3.67 \(\int \frac{A+B x^3}{x^5 \left (a+b x^3\right )} \, dx\)

Optimal. Leaf size=165 \[ \frac{\sqrt [3]{b} (A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{7/3}}-\frac{\sqrt [3]{b} (A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{7/3}}-\frac{\sqrt [3]{b} (A b-a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{7/3}}+\frac{A b-a B}{a^2 x}-\frac{A}{4 a x^4} \]

[Out]

-A/(4*a*x^4) + (A*b - a*B)/(a^2*x) - (b^(1/3)*(A*b - a*B)*ArcTan[(a^(1/3) - 2*b^
(1/3)*x)/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(7/3)) - (b^(1/3)*(A*b - a*B)*Log[a^(1/3
) + b^(1/3)*x])/(3*a^(7/3)) + (b^(1/3)*(A*b - a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)
*x + b^(2/3)*x^2])/(6*a^(7/3))

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Rubi [A]  time = 0.286374, antiderivative size = 165, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ \frac{\sqrt [3]{b} (A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{7/3}}-\frac{\sqrt [3]{b} (A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{7/3}}-\frac{\sqrt [3]{b} (A b-a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{7/3}}+\frac{A b-a B}{a^2 x}-\frac{A}{4 a x^4} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x^3)/(x^5*(a + b*x^3)),x]

[Out]

-A/(4*a*x^4) + (A*b - a*B)/(a^2*x) - (b^(1/3)*(A*b - a*B)*ArcTan[(a^(1/3) - 2*b^
(1/3)*x)/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(7/3)) - (b^(1/3)*(A*b - a*B)*Log[a^(1/3
) + b^(1/3)*x])/(3*a^(7/3)) + (b^(1/3)*(A*b - a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)
*x + b^(2/3)*x^2])/(6*a^(7/3))

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Rubi in Sympy [A]  time = 37.821, size = 150, normalized size = 0.91 \[ - \frac{A}{4 a x^{4}} + \frac{A b - B a}{a^{2} x} - \frac{\sqrt [3]{b} \left (A b - B a\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 a^{\frac{7}{3}}} + \frac{\sqrt [3]{b} \left (A b - B a\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 a^{\frac{7}{3}}} - \frac{\sqrt{3} \sqrt [3]{b} \left (A b - B a\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 a^{\frac{7}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x**3+A)/x**5/(b*x**3+a),x)

[Out]

-A/(4*a*x**4) + (A*b - B*a)/(a**2*x) - b**(1/3)*(A*b - B*a)*log(a**(1/3) + b**(1
/3)*x)/(3*a**(7/3)) + b**(1/3)*(A*b - B*a)*log(a**(2/3) - a**(1/3)*b**(1/3)*x +
b**(2/3)*x**2)/(6*a**(7/3)) - sqrt(3)*b**(1/3)*(A*b - B*a)*atan(sqrt(3)*(a**(1/3
)/3 - 2*b**(1/3)*x/3)/a**(1/3))/(3*a**(7/3))

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Mathematica [A]  time = 0.233049, size = 154, normalized size = 0.93 \[ \frac{2 \sqrt [3]{b} (A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-\frac{3 a^{4/3} A}{x^4}+\frac{12 \sqrt [3]{a} (A b-a B)}{x}+4 \sqrt [3]{b} (a B-A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-4 \sqrt{3} \sqrt [3]{b} (A b-a B) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{12 a^{7/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x^3)/(x^5*(a + b*x^3)),x]

[Out]

((-3*a^(4/3)*A)/x^4 + (12*a^(1/3)*(A*b - a*B))/x - 4*Sqrt[3]*b^(1/3)*(A*b - a*B)
*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 4*b^(1/3)*(-(A*b) + a*B)*Log[a^(1
/3) + b^(1/3)*x] + 2*b^(1/3)*(A*b - a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/
3)*x^2])/(12*a^(7/3))

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Maple [A]  time = 0.009, size = 216, normalized size = 1.3 \[ -{\frac{A}{4\,a{x}^{4}}}+{\frac{Ab}{x{a}^{2}}}-{\frac{B}{ax}}-{\frac{Ab}{3\,{a}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{B}{3\,a}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{Ab}{6\,{a}^{2}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{B}{6\,a}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{b\sqrt{3}A}{3\,{a}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{\sqrt{3}B}{3\,a}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^3+A)/x^5/(b*x^3+a),x)

[Out]

-1/4*A/a/x^4+1/a^2/x*A*b-B/a/x-1/3*b/a^2/(a/b)^(1/3)*ln(x+(a/b)^(1/3))*A+1/3/a/(
a/b)^(1/3)*ln(x+(a/b)^(1/3))*B+1/6*b/a^2/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^
(2/3))*A-1/6/a/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*B+1/3*b/a^2*3^(1/2)
/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*A-1/3/a*3^(1/2)/(a/b)^(1/3)
*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*B

