3.71 \(\int \frac{x^9 \left (A+B x^3\right )}{\left (a+b x^3\right )^2} \, dx\)

Optimal. Leaf size=233 \[ -\frac{a^{4/3} (7 A b-10 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 b^{13/3}}+\frac{a^{4/3} (7 A b-10 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 b^{13/3}}-\frac{a^{4/3} (7 A b-10 a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} b^{13/3}}-\frac{a x (7 A b-10 a B)}{3 b^4}+\frac{x^4 (7 A b-10 a B)}{12 b^3}-\frac{x^7 (7 A b-10 a B)}{21 a b^2}+\frac{x^{10} (A b-a B)}{3 a b \left (a+b x^3\right )} \]

[Out]

-(a*(7*A*b - 10*a*B)*x)/(3*b^4) + ((7*A*b - 10*a*B)*x^4)/(12*b^3) - ((7*A*b - 10
*a*B)*x^7)/(21*a*b^2) + ((A*b - a*B)*x^10)/(3*a*b*(a + b*x^3)) - (a^(4/3)*(7*A*b
- 10*a*B)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*b^(13/3
)) + (a^(4/3)*(7*A*b - 10*a*B)*Log[a^(1/3) + b^(1/3)*x])/(9*b^(13/3)) - (a^(4/3)
*(7*A*b - 10*a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*b^(13/3))

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Rubi [A]  time = 0.385021, antiderivative size = 233, 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 \[ -\frac{a^{4/3} (7 A b-10 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 b^{13/3}}+\frac{a^{4/3} (7 A b-10 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 b^{13/3}}-\frac{a^{4/3} (7 A b-10 a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} b^{13/3}}-\frac{a x (7 A b-10 a B)}{3 b^4}+\frac{x^4 (7 A b-10 a B)}{12 b^3}-\frac{x^7 (7 A b-10 a B)}{21 a b^2}+\frac{x^{10} (A b-a B)}{3 a b \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]  Int[(x^9*(A + B*x^3))/(a + b*x^3)^2,x]

[Out]

-(a*(7*A*b - 10*a*B)*x)/(3*b^4) + ((7*A*b - 10*a*B)*x^4)/(12*b^3) - ((7*A*b - 10
*a*B)*x^7)/(21*a*b^2) + ((A*b - a*B)*x^10)/(3*a*b*(a + b*x^3)) - (a^(4/3)*(7*A*b
- 10*a*B)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*b^(13/3
)) + (a^(4/3)*(7*A*b - 10*a*B)*Log[a^(1/3) + b^(1/3)*x])/(9*b^(13/3)) - (a^(4/3)
*(7*A*b - 10*a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*b^(13/3))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{a^{\frac{4}{3}} \left (7 A b - 10 B a\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{9 b^{\frac{13}{3}}} - \frac{a^{\frac{4}{3}} \left (7 A b - 10 B a\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{18 b^{\frac{13}{3}}} - \frac{\sqrt{3} a^{\frac{4}{3}} \left (7 A b - 10 B a\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{9 b^{\frac{13}{3}}} + \frac{x^{4} \left (7 A b - 10 B a\right )}{12 b^{3}} + \frac{x^{10} \left (A b - B a\right )}{3 a b \left (a + b x^{3}\right )} - \frac{x^{7} \left (7 A b - 10 B a\right )}{21 a b^{2}} - \frac{\left (7 A b - 10 B a\right ) \int a^{2}\, dx}{3 a b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**9*(B*x**3+A)/(b*x**3+a)**2,x)

[Out]

a**(4/3)*(7*A*b - 10*B*a)*log(a**(1/3) + b**(1/3)*x)/(9*b**(13/3)) - a**(4/3)*(7
*A*b - 10*B*a)*log(a**(2/3) - a**(1/3)*b**(1/3)*x + b**(2/3)*x**2)/(18*b**(13/3)
) - sqrt(3)*a**(4/3)*(7*A*b - 10*B*a)*atan(sqrt(3)*(a**(1/3)/3 - 2*b**(1/3)*x/3)
/a**(1/3))/(9*b**(13/3)) + x**4*(7*A*b - 10*B*a)/(12*b**3) + x**10*(A*b - B*a)/(
3*a*b*(a + b*x**3)) - x**7*(7*A*b - 10*B*a)/(21*a*b**2) - (7*A*b - 10*B*a)*Integ
ral(a**2, x)/(3*a*b**4)

