3.68 \(\int \frac{A+B x^3}{x^6 \left (a+b x^3\right )} \, dx\)

Optimal. Leaf size=168 \[ -\frac{b^{2/3} (A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{8/3}}+\frac{b^{2/3} (A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{8/3}}-\frac{b^{2/3} (A b-a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{8/3}}+\frac{A b-a B}{2 a^2 x^2}-\frac{A}{5 a x^5} \]

[Out]

-A/(5*a*x^5) + (A*b - a*B)/(2*a^2*x^2) - (b^(2/3)*(A*b - a*B)*ArcTan[(a^(1/3) -
2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(8/3)) + (b^(2/3)*(A*b - a*B)*Log[a^
(1/3) + b^(1/3)*x])/(3*a^(8/3)) - (b^(2/3)*(A*b - a*B)*Log[a^(2/3) - a^(1/3)*b^(
1/3)*x + b^(2/3)*x^2])/(6*a^(8/3))

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Rubi [A]  time = 0.28673, antiderivative size = 168, 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 \[ -\frac{b^{2/3} (A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{8/3}}+\frac{b^{2/3} (A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{8/3}}-\frac{b^{2/3} (A b-a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{8/3}}+\frac{A b-a B}{2 a^2 x^2}-\frac{A}{5 a x^5} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x^3)/(x^6*(a + b*x^3)),x]

[Out]

-A/(5*a*x^5) + (A*b - a*B)/(2*a^2*x^2) - (b^(2/3)*(A*b - a*B)*ArcTan[(a^(1/3) -
2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(8/3)) + (b^(2/3)*(A*b - a*B)*Log[a^
(1/3) + b^(1/3)*x])/(3*a^(8/3)) - (b^(2/3)*(A*b - a*B)*Log[a^(2/3) - a^(1/3)*b^(
1/3)*x + b^(2/3)*x^2])/(6*a^(8/3))

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Rubi in Sympy [A]  time = 38.7188, size = 153, normalized size = 0.91 \[ - \frac{A}{5 a x^{5}} + \frac{A b - B a}{2 a^{2} x^{2}} + \frac{b^{\frac{2}{3}} \left (A b - B a\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 a^{\frac{8}{3}}} - \frac{b^{\frac{2}{3}} \left (A b - B a\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 a^{\frac{8}{3}}} - \frac{\sqrt{3} b^{\frac{2}{3}} \left (A b - B a\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 a^{\frac{8}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x**3+A)/x**6/(b*x**3+a),x)

[Out]

-A/(5*a*x**5) + (A*b - B*a)/(2*a**2*x**2) + b**(2/3)*(A*b - B*a)*log(a**(1/3) +
b**(1/3)*x)/(3*a**(8/3)) - b**(2/3)*(A*b - B*a)*log(a**(2/3) - a**(1/3)*b**(1/3)
*x + b**(2/3)*x**2)/(6*a**(8/3)) - sqrt(3)*b**(2/3)*(A*b - B*a)*atan(sqrt(3)*(a*
*(1/3)/3 - 2*b**(1/3)*x/3)/a**(1/3))/(3*a**(8/3))

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Mathematica [A]  time = 0.2404, size = 154, normalized size = 0.92 \[ \frac{5 b^{2/3} (a B-A b) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+\frac{15 a^{2/3} (A b-a B)}{x^2}-\frac{6 a^{5/3} A}{x^5}+10 b^{2/3} (A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-10 \sqrt{3} b^{2/3} (A b-a B) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{30 a^{8/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x^3)/(x^6*(a + b*x^3)),x]

[Out]

((-6*a^(5/3)*A)/x^5 + (15*a^(2/3)*(A*b - a*B))/x^2 - 10*Sqrt[3]*b^(2/3)*(A*b - a
*B)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 10*b^(2/3)*(A*b - a*B)*Log[a^(
1/3) + b^(1/3)*x] + 5*b^(2/3)*(-(A*b) + a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b
^(2/3)*x^2])/(30*a^(8/3))

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Maple [A]  time = 0.011, size = 217, normalized size = 1.3 \[ -{\frac{A}{5\,a{x}^{5}}}+{\frac{Ab}{2\,{a}^{2}{x}^{2}}}-{\frac{B}{2\,a{x}^{2}}}+{\frac{Ab}{3\,{a}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{B}{3\,a}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{Ab}{6\,{a}^{2}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{B}{6\,a}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{b\sqrt{3}A}{3\,{a}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{\sqrt{3}B}{3\,a}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^3+A)/x^6/(b*x^3+a),x)

[Out]

