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

Optimal. Leaf size=246 \[ \frac{7 \sqrt [3]{b} (5 A b-2 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{13/3}}-\frac{7 \sqrt [3]{b} (5 A b-2 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{13/3}}-\frac{7 \sqrt [3]{b} (5 A b-2 a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{13/3}}+\frac{7 (5 A b-2 a B)}{9 a^4 x}-\frac{7 (5 A b-2 a B)}{36 a^3 b x^4}+\frac{5 A b-2 a B}{9 a^2 b x^4 \left (a+b x^3\right )}+\frac{A b-a B}{6 a b x^4 \left (a+b x^3\right )^2} \]

[Out]

(-7*(5*A*b - 2*a*B))/(36*a^3*b*x^4) + (7*(5*A*b - 2*a*B))/(9*a^4*x) + (A*b - a*B
)/(6*a*b*x^4*(a + b*x^3)^2) + (5*A*b - 2*a*B)/(9*a^2*b*x^4*(a + b*x^3)) - (7*b^(
1/3)*(5*A*b - 2*a*B)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[
3]*a^(13/3)) - (7*b^(1/3)*(5*A*b - 2*a*B)*Log[a^(1/3) + b^(1/3)*x])/(27*a^(13/3)
) + (7*b^(1/3)*(5*A*b - 2*a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(
54*a^(13/3))

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Rubi [A]  time = 0.424296, antiderivative size = 246, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.45 \[ \frac{7 \sqrt [3]{b} (5 A b-2 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{13/3}}-\frac{7 \sqrt [3]{b} (5 A b-2 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{13/3}}-\frac{7 \sqrt [3]{b} (5 A b-2 a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{13/3}}+\frac{7 (5 A b-2 a B)}{9 a^4 x}-\frac{7 (5 A b-2 a B)}{36 a^3 b x^4}+\frac{5 A b-2 a B}{9 a^2 b x^4 \left (a+b x^3\right )}+\frac{A b-a B}{6 a b x^4 \left (a+b x^3\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

(-7*(5*A*b - 2*a*B))/(36*a^3*b*x^4) + (7*(5*A*b - 2*a*B))/(9*a^4*x) + (A*b - a*B
)/(6*a*b*x^4*(a + b*x^3)^2) + (5*A*b - 2*a*B)/(9*a^2*b*x^4*(a + b*x^3)) - (7*b^(
1/3)*(5*A*b - 2*a*B)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[
3]*a^(13/3)) - (7*b^(1/3)*(5*A*b - 2*a*B)*Log[a^(1/3) + b^(1/3)*x])/(27*a^(13/3)
) + (7*b^(1/3)*(5*A*b - 2*a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(
54*a^(13/3))

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Rubi in Sympy [A]  time = 52.9256, size = 233, normalized size = 0.95 \[ \frac{A b - B a}{6 a b x^{4} \left (a + b x^{3}\right )^{2}} + \frac{5 A b - 2 B a}{9 a^{2} b x^{4} \left (a + b x^{3}\right )} - \frac{7 \left (5 A b - 2 B a\right )}{36 a^{3} b x^{4}} + \frac{7 \left (5 A b - 2 B a\right )}{9 a^{4} x} - \frac{7 \sqrt [3]{b} \left (5 A b - 2 B a\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{27 a^{\frac{13}{3}}} + \frac{7 \sqrt [3]{b} \left (5 A b - 2 B a\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{54 a^{\frac{13}{3}}} - \frac{7 \sqrt{3} \sqrt [3]{b} \left (5 A b - 2 B a\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{27 a^{\frac{13}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(A*b - B*a)/(6*a*b*x**4*(a + b*x**3)**2) + (5*A*b - 2*B*a)/(9*a**2*b*x**4*(a + b
*x**3)) - 7*(5*A*b - 2*B*a)/(36*a**3*b*x**4) + 7*(5*A*b - 2*B*a)/(9*a**4*x) - 7*
b**(1/3)*(5*A*b - 2*B*a)*log(a**(1/3) + b**(1/3)*x)/(27*a**(13/3)) + 7*b**(1/3)*
(5*A*b - 2*B*a)*log(a**(2/3) - a**(1/3)*b**(1/3)*x + b**(2/3)*x**2)/(54*a**(13/3
)) - 7*sqrt(3)*b**(1/3)*(5*A*b - 2*B*a)*atan(sqrt(3)*(a**(1/3)/3 - 2*b**(1/3)*x/
3)/a**(1/3))/(27*a**(13/3))

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Mathematica [A]  time = 0.367947, size = 214, normalized size = 0.87 \[ \frac{14 \sqrt [3]{b} (5 A b-2 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-\frac{18 a^{4/3} b x^2 (a B-A b)}{\left (a+b x^3\right )^2}-\frac{27 a^{4/3} A}{x^4}-\frac{12 \sqrt [3]{a} b x^2 (5 a B-8 A b)}{a+b x^3}-\frac{108 \sqrt [3]{a} (a B-3 A b)}{x}+28 \sqrt [3]{b} (2 a B-5 A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-28 \sqrt{3} \sqrt [3]{b} (5 A b-2 a B) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{108 a^{13/3}} \]

Antiderivative was successfully verified.

