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 {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 {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 {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 = 0.18, 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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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 66.08, size = 439, normalized size = 3.51 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\text {DirectedInfinity}\left [x^{\frac {1}{3}}\right ],a\text {==}0\text {\&\&}b\text {==}0\right \},\left \{\frac {3 x^{\frac {7}{3}}}{7 a^2},b\text {==}0\right \},\left \{\frac {3 x^{\frac {1}{3}}}{b^2},a\text {==}0\right \}\right \},\frac {-4 \sqrt {3} a \text {ArcTan}\left [\frac {\sqrt {3}}{3}+\frac {2 \sqrt {3} x^{\frac {1}{3}}}{3 \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right ] \left (-\frac {a}{b}\right )^{\frac {1}{3}}}{3 a b^2+3 b^3 x}-\frac {2 a \text {Log}\left [4 x^{\frac {1}{3}} \left (-\frac {a}{b}\right )^{\frac {1}{3}}+4 x^{\frac {2}{3}}+4 \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right ] \left (-\frac {a}{b}\right )^{\frac {1}{3}}}{3 a b^2+3 b^3 x}+\frac {4 a \text {Log}\left [2\right ] \left (-\frac {a}{b}\right )^{\frac {1}{3}}}{3 a b^2+3 b^3 x}+\frac {4 a \text {Log}\left [x^{\frac {1}{3}}-\left (-\frac {a}{b}\right )^{\frac {1}{3}}\right ] \left (-\frac {a}{b}\right )^{\frac {1}{3}}}{3 a b^2+3 b^3 x}+\frac {12 a x^{\frac {1}{3}}}{3 a b^2+3 b^3 x}-\frac {4 \sqrt {3} b x \text {ArcTan}\left [\frac {\sqrt {3}}{3}+\frac {2 \sqrt {3} x^{\frac {1}{3}}}{3 \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right ] \left (-\frac {a}{b}\right )^{\frac {1}{3}}}{3 a b^2+3 b^3 x}-\frac {2 b x \text {Log}\left [4 x^{\frac {1}{3}} \left (-\frac {a}{b}\right )^{\frac {1}{3}}+4 x^{\frac {2}{3}}+4 \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right ] \left (-\frac {a}{b}\right )^{\frac {1}{3}}}{3 a b^2+3 b^3 x}+\frac {4 b x \text {Log}\left [2\right ] \left (-\frac {a}{b}\right )^{\frac {1}{3}}}{3 a b^2+3 b^3 x}+\frac {4 b x \text {Log}\left [x^{\frac {1}{3}}-\left (-\frac {a}{b}\right )^{\frac {1}{3}}\right ] \left (-\frac {a}{b}\right )^{\frac {1}{3}}}{3 a b^2+3 b^3 x}+\frac {9 b x^{\frac {4}{3}}}{3 a b^2+3 b^3 x}\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[x^(4/3)/(a + b*x)^2,x]')

[Out]

Piecewise[{{DirectedInfinity[x ^ (1 / 3)], a == 0 && b == 0}, {3 x ^ (7 / 3) / (7 a ^ 2), b == 0}, {3 x ^ (1 /
3) / b ^ 2, a == 0}}, -4 Sqrt[3] a ArcTan[Sqrt[3] / 3 + 2 Sqrt[3] x ^ (1 / 3) / (3 (-a / b) ^ (1 / 3))] (-a /
b) ^ (1 / 3) / (3 a b ^ 2 + 3 b ^ 3 x) - 2 a Log[4 x ^ (1 / 3) (-a / b) ^ (1 / 3) + 4 x ^ (2 / 3) + 4 (-a / b
) ^ (2 / 3)] (-a / b) ^ (1 / 3) / (3 a b ^ 2 + 3 b ^ 3 x) + 4 a Log[2] (-a / b) ^ (1 / 3) / (3 a b ^ 2 + 3 b ^
3 x) + 4 a Log[x ^ (1 / 3) - (-a / b) ^ (1 / 3)] (-a / b) ^ (1 / 3) / (3 a b ^ 2 + 3 b ^ 3 x) + 12 a x ^ (1 /
3) / (3 a b ^ 2 + 3 b ^ 3 x) - 4 Sqrt[3] b x ArcTan[Sqrt[3] / 3 + 2 Sqrt[3] x ^ (1 / 3) / (3 (-a / b) ^ (1 /
3))] (-a / b) ^ (1 / 3) / (3 a b ^ 2 + 3 b ^ 3 x) - 2 b x Log[4 x ^ (1 / 3) (-a / b) ^ (1 / 3) + 4 x ^ (2 / 3)
+ 4 (-a / b) ^ (2 / 3)] (-a / b) ^ (1 / 3) / (3 a b ^ 2 + 3 b ^ 3 x) + 4 b x Log[2] (-a / b) ^ (1 / 3) / (3 a
b ^ 2 + 3 b ^ 3 x) + 4 b x Log[x ^ (1 / 3) - (-a / b) ^ (1 / 3)] (-a / b) ^ (1 / 3) / (3 a b ^ 2 + 3 b ^ 3 x)
+ 9 b x ^ (4 / 3) / (3 a b ^ 2 + 3 b ^ 3 x)]

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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 = 0.35, 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 = 0.33, 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 [F(-1)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]
time = 0.01, size = 196, normalized size = 1.57 \begin {gather*} 3 \left (\frac {x^{\frac {1}{3}}}{b^{2}}+\frac {\frac {1}{3} x^{\frac {1}{3}} a}{b^{2} \left (x b+a\right )}-\frac {2 \left (-a b^{2}\right )^{\frac {1}{3}} \ln \left (\left (x^{\frac {1}{3}}\right )^{2}+\left (-\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (-\frac {a}{b}\right )^{\frac {1}{3}} \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 b^{3}}-\frac {\frac {1}{3}\cdot 4 \left (-a b^{2}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \left (x^{\frac {1}{3}}+\frac {\left (-\frac {a}{b}\right )^{\frac {1}{3}}}{2}\right )}{\sqrt {3} \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{\sqrt {3} b^{3}}+\frac {4 a \left (-\frac {a}{b}\right )^{\frac {1}{3}} \ln \left |x^{\frac {1}{3}}-\left (-\frac {a}{b}\right )^{\frac {1}{3}}\right |}{3\cdot 3 b^{2} a}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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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