^{2/3} \sqrt [3]{a^3 x+b^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b^2*x^2 + a^3*x^3)^(1/3)/a + ((3*a^2 - b)*b*(b^2*x^2 + a^3*x^3)^(1/3)*ArcTan[1/Sqrt[3] + (2*a*x^(1/3))/(Sqrt[
3]*(b^2 + a^3*x)^(1/3))])/(Sqrt[3]*a^3*x^(2/3)*(b^2 + a^3*x)^(1/3)) - (Sqrt[3]*(a^2 - b)^(1/3)*b*(b^2*x^2 + a^
3*x^3)^(1/3)*ArcTan[1/Sqrt[3] + (2*a^(1/3)*(a^2 - b)^(1/3)*x^(1/3))/(Sqrt[3]*(b^2 + a^3*x)^(1/3))])/(a^(5/3)*x
^(2/3)*(b^2 + a^3*x)^(1/3)) + ((a^2 - b)^(1/3)*b*(b^2*x^2 + a^3*x^3)^(1/3)*Log[b + a*x])/(2*a^(5/3)*x^(2/3)*(b
^2 + a^3*x)^(1/3)) + ((3*a^2 - b)*b*(b^2*x^2 + a^3*x^3)^(1/3)*Log[b^2 + a^3*x])/(6*a^3*x^(2/3)*(b^2 + a^3*x)^(
1/3)) + ((3*a^2 - b)*b*(b^2*x^2 + a^3*x^3)^(1/3)*Log[-1 + (a*x^(1/3))/(b^2 + a^3*x)^(1/3)])/(2*a^3*x^(2/3)*(b^
2 + a^3*x)^(1/3)) - (3*(a^2 - b)^(1/3)*b*(b^2*x^2 + a^3*x^3)^(1/3)*Log[a^(1/3)*(a^2 - b)^(1/3)*x^(1/3) - (b^2
+ a^3*x)^(1/3)])/(2*a^(5/3)*x^(2/3)*(b^2 + a^3*x)^(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 101

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((a +
b*x)^m*(c + d*x)^n*(e + f*x)^(p + 1))/(f*(m + n + p + 1)), x] - Dist[1/(f*(m + n + p + 1)), Int[(a + b*x)^(m -
1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[c*m*(b*e - a*f) + a*n*(d*e - c*f) + (d*m*(b*e - a*f) + b*n*(d*e - c*f))
*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[m, 0] && GtQ[n, 0] && NeQ[m + n + p + 1, 0] && (Integ
ersQ[2*m, 2*n, 2*p] || (IntegersQ[m, n + p] || IntegersQ[p, m + n]))

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

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Mathematica [C]  time = 0.15, size = 162, normalized size = 0.31 \begin {gather*} \frac {3 x^2 \left (\left (a^3 b x+b^3\right ) \, _2F_1\left (-\frac {1}{3},\frac {2}{3};\frac {5}{3};-\frac {a^3 x}{b^2}\right )-\left (a^2 \left (a^3 x+b^2\right ) \, _2F_1\left (\frac {2}{3},\frac {2}{3};\frac {5}{3};-\frac {a^3 x}{b^2}\right )\right )+b^2 \left (a^2-b\right ) \sqrt [3]{\frac {a^3 x}{b^2}+1} \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {\left (a^3-a b\right ) x}{x a^3+b^2}\right )\right )}{2 a b \left (x^2 \left (a^3 x+b^2\right )\right )^{2/3} \sqrt [3]{\frac {a^3 x}{b^2}+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*x^2*((b^3 + a^3*b*x)*Hypergeometric2F1[-1/3, 2/3, 5/3, -((a^3*x)/b^2)] - a^2*(b^2 + a^3*x)*Hypergeometric2F
1[2/3, 2/3, 5/3, -((a^3*x)/b^2)] + (a^2 - b)*b^2*(1 + (a^3*x)/b^2)^(1/3)*Hypergeometric2F1[2/3, 1, 5/3, ((a^3
- a*b)*x)/(b^2 + a^3*x)]))/(2*a*b*(x^2*(b^2 + a^3*x))^(2/3)*(1 + (a^3*x)/b^2)^(1/3))

