3.4.89 \(\int \frac {(a+b x)^{4/3}}{x} \, dx\)

Optimal. Leaf size=105 \[ \frac {3}{2} a^{4/3} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )-\sqrt {3} a^{4/3} \tan ^{-1}\left (\frac {2 \sqrt [3]{a+b x}+\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )-\frac {1}{2} a^{4/3} \log (x)+3 a \sqrt [3]{a+b x}+\frac {3}{4} (a+b x)^{4/3} \]

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Rubi [A]  time = 0.04, antiderivative size = 105, 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, 57, 617, 204, 31} \begin {gather*} \frac {3}{2} a^{4/3} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )-\sqrt {3} a^{4/3} \tan ^{-1}\left (\frac {2 \sqrt [3]{a+b x}+\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )-\frac {1}{2} a^{4/3} \log (x)+3 a \sqrt [3]{a+b x}+\frac {3}{4} (a+b x)^{4/3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^(4/3)/x,x]

[Out]

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

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 57

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/b, 3]}, -Simp[L
og[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]
&& PosQ[(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 {(a+b x)^{4/3}}{x} \, dx &=\frac {3}{4} (a+b x)^{4/3}+a \int \frac {\sqrt [3]{a+b x}}{x} \, dx\\ &=3 a \sqrt [3]{a+b x}+\frac {3}{4} (a+b x)^{4/3}+a^2 \int \frac {1}{x (a+b x)^{2/3}} \, dx\\ &=3 a \sqrt [3]{a+b x}+\frac {3}{4} (a+b x)^{4/3}-\frac {1}{2} a^{4/3} \log (x)-\frac {1}{2} \left (3 a^{4/3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+b x}\right )-\frac {1}{2} \left (3 a^{5/3}\right ) \operatorname {Subst}\left (\int \frac {1}{a^{2/3}+\sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{a+b x}\right )\\ &=3 a \sqrt [3]{a+b x}+\frac {3}{4} (a+b x)^{4/3}-\frac {1}{2} a^{4/3} \log (x)+\frac {3}{2} a^{4/3} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )+\left (3 a^{4/3}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}\right )\\ &=3 a \sqrt [3]{a+b x}+\frac {3}{4} (a+b x)^{4/3}-\sqrt {3} a^{4/3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )-\frac {1}{2} a^{4/3} \log (x)+\frac {3}{2} a^{4/3} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 130, normalized size = 1.24 \begin {gather*} \frac {1}{4} \left (4 a^{4/3} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )-2 a^{4/3} \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x}+(a+b x)^{2/3}\right )-4 \sqrt {3} a^{4/3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}+1}{\sqrt {3}}\right )+15 a \sqrt [3]{a+b x}+3 b x \sqrt [3]{a+b x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^(4/3)/x,x]

[Out]

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

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IntegrateAlgebraic [A]  time = 0.06, size = 131, normalized size = 1.25 \begin {gather*} a^{4/3} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )-\frac {1}{2} a^{4/3} \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x}+(a+b x)^{2/3}\right )-\sqrt {3} a^{4/3} \tan ^{-1}\left (\frac {2 \sqrt [3]{a+b x}}{\sqrt {3} \sqrt [3]{a}}+\frac {1}{\sqrt {3}}\right )+\frac {3}{4} \left ((a+b x)^{4/3}+4 a \sqrt [3]{a+b x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a + b*x)^(4/3)/x,x]

[Out]

