3.7.76 \(\int \frac {x^{2/3}}{a+b x} \, dx\) [676]

Optimal. Leaf size=111 \[ \frac {3 x^{2/3}}{2 b}+\frac {\sqrt {3} a^{2/3} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a}}\right )}{b^{5/3}}+\frac {3 a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 b^{5/3}}-\frac {a^{2/3} \log (a+b x)}{2 b^{5/3}} \]

[Out]

3/2*x^(2/3)/b+3/2*a^(2/3)*ln(a^(1/3)+b^(1/3)*x^(1/3))/b^(5/3)-1/2*a^(2/3)*ln(b*x+a)/b^(5/3)+a^(2/3)*arctan(1/3
*(a^(1/3)-2*b^(1/3)*x^(1/3))/a^(1/3)*3^(1/2))*3^(1/2)/b^(5/3)

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Rubi [A]
time = 0.03, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {52, 58, 631, 210, 31} \begin {gather*} \frac {\sqrt {3} a^{2/3} \text {ArcTan}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a}}\right )}{b^{5/3}}+\frac {3 a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 b^{5/3}}-\frac {a^{2/3} \log (a+b x)}{2 b^{5/3}}+\frac {3 x^{2/3}}{2 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^(2/3)/(a + b*x),x]

[Out]

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

Rule 31

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

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 58

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[{q = Rt[-(b*c - a*d)/b, 3]}, Simp[L
og[RemoveContent[a + b*x, x]]/(2*b*q), x] + (Dist[3/(2*b), Subst[Int[1/(q^2 - q*x + x^2), x], x, (c + d*x)^(1/
3)], x] - Dist[3/(2*b*q), 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^{2/3}}{a+b x} \, dx &=\frac {3 x^{2/3}}{2 b}-\frac {a \int \frac {1}{\sqrt [3]{x} (a+b x)} \, dx}{b}\\ &=\frac {3 x^{2/3}}{2 b}-\frac {a^{2/3} \log (a+b x)}{2 b^{5/3}}-\frac {(3 a) \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 )}{2 b^2}+\frac {\left (3 a^{2/3}\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt [3]{a}}{\sqrt [3]{b}}+x} \, dx,x,\sqrt [3]{x}\right )}{2 b^{5/3}}\\ &=\frac {3 x^{2/3}}{2 b}+\frac {3 a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 b^{5/3}}-\frac {a^{2/3} \log (a+b x)}{2 b^{5/3}}-\frac {\left (3 a^{2/3}\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^{5/3}}\\ &=\frac {3 x^{2/3}}{2 b}+\frac {\sqrt {3} a^{2/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{b^{5/3}}+\frac {3 a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 b^{5/3}}-\frac {a^{2/3} \log (a+b x)}{2 b^{5/3}}\\ \end {align*}

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Mathematica [A]
time = 0.07, size = 127, normalized size = 1.14 \begin {gather*} \frac {3 b^{2/3} x^{2/3}+2 \sqrt {3} a^{2/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )+2 a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )-a^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{x}+b^{2/3} x^{2/3}\right )}{2 b^{5/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^(2/3)/(a + b*x),x]

[Out]

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

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Maple [A]
time = 0.11, size = 112, normalized size = 1.01

method	result	size
risch	\(\frac {3 x^{\frac {2}{3}}}{2 b}+\frac {a \ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{b^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}}-\frac {a \ln \left (x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{2 b^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}}-\frac {a \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{b^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}}\)	\(107\)
derivativedivides	\(\frac {3 x^{\frac {2}{3}}}{2 b}-\frac {3 \left (-\frac {\ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right ) a}{b}\)	\(112\)
default	\(\frac {3 x^{\frac {2}{3}}}{2 b}-\frac {3 \left (-\frac {\ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right ) a}{b}\)	\(112\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*x^(2/3)/b-3*(-1/3/b/(a/b)^(1/3)*ln(x^(1/3)+(a/b)^(1/3))+1/6/b/(a/b)^(1/3)*ln(x^(2/3)-(a/b)^(1/3)*x^(1/3)+(
a/b)^(2/3))+1/3*3^(1/2)/b/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x^(1/3)-1)))*a/b