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Mathematica [A]  time = 0.304711, size = 203, normalized size = 0.87 \[ \frac{14 a^{4/3} (10 a B-7 A b) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-28 a^{4/3} (10 a B-7 A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+28 \sqrt{3} a^{4/3} (10 a B-7 A b) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )+\frac{84 a^2 \sqrt [3]{b} x (a B-A b)}{a+b x^3}+63 b^{4/3} x^4 (A b-2 a B)+252 a \sqrt [3]{b} x (3 a B-2 A b)+36 b^{7/3} B x^7}{252 b^{13/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^9*(A + B*x^3))/(a + b*x^3)^2,x]

[Out]

(252*a*b^(1/3)*(-2*A*b + 3*a*B)*x + 63*b^(4/3)*(A*b - 2*a*B)*x^4 + 36*b^(7/3)*B*
x^7 + (84*a^2*b^(1/3)*(-(A*b) + a*B)*x)/(a + b*x^3) + 28*Sqrt[3]*a^(4/3)*(-7*A*b
+ 10*a*B)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] - 28*a^(4/3)*(-7*A*b + 10
*a*B)*Log[a^(1/3) + b^(1/3)*x] + 14*a^(4/3)*(-7*A*b + 10*a*B)*Log[a^(2/3) - a^(1
/3)*b^(1/3)*x + b^(2/3)*x^2])/(252*b^(13/3))

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Maple [A]  time = 0.014, size = 288, normalized size = 1.2 \[{\frac{B{x}^{7}}{7\,{b}^{2}}}+{\frac{A{x}^{4}}{4\,{b}^{2}}}-{\frac{B{x}^{4}a}{2\,{b}^{3}}}-2\,{\frac{aAx}{{b}^{3}}}+3\,{\frac{Bx{a}^{2}}{{b}^{4}}}-{\frac{{a}^{2}Ax}{3\,{b}^{3} \left ( b{x}^{3}+a \right ) }}+{\frac{{a}^{3}xB}{3\,{b}^{4} \left ( b{x}^{3}+a \right ) }}+{\frac{7\,A{a}^{2}}{9\,{b}^{4}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{7\,A{a}^{2}}{18\,{b}^{4}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{7\,A{a}^{2}\sqrt{3}}{9\,{b}^{4}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{10\,B{a}^{3}}{9\,{b}^{5}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{5\,B{a}^{3}}{9\,{b}^{5}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{10\,B{a}^{3}\sqrt{3}}{9\,{b}^{5}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^9*(B*x^3+A)/(b*x^3+a)^2,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.238266, size = 385, normalized size = 1.65 \[ \frac{\sqrt{3}{\left (14 \, \sqrt{3}{\left (10 \, B a^{3} - 7 \, A a^{2} b +{\left (10 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{3}\right )} \left (\frac{a}{b}\right )^{\frac{1}{3}} \log \left (x^{2} - x \left (\frac{a}{b}\right )^{\frac{1}{3}} + \left (\frac{a}{b}\right )^{\frac{2}{3}}\right ) - 28 \, \sqrt{3}{\left (10 \, B a^{3} - 7 \, A a^{2} b +{\left (10 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{3}\right )} \left (\frac{a}{b}\right )^{\frac{1}{3}} \log \left (x + \left (\frac{a}{b}\right )^{\frac{1}{3}}\right ) + 84 \,{\left (10 \, B a^{3} - 7 \, A a^{2} b +{\left (10 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{3}\right )} \left (\frac{a}{b}\right )^{\frac{1}{3}} \arctan \left (-\frac{2 \, \sqrt{3} x - \sqrt{3} \left (\frac{a}{b}\right )^{\frac{1}{3}}}{3 \, \left (\frac{a}{b}\right )^{\frac{1}{3}}}\right ) + 3 \, \sqrt{3}{\left (12 \, B b^{3} x^{10} - 3 \,{\left (10 \, B a b^{2} - 7 \, A b^{3}\right )} x^{7} + 21 \,{\left (10 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{4} + 28 \,{\left (10 \, B a^{3} - 7 \, A a^{2} b\right )} x\right )}\right )}}{756 \,{\left (b^{5} x^{3} + a b^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*x^9/(b*x^3 + a)^2,x, algorithm="fricas")