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

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fricas [A]  time = 0.66, size = 98, normalized size = 0.93 \begin {gather*} -\sqrt {3} a^{\frac {4}{3}} \arctan \left (\frac {2 \, \sqrt {3} {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {2}{3}} + \sqrt {3} a}{3 \, a}\right ) - \frac {1}{2} \, a^{\frac {4}{3}} \log \left ({\left (b x + a\right )}^{\frac {2}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right ) + a^{\frac {4}{3}} \log \left ({\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right ) + \frac {3}{4} \, {\left (b x + 5 \, a\right )} {\left (b x + a\right )}^{\frac {1}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 2.02, size = 97, normalized size = 0.92 \begin {gather*} -\sqrt {3} a^{\frac {4}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right ) - \frac {1}{2} \, a^{\frac {4}{3}} \log \left ({\left (b x + a\right )}^{\frac {2}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right ) + a^{\frac {4}{3}} \log \left ({\left | {\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}} \right |}\right ) + \frac {3}{4} \, {\left (b x + a\right )}^{\frac {4}{3}} + 3 \, {\left (b x + a\right )}^{\frac {1}{3}} a \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.01, size = 95, normalized size = 0.90 \begin {gather*} -\sqrt {3}\, a^{\frac {4}{3}} \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b x +a \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )+a^{\frac {4}{3}} \ln \left (-a^{\frac {1}{3}}+\left (b x +a \right )^{\frac {1}{3}}\right )-\frac {a^{\frac {4}{3}} \ln \left (a^{\frac {2}{3}}+\left (b x +a \right )^{\frac {1}{3}} a^{\frac {1}{3}}+\left (b x +a \right )^{\frac {2}{3}}\right )}{2}+3 \left (b x +a \right )^{\frac {1}{3}} a +\frac {3 \left (b x +a \right )^{\frac {4}{3}}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 3.03, size = 96, normalized size = 0.91 \begin {gather*} -\sqrt {3} a^{\frac {4}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right ) - \frac {1}{2} \, a^{\frac {4}{3}} \log \left ({\left (b x + a\right )}^{\frac {2}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right ) + a^{\frac {4}{3}} \log \left ({\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right ) + \frac {3}{4} \, {\left (b x + a\right )}^{\frac {4}{3}} + 3 \, {\left (b x + a\right )}^{\frac {1}{3}} a \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.06, size = 123, normalized size = 1.17 \begin {gather*} 3\,a\,{\left (a+b\,x\right )}^{1/3}+\frac {3\,{\left (a+b\,x\right )}^{4/3}}{4}+a^{4/3}\,\ln \left (9\,a^2\,{\left (a+b\,x\right )}^{1/3}-9\,a^{7/3}\right )+\frac {a^{4/3}\,\ln \left (\frac {9\,a^{7/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}-9\,a^2\,{\left (a+b\,x\right )}^{1/3}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}-\frac {a^{4/3}\,\ln \left (\frac {9\,a^{7/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}+9\,a^2\,{\left (a+b\,x\right )}^{1/3}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^(4/3)/x,x)

[Out]

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

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sympy [C]  time = 2.39, size = 209, normalized size = 1.99 \begin {gather*} \frac {7 a^{\frac {4}{3}} \log {\left (1 - \frac {\sqrt [3]{b} \sqrt [3]{\frac {a}{b} + x}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac {7}{3}\right )}{3 \Gamma \left (\frac {10}{3}\right )} + \frac {7 a^{\frac {4}{3}} e^{- \frac {2 i \pi }{3}} \log {\left (1 - \frac {\sqrt [3]{b} \sqrt [3]{\frac {a}{b} + x} e^{\frac {2 i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac {7}{3}\right )}{3 \Gamma \left (\frac {10}{3}\right )} + \frac {7 a^{\frac {4}{3}} e^{\frac {2 i \pi }{3}} \log {\left (1 - \frac {\sqrt [3]{b} \sqrt [3]{\frac {a}{b} + x} e^{\frac {4 i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac {7}{3}\right )}{3 \Gamma \left (\frac {10}{3}\right )} + \frac {7 a \sqrt [3]{b} \sqrt [3]{\frac {a}{b} + x} \Gamma \left (\frac {7}{3}\right )}{\Gamma \left (\frac {10}{3}\right )} + \frac {7 b^{\frac {4}{3}} \left (\frac {a}{b} + x\right )^{\frac {4}{3}} \Gamma \left (\frac {7}{3}\right )}{4 \Gamma \left (\frac {10}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

7*a**(4/3)*log(1 - b**(1/3)*(a/b + x)**(1/3)/a**(1/3))*gamma(7/3)/(3*gamma(10/3)) + 7*a**(4/3)*exp(-2*I*pi/3)*
log(1 - b**(1/3)*(a/b + x)**(1/3)*exp_polar(2*I*pi/3)/a**(1/3))*gamma(7/3)/(3*gamma(10/3)) + 7*a**(4/3)*exp(2*
I*pi/3)*log(1 - b**(1/3)*(a/b + x)**(1/3)*exp_polar(4*I*pi/3)/a**(1/3))*gamma(7/3)/(3*gamma(10/3)) + 7*a*b**(1
/3)*(a/b + x)**(1/3)*gamma(7/3)/gamma(10/3) + 7*b**(4/3)*(a/b + x)**(4/3)*gamma(7/3)/(4*gamma(10/3))

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