3.7.83 \(\int \frac {x^{4/3}}{(a+b x)^2} \, dx\) [683]

Optimal. Leaf size=125 \[ \frac {4 \sqrt [3]{x}}{b^2}-\frac {x^{4/3}}{b (a+b x)}+\frac {4 \sqrt [3]{a} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} b^{7/3}}-\frac {2 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{b^{7/3}}+\frac {2 \sqrt [3]{a} \log (a+b x)}{3 b^{7/3}} \]

[Out]

4*x^(1/3)/b^2-x^(4/3)/b/(b*x+a)-2*a^(1/3)*ln(a^(1/3)+b^(1/3)*x^(1/3))/b^(7/3)+2/3*a^(1/3)*ln(b*x+a)/b^(7/3)+4/
3*a^(1/3)*arctan(1/3*(a^(1/3)-2*b^(1/3)*x^(1/3))/a^(1/3)*3^(1/2))/b^(7/3)*3^(1/2)

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Rubi [A]
time = 0.03, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {43, 52, 60, 631, 210, 31} \begin {gather*} \frac {4 \sqrt [3]{a} \text {ArcTan}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} b^{7/3}}-\frac {2 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{b^{7/3}}+\frac {2 \sqrt [3]{a} \log (a+b x)}{3 b^{7/3}}-\frac {x^{4/3}}{b (a+b x)}+\frac {4 \sqrt [3]{x}}{b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^(4/3)/(a + b*x)^2,x]

[Out]

(4*x^(1/3))/b^2 - x^(4/3)/(b*(a + b*x)) + (4*a^(1/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*x^(1/3))/(Sqrt[3]*a^(1/3))])/
(Sqrt[3]*b^(7/3)) - (2*a^(1/3)*Log[a^(1/3) + b^(1/3)*x^(1/3)])/b^(7/3) + (2*a^(1/3)*Log[a + b*x])/(3*b^(7/3))

Rule 31

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

Rule 43

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

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 60

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

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 631

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]

Rubi steps

\begin {align*} \int \frac {x^{4/3}}{(a+b x)^2} \, dx &=-\frac {x^{4/3}}{b (a+b x)}+\frac {4 \int \frac {\sqrt [3]{x}}{a+b x} \, dx}{3 b}\\ &=\frac {4 \sqrt [3]{x}}{b^2}-\frac {x^{4/3}}{b (a+b x)}-\frac {(4 a) \int \frac {1}{x^{2/3} (a+b x)} \, dx}{3 b^2}\\ &=\frac {4 \sqrt [3]{x}}{b^2}-\frac {x^{4/3}}{b (a+b x)}+\frac {2 \sqrt [3]{a} \log (a+b x)}{3 b^{7/3}}-\frac {\left (2 a^{2/3}\right ) \text {Subst}\left (\int \frac {1}{\frac {a^{2/3}}{b^{2/3}}-\frac {\sqrt [3]{a} x}{\sqrt [3]{b}}+x^2} \, dx,x,\sqrt [3]{x}\right )}{b^{8/3}}-\frac {\left (2 \sqrt [3]{a}\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt [3]{a}}{\sqrt [3]{b}}+x} \, dx,x,\sqrt [3]{x}\right )}{b^{7/3}}\\ &=\frac {4 \sqrt [3]{x}}{b^2}-\frac {x^{4/3}}{b (a+b x)}-\frac {2 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{b^{7/3}}+\frac {2 \sqrt [3]{a} \log (a+b x)}{3 b^{7/3}}-\frac {\left (4 \sqrt [3]{a}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}\right )}{b^{7/3}}\\ &=\frac {4 \sqrt [3]{x}}{b^2}-\frac {x^{4/3}}{b (a+b x)}+\frac {4 \sqrt [3]{a} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt {3} b^{7/3}}-\frac {2 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{b^{7/3}}+\frac {2 \sqrt [3]{a} \log (a+b x)}{3 b^{7/3}}\\ \end {align*}

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Mathematica [A]
time = 0.20, size = 142, normalized size = 1.14 \begin {gather*} \frac {\frac {3 \sqrt [3]{b} \sqrt [3]{x} (4 a+3 b x)}{a+b x}+4 \sqrt {3} \sqrt [3]{a} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )-4 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )+2 \sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{x}+b^{2/3} x^{2/3}\right )}{3 b^{7/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^(4/3)/(a + b*x)^2,x]

[Out]

((3*b^(1/3)*x^(1/3)*(4*a + 3*b*x))/(a + b*x) + 4*Sqrt[3]*a^(1/3)*ArcTan[(1 - (2*b^(1/3)*x^(1/3))/a^(1/3))/Sqrt
[3]] - 4*a^(1/3)*Log[a^(1/3) + b^(1/3)*x^(1/3)] + 2*a^(1/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x^(1/3) + b^(2/3)*x^
(2/3)])/(3*b^(7/3))

