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

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

[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

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Rubi [A]  time = 0.0406306, antiderivative size = 105, normalized size of antiderivative = 1., 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} \[ \frac{3}{2} a^{4/3} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )-\sqrt{3} a^{4/3} \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )-\frac{1}{2} a^{4/3} \log (x)+3 a \sqrt [3]{a+b x}+\frac{3}{4} (a+b x)^{4/3} \]

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 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 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]

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 31

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

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*}

Mathematica [A]  time = 0.0687591, size = 130, normalized size = 1.24 \[ \frac{1}{4} \left (4 a^{4/3} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )-2 a^{4/3} \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x}+(a+b x)^{2/3}\right )-4 \sqrt{3} a^{4/3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}+1}{\sqrt{3}}\right )+15 a \sqrt [3]{a+b x}+3 b x \sqrt [3]{a+b x}\right ) \]

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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Maple [A]  time = 0.004, size = 95, normalized size = 0.9 \begin{align*}{\frac{3}{4} \left ( bx+a \right ) ^{{\frac{4}{3}}}}+3\,a\sqrt [3]{bx+a}+{a}^{{\frac{4}{3}}}\ln \left ( \sqrt [3]{bx+a}-\sqrt [3]{a} \right ) -{\frac{1}{2}{a}^{{\frac{4}{3}}}\ln \left ( \left ( bx+a \right ) ^{{\frac{2}{3}}}+\sqrt [3]{a}\sqrt [3]{bx+a}+{a}^{{\frac{2}{3}}} \right ) }-{a}^{{\frac{4}{3}}}\sqrt{3}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\frac{\sqrt [3]{bx+a}}{\sqrt [3]{a}}}+1 \right ) } \right ) \end{align*}

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.67614, size = 305, normalized size = 2.9 \begin{align*} -\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{align*}

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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Sympy [C]  time = 3.15185, size = 209, normalized size = 1.99 \begin{align*} \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{align*}

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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Giac [A]  time = 2.10626, size = 131, normalized size = 1.25 \begin{align*} -\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{align*}

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