3.95 \(\int \frac{x^{10} \left (A+B x^3\right )}{\left (a+b x^3\right )^3} \, dx\)

Optimal. Leaf size=246 \[ -\frac{2 a^{2/3} (5 A b-11 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{27 b^{14/3}}+\frac{4 a^{2/3} (5 A b-11 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 b^{14/3}}+\frac{4 a^{2/3} (5 A b-11 a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} b^{14/3}}+\frac{2 x^2 (5 A b-11 a B)}{9 b^4}-\frac{4 x^5 (5 A b-11 a B)}{45 a b^3}+\frac{x^8 (5 A b-11 a B)}{18 a b^2 \left (a+b x^3\right )}+\frac{x^{11} (A b-a B)}{6 a b \left (a+b x^3\right )^2} \]

[Out]

(2*(5*A*b - 11*a*B)*x^2)/(9*b^4) - (4*(5*A*b - 11*a*B)*x^5)/(45*a*b^3) + ((A*b -
a*B)*x^11)/(6*a*b*(a + b*x^3)^2) + ((5*A*b - 11*a*B)*x^8)/(18*a*b^2*(a + b*x^3)
) + (4*a^(2/3)*(5*A*b - 11*a*B)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))
])/(9*Sqrt[3]*b^(14/3)) + (4*a^(2/3)*(5*A*b - 11*a*B)*Log[a^(1/3) + b^(1/3)*x])/
(27*b^(14/3)) - (2*a^(2/3)*(5*A*b - 11*a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^
(2/3)*x^2])/(27*b^(14/3))

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Rubi [A]  time = 0.448386, 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 \[ -\frac{2 a^{2/3} (5 A b-11 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{27 b^{14/3}}+\frac{4 a^{2/3} (5 A b-11 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 b^{14/3}}+\frac{4 a^{2/3} (5 A b-11 a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} b^{14/3}}+\frac{2 x^2 (5 A b-11 a B)}{9 b^4}-\frac{4 x^5 (5 A b-11 a B)}{45 a b^3}+\frac{x^8 (5 A b-11 a B)}{18 a b^2 \left (a+b x^3\right )}+\frac{x^{11} (A b-a B)}{6 a b \left (a+b x^3\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[(x^10*(A + B*x^3))/(a + b*x^3)^3,x]

[Out]

(2*(5*A*b - 11*a*B)*x^2)/(9*b^4) - (4*(5*A*b - 11*a*B)*x^5)/(45*a*b^3) + ((A*b -
a*B)*x^11)/(6*a*b*(a + b*x^3)^2) + ((5*A*b - 11*a*B)*x^8)/(18*a*b^2*(a + b*x^3)
) + (4*a^(2/3)*(5*A*b - 11*a*B)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))
])/(9*Sqrt[3]*b^(14/3)) + (4*a^(2/3)*(5*A*b - 11*a*B)*Log[a^(1/3) + b^(1/3)*x])/
(27*b^(14/3)) - (2*a^(2/3)*(5*A*b - 11*a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^
(2/3)*x^2])/(27*b^(14/3))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{4 a^{\frac{2}{3}} \left (5 A b - 11 B a\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{27 b^{\frac{14}{3}}} - \frac{2 a^{\frac{2}{3}} \left (5 A b - 11 B a\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{27 b^{\frac{14}{3}}} + \frac{4 \sqrt{3} a^{\frac{2}{3}} \left (5 A b - 11 B a\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{27 b^{\frac{14}{3}}} + \frac{4 \left (5 A b - 11 B a\right ) \int x\, dx}{9 b^{4}} + \frac{x^{11} \left (A b - B a\right )}{6 a b \left (a + b x^{3}\right )^{2}} + \frac{x^{8} \left (5 A b - 11 B a\right )}{18 a b^{2} \left (a + b x^{3}\right )} - \frac{4 x^{5} \left (5 A b - 11 B a\right )}{45 a b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**10*(B*x**3+A)/(b*x**3+a)**3,x)

[Out]

4*a**(2/3)*(5*A*b - 11*B*a)*log(a**(1/3) + b**(1/3)*x)/(27*b**(14/3)) - 2*a**(2/
3)*(5*A*b - 11*B*a)*log(a**(2/3) - a**(1/3)*b**(1/3)*x + b**(2/3)*x**2)/(27*b**(
14/3)) + 4*sqrt(3)*a**(2/3)*(5*A*b - 11*B*a)*atan(sqrt(3)*(a**(1/3)/3 - 2*b**(1/
3)*x/3)/a**(1/3))/(27*b**(14/3)) + 4*(5*A*b - 11*B*a)*Integral(x, x)/(9*b**4) +
x**11*(A*b - B*a)/(6*a*b*(a + b*x**3)**2) + x**8*(5*A*b - 11*B*a)/(18*a*b**2*(a
+ b*x**3)) - 4*x**5*(5*A*b - 11*B*a)/(45*a*b**3)

