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 = 0.103591, 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 \[ \frac{3}{2} a^{4/3} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )-\sqrt{3} a^{4/3} \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )-\frac{1}{2} a^{4/3} \log (x)+3 a \sqrt [3]{a+b x}+\frac{3}{4} (a+b x)^{4/3} \]

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

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Rubi in Sympy [A]  time = 9.63422, size = 97, normalized size = 0.92 \[ - \frac{a^{\frac{4}{3}} \log{\left (x \right )}}{2} + \frac{3 a^{\frac{4}{3}} \log{\left (\sqrt [3]{a} - \sqrt [3]{a + b x} \right )}}{2} - \sqrt{3} a^{\frac{4}{3}} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} + \frac{2 \sqrt [3]{a + b x}}{3}\right )}{\sqrt [3]{a}} \right )} + 3 a \sqrt [3]{a + b x} + \frac{3 \left (a + b x\right )^{\frac{4}{3}}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(4/3)/x,x)

[Out]

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

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Mathematica [C]  time = 0.0373007, size = 74, normalized size = 0.7 \[ \left (\frac{15 a}{4}+\frac{3 b x}{4}\right ) \sqrt [3]{a+b x}-\frac{3 a^2 \left (\frac{a+b x}{b x}\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{2}{3};\frac{5}{3};-\frac{a}{b x}\right )}{2 (a+b x)^{2/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^(4/3)/x,x]

[Out]

((15*a)/4 + (3*b*x)/4)*(a + b*x)^(1/3) - (3*a^2*((a + b*x)/(b*x))^(2/3)*Hypergeo
metric2F1[2/3, 2/3, 5/3, -(a/(b*x))])/(2*(a + b*x)^(2/3))

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Maple [A]  time = 0.007, size = 95, normalized size = 0.9 \[{\frac{3}{4} \left ( bx+a \right ) ^{{\frac{4}{3}}}}+3\,a\sqrt [3]{bx+a}+{a}^{{\frac{4}{3}}}\ln \left ( \sqrt [3]{bx+a}-\sqrt [3]{a} \right ) -{\frac{1}{2}{a}^{{\frac{4}{3}}}\ln \left ( \left ( bx+a \right ) ^{{\frac{2}{3}}}+\sqrt [3]{bx+a}\sqrt [3]{a}+{a}^{{\frac{2}{3}}} \right ) }-{a}^{{\frac{4}{3}}}\sqrt{3}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\frac{\sqrt [3]{bx+a}}{\sqrt [3]{a}}}+1 \right ) } \right ) \]

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)+(b*x+a)^(1/3)*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]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

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 = 0.236972, size = 126, normalized size = 1.2 \[ -\sqrt{3} a^{\frac{4}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \,{\left (b x + a\right )}^{\frac{1}{3}} + a^{\frac{1}{3}}\right )}}{3 \, a^{\frac{1}{3}}}\right ) - \frac{1}{2} \, a^{\frac{4}{3}} \log \left ({\left (b x + a\right )}^{\frac{2}{3}} +{\left (b x + a\right )}^{\frac{1}{3}} a^{\frac{1}{3}} + a^{\frac{2}{3}}\right ) + a^{\frac{4}{3}} \log \left ({\left (b x + a\right )}^{\frac{1}{3}} - a^{\frac{1}{3}}\right ) + \frac{3}{4} \,{\left (b x + 5 \, a\right )}{\left (b x + a\right )}^{\frac{1}{3}} \]

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*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 + 5*a)*(b*x + a)^(1/3)

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Sympy [A]  time = 7.56517, size = 209, normalized size = 1.99 \[ \frac{7 a^{\frac{4}{3}} \log{\left (1 - \frac{\sqrt [3]{b} \sqrt [3]{\frac{a}{b} + x}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac{7}{3}\right )}{3 \Gamma \left (\frac{10}{3}\right )} + \frac{7 a^{\frac{4}{3}} e^{\frac{4 i \pi }{3}} \log{\left (1 - \frac{\sqrt [3]{b} \sqrt [3]{\frac{a}{b} + x} e^{\frac{2 i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac{7}{3}\right )}{3 \Gamma \left (\frac{10}{3}\right )} + \frac{7 a^{\frac{4}{3}} e^{\frac{2 i \pi }{3}} \log{\left (1 - \frac{\sqrt [3]{b} \sqrt [3]{\frac{a}{b} + x} e^{\frac{4 i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac{7}{3}\right )}{3 \Gamma \left (\frac{10}{3}\right )} + \frac{7 a \sqrt [3]{b} \sqrt [3]{\frac{a}{b} + x} \Gamma \left (\frac{7}{3}\right )}{\Gamma \left (\frac{10}{3}\right )} + \frac{7 b^{\frac{4}{3}} \left (\frac{a}{b} + x\right )^{\frac{4}{3}} \Gamma \left (\frac{7}{3}\right )}{4 \Gamma \left (\frac{10}{3}\right )} \]

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(4*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/XCAS [A]  time = 0.510789, size = 131, normalized size = 1.25 \[ -\sqrt{3} a^{\frac{4}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \,{\left (b x + a\right )}^{\frac{1}{3}} + a^{\frac{1}{3}}\right )}}{3 \, a^{\frac{1}{3}}}\right ) - \frac{1}{2} \, a^{\frac{4}{3}}{\rm ln}\left ({\left (b x + a\right )}^{\frac{2}{3}} +{\left (b x + a\right )}^{\frac{1}{3}} a^{\frac{1}{3}} + a^{\frac{2}{3}}\right ) + a^{\frac{4}{3}}{\rm ln}\left ({\left |{\left (b x + a\right )}^{\frac{1}{3}} - a^{\frac{1}{3}} \right |}\right ) + \frac{3}{4} \,{\left (b x + a\right )}^{\frac{4}{3}} + 3 \,{\left (b x + a\right )}^{\frac{1}{3}} a \]

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