3.7.74 \(\int \frac {x^{5/3}}{a+b x} \, dx\)

Optimal. Leaf size=125 \[ -\frac {3 a^{5/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 b^{8/3}}+\frac {a^{5/3} \log (a+b x)}{2 b^{8/3}}-\frac {\sqrt {3} a^{5/3} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a}}\right )}{b^{8/3}}-\frac {3 a x^{2/3}}{2 b^2}+\frac {3 x^{5/3}}{5 b} \]

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Rubi [A]  time = 0.07, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {50, 56, 617, 204, 31} \begin {gather*} -\frac {3 a^{5/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 b^{8/3}}+\frac {a^{5/3} \log (a+b x)}{2 b^{8/3}}-\frac {\sqrt {3} a^{5/3} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a}}\right )}{b^{8/3}}-\frac {3 a x^{2/3}}{2 b^2}+\frac {3 x^{5/3}}{5 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^(5/3)/(a + b*x),x]

[Out]

(-3*a*x^(2/3))/(2*b^2) + (3*x^(5/3))/(5*b) - (Sqrt[3]*a^(5/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*x^(1/3))/(Sqrt[3]*a^
(1/3))])/b^(8/3) - (3*a^(5/3)*Log[a^(1/3) + b^(1/3)*x^(1/3)])/(2*b^(8/3)) + (a^(5/3)*Log[a + b*x])/(2*b^(8/3))

Rule 31

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

Rule 50

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 56

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[{q = Rt[-((b*c - a*d)/b), 3]}, Simp
[Log[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] && Ne
gQ[(b*c - a*d)/b]

Rule 204

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

Rule 617

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^{5/3}}{a+b x} \, dx &=\frac {3 x^{5/3}}{5 b}-\frac {a \int \frac {x^{2/3}}{a+b x} \, dx}{b}\\ &=-\frac {3 a x^{2/3}}{2 b^2}+\frac {3 x^{5/3}}{5 b}+\frac {a^2 \int \frac {1}{\sqrt [3]{x} (a+b x)} \, dx}{b^2}\\ &=-\frac {3 a x^{2/3}}{2 b^2}+\frac {3 x^{5/3}}{5 b}+\frac {a^{5/3} \log (a+b x)}{2 b^{8/3}}+\frac {\left (3 a^2\right ) \operatorname {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^3}-\frac {\left (3 a^{5/3}\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt [3]{a}}{\sqrt [3]{b}}+x} \, dx,x,\sqrt [3]{x}\right )}{2 b^{8/3}}\\ &=-\frac {3 a x^{2/3}}{2 b^2}+\frac {3 x^{5/3}}{5 b}-\frac {3 a^{5/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 b^{8/3}}+\frac {a^{5/3} \log (a+b x)}{2 b^{8/3}}+\frac {\left (3 a^{5/3}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}\right )}{b^{8/3}}\\ &=-\frac {3 a x^{2/3}}{2 b^2}+\frac {3 x^{5/3}}{5 b}-\frac {\sqrt {3} a^{5/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{b^{8/3}}-\frac {3 a^{5/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 b^{8/3}}+\frac {a^{5/3} \log (a+b x)}{2 b^{8/3}}\\ \end {align*}

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Mathematica [C]  time = 0.01, size = 38, normalized size = 0.30 \begin {gather*} \frac {3 x^{2/3} \left (5 a \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};-\frac {b x}{a}\right )-5 a+2 b x\right )}{10 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^(5/3)/(a + b*x),x]

[Out]

(3*x^(2/3)*(-5*a + 2*b*x + 5*a*Hypergeometric2F1[2/3, 1, 5/3, -((b*x)/a)]))/(10*b^2)

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IntegrateAlgebraic [A]  time = 0.10, size = 150, normalized size = 1.20 \begin {gather*} \frac {a^{5/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{x}+b^{2/3} x^{2/3}\right )}{2 b^{8/3}}-\frac {a^{5/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{b^{8/3}}-\frac {\sqrt {3} a^{5/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a}}\right )}{b^{8/3}}+\frac {3 \left (2 b x^{5/3}-5 a x^{2/3}\right )}{10 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^(5/3)/(a + b*x),x]

[Out]