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Mathematica [A]  time = 0.335251, size = 216, normalized size = 0.88 \[ \frac{20 a^{2/3} (11 a B-5 A b) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-40 a^{2/3} (11 a B-5 A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-40 \sqrt{3} a^{2/3} (11 a B-5 A b) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )+\frac{45 a^2 b^{2/3} x^2 (a B-A b)}{\left (a+b x^3\right )^2}+135 b^{2/3} x^2 (A b-3 a B)+\frac{30 a b^{2/3} x^2 (7 A b-10 a B)}{a+b x^3}+54 b^{5/3} B x^5}{270 b^{14/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^10*(A + B*x^3))/(a + b*x^3)^3,x]

[Out]

(135*b^(2/3)*(A*b - 3*a*B)*x^2 + 54*b^(5/3)*B*x^5 + (45*a^2*b^(2/3)*(-(A*b) + a*
B)*x^2)/(a + b*x^3)^2 + (30*a*b^(2/3)*(7*A*b - 10*a*B)*x^2)/(a + b*x^3) - 40*Sqr
t[3]*a^(2/3)*(-5*A*b + 11*a*B)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] - 40*
a^(2/3)*(-5*A*b + 11*a*B)*Log[a^(1/3) + b^(1/3)*x] + 20*a^(2/3)*(-5*A*b + 11*a*B
)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(270*b^(14/3))

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Maple [A]  time = 0.019, size = 308, normalized size = 1.3 \[{\frac{B{x}^{5}}{5\,{b}^{3}}}+{\frac{A{x}^{2}}{2\,{b}^{3}}}-{\frac{3\,B{x}^{2}a}{2\,{b}^{4}}}+{\frac{7\,aA{x}^{5}}{9\,{b}^{2} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{10\,{a}^{2}B{x}^{5}}{9\,{b}^{3} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{11\,{a}^{2}A{x}^{2}}{18\,{b}^{3} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{17\,B{a}^{3}{x}^{2}}{18\,{b}^{4} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{20\,aA}{27\,{b}^{4}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{10\,aA}{27\,{b}^{4}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{20\,aA\sqrt{3}}{27\,{b}^{4}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{44\,{a}^{2}B}{27\,{b}^{5}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{22\,{a}^{2}B}{27\,{b}^{5}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{44\,{a}^{2}B\sqrt{3}}{27\,{b}^{5}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^10*(B*x^3+A)/(b*x^3+a)^3,x)

[Out]

1/5/b^3*B*x^5+1/2/b^3*A*x^2-3/2/b^4*B*x^2*a+7/9*a/b^2/(b*x^3+a)^2*A*x^5-10/9*a^2
/b^3/(b*x^3+a)^2*B*x^5+11/18*a^2/b^3/(b*x^3+a)^2*A*x^2-17/18*a^3/b^4/(b*x^3+a)^2
*B*x^2+20/27*a/b^4*A/(a/b)^(1/3)*ln(x+(a/b)^(1/3))-10/27*a/b^4*A/(a/b)^(1/3)*ln(
x^2-x*(a/b)^(1/3)+(a/b)^(2/3))-20/27*a/b^4*A*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1
/2)*(2/(a/b)^(1/3)*x-1))-44/27*a^2/b^5*B/(a/b)^(1/3)*ln(x+(a/b)^(1/3))+22/27*a^2
/b^5*B/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))+44/27*a^2/b^5*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)*x^10/(b*x^3 + a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.240893, size = 520, normalized size = 2.11 \[ \frac{\sqrt{3}{\left (20 \, \sqrt{3}{\left ({\left (11 \, B a b^{2} - 5 \, A b^{3}\right )} x^{6} + 11 \, B a^{3} - 5 \, A a^{2} b + 2 \,{\left (11 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{3}\right )} \left (\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \log \left (a x^{2} - b x \left (\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}} + a \left (\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}}\right ) - 40 \, \sqrt{3}{\left ({\left (11 \, B a b^{2} - 5 \, A b^{3}\right )} x^{6} + 11 \, B a^{3} - 5 \, A a^{2} b + 2 \,{\left (11 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{3}\right )} \left (\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \log \left (a x + b \left (\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}}\right ) - 120 \,{\left ({\left (11 \, B a b^{2} - 5 \, A b^{3}\right )} x^{6} + 11 \, B a^{3} - 5 \, A a^{2} b + 2 \,{\left (11 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{3}\right )} \left (\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \arctan \left (-\frac{2 \, \sqrt{3} a x - \sqrt{3} b \left (\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}}}{3 \, b \left (\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}}}\right ) + 3 \, \sqrt{3}{\left (18 \, B b^{3} x^{11} - 9 \,{\left (11 \, B a b^{2} - 5 \, A b^{3}\right )} x^{8} - 32 \,{\left (11 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{5} - 20 \,{\left (11 \, B a^{3} - 5 \, A a^{2} b\right )} x^{2}\right )}\right )}}{810 \,{\left (b^{6} x^{6} + 2 \, a b^{5} x^{3} + a^{2} b^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*x^10/(b*x^3 + a)^3,x, algorithm="fricas")

[Out]