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Maple [A]
time = 0.26, size = 124, normalized size = 0.99

method	result	size
derivativedivides	\(\frac {3 x^{\frac {1}{3}}}{b^{2}}-\frac {3 a \left (-\frac {x^{\frac {1}{3}}}{3 \left (b x +a \right )}+\frac {4 \ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {2 \ln \left (x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {4 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{b^{2}}\)	\(124\)
default	\(\frac {3 x^{\frac {1}{3}}}{b^{2}}-\frac {3 a \left (-\frac {x^{\frac {1}{3}}}{3 \left (b x +a \right )}+\frac {4 \ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {2 \ln \left (x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {4 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{b^{2}}\)	\(124\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(4/3)/(b*x+a)^2,x,method=_RETURNVERBOSE)

[Out]

3*x^(1/3)/b^2-3*a/b^2*(-1/3*x^(1/3)/(b*x+a)+4/9/b/(a/b)^(2/3)*ln(x^(1/3)+(a/b)^(1/3))-2/9/b/(a/b)^(2/3)*ln(x^(
2/3)-(a/b)^(1/3)*x^(1/3)+(a/b)^(2/3))+4/9/b/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x^(1/3)-1)))

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Maxima [A]
time = 0.48, size = 133, normalized size = 1.06 \begin {gather*} \frac {a x^{\frac {1}{3}}}{b^{3} x + a b^{2}} - \frac {4 \, \sqrt {3} a \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{\frac {1}{3}} - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {3 \, x^{\frac {1}{3}}}{b^{2}} + \frac {2 \, a \log \left (x^{\frac {2}{3}} - x^{\frac {1}{3}} \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{3 \, b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {4 \, a \log \left (x^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(4/3)/(b*x+a)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.81, size = 147, normalized size = 1.18 \begin {gather*} \frac {4 \, \sqrt {3} {\left (b x + a\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} b x^{\frac {1}{3}} \left (-\frac {a}{b}\right )^{\frac {2}{3}} - \sqrt {3} a}{3 \, a}\right ) - 2 \, {\left (b x + a\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left (x^{\frac {2}{3}} + x^{\frac {1}{3}} \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right ) + 4 \, {\left (b x + a\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left (x^{\frac {1}{3}} - \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right ) + 3 \, {\left (3 \, b x + 4 \, a\right )} x^{\frac {1}{3}}}{3 \, {\left (b^{3} x + a b^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(4/3)/(b*x+a)^2,x, algorithm="fricas")

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 457 vs. \(2 (121) = 242\).
time = 68.81, size = 457, normalized size = 3.66 \begin {gather*} \begin {cases} \tilde {\infty } \sqrt [3]{x} & \text {for}\: a = 0 \wedge b = 0 \\\frac {3 x^{\frac {7}{3}}}{7 a^{2}} & \text {for}\: b = 0 \\\frac {3 \sqrt [3]{x}}{b^{2}} & \text {for}\: a = 0 \\\frac {12 a \sqrt [3]{x}}{3 a b^{2} + 3 b^{3} x} + \frac {4 a \sqrt [3]{- \frac {a}{b}} \log {\left (\sqrt [3]{x} - \sqrt [3]{- \frac {a}{b}} \right )}}{3 a b^{2} + 3 b^{3} x} - \frac {2 a \sqrt [3]{- \frac {a}{b}} \log {\left (4 x^{\frac {2}{3}} + 4 \sqrt [3]{x} \sqrt [3]{- \frac {a}{b}} + 4 \left (- \frac {a}{b}\right )^{\frac {2}{3}} \right )}}{3 a b^{2} + 3 b^{3} x} - \frac {4 \sqrt {3} a \sqrt [3]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt [3]{x}}{3 \sqrt [3]{- \frac {a}{b}}} + \frac {\sqrt {3}}{3} \right )}}{3 a b^{2} + 3 b^{3} x} + \frac {4 a \sqrt [3]{- \frac {a}{b}} \log {\left (2 \right )}}{3 a b^{2} + 3 b^{3} x} + \frac {9 b x^{\frac {4}{3}}}{3 a b^{2} + 3 b^{3} x} + \frac {4 b x \sqrt [3]{- \frac {a}{b}} \log {\left (\sqrt [3]{x} - \sqrt [3]{- \frac {a}{b}} \right )}}{3 a b^{2} + 3 b^{3} x} - \frac {2 b x \sqrt [3]{- \frac {a}{b}} \log {\left (4 x^{\frac {2}{3}} + 4 \sqrt [3]{x} \sqrt [3]{- \frac {a}{b}} + 4 \left (- \frac {a}{b}\right )^{\frac {2}{3}} \right )}}{3 a b^{2} + 3 b^{3} x} - \frac {4 \sqrt {3} b x \sqrt [3]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt [3]{x}}{3 \sqrt [3]{- \frac {a}{b}}} + \frac {\sqrt {3}}{3} \right )}}{3 a b^{2} + 3 b^{3} x} + \frac {4 b x \sqrt [3]{- \frac {a}{b}} \log {\left (2 \right )}}{3 a b^{2} + 3 b^{3} x} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(4/3)/(b*x+a)**2,x)

