3.70 \(\int \frac{A+B x^3}{x^8 \left (a+b x^3\right )} \, dx\)

Optimal. Leaf size=184 \[ -\frac{b^{4/3} (A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{10/3}}+\frac{b^{4/3} (A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{10/3}}+\frac{b^{4/3} (A b-a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{10/3}}-\frac{b (A b-a B)}{a^3 x}+\frac{A b-a B}{4 a^2 x^4}-\frac{A}{7 a x^7} \]

[Out]

-A/(7*a*x^7) + (A*b - a*B)/(4*a^2*x^4) - (b*(A*b - a*B))/(a^3*x) + (b^(4/3)*(A*b
- a*B)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(10/3)) +
(b^(4/3)*(A*b - a*B)*Log[a^(1/3) + b^(1/3)*x])/(3*a^(10/3)) - (b^(4/3)*(A*b - a*
B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*a^(10/3))

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Rubi [A]  time = 0.346352, antiderivative size = 184, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ -\frac{b^{4/3} (A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{10/3}}+\frac{b^{4/3} (A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{10/3}}+\frac{b^{4/3} (A b-a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{10/3}}-\frac{b (A b-a B)}{a^3 x}+\frac{A b-a B}{4 a^2 x^4}-\frac{A}{7 a x^7} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x^3)/(x^8*(a + b*x^3)),x]

[Out]

-A/(7*a*x^7) + (A*b - a*B)/(4*a^2*x^4) - (b*(A*b - a*B))/(a^3*x) + (b^(4/3)*(A*b
- a*B)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(10/3)) +
(b^(4/3)*(A*b - a*B)*Log[a^(1/3) + b^(1/3)*x])/(3*a^(10/3)) - (b^(4/3)*(A*b - a*
B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*a^(10/3))

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Rubi in Sympy [A]  time = 43.7329, size = 167, normalized size = 0.91 \[ - \frac{A}{7 a x^{7}} + \frac{A b - B a}{4 a^{2} x^{4}} - \frac{b \left (A b - B a\right )}{a^{3} x} + \frac{b^{\frac{4}{3}} \left (A b - B a\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 a^{\frac{10}{3}}} - \frac{b^{\frac{4}{3}} \left (A b - B a\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 a^{\frac{10}{3}}} + \frac{\sqrt{3} b^{\frac{4}{3}} \left (A b - B a\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 a^{\frac{10}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x**3+A)/x**8/(b*x**3+a),x)

[Out]

-A/(7*a*x**7) + (A*b - B*a)/(4*a**2*x**4) - b*(A*b - B*a)/(a**3*x) + b**(4/3)*(A
*b - B*a)*log(a**(1/3) + b**(1/3)*x)/(3*a**(10/3)) - b**(4/3)*(A*b - B*a)*log(a*
*(2/3) - a**(1/3)*b**(1/3)*x + b**(2/3)*x**2)/(6*a**(10/3)) + sqrt(3)*b**(4/3)*(
A*b - B*a)*atan(sqrt(3)*(a**(1/3)/3 - 2*b**(1/3)*x/3)/a**(1/3))/(3*a**(10/3))

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Mathematica [A]  time = 0.330165, size = 173, normalized size = 0.94 \[ \frac{14 b^{4/3} (a B-A b) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+\frac{21 a^{4/3} (A b-a B)}{x^4}-\frac{12 a^{7/3} A}{x^7}+28 b^{4/3} (A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+28 \sqrt{3} b^{4/3} (A b-a B) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )+\frac{84 \sqrt [3]{a} b (a B-A b)}{x}}{84 a^{10/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x^3)/(x^8*(a + b*x^3)),x]

[Out]

((-12*a^(7/3)*A)/x^7 + (21*a^(4/3)*(A*b - a*B))/x^4 + (84*a^(1/3)*b*(-(A*b) + a*
B))/x + 28*Sqrt[3]*b^(4/3)*(A*b - a*B)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3
]] + 28*b^(4/3)*(A*b - a*B)*Log[a^(1/3) + b^(1/3)*x] + 14*b^(4/3)*(-(A*b) + a*B)
*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(84*a^(10/3))

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Maple [A]  time = 0.01, size = 247, normalized size = 1.3 \[ -{\frac{A}{7\,a{x}^{7}}}+{\frac{Ab}{4\,{x}^{4}{a}^{2}}}-{\frac{B}{4\,a{x}^{4}}}-{\frac{{b}^{2}A}{{a}^{3}x}}+{\frac{Bb}{x{a}^{2}}}+{\frac{{b}^{2}A}{3\,{a}^{3}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{Bb}{3\,{a}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{{b}^{2}A}{6\,{a}^{3}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{Bb}{6\,{a}^{2}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{{b}^{2}\sqrt{3}A}{3\,{a}^{3}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{b\sqrt{3}B}{3\,{a}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^3+A)/x^8/(b*x^3+a),x)

[Out]

