3.85 \(\int \frac{A+B x^3}{x^5 \left (a+b x^3\right )^2} \, dx\)

Optimal. Leaf size=215 \[ \frac{\sqrt [3]{b} (7 A b-4 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{10/3}}-\frac{\sqrt [3]{b} (7 A b-4 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{10/3}}-\frac{\sqrt [3]{b} (7 A b-4 a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{10/3}}+\frac{7 A b-4 a B}{3 a^3 x}-\frac{7 A b-4 a B}{12 a^2 b x^4}+\frac{A b-a B}{3 a b x^4 \left (a+b x^3\right )} \]

[Out]

-(7*A*b - 4*a*B)/(12*a^2*b*x^4) + (7*A*b - 4*a*B)/(3*a^3*x) + (A*b - a*B)/(3*a*b
*x^4*(a + b*x^3)) - (b^(1/3)*(7*A*b - 4*a*B)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqr
t[3]*a^(1/3))])/(3*Sqrt[3]*a^(10/3)) - (b^(1/3)*(7*A*b - 4*a*B)*Log[a^(1/3) + b^
(1/3)*x])/(9*a^(10/3)) + (b^(1/3)*(7*A*b - 4*a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*
x + b^(2/3)*x^2])/(18*a^(10/3))

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Rubi [A]  time = 0.346642, antiderivative size = 215, 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 \[ \frac{\sqrt [3]{b} (7 A b-4 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{10/3}}-\frac{\sqrt [3]{b} (7 A b-4 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{10/3}}-\frac{\sqrt [3]{b} (7 A b-4 a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{10/3}}+\frac{7 A b-4 a B}{3 a^3 x}-\frac{7 A b-4 a B}{12 a^2 b x^4}+\frac{A b-a B}{3 a b x^4 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x^3)/(x^5*(a + b*x^3)^2),x]

[Out]

-(7*A*b - 4*a*B)/(12*a^2*b*x^4) + (7*A*b - 4*a*B)/(3*a^3*x) + (A*b - a*B)/(3*a*b
*x^4*(a + b*x^3)) - (b^(1/3)*(7*A*b - 4*a*B)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqr
t[3]*a^(1/3))])/(3*Sqrt[3]*a^(10/3)) - (b^(1/3)*(7*A*b - 4*a*B)*Log[a^(1/3) + b^
(1/3)*x])/(9*a^(10/3)) + (b^(1/3)*(7*A*b - 4*a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*
x + b^(2/3)*x^2])/(18*a^(10/3))

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Rubi in Sympy [A]  time = 46.2675, size = 199, normalized size = 0.93 \[ \frac{A b - B a}{3 a b x^{4} \left (a + b x^{3}\right )} - \frac{7 A b - 4 B a}{12 a^{2} b x^{4}} + \frac{7 A b - 4 B a}{3 a^{3} x} - \frac{\sqrt [3]{b} \left (7 A b - 4 B a\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{9 a^{\frac{10}{3}}} + \frac{\sqrt [3]{b} \left (7 A b - 4 B a\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{18 a^{\frac{10}{3}}} - \frac{\sqrt{3} \sqrt [3]{b} \left (7 A b - 4 B a\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{9 a^{\frac{10}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x**3+A)/x**5/(b*x**3+a)**2,x)

[Out]

(A*b - B*a)/(3*a*b*x**4*(a + b*x**3)) - (7*A*b - 4*B*a)/(12*a**2*b*x**4) + (7*A*
b - 4*B*a)/(3*a**3*x) - b**(1/3)*(7*A*b - 4*B*a)*log(a**(1/3) + b**(1/3)*x)/(9*a
**(10/3)) + b**(1/3)*(7*A*b - 4*B*a)*log(a**(2/3) - a**(1/3)*b**(1/3)*x + b**(2/
3)*x**2)/(18*a**(10/3)) - sqrt(3)*b**(1/3)*(7*A*b - 4*B*a)*atan(sqrt(3)*(a**(1/3
)/3 - 2*b**(1/3)*x/3)/a**(1/3))/(9*a**(10/3))