[Out]

Piecewise((zoo*x**(1/3), Eq(a, 0) & Eq(b, 0)), (3*x**(7/3)/(7*a**2), Eq(b, 0)), (3*x**(1/3)/b**2, Eq(a, 0)), (
12*a*x**(1/3)/(3*a*b**2 + 3*b**3*x) + 4*a*(-a/b)**(1/3)*log(x**(1/3) - (-a/b)**(1/3))/(3*a*b**2 + 3*b**3*x) -
2*a*(-a/b)**(1/3)*log(4*x**(2/3) + 4*x**(1/3)*(-a/b)**(1/3) + 4*(-a/b)**(2/3))/(3*a*b**2 + 3*b**3*x) - 4*sqrt(
3)*a*(-a/b)**(1/3)*atan(2*sqrt(3)*x**(1/3)/(3*(-a/b)**(1/3)) + sqrt(3)/3)/(3*a*b**2 + 3*b**3*x) + 4*a*(-a/b)**
(1/3)*log(2)/(3*a*b**2 + 3*b**3*x) + 9*b*x**(4/3)/(3*a*b**2 + 3*b**3*x) + 4*b*x*(-a/b)**(1/3)*log(x**(1/3) - (
-a/b)**(1/3))/(3*a*b**2 + 3*b**3*x) - 2*b*x*(-a/b)**(1/3)*log(4*x**(2/3) + 4*x**(1/3)*(-a/b)**(1/3) + 4*(-a/b)
**(2/3))/(3*a*b**2 + 3*b**3*x) - 4*sqrt(3)*b*x*(-a/b)**(1/3)*atan(2*sqrt(3)*x**(1/3)/(3*(-a/b)**(1/3)) + sqrt(
3)/3)/(3*a*b**2 + 3*b**3*x) + 4*b*x*(-a/b)**(1/3)*log(2)/(3*a*b**2 + 3*b**3*x), True))

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Giac [A]
time = 0.86, size = 135, normalized size = 1.08 \begin {gather*} \frac {4 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x^{\frac {1}{3}} - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, b^{2}} - \frac {4 \, \sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, b^{3}} + \frac {a x^{\frac {1}{3}}}{{\left (b x + a\right )} b^{2}} + \frac {3 \, x^{\frac {1}{3}}}{b^{2}} - \frac {2 \, \left (-a b^{2}\right )^{\frac {1}{3}} \log \left (x^{\frac {2}{3}} + x^{\frac {1}{3}} \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{3 \, b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(4/3)/(b*x+a)^2,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.15, size = 142, normalized size = 1.14 \begin {gather*} \frac {3\,x^{1/3}}{b^2}+\frac {a\,x^{1/3}}{x\,b^3+a\,b^2}+\frac {4\,{\left (-a\right )}^{1/3}\,\ln \left (\frac {12\,{\left (-a\right )}^{4/3}}{b^{1/3}}+12\,a\,x^{1/3}\right )}{3\,b^{7/3}}-\frac {4\,{\left (-a\right )}^{1/3}\,\ln \left (12\,a\,x^{1/3}-\frac {12\,{\left (-a\right )}^{4/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{b^{1/3}}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{3\,b^{7/3}}+\frac {{\left (-a\right )}^{1/3}\,\ln \left (12\,a\,x^{1/3}+\frac {9\,{\left (-a\right )}^{4/3}\,\left (-\frac {2}{3}+\frac {\sqrt {3}\,2{}\mathrm {i}}{3}\right )}{b^{1/3}}\right )\,\left (-\frac {2}{3}+\frac {\sqrt {3}\,2{}\mathrm {i}}{3}\right )}{b^{7/3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(4/3)/(a + b*x)^2,x)

[Out]

(3*x^(1/3))/b^2 + (a*x^(1/3))/(a*b^2 + b^3*x) + (4*(-a)^(1/3)*log((12*(-a)^(4/3))/b^(1/3) + 12*a*x^(1/3)))/(3*
b^(7/3)) - (4*(-a)^(1/3)*log(12*a*x^(1/3) - (12*(-a)^(4/3)*((3^(1/2)*1i)/2 + 1/2))/b^(1/3))*((3^(1/2)*1i)/2 +
1/2))/(3*b^(7/3)) + ((-a)^(1/3)*log(12*a*x^(1/3) + (9*(-a)^(4/3)*((3^(1/2)*2i)/3 - 2/3))/b^(1/3))*((3^(1/2)*2i
)/3 - 2/3))/b^(7/3)

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