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

[Out]

((-27*a^(4/3)*A)/x^4 - (108*a^(1/3)*(-3*A*b + a*B))/x - (18*a^(4/3)*b*(-(A*b) +
a*B)*x^2)/(a + b*x^3)^2 - (12*a^(1/3)*b*(-8*A*b + 5*a*B)*x^2)/(a + b*x^3) - 28*S
qrt[3]*b^(1/3)*(5*A*b - 2*a*B)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 28*
b^(1/3)*(-5*A*b + 2*a*B)*Log[a^(1/3) + b^(1/3)*x] + 14*b^(1/3)*(5*A*b - 2*a*B)*L
og[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(108*a^(13/3))

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Maple [A]  time = 0.025, size = 299, normalized size = 1.2 \[ -{\frac{A}{4\,{a}^{3}{x}^{4}}}+3\,{\frac{Ab}{x{a}^{4}}}-{\frac{B}{{a}^{3}x}}+{\frac{8\,{b}^{3}A{x}^{5}}{9\,{a}^{4} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{5\,{b}^{2}B{x}^{5}}{9\,{a}^{3} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{19\,A{x}^{2}{b}^{2}}{18\,{a}^{3} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{13\,bB{x}^{2}}{18\,{a}^{2} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{35\,Ab}{27\,{a}^{4}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{35\,Ab}{54\,{a}^{4}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{35\,Ab\sqrt{3}}{27\,{a}^{4}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{14\,B}{27\,{a}^{3}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{7\,B}{27\,{a}^{3}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{14\,B\sqrt{3}}{27\,{a}^{3}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*A/a^3/x^4+3/x/a^4*A*b-B/a^3/x+8/9/a^4*b^3/(b*x^3+a)^2*A*x^5-5/9/a^3*b^2/(b*
x^3+a)^2*B*x^5+19/18/a^3*b^2/(b*x^3+a)^2*A*x^2-13/18/a^2*b/(b*x^3+a)^2*B*x^2-35/
27/a^4*b*A/(a/b)^(1/3)*ln(x+(a/b)^(1/3))+35/54/a^4*b*A/(a/b)^(1/3)*ln(x^2-x*(a/b
)^(1/3)+(a/b)^(2/3))+35/27/a^4*b*A*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/
b)^(1/3)*x-1))+14/27/a^3*B/(a/b)^(1/3)*ln(x+(a/b)^(1/3))-7/27/a^3*B/(a/b)^(1/3)*
ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))-14/27/a^3*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)^3*x^5),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.240957, size = 531, normalized size = 2.16 \[ \frac{\sqrt{3}{\left (14 \, \sqrt{3}{\left ({\left (2 \, B a b^{2} - 5 \, A b^{3}\right )} x^{10} + 2 \,{\left (2 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{7} +{\left (2 \, B a^{3} - 5 \, A a^{2} b\right )} x^{4}\right )} \left (-\frac{b}{a}\right )^{\frac{1}{3}} \log \left (b x^{2} - a x \left (-\frac{b}{a}\right )^{\frac{2}{3}} - a \left (-\frac{b}{a}\right )^{\frac{1}{3}}\right ) - 28 \, \sqrt{3}{\left ({\left (2 \, B a b^{2} - 5 \, A b^{3}\right )} x^{10} + 2 \,{\left (2 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{7} +{\left (2 \, B a^{3} - 5 \, A a^{2} b\right )} x^{4}\right )} \left (-\frac{b}{a}\right )^{\frac{1}{3}} \log \left (b x + a \left (-\frac{b}{a}\right )^{\frac{2}{3}}\right ) - 84 \,{\left ({\left (2 \, B a b^{2} - 5 \, A b^{3}\right )} x^{10} + 2 \,{\left (2 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{7} +{\left (2 \, B a^{3} - 5 \, A a^{2} b\right )} x^{4}\right )} \left (-\frac{b}{a}\right )^{\frac{1}{3}} \arctan \left (-\frac{2 \, \sqrt{3} b x - \sqrt{3} a \left (-\frac{b}{a}\right )^{\frac{2}{3}}}{3 \, a \left (-\frac{b}{a}\right )^{\frac{2}{3}}}\right ) - 3 \, \sqrt{3}{\left (28 \,{\left (2 \, B a b^{2} - 5 \, A b^{3}\right )} x^{9} + 49 \,{\left (2 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{6} + 9 \, A a^{3} + 18 \,{\left (2 \, B a^{3} - 5 \, A a^{2} b\right )} x^{3}\right )}\right )}}{324 \,{\left (a^{4} b^{2} x^{10} + 2 \, a^{5} b x^{7} + a^{6} x^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/324*sqrt(3)*(14*sqrt(3)*((2*B*a*b^2 - 5*A*b^3)*x^10 + 2*(2*B*a^2*b - 5*A*a*b^2
)*x^7 + (2*B*a^3 - 5*A*a^2*b)*x^4)*(-b/a)^(1/3)*log(b*x^2 - a*x*(-b/a)^(2/3) - a
*(-b/a)^(1/3)) - 28*sqrt(3)*((2*B*a*b^2 - 5*A*b^3)*x^10 + 2*(2*B*a^2*b - 5*A*a*b
^2)*x^7 + (2*B*a^3 - 5*A*a^2*b)*x^4)*(-b/a)^(1/3)*log(b*x + a*(-b/a)^(2/3)) - 84
*((2*B*a*b^2 - 5*A*b^3)*x^10 + 2*(2*B*a^2*b - 5*A*a*b^2)*x^7 + (2*B*a^3 - 5*A*a^
2*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)*(28*(2*B*a*b^2 - 5*A*b^3)*x^9 + 49*(2*B*a^2*b - 5*A*a*
b^2)*x^6 + 9*A*a^3 + 18*(2*B*a^3 - 5*A*a^2*b)*x^3))/(a^4*b^2*x^10 + 2*a^5*b*x^7
+ a^6*x^4)