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.230054, size = 248, normalized size = 1.5 \[ \frac{\sqrt{3}{\left (2 \, \sqrt{3}{\left (B a - A b\right )} x^{4} \left (-\frac{b}{a}\right )^{\frac{1}{3}} \log \left (b x^{2} - a x \left (-\frac{b}{a}\right )^{\frac{2}{3}} - a \left (-\frac{b}{a}\right )^{\frac{1}{3}}\right ) - 4 \, \sqrt{3}{\left (B a - A b\right )} x^{4} \left (-\frac{b}{a}\right )^{\frac{1}{3}} \log \left (b x + a \left (-\frac{b}{a}\right )^{\frac{2}{3}}\right ) - 12 \,{\left (B a - A b\right )} x^{4} \left (-\frac{b}{a}\right )^{\frac{1}{3}} \arctan \left (-\frac{2 \, \sqrt{3} b x - \sqrt{3} a \left (-\frac{b}{a}\right )^{\frac{2}{3}}}{3 \, a \left (-\frac{b}{a}\right )^{\frac{2}{3}}}\right ) - 3 \, \sqrt{3}{\left (4 \,{\left (B a - A b\right )} x^{3} + A a\right )}\right )}}{36 \, a^{2} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)/((b*x^3 + a)*x^5),x, algorithm="fricas")

[Out]

1/36*sqrt(3)*(2*sqrt(3)*(B*a - A*b)*x^4*(-b/a)^(1/3)*log(b*x^2 - a*x*(-b/a)^(2/3
) - a*(-b/a)^(1/3)) - 4*sqrt(3)*(B*a - A*b)*x^4*(-b/a)^(1/3)*log(b*x + a*(-b/a)^
(2/3)) - 12*(B*a - A*b)*x^4*(-b/a)^(1/3)*arctan(-1/3*(2*sqrt(3)*b*x - sqrt(3)*a*
(-b/a)^(2/3))/(a*(-b/a)^(2/3))) - 3*sqrt(3)*(4*(B*a - A*b)*x^3 + A*a))/(a^2*x^4)

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Sympy [A]  time = 2.64998, size = 112, normalized size = 0.68 \[ \operatorname{RootSum}{\left (27 t^{3} a^{7} + A^{3} b^{4} - 3 A^{2} B a b^{3} + 3 A B^{2} a^{2} b^{2} - B^{3} a^{3} b, \left ( t \mapsto t \log{\left (\frac{9 t^{2} a^{5}}{A^{2} b^{3} - 2 A B a b^{2} + B^{2} a^{2} b} + x \right )} \right )\right )} - \frac{A a + x^{3} \left (- 4 A b + 4 B a\right )}{4 a^{2} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x**3+A)/x**5/(b*x**3+a),x)

[Out]

RootSum(27*_t**3*a**7 + A**3*b**4 - 3*A**2*B*a*b**3 + 3*A*B**2*a**2*b**2 - B**3*
a**3*b, Lambda(_t, _t*log(9*_t**2*a**5/(A**2*b**3 - 2*A*B*a*b**2 + B**2*a**2*b)
+ x))) - (A*a + x**3*(-4*A*b + 4*B*a))/(4*a**2*x**4)

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GIAC/XCAS [A]  time = 0.221523, size = 266, normalized size = 1.61 \[ \frac{{\left (B a b \left (-\frac{a}{b}\right )^{\frac{1}{3}} - A b^{2} \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a^{3}} + \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{2}{3}} B a - \left (-a b^{2}\right )^{\frac{2}{3}} A b\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, a^{3} b} - \frac{{\left (\left (-a b^{2}\right )^{\frac{2}{3}} B a - \left (-a b^{2}\right )^{\frac{2}{3}} A b\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, a^{3} b} - \frac{4 \, B a x^{3} - 4 \, A b x^{3} + A a}{4 \, a^{2} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)/((b*x^3 + a)*x^5),x, algorithm="giac")

[Out]

1/3*(B*a*b*(-a/b)^(1/3) - A*b^2*(-a/b)^(1/3))*(-a/b)^(1/3)*ln(abs(x - (-a/b)^(1/
3)))/a^3 + 1/3*sqrt(3)*((-a*b^2)^(2/3)*B*a - (-a*b^2)^(2/3)*A*b)*arctan(1/3*sqrt
(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a^3*b) - 1/6*((-a*b^2)^(2/3)*B*a - (-a*b
^2)^(2/3)*A*b)*ln(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^3*b) - 1/4*(4*B*a*x^3
- 4*A*b*x^3 + A*a)/(a^2*x^4)