1/810*sqrt(3)*(20*sqrt(3)*((11*B*a*b^2 - 5*A*b^3)*x^6 + 11*B*a^3 - 5*A*a^2*b + 2
*(11*B*a^2*b - 5*A*a*b^2)*x^3)*(a^2/b^2)^(1/3)*log(a*x^2 - b*x*(a^2/b^2)^(2/3) +
a*(a^2/b^2)^(1/3)) - 40*sqrt(3)*((11*B*a*b^2 - 5*A*b^3)*x^6 + 11*B*a^3 - 5*A*a^
2*b + 2*(11*B*a^2*b - 5*A*a*b^2)*x^3)*(a^2/b^2)^(1/3)*log(a*x + b*(a^2/b^2)^(2/3
)) - 120*((11*B*a*b^2 - 5*A*b^3)*x^6 + 11*B*a^3 - 5*A*a^2*b + 2*(11*B*a^2*b - 5*
A*a*b^2)*x^3)*(a^2/b^2)^(1/3)*arctan(-1/3*(2*sqrt(3)*a*x - sqrt(3)*b*(a^2/b^2)^(
2/3))/(b*(a^2/b^2)^(2/3))) + 3*sqrt(3)*(18*B*b^3*x^11 - 9*(11*B*a*b^2 - 5*A*b^3)
*x^8 - 32*(11*B*a^2*b - 5*A*a*b^2)*x^5 - 20*(11*B*a^3 - 5*A*a^2*b)*x^2))/(b^6*x^
6 + 2*a*b^5*x^3 + a^2*b^4)

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Sympy [A]  time = 12.6246, size = 189, normalized size = 0.77 \[ \frac{B x^{5}}{5 b^{3}} - \frac{x^{5} \left (- 14 A a b^{2} + 20 B a^{2} b\right ) + x^{2} \left (- 11 A a^{2} b + 17 B a^{3}\right )}{18 a^{2} b^{4} + 36 a b^{5} x^{3} + 18 b^{6} x^{6}} + \operatorname{RootSum}{\left (19683 t^{3} b^{14} - 8000 A^{3} a^{2} b^{3} + 52800 A^{2} B a^{3} b^{2} - 116160 A B^{2} a^{4} b + 85184 B^{3} a^{5}, \left ( t \mapsto t \log{\left (\frac{729 t^{2} b^{9}}{400 A^{2} a b^{2} - 1760 A B a^{2} b + 1936 B^{2} a^{3}} + x \right )} \right )\right )} - \frac{x^{2} \left (- A b + 3 B a\right )}{2 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**10*(B*x**3+A)/(b*x**3+a)**3,x)

[Out]

B*x**5/(5*b**3) - (x**5*(-14*A*a*b**2 + 20*B*a**2*b) + x**2*(-11*A*a**2*b + 17*B
*a**3))/(18*a**2*b**4 + 36*a*b**5*x**3 + 18*b**6*x**6) + RootSum(19683*_t**3*b**
14 - 8000*A**3*a**2*b**3 + 52800*A**2*B*a**3*b**2 - 116160*A*B**2*a**4*b + 85184
*B**3*a**5, Lambda(_t, _t*log(729*_t**2*b**9/(400*A**2*a*b**2 - 1760*A*B*a**2*b
+ 1936*B**2*a**3) + x))) - x**2*(-A*b + 3*B*a)/(2*b**4)

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GIAC/XCAS [A]  time = 0.223844, size = 350, normalized size = 1.42 \[ -\frac{4 \,{\left (11 \, B a^{2} \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 5 \, A a b \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{27 \, a b^{4}} - \frac{4 \, \sqrt{3}{\left (11 \, \left (-a b^{2}\right )^{\frac{2}{3}} B a - 5 \, \left (-a b^{2}\right )^{\frac{2}{3}} A b\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{27 \, b^{6}} + \frac{2 \,{\left (11 \, \left (-a b^{2}\right )^{\frac{2}{3}} B a - 5 \, \left (-a b^{2}\right )^{\frac{2}{3}} A b\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{27 \, b^{6}} - \frac{20 \, B a^{2} b x^{5} - 14 \, A a b^{2} x^{5} + 17 \, B a^{3} x^{2} - 11 \, A a^{2} b x^{2}}{18 \,{\left (b x^{3} + a\right )}^{2} b^{4}} + \frac{2 \, B b^{12} x^{5} - 15 \, B a b^{11} x^{2} + 5 \, A b^{12} x^{2}}{10 \, b^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*x^10/(b*x^3 + a)^3,x, algorithm="giac")

[Out]

-4/27*(11*B*a^2*(-a/b)^(1/3) - 5*A*a*b*(-a/b)^(1/3))*(-a/b)^(1/3)*ln(abs(x - (-a
/b)^(1/3)))/(a*b^4) - 4/27*sqrt(3)*(11*(-a*b^2)^(2/3)*B*a - 5*(-a*b^2)^(2/3)*A*b
)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/b^6 + 2/27*(11*(-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))/b^6 -
1/18*(20*B*a^2*b*x^5 - 14*A*a*b^2*x^5 + 17*B*a^3*x^2 - 11*A*a^2*b*x^2)/((b*x^3 +
a)^2*b^4) + 1/10*(2*B*b^12*x^5 - 15*B*a*b^11*x^2 + 5*A*b^12*x^2)/b^15