(3*(-5*a*x^(2/3) + 2*b*x^(5/3)))/(10*b^2) - (Sqrt[3]*a^(5/3)*ArcTan[1/Sqrt[3] - (2*b^(1/3)*x^(1/3))/(Sqrt[3]*a
^(1/3))])/b^(8/3) - (a^(5/3)*Log[a^(1/3) + b^(1/3)*x^(1/3)])/b^(8/3) + (a^(5/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*
x^(1/3) + b^(2/3)*x^(2/3)])/(2*b^(8/3))

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fricas [A]  time = 0.85, size = 147, normalized size = 1.18 \begin {gather*} \frac {10 \, \sqrt {3} a \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 ) - 5 \, a \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 ) + 10 \, a \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 \, {\left (2 \, b x - 5 \, a\right )} x^{\frac {2}{3}}}{10 \, b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/3)/(b*x+a),x, algorithm="fricas")

[Out]

1/10*(10*sqrt(3)*a*(-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) - 5*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)) + 10*a*(-a^2/b^2)^(1/3)*log(b*
(-a^2/b^2)^(2/3) + a*x^(1/3)) + 3*(2*b*x - 5*a)*x^(2/3))/b^2

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giac [A]  time = 1.04, size = 138, normalized size = 1.10 \begin {gather*} -\frac {a \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^{2}} - \frac {\sqrt {3} \left (-a b^{2}\right )^{\frac {2}{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^{4}} + \frac {\left (-a b^{2}\right )^{\frac {2}{3}} 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^{4}} + \frac {3 \, {\left (2 \, b^{4} x^{\frac {5}{3}} - 5 \, a b^{3} x^{\frac {2}{3}}\right )}}{10 \, b^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/3)/(b*x+a),x, algorithm="giac")

[Out]

-a*(-a/b)^(2/3)*log(abs(x^(1/3) - (-a/b)^(1/3)))/b^2 - sqrt(3)*(-a*b^2)^(2/3)*a*arctan(1/3*sqrt(3)*(2*x^(1/3)
+ (-a/b)^(1/3))/(-a/b)^(1/3))/b^4 + 1/2*(-a*b^2)^(2/3)*a*log(x^(2/3) + x^(1/3)*(-a/b)^(1/3) + (-a/b)^(2/3))/b^
4 + 3/10*(2*b^4*x^(5/3) - 5*a*b^3*x^(2/3))/b^5

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maple [A]  time = 0.01, size = 122, normalized size = 0.98 \begin {gather*} \frac {3 x^{\frac {5}{3}}}{5 b}+\frac {\sqrt {3}\, a^{2} \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}} b^{3}}-\frac {a^{2} \ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}} b^{3}}+\frac {a^{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 )}{2 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{3}}-\frac {3 a \,x^{\frac {2}{3}}}{2 b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(5/3)/(b*x+a),x)

[Out]

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

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maxima [A]  time = 3.00, size = 130, normalized size = 1.04 \begin {gather*} \frac {\sqrt {3} a^{2} \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} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {a^{2} \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} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {a^{2} \log \left (x^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{b^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {3 \, {\left (2 \, b x^{\frac {5}{3}} - 5 \, a x^{\frac {2}{3}}\right )}}{10 \, b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/3)/(b*x+a),x, algorithm="maxima")

[Out]

sqrt(3)*a^2*arctan(1/3*sqrt(3)*(2*x^(1/3) - (a/b)^(1/3))/(a/b)^(1/3))/(b^3*(a/b)^(1/3)) + 1/2*a^2*log(x^(2/3)
- x^(1/3)*(a/b)^(1/3) + (a/b)^(2/3))/(b^3*(a/b)^(1/3)) - a^2*log(x^(1/3) + (a/b)^(1/3))/(b^3*(a/b)^(1/3)) + 3/
10*(2*b*x^(5/3) - 5*a*x^(2/3))/b^2

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mupad [B]  time = 0.24, size = 151, normalized size = 1.21 \begin {gather*} \frac {3\,x^{5/3}}{5\,b}+\frac {{\left (-a\right )}^{5/3}\,\ln \left (\frac {9\,a^4\,x^{1/3}}{b^3}-\frac {9\,{\left (-a\right )}^{13/3}}{b^{10/3}}\right )}{b^{8/3}}-\frac {3\,a\,x^{2/3}}{2\,b^2}+\frac {{\left (-a\right )}^{5/3}\,\ln \left (\frac {9\,a^4\,x^{1/3}}{b^3}-\frac {9\,{\left (-a\right )}^{13/3}\,{\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2}{b^{10/3}}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{b^{8/3}}-\frac {{\left (-a\right )}^{5/3}\,\ln \left (\frac {9\,a^4\,x^{1/3}}{b^3}-\frac {9\,{\left (-a\right )}^{13/3}\,{\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2}{b^{10/3}}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{b^{8/3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(5/3)/(a + b*x),x)