-1/5*A/a/x^5+1/2/a^2/x^2*A*b-1/2/a/x^2*B+1/3*b/a^2/(a/b)^(2/3)*ln(x+(a/b)^(1/3))
*A-1/3/a/(a/b)^(2/3)*ln(x+(a/b)^(1/3))*B-1/6*b/a^2/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1
/3)+(a/b)^(2/3))*A+1/6/a/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*B+1/3*b/a
^2/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*A-1/3/a/(a/b)^(2/
3)*3^(1/2)*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^6),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.236858, size = 266, normalized size = 1.58 \[ \frac{\sqrt{3}{\left (5 \, \sqrt{3}{\left (B a - A b\right )} x^{5} \left (\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (b^{2} x^{2} - a b x \left (\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} + a^{2} \left (\frac{b^{2}}{a^{2}}\right )^{\frac{2}{3}}\right ) - 10 \, \sqrt{3}{\left (B a - A b\right )} x^{5} \left (\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (b x + a \left (\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}\right ) + 30 \,{\left (B a - A b\right )} x^{5} \left (\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \arctan \left (-\frac{2 \, \sqrt{3} b x - \sqrt{3} a \left (\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}}{3 \, a \left (\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}}\right ) - 3 \, \sqrt{3}{\left (5 \,{\left (B a - A b\right )} x^{3} + 2 \, A a\right )}\right )}}{90 \, a^{2} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)/((b*x^3 + a)*x^6),x, algorithm="fricas")

[Out]

1/90*sqrt(3)*(5*sqrt(3)*(B*a - A*b)*x^5*(b^2/a^2)^(1/3)*log(b^2*x^2 - a*b*x*(b^2
/a^2)^(1/3) + a^2*(b^2/a^2)^(2/3)) - 10*sqrt(3)*(B*a - A*b)*x^5*(b^2/a^2)^(1/3)*
log(b*x + a*(b^2/a^2)^(1/3)) + 30*(B*a - A*b)*x^5*(b^2/a^2)^(1/3)*arctan(-1/3*(2
*sqrt(3)*b*x - sqrt(3)*a*(b^2/a^2)^(1/3))/(a*(b^2/a^2)^(1/3))) - 3*sqrt(3)*(5*(B
*a - A*b)*x^3 + 2*A*a))/(a^2*x^5)

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Sympy [A]  time = 3.18139, size = 99, normalized size = 0.59 \[ \operatorname{RootSum}{\left (27 t^{3} a^{8} - A^{3} b^{5} + 3 A^{2} B a b^{4} - 3 A B^{2} a^{2} b^{3} + B^{3} a^{3} b^{2}, \left ( t \mapsto t \log{\left (- \frac{3 t a^{3}}{- A b^{2} + B a b} + x \right )} \right )\right )} - \frac{2 A a + x^{3} \left (- 5 A b + 5 B a\right )}{10 a^{2} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x**3+A)/x**6/(b*x**3+a),x)

[Out]

RootSum(27*_t**3*a**8 - A**3*b**5 + 3*A**2*B*a*b**4 - 3*A*B**2*a**2*b**3 + B**3*
a**3*b**2, Lambda(_t, _t*log(-3*_t*a**3/(-A*b**2 + B*a*b) + x))) - (2*A*a + x**3
*(-5*A*b + 5*B*a))/(10*a**2*x**5)

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GIAC/XCAS [A]  time = 0.216897, size = 238, normalized size = 1.42 \[ -\frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} B a - \left (-a b^{2}\right )^{\frac{1}{3}} A b\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, a^{3}} + \frac{{\left (B a b - A b^{2}\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a^{3}} - \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} B a - \left (-a b^{2}\right )^{\frac{1}{3}} A b\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, a^{3}} - \frac{5 \, B a x^{3} - 5 \, A b x^{3} + 2 \, A a}{10 \, a^{2} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)/((b*x^3 + a)*x^6),x, algorithm="giac")

[Out]

-1/3*sqrt(3)*((-a*b^2)^(1/3)*B*a - (-a*b^2)^(1/3)*A*b)*arctan(1/3*sqrt(3)*(2*x +
(-a/b)^(1/3))/(-a/b)^(1/3))/a^3 + 1/3*(B*a*b - A*b^2)*(-a/b)^(1/3)*ln(abs(x - (
-a/b)^(1/3)))/a^3 - 1/6*((-a*b^2)^(1/3)*B*a - (-a*b^2)^(1/3)*A*b)*ln(x^2 + x*(-a
/b)^(1/3) + (-a/b)^(2/3))/a^3 - 1/10*(5*B*a*x^3 - 5*A*b*x^3 + 2*A*a)/(a^2*x^5)