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Mathematica [A]  time = 0.313893, size = 185, normalized size = 0.86 \[ \frac{2 \sqrt [3]{b} (7 A b-4 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-\frac{9 a^{4/3} A}{x^4}-\frac{12 \sqrt [3]{a} b x^2 (a B-A b)}{a+b x^3}-\frac{36 \sqrt [3]{a} (a B-2 A b)}{x}+4 \sqrt [3]{b} (4 a B-7 A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-4 \sqrt{3} \sqrt [3]{b} (7 A b-4 a B) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{36 a^{10/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x^3)/(x^5*(a + b*x^3)^2),x]

[Out]

((-9*a^(4/3)*A)/x^4 - (36*a^(1/3)*(-2*A*b + a*B))/x - (12*a^(1/3)*b*(-(A*b) + a*
B)*x^2)/(a + b*x^3) - 4*Sqrt[3]*b^(1/3)*(7*A*b - 4*a*B)*ArcTan[(1 - (2*b^(1/3)*x
)/a^(1/3))/Sqrt[3]] + 4*b^(1/3)*(-7*A*b + 4*a*B)*Log[a^(1/3) + b^(1/3)*x] + 2*b^
(1/3)*(7*A*b - 4*a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(36*a^(10/
3))

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Maple [A]  time = 0.019, size = 257, normalized size = 1.2 \[ -{\frac{A}{4\,{x}^{4}{a}^{2}}}+2\,{\frac{Ab}{{a}^{3}x}}-{\frac{B}{x{a}^{2}}}+{\frac{A{x}^{2}{b}^{2}}{3\,{a}^{3} \left ( b{x}^{3}+a \right ) }}-{\frac{bB{x}^{2}}{3\,{a}^{2} \left ( b{x}^{3}+a \right ) }}-{\frac{7\,Ab}{9\,{a}^{3}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{7\,Ab}{18\,{a}^{3}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{7\,Ab\sqrt{3}}{9\,{a}^{3}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{4\,B}{9\,{a}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{2\,B}{9\,{a}^{2}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{4\,B\sqrt{3}}{9\,{a}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^3+A)/x^5/(b*x^3+a)^2,x)

[Out]

-1/4*A/a^2/x^4+2/a^3/x*A*b-B/a^2/x+1/3*b^2/a^3*x^2/(b*x^3+a)*A-1/3*b/a^2*x^2/(b*
x^3+a)*B-7/9*b/a^3*A/(a/b)^(1/3)*ln(x+(a/b)^(1/3))+7/18*b/a^3*A/(a/b)^(1/3)*ln(x
^2-x*(a/b)^(1/3)+(a/b)^(2/3))+7/9*b/a^3*A*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)
*(2/(a/b)^(1/3)*x-1))+4/9/a^2*B/(a/b)^(1/3)*ln(x+(a/b)^(1/3))-2/9/a^2*B/(a/b)^(1
/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))-4/9/a^2*B*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3
^(1/2)*(2/(a/b)^(1/3)*x-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^3 + A)/((b*x^3 + a)^2*x^5),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.23171, size = 386, normalized size = 1.8 \[ \frac{\sqrt{3}{\left (2 \, \sqrt{3}{\left ({\left (4 \, B a b - 7 \, A b^{2}\right )} x^{7} +{\left (4 \, B a^{2} - 7 \, A a b\right )} x^{4}\right )} \left (-\frac{b}{a}\right )^{\frac{1}{3}} \log \left (b x^{2} - a x \left (-\frac{b}{a}\right )^{\frac{2}{3}} - a \left (-\frac{b}{a}\right )^{\frac{1}{3}}\right ) - 4 \, \sqrt{3}{\left ({\left (4 \, B a b - 7 \, A b^{2}\right )} x^{7} +{\left (4 \, B a^{2} - 7 \, A a b\right )} x^{4}\right )} \left (-\frac{b}{a}\right )^{\frac{1}{3}} \log \left (b x + a \left (-\frac{b}{a}\right )^{\frac{2}{3}}\right ) - 12 \,{\left ({\left (4 \, B a b - 7 \, A b^{2}\right )} x^{7} +{\left (4 \, B a^{2} - 7 \, A a b\right )} x^{4}\right )} \left (-\frac{b}{a}\right )^{\frac{1}{3}} \arctan \left (-\frac{2 \, \sqrt{3} b x - \sqrt{3} a \left (-\frac{b}{a}\right )^{\frac{2}{3}}}{3 \, a \left (-\frac{b}{a}\right )^{\frac{2}{3}}}\right ) - 3 \, \sqrt{3}{\left (4 \,{\left (4 \, B a b - 7 \, A b^{2}\right )} x^{6} + 3 \,{\left (4 \, B a^{2} - 7 \, A a b\right )} x^{3} + 3 \, A a^{2}\right )}\right )}}{108 \,{\left (a^{3} b x^{7} + a^{4} x^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)/((b*x^3 + a)^2*x^5),x, algorithm="fricas")