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Sympy [A]  time = 18.2187, size = 189, normalized size = 0.77 \[ \operatorname{RootSum}{\left (19683 t^{3} a^{13} + 42875 A^{3} b^{4} - 51450 A^{2} B a b^{3} + 20580 A B^{2} a^{2} b^{2} - 2744 B^{3} a^{3} b, \left ( t \mapsto t \log{\left (\frac{729 t^{2} a^{9}}{1225 A^{2} b^{3} - 980 A B a b^{2} + 196 B^{2} a^{2} b} + x \right )} \right )\right )} - \frac{9 A a^{3} + x^{9} \left (- 140 A b^{3} + 56 B a b^{2}\right ) + x^{6} \left (- 245 A a b^{2} + 98 B a^{2} b\right ) + x^{3} \left (- 90 A a^{2} b + 36 B a^{3}\right )}{36 a^{6} x^{4} + 72 a^{5} b x^{7} + 36 a^{4} b^{2} x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

RootSum(19683*_t**3*a**13 + 42875*A**3*b**4 - 51450*A**2*B*a*b**3 + 20580*A*B**2
*a**2*b**2 - 2744*B**3*a**3*b, Lambda(_t, _t*log(729*_t**2*a**9/(1225*A**2*b**3
- 980*A*B*a*b**2 + 196*B**2*a**2*b) + x))) - (9*A*a**3 + x**9*(-140*A*b**3 + 56*
B*a*b**2) + x**6*(-245*A*a*b**2 + 98*B*a**2*b) + x**3*(-90*A*a**2*b + 36*B*a**3)
)/(36*a**6*x**4 + 72*a**5*b*x**7 + 36*a**4*b**2*x**10)

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GIAC/XCAS [A]  time = 0.220985, size = 343, normalized size = 1.39 \[ \frac{7 \,{\left (2 \, B a b \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 5 \, A b^{2} \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{27 \, a^{5}} + \frac{7 \, \sqrt{3}{\left (2 \, \left (-a b^{2}\right )^{\frac{2}{3}} B a - 5 \, \left (-a b^{2}\right )^{\frac{2}{3}} A b\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{27 \, a^{5} b} - \frac{7 \,{\left (2 \, \left (-a b^{2}\right )^{\frac{2}{3}} B a - 5 \, \left (-a b^{2}\right )^{\frac{2}{3}} A b\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{54 \, a^{5} b} - \frac{10 \, B a b^{2} x^{5} - 16 \, A b^{3} x^{5} + 13 \, B a^{2} b x^{2} - 19 \, A a b^{2} x^{2}}{18 \,{\left (b x^{3} + a\right )}^{2} a^{4}} - \frac{4 \, B a x^{3} - 12 \, A b x^{3} + A a}{4 \, a^{4} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

7/27*(2*B*a*b*(-a/b)^(1/3) - 5*A*b^2*(-a/b)^(1/3))*(-a/b)^(1/3)*ln(abs(x - (-a/b
)^(1/3)))/a^5 + 7/27*sqrt(3)*(2*(-a*b^2)^(2/3)*B*a - 5*(-a*b^2)^(2/3)*A*b)*arcta
n(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a^5*b) - 7/54*(2*(-a*b^2)^(2/3
)*B*a - 5*(-a*b^2)^(2/3)*A*b)*ln(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^5*b) -
1/18*(10*B*a*b^2*x^5 - 16*A*b^3*x^5 + 13*B*a^2*b*x^2 - 19*A*a*b^2*x^2)/((b*x^3 +
a)^2*a^4) - 1/4*(4*B*a*x^3 - 12*A*b*x^3 + A*a)/(a^4*x^4)