[Out]

1/756*sqrt(3)*(14*sqrt(3)*(10*B*a^3 - 7*A*a^2*b + (10*B*a^2*b - 7*A*a*b^2)*x^3)*
(a/b)^(1/3)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3)) - 28*sqrt(3)*(10*B*a^3 - 7*A*
a^2*b + (10*B*a^2*b - 7*A*a*b^2)*x^3)*(a/b)^(1/3)*log(x + (a/b)^(1/3)) + 84*(10*
B*a^3 - 7*A*a^2*b + (10*B*a^2*b - 7*A*a*b^2)*x^3)*(a/b)^(1/3)*arctan(-1/3*(2*sqr
t(3)*x - sqrt(3)*(a/b)^(1/3))/(a/b)^(1/3)) + 3*sqrt(3)*(12*B*b^3*x^10 - 3*(10*B*
a*b^2 - 7*A*b^3)*x^7 + 21*(10*B*a^2*b - 7*A*a*b^2)*x^4 + 28*(10*B*a^3 - 7*A*a^2*
b)*x))/(b^5*x^3 + a*b^4)

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Sympy [A]  time = 4.49415, size = 153, normalized size = 0.66 \[ \frac{B x^{7}}{7 b^{2}} + \frac{x \left (- A a^{2} b + B a^{3}\right )}{3 a b^{4} + 3 b^{5} x^{3}} + \operatorname{RootSum}{\left (729 t^{3} b^{13} - 343 A^{3} a^{4} b^{3} + 1470 A^{2} B a^{5} b^{2} - 2100 A B^{2} a^{6} b + 1000 B^{3} a^{7}, \left ( t \mapsto t \log{\left (- \frac{9 t b^{4}}{- 7 A a b + 10 B a^{2}} + x \right )} \right )\right )} - \frac{x^{4} \left (- A b + 2 B a\right )}{4 b^{3}} + \frac{x \left (- 2 A a b + 3 B a^{2}\right )}{b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**9*(B*x**3+A)/(b*x**3+a)**2,x)

[Out]

B*x**7/(7*b**2) + x*(-A*a**2*b + B*a**3)/(3*a*b**4 + 3*b**5*x**3) + RootSum(729*
_t**3*b**13 - 343*A**3*a**4*b**3 + 1470*A**2*B*a**5*b**2 - 2100*A*B**2*a**6*b +
1000*B**3*a**7, Lambda(_t, _t*log(-9*_t*b**4/(-7*A*a*b + 10*B*a**2) + x))) - x**
4*(-A*b + 2*B*a)/(4*b**3) + x*(-2*A*a*b + 3*B*a**2)/b**4

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GIAC/XCAS [A]  time = 0.218661, size = 329, normalized size = 1.41 \[ -\frac{\sqrt{3}{\left (10 \, \left (-a b^{2}\right )^{\frac{1}{3}} B a^{2} - 7 \, \left (-a b^{2}\right )^{\frac{1}{3}} A a b\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{9 \, b^{5}} + \frac{{\left (10 \, B a^{3} - 7 \, A a^{2} b\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a b^{4}} - \frac{{\left (10 \, \left (-a b^{2}\right )^{\frac{1}{3}} B a^{2} - 7 \, \left (-a b^{2}\right )^{\frac{1}{3}} A a b\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{18 \, b^{5}} + \frac{B a^{3} x - A a^{2} b x}{3 \,{\left (b x^{3} + a\right )} b^{4}} + \frac{4 \, B b^{12} x^{7} - 14 \, B a b^{11} x^{4} + 7 \, A b^{12} x^{4} + 84 \, B a^{2} b^{10} x - 56 \, A a b^{11} x}{28 \, b^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*x^9/(b*x^3 + a)^2,x, algorithm="giac")

[Out]

-1/9*sqrt(3)*(10*(-a*b^2)^(1/3)*B*a^2 - 7*(-a*b^2)^(1/3)*A*a*b)*arctan(1/3*sqrