3.688 \(\int \frac{1}{x^{4/3} (a+b x)^2} \, dx\)

Optimal. Leaf size=124 \[ \frac{2 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{a^{7/3}}-\frac{2 \sqrt [3]{b} \log (a+b x)}{3 a^{7/3}}+\frac{4 \sqrt [3]{b} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{7/3}}-\frac{4}{a^2 \sqrt [3]{x}}+\frac{1}{a \sqrt [3]{x} (a+b x)} \]

[Out]

-4/(a^2*x^(1/3)) + 1/(a*x^(1/3)*(a + b*x)) + (4*b^(1/3)*ArcTan[(a^(1/3) - 2*b^(1
/3)*x^(1/3))/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(7/3)) + (2*b^(1/3)*Log[a^(1/3) + b^
(1/3)*x^(1/3)])/a^(7/3) - (2*b^(1/3)*Log[a + b*x])/(3*a^(7/3))

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Rubi [A]  time = 0.114734, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.462 \[ \frac{2 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{a^{7/3}}-\frac{2 \sqrt [3]{b} \log (a+b x)}{3 a^{7/3}}+\frac{4 \sqrt [3]{b} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{7/3}}-\frac{4}{a^2 \sqrt [3]{x}}+\frac{1}{a \sqrt [3]{x} (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^(4/3)*(a + b*x)^2),x]

[Out]

-4/(a^2*x^(1/3)) + 1/(a*x^(1/3)*(a + b*x)) + (4*b^(1/3)*ArcTan[(a^(1/3) - 2*b^(1
/3)*x^(1/3))/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(7/3)) + (2*b^(1/3)*Log[a^(1/3) + b^
(1/3)*x^(1/3)])/a^(7/3) - (2*b^(1/3)*Log[a + b*x])/(3*a^(7/3))

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Rubi in Sympy [A]  time = 15.1807, size = 122, normalized size = 0.98 \[ \frac{1}{a \sqrt [3]{x} \left (a + b x\right )} - \frac{4}{a^{2} \sqrt [3]{x}} + \frac{2 \sqrt [3]{b} \log{\left (\sqrt [3]{a} + \sqrt [3]{b} \sqrt [3]{x} \right )}}{a^{\frac{7}{3}}} - \frac{2 \sqrt [3]{b} \log{\left (a + b x \right )}}{3 a^{\frac{7}{3}}} + \frac{4 \sqrt{3} \sqrt [3]{b} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} \sqrt [3]{x}}{3}\right )}{\sqrt [3]{a}} \right )}}{3 a^{\frac{7}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**(4/3)/(b*x+a)**2,x)

[Out]

1/(a*x**(1/3)*(a + b*x)) - 4/(a**2*x**(1/3)) + 2*b**(1/3)*log(a**(1/3) + b**(1/3
)*x**(1/3))/a**(7/3) - 2*b**(1/3)*log(a + b*x)/(3*a**(7/3)) + 4*sqrt(3)*b**(1/3)
*atan(sqrt(3)*(a**(1/3)/3 - 2*b**(1/3)*x**(1/3)/3)/a**(1/3))/(3*a**(7/3))

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Mathematica [A]  time = 0.201982, size = 147, normalized size = 1.19 \[ \frac{-2 \sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{x}+b^{2/3} x^{2/3}\right )-\frac{3 \sqrt [3]{a} b x^{2/3}}{a+b x}+4 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )+4 \sqrt{3} \sqrt [3]{b} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt{3}}\right )-\frac{9 \sqrt [3]{a}}{\sqrt [3]{x}}}{3 a^{7/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^(4/3)*(a + b*x)^2),x]

[Out]

((-9*a^(1/3))/x^(1/3) - (3*a^(1/3)*b*x^(2/3))/(a + b*x) + 4*Sqrt[3]*b^(1/3)*ArcT
an[(1 - (2*b^(1/3)*x^(1/3))/a^(1/3))/Sqrt[3]] + 4*b^(1/3)*Log[a^(1/3) + b^(1/3)*
x^(1/3)] - 2*b^(1/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x^(1/3) + b^(2/3)*x^(2/3)])/(
3*a^(7/3))

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Maple [A]  time = 0.019, size = 121, normalized size = 1. \[ -3\,{\frac{1}{{a}^{2}\sqrt [3]{x}}}-{\frac{b}{{a}^{2} \left ( bx+a \right ) }{x}^{{\frac{2}{3}}}}+{\frac{4}{3\,{a}^{2}}\ln \left ( \sqrt [3]{x}+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{2}{3\,{a}^{2}}\ln \left ({x}^{{\frac{2}{3}}}-\sqrt [3]{x}\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{4\,\sqrt{3}}{3\,{a}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\sqrt [3]{x}{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(4/3)/(b*x+a)^2,x)

[Out]