[Out]

(3*x^(5/3))/(5*b) + ((-a)^(5/3)*log((9*a^4*x^(1/3))/b^3 - (9*(-a)^(13/3))/b^(10/3)))/b^(8/3) - (3*a*x^(2/3))/(
2*b^2) + ((-a)^(5/3)*log((9*a^4*x^(1/3))/b^3 - (9*(-a)^(13/3)*((3^(1/2)*1i)/2 - 1/2)^2)/b^(10/3))*((3^(1/2)*1i
)/2 - 1/2))/b^(8/3) - ((-a)^(5/3)*log((9*a^4*x^(1/3))/b^3 - (9*(-a)^(13/3)*((3^(1/2)*1i)/2 + 1/2)^2)/b^(10/3))
*((3^(1/2)*1i)/2 + 1/2))/b^(8/3)

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sympy [A]  time = 47.12, size = 241, normalized size = 1.93 \begin {gather*} \begin {cases} \tilde {\infty } x^{\frac {5}{3}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {3 x^{\frac {8}{3}}}{8 a} & \text {for}\: b = 0 \\\frac {3 x^{\frac {5}{3}}}{5 b} & \text {for}\: a = 0 \\- \frac {\left (-1\right )^{\frac {2}{3}} a^{\frac {5}{3}} \log {\left (- \sqrt [3]{-1} \sqrt [3]{a} \sqrt [3]{\frac {1}{b}} + \sqrt [3]{x} \right )}}{b^{4} \left (\frac {1}{b}\right )^{\frac {4}{3}}} + \frac {\left (-1\right )^{\frac {2}{3}} a^{\frac {5}{3}} \log {\left (4 \left (-1\right )^{\frac {2}{3}} a^{\frac {2}{3}} \left (\frac {1}{b}\right )^{\frac {2}{3}} + 4 \sqrt [3]{-1} \sqrt [3]{a} \sqrt [3]{x} \sqrt [3]{\frac {1}{b}} + 4 x^{\frac {2}{3}} \right )}}{2 b^{4} \left (\frac {1}{b}\right )^{\frac {4}{3}}} - \frac {\left (-1\right )^{\frac {2}{3}} \sqrt {3} a^{\frac {5}{3}} \operatorname {atan}{\left (\frac {\sqrt {3}}{3} - \frac {2 \left (-1\right )^{\frac {2}{3}} \sqrt {3} \sqrt [3]{x}}{3 \sqrt [3]{a} \sqrt [3]{\frac {1}{b}}} \right )}}{b^{4} \left (\frac {1}{b}\right )^{\frac {4}{3}}} - \frac {3 a x^{\frac {2}{3}}}{2 b^{2}} + \frac {3 x^{\frac {5}{3}}}{5 b} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(5/3)/(b*x+a),x)

[Out]

Piecewise((zoo*x**(5/3), Eq(a, 0) & Eq(b, 0)), (3*x**(8/3)/(8*a), Eq(b, 0)), (3*x**(5/3)/(5*b), Eq(a, 0)), (-(
-1)**(2/3)*a**(5/3)*log(-(-1)**(1/3)*a**(1/3)*(1/b)**(1/3) + x**(1/3))/(b**4*(1/b)**(4/3)) + (-1)**(2/3)*a**(5
/3)*log(4*(-1)**(2/3)*a**(2/3)*(1/b)**(2/3) + 4*(-1)**(1/3)*a**(1/3)*x**(1/3)*(1/b)**(1/3) + 4*x**(2/3))/(2*b*
*4*(1/b)**(4/3)) - (-1)**(2/3)*sqrt(3)*a**(5/3)*atan(sqrt(3)/3 - 2*(-1)**(2/3)*sqrt(3)*x**(1/3)/(3*a**(1/3)*(1
/b)**(1/3)))/(b**4*(1/b)**(4/3)) - 3*a*x**(2/3)/(2*b**2) + 3*x**(5/3)/(5*b), True))

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