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IntegrateAlgebraic [A]  time = 4.15, size = 571, normalized size = 1.09 \begin {gather*} \frac {\sqrt [3]{b^2 x^2+a^3 x^3}}{a}+\frac {\left (3 a^2 b-b^2\right ) \tan ^{-1}\left (\frac {\sqrt {3} a x}{a x+2 \sqrt [3]{b^2 x^2+a^3 x^3}}\right )}{\sqrt {3} a^3}+\frac {\sqrt {-3-3 i \sqrt {3}} \sqrt [3]{a^2-b} b \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 {2} a^{5/3}}+\frac {\left (3 a^2 b-b^2\right ) \log \left (-a x+\sqrt [3]{b^2 x^2+a^3 x^3}\right )}{3 a^3}+\frac {\left (\sqrt [3]{a^2-b} b-i \sqrt {3} \sqrt [3]{a^2-b} b\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 )}{2 a^{5/3}}+\frac {\left (-3 a^2 b+b^2\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 )}{6 a^3}+\frac {i \left (i \sqrt [3]{a^2-b} b+\sqrt {3} \sqrt [3]{a^2-b} b\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 )}{4 a^{5/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.52, size = 437, normalized size = 0.83 \begin {gather*} \frac {6 \, \sqrt {3} a^{2} b \left (-\frac {a^{2} - b}{a^{2}}\right )^{\frac {1}{3}} \arctan \left (-\frac {\sqrt {3} {\left (a^{2} - b\right )} x + 2 \, \sqrt {3} {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}} a \left (-\frac {a^{2} - b}{a^{2}}\right )^{\frac {2}{3}}}{3 \, {\left (a^{2} - b\right )} x}\right ) + 6 \, a^{2} b \left (-\frac {a^{2} - b}{a^{2}}\right )^{\frac {1}{3}} \log \left (\frac {a x \left (-\frac {a^{2} - b}{a^{2}}\right )^{\frac {1}{3}} + {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - 3 \, a^{2} b \left (-\frac {a^{2} - b}{a^{2}}\right )^{\frac {1}{3}} \log \left (\frac {a^{2} x^{2} \left (-\frac {a^{2} - b}{a^{2}}\right )^{\frac {2}{3}} - {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}} a x \left (-\frac {a^{2} - b}{a^{2}}\right )^{\frac {1}{3}} + {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {2}{3}}}{x^{2}}\right ) - 2 \, \sqrt {3} {\left (3 \, a^{2} b - b^{2}\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 ) + 6 \, {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}} a^{2} + 2 \, {\left (3 \, a^{2} b - b^{2}\right )} \log \left (-\frac {a x - {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - {\left (3 \, a^{2} b - b^{2}\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 )}{6 \, a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(6*sqrt(3)*a^2*b*(-(a^2 - b)/a^2)^(1/3)*arctan(-1/3*(sqrt(3)*(a^2 - b)*x + 2*sqrt(3)*(a^3*x^3 + b^2*x^2)^(
1/3)*a*(-(a^2 - b)/a^2)^(2/3))/((a^2 - b)*x)) + 6*a^2*b*(-(a^2 - b)/a^2)^(1/3)*log((a*x*(-(a^2 - b)/a^2)^(1/3)
+ (a^3*x^3 + b^2*x^2)^(1/3))/x) - 3*a^2*b*(-(a^2 - b)/a^2)^(1/3)*log((a^2*x^2*(-(a^2 - b)/a^2)^(2/3) - (a^3*x
^3 + b^2*x^2)^(1/3)*a*x*(-(a^2 - b)/a^2)^(1/3) + (a^3*x^3 + b^2*x^2)^(2/3))/x^2) - 2*sqrt(3)*(3*a^2*b - b^2)*a
rctan(1/3*(sqrt(3)*a*x + 2*sqrt(3)*(a^3*x^3 + b^2*x^2)^(1/3))/(a*x)) + 6*(a^3*x^3 + b^2*x^2)^(1/3)*a^2 + 2*(3*
a^2*b - b^2)*log(-(a*x - (a^3*x^3 + b^2*x^2)^(1/3))/x) - (3*a^2*b - b^2)*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))/a^3

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giac [A]  time = 126.67, size = 339, normalized size = 0.64 \begin {gather*} -\frac {{\left (a^{3} - a b\right )}^{\frac {1}{3}} {\left (a^{2} b - b^{2}\right )} \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 )}{a^{4} - a^{2} b} + \frac {\sqrt {3} {\left (a^{3} - a b\right )}^{\frac {1}{3}} b \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 )}{a^{2}} + \frac {{\left (a^{3} + \frac {b^{2}}{x}\right )}^{\frac {1}{3}} x}{a} + \frac {{\left (a^{3} - a b\right )}^{\frac {1}{3}} b \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 )}{2 \, a^{2}} - \frac {\sqrt {3} {\left (3 \, a^{2} b - b^{2}\right )} \arctan \left (\frac {\sqrt {3} {\left (a + 2 \, {\left (a^{3} + \frac {b^{2}}{x}\right )}^{\frac {1}{3}}\right )}}{3 \, a}\right )}{3 \, a^{3}} - \frac {{\left (3 \, a^{2} b - b^{2}\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 )}{6 \, a^{3}} + \frac {{\left (3 \, a^{2} b - b^{2}\right )} \log \left ({\left | -a + {\left (a^{3} + \frac {b^{2}}{x}\right )}^{\frac {1}{3}} \right |}\right )}{3 \, a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((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 {\sqrt [3]{x^{2} \left (a^{3} x + b^{2}\right )}}{a x + b}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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