-3/a^2/x^(1/3)-1/a^2*b*x^(2/3)/(b*x+a)+4/3/a^2/(a/b)^(1/3)*ln(x^(1/3)+(a/b)^(1/3
))-2/3/a^2/(a/b)^(1/3)*ln(x^(2/3)-x^(1/3)*(a/b)^(1/3)+(a/b)^(2/3))-4/3/a^2*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 [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^2*x^(4/3)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.224089, size = 231, normalized size = 1.86 \[ -\frac{\sqrt{3}{\left (2 \, \sqrt{3}{\left (b x + a\right )} x^{\frac{1}{3}} \left (\frac{b}{a}\right )^{\frac{1}{3}} \log \left (-a x^{\frac{1}{3}} \left (\frac{b}{a}\right )^{\frac{2}{3}} + b x^{\frac{2}{3}} + a \left (\frac{b}{a}\right )^{\frac{1}{3}}\right ) - 4 \, \sqrt{3}{\left (b x + a\right )} x^{\frac{1}{3}} \left (\frac{b}{a}\right )^{\frac{1}{3}} \log \left (a \left (\frac{b}{a}\right )^{\frac{2}{3}} + b x^{\frac{1}{3}}\right ) + 12 \,{\left (b x + a\right )} x^{\frac{1}{3}} \left (\frac{b}{a}\right )^{\frac{1}{3}} \arctan \left (-\frac{\sqrt{3} a \left (\frac{b}{a}\right )^{\frac{2}{3}} - 2 \, \sqrt{3} b x^{\frac{1}{3}}}{3 \, a \left (\frac{b}{a}\right )^{\frac{2}{3}}}\right ) + 3 \, \sqrt{3}{\left (4 \, b x + 3 \, a\right )}\right )}}{9 \,{\left (a^{2} b x + a^{3}\right )} x^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^2*x^(4/3)),x, algorithm="fricas")

[Out]

-1/9*sqrt(3)*(2*sqrt(3)*(b*x + a)*x^(1/3)*(b/a)^(1/3)*log(-a*x^(1/3)*(b/a)^(2/3)
+ b*x^(2/3) + a*(b/a)^(1/3)) - 4*sqrt(3)*(b*x + a)*x^(1/3)*(b/a)^(1/3)*log(a*(b
/a)^(2/3) + b*x^(1/3)) + 12*(b*x + a)*x^(1/3)*(b/a)^(1/3)*arctan(-1/3*(sqrt(3)*a
*(b/a)^(2/3) - 2*sqrt(3)*b*x^(1/3))/(a*(b/a)^(2/3))) + 3*sqrt(3)*(4*b*x + 3*a))/
((a^2*b*x + a^3)*x^(1/3))

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Sympy [A]  time = 5.47909, size = 619, normalized size = 4.99 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**(4/3)/(b*x+a)**2,x)

[Out]

9*a**(4/3)*gamma(-1/3)/(9*a**(10/3)*x**(1/3)*gamma(2/3) + 9*a**(7/3)*b*x**(4/3)*
gamma(2/3)) + 12*a**(1/3)*b*x*gamma(-1/3)/(9*a**(10/3)*x**(1/3)*gamma(2/3) + 9*a
**(7/3)*b*x**(4/3)*gamma(2/3)) - 4*a*b**(1/3)*x**(1/3)*exp(10*I*pi/3)*log(1 - b*
*(1/3)*x**(1/3)*exp_polar(I*pi/3)/a**(1/3))*gamma(-1/3)/(9*a**(10/3)*x**(1/3)*ga
mma(2/3) + 9*a**(7/3)*b*x**(4/3)*gamma(2/3)) - 4*a*b**(1/3)*x**(1/3)*log(1 - b**
(1/3)*x**(1/3)*exp_polar(I*pi)/a**(1/3))*gamma(-1/3)/(9*a**(10/3)*x**(1/3)*gamma
(2/3) + 9*a**(7/3)*b*x**(4/3)*gamma(2/3)) - 4*a*b**(1/3)*x**(1/3)*exp(2*I*pi/3)*
log(1 - b**(1/3)*x**(1/3)*exp_polar(5*I*pi/3)/a**(1/3))*gamma(-1/3)/(9*a**(10/3)
*x**(1/3)*gamma(2/3) + 9*a**(7/3)*b*x**(4/3)*gamma(2/3)) - 4*b**(4/3)*x**(4/3)*e
xp(10*I*pi/3)*log(1 - b**(1/3)*x**(1/3)*exp_polar(I*pi/3)/a**(1/3))*gamma(-1/3)/
(9*a**(10/3)*x**(1/3)*gamma(2/3) + 9*a**(7/3)*b*x**(4/3)*gamma(2/3)) - 4*b**(4/3
)*x**(4/3)*log(1 - b**(1/3)*x**(1/3)*exp_polar(I*pi)/a**(1/3))*gamma(-1/3)/(9*a*
*(10/3)*x**(1/3)*gamma(2/3) + 9*a**(7/3)*b*x**(4/3)*gamma(2/3)) - 4*b**(4/3)*x**
(4/3)*exp(2*I*pi/3)*log(1 - b**(1/3)*x**(1/3)*exp_polar(5*I*pi/3)/a**(1/3))*gamm
a(-1/3)/(9*a**(10/3)*x**(1/3)*gamma(2/3) + 9*a**(7/3)*b*x**(4/3)*gamma(2/3))

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GIAC/XCAS [A]  time = 0.221491, size = 196, normalized size = 1.58 \[ \frac{4 \, b \left (-\frac{a}{b}\right )^{\frac{2}{3}}{\rm ln}\left ({\left | x^{\frac{1}{3}} - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a^{3}} + \frac{4 \, \sqrt{3} \left (-a b^{2}\right )^{\frac{2}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, a^{3} b} - \frac{4 \, b x + 3 \, a}{{\left (b x^{\frac{4}{3}} + a x^{\frac{1}{3}}\right )} a^{2}} - \frac{2 \, \left (-a b^{2}\right )^{\frac{2}{3}}{\rm ln}\left (x^{\frac{2}{3}} + x^{\frac{1}{3}} \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{3 \, a^{3} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^2*x^(4/3)),x, algorithm="giac")

[Out]

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