-1/7*A/a/x^7+1/4/a^2/x^4*A*b-1/4/a/x^4*B-1/a^3*b^2/x*A+1/a^2*b/x*B+1/3/a^3*b^2/(
a/b)^(1/3)*ln(x+(a/b)^(1/3))*A-1/3/a^2*b/(a/b)^(1/3)*ln(x+(a/b)^(1/3))*B-1/6/a^3
*b^2/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*A+1/6/a^2*b/(a/b)^(1/3)*ln(x^
2-x*(a/b)^(1/3)+(a/b)^(2/3))*B-1/3/a^3*b^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^2*b*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^8),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.230142, size = 278, normalized size = 1.51 \[ \frac{\sqrt{3}{\left (14 \, \sqrt{3}{\left (B a b - A b^{2}\right )} x^{7} \left (\frac{b}{a}\right )^{\frac{1}{3}} \log \left (b x^{2} - a x \left (\frac{b}{a}\right )^{\frac{2}{3}} + a \left (\frac{b}{a}\right )^{\frac{1}{3}}\right ) - 28 \, \sqrt{3}{\left (B a b - A b^{2}\right )} x^{7} \left (\frac{b}{a}\right )^{\frac{1}{3}} \log \left (b x + a \left (\frac{b}{a}\right )^{\frac{2}{3}}\right ) - 84 \,{\left (B a b - A b^{2}\right )} x^{7} \left (\frac{b}{a}\right )^{\frac{1}{3}} \arctan \left (-\frac{2 \, \sqrt{3} b x - \sqrt{3} a \left (\frac{b}{a}\right )^{\frac{2}{3}}}{3 \, a \left (\frac{b}{a}\right )^{\frac{2}{3}}}\right ) + 3 \, \sqrt{3}{\left (28 \,{\left (B a b - A b^{2}\right )} x^{6} - 7 \,{\left (B a^{2} - A a b\right )} x^{3} - 4 \, A a^{2}\right )}\right )}}{252 \, a^{3} x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)/((b*x^3 + a)*x^8),x, algorithm="fricas")

[Out]

1/252*sqrt(3)*(14*sqrt(3)*(B*a*b - A*b^2)*x^7*(b/a)^(1/3)*log(b*x^2 - a*x*(b/a)^
(2/3) + a*(b/a)^(1/3)) - 28*sqrt(3)*(B*a*b - A*b^2)*x^7*(b/a)^(1/3)*log(b*x + a*
(b/a)^(2/3)) - 84*(B*a*b - A*b^2)*x^7*(b/a)^(1/3)*arctan(-1/3*(2*sqrt(3)*b*x - s
qrt(3)*a*(b/a)^(2/3))/(a*(b/a)^(2/3))) + 3*sqrt(3)*(28*(B*a*b - A*b^2)*x^6 - 7*(
B*a^2 - A*a*b)*x^3 - 4*A*a^2))/(a^3*x^7)

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Sympy [A]  time = 3.9306, size = 139, normalized size = 0.76 \[ \operatorname{RootSum}{\left (27 t^{3} a^{10} - A^{3} b^{7} + 3 A^{2} B a b^{6} - 3 A B^{2} a^{2} b^{5} + B^{3} a^{3} b^{4}, \left ( t \mapsto t \log{\left (\frac{9 t^{2} a^{7}}{A^{2} b^{5} - 2 A B a b^{4} + B^{2} a^{2} b^{3}} + x \right )} \right )\right )} + \frac{- 4 A a^{2} + x^{6} \left (- 28 A b^{2} + 28 B a b\right ) + x^{3} \left (7 A a b - 7 B a^{2}\right )}{28 a^{3} x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x**3+A)/x**8/(b*x**3+a),x)

[Out]

RootSum(27*_t**3*a**10 - A**3*b**7 + 3*A**2*B*a*b**6 - 3*A*B**2*a**2*b**5 + B**3
*a**3*b**4, Lambda(_t, _t*log(9*_t**2*a**7/(A**2*b**5 - 2*A*B*a*b**4 + B**2*a**2
*b**3) + x))) + (-4*A*a**2 + x**6*(-28*A*b**2 + 28*B*a*b) + x**3*(7*A*a*b - 7*B*
a**2))/(28*a**3*x**7)

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GIAC/XCAS [A]  time = 0.21815, size = 292, normalized size = 1.59 \[ -\frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{2}{3}} B a - \left (-a b^{2}\right )^{\frac{2}{3}} A b\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, a^{4}} - \frac{{\left (B a b^{2} \left (-\frac{a}{b}\right )^{\frac{1}{3}} - A b^{3} \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a^{4}} + \frac{{\left (\left (-a b^{2}\right )^{\frac{2}{3}} B a - \left (-a b^{2}\right )^{\frac{2}{3}} A b\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, a^{4}} + \frac{28 \, B a b x^{6} - 28 \, A b^{2} x^{6} - 7 \, B a^{2} x^{3} + 7 \, A a b x^{3} - 4 \, A a^{2}}{28 \, a^{3} x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)/((b*x^3 + a)*x^8),x, algorithm="giac")

[Out]

-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^4 - 1/3*(B*a*b^2*(-a/b)^(1/3) - A*b^3*(-a/b)^(1/3
))*(-a/b)^(1/3)*ln(abs(x - (-a/b)^(1/3)))/a^4 + 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^4 + 1/28*(28*B*a*b*x^6 -
28*A*b^2*x^6 - 7*B*a^2*x^3 + 7*A*a*b*x^3 - 4*A*a^2)/(a^3*x^7)