[Out]

1/108*sqrt(3)*(2*sqrt(3)*((4*B*a*b - 7*A*b^2)*x^7 + (4*B*a^2 - 7*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)*((4*B*a*b -
7*A*b^2)*x^7 + (4*B*a^2 - 7*A*a*b)*x^4)*(-b/a)^(1/3)*log(b*x + a*(-b/a)^(2/3))
- 12*((4*B*a*b - 7*A*b^2)*x^7 + (4*B*a^2 - 7*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*(4*B
*a*b - 7*A*b^2)*x^6 + 3*(4*B*a^2 - 7*A*a*b)*x^3 + 3*A*a^2))/(a^3*b*x^7 + a^4*x^4
)

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Sympy [A]  time = 5.18664, size = 153, normalized size = 0.71 \[ \operatorname{RootSum}{\left (729 t^{3} a^{10} + 343 A^{3} b^{4} - 588 A^{2} B a b^{3} + 336 A B^{2} a^{2} b^{2} - 64 B^{3} a^{3} b, \left ( t \mapsto t \log{\left (\frac{81 t^{2} a^{7}}{49 A^{2} b^{3} - 56 A B a b^{2} + 16 B^{2} a^{2} b} + x \right )} \right )\right )} - \frac{3 A a^{2} + x^{6} \left (- 28 A b^{2} + 16 B a b\right ) + x^{3} \left (- 21 A a b + 12 B a^{2}\right )}{12 a^{4} x^{4} + 12 a^{3} b x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x**3+A)/x**5/(b*x**3+a)**2,x)

[Out]

RootSum(729*_t**3*a**10 + 343*A**3*b**4 - 588*A**2*B*a*b**3 + 336*A*B**2*a**2*b*
*2 - 64*B**3*a**3*b, Lambda(_t, _t*log(81*_t**2*a**7/(49*A**2*b**3 - 56*A*B*a*b*
*2 + 16*B**2*a**2*b) + x))) - (3*A*a**2 + x**6*(-28*A*b**2 + 16*B*a*b) + x**3*(-
21*A*a*b + 12*B*a**2))/(12*a**4*x**4 + 12*a**3*b*x**7)

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GIAC/XCAS [A]  time = 0.220889, size = 312, normalized size = 1.45 \[ \frac{{\left (4 \, B a b \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 7 \, A b^{2} \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a^{4}} + \frac{\sqrt{3}{\left (4 \, \left (-a b^{2}\right )^{\frac{2}{3}} B a - 7 \, \left (-a b^{2}\right )^{\frac{2}{3}} A b\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{9 \, a^{4} b} - \frac{B a b x^{2} - A b^{2} x^{2}}{3 \,{\left (b x^{3} + a\right )} a^{3}} - \frac{{\left (4 \, \left (-a b^{2}\right )^{\frac{2}{3}} B a - 7 \, \left (-a b^{2}\right )^{\frac{2}{3}} A b\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{18 \, a^{4} b} - \frac{4 \, B a x^{3} - 8 \, A b x^{3} + A a}{4 \, a^{3} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)/((b*x^3 + a)^2*x^5),x, algorithm="giac")

[Out]

1/9*(4*B*a*b*(-a/b)^(1/3) - 7*A*b^2*(-a/b)^(1/3))*(-a/b)^(1/3)*ln(abs(x - (-a/b)
^(1/3)))/a^4 + 1/9*sqrt(3)*(4*(-a*b^2)^(2/3)*B*a - 7*(-a*b^2)^(2/3)*A*b)*arctan(
1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a^4*b) - 1/3*(B*a*b*x^2 - A*b^2*
x^2)/((b*x^3 + a)*a^3) - 1/18*(4*(-a*b^2)^(2/3)*B*a - 7*(-a*b^2)^(2/3)*A*b)*ln(x
^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^4*b) - 1/4*(4*B*a*x^3 - 8*A*b*x^3 + A*a)/
(a^3*x^4)