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Maxima [A]
time = 0.48, size = 114, normalized size = 1.03 \begin {gather*} -\frac {\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 )}{b^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {3 \, x^{\frac {2}{3}}}{2 \, b} - \frac {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 )}{2 \, b^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {a \log \left (x^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{b^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.45, size = 128, normalized size = 1.15 \begin {gather*} -\frac {2 \, \sqrt {3} \left (\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} b x^{\frac {1}{3}} \left (\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} - \sqrt {3} a}{3 \, a}\right ) + \left (\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} \log \left (-b x^{\frac {1}{3}} \left (\frac {a^{2}}{b^{2}}\right )^{\frac {2}{3}} + a x^{\frac {2}{3}} + a \left (\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}}\right ) - 2 \, \left (\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} \log \left (b \left (\frac {a^{2}}{b^{2}}\right )^{\frac {2}{3}} + a x^{\frac {1}{3}}\right ) - 3 \, x^{\frac {2}{3}}}{2 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 3.15, size = 162, normalized size = 1.46 \begin {gather*} \begin {cases} \tilde {\infty } x^{\frac {2}{3}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {3 x^{\frac {5}{3}}}{5 a} & \text {for}\: b = 0 \\\frac {3 x^{\frac {2}{3}}}{2 b} & \text {for}\: a = 0 \\- \frac {a \log {\left (\sqrt [3]{x} - \sqrt [3]{- \frac {a}{b}} \right )}}{b^{2} \sqrt [3]{- \frac {a}{b}}} + \frac {a \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 )}}{2 b^{2} \sqrt [3]{- \frac {a}{b}}} - \frac {\sqrt {3} a \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt [3]{x}}{3 \sqrt [3]{- \frac {a}{b}}} + \frac {\sqrt {3}}{3} \right )}}{b^{2} \sqrt [3]{- \frac {a}{b}}} + \frac {3 x^{\frac {2}{3}}}{2 b} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo*x**(2/3), Eq(a, 0) & Eq(b, 0)), (3*x**(5/3)/(5*a), Eq(b, 0)), (3*x**(2/3)/(2*b), Eq(a, 0)), (-a
*log(x**(1/3) - (-a/b)**(1/3))/(b**2*(-a/b)**(1/3)) + a*log(4*x**(2/3) + 4*x**(1/3)*(-a/b)**(1/3) + 4*(-a/b)**
(2/3))/(2*b**2*(-a/b)**(1/3)) - sqrt(3)*a*atan(2*sqrt(3)*x**(1/3)/(3*(-a/b)**(1/3)) + sqrt(3)/3)/(b**2*(-a/b)*
*(1/3)) + 3*x**(2/3)/(2*b), True))

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Giac [A]
time = 1.18, size = 118, normalized size = 1.06 \begin {gather*} \frac {\left (-\frac {a}{b}\right )^{\frac {2}{3}} \log \left ({\left | x^{\frac {1}{3}} - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{b} + \frac {3 \, x^{\frac {2}{3}}}{2 \, b} + \frac {\sqrt {3} \left (-a b^{2}\right )^{\frac {2}{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 )}{b^{3}} - \frac {\left (-a b^{2}\right )^{\frac {2}{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 )}{2 \, b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-a/b)^(2/3)*log(abs(x^(1/3) - (-a/b)^(1/3)))/b + 3/2*x^(2/3)/b + sqrt(3)*(-a*b^2)^(2/3)*arctan(1/3*sqrt(3)*(2
*x^(1/3) + (-a/b)^(1/3))/(-a/b)^(1/3))/b^3 - 1/2*(-a*b^2)^(2/3)*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 = 130, normalized size = 1.17 \begin {gather*} \frac {3\,x^{2/3}}{2\,b}+\frac {a^{2/3}\,\ln \left (\frac {9\,a^{7/3}}{b^{4/3}}+\frac {9\,a^2\,x^{1/3}}{b}\right )}{b^{5/3}}+\frac {a^{2/3}\,\ln \left (\frac {9\,a^{7/3}\,{\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2}{b^{4/3}}+\frac {9\,a^2\,x^{1/3}}{b}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{b^{5/3}}-\frac {a^{2/3}\,\ln \left (\frac {9\,a^{7/3}\,{\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2}{b^{4/3}}+\frac {9\,a^2\,x^{1/3}}{b}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{b^{5/3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(2/3)/(a + b*x),x)

[Out]

(3*x^(2/3))/(2*b) + (a^(2/3)*log((9*a^(7/3))/b^(4/3) + (9*a^2*x^(1/3))/b))/b^(5/3) + (a^(2/3)*log((9*a^(7/3)*(
(3^(1/2)*1i)/2 - 1/2)^2)/b^(4/3) + (9*a^2*x^(1/3))/b)*((3^(1/2)*1i)/2 - 1/2))/b^(5/3) - (a^(2/3)*log((9*a^(7/3
)*((3^(1/2)*1i)/2 + 1/2)^2)/b^(4/3) + (9*a^2*x^(1/3))/b)*((3^(1/2)*1i)/2 + 1/2))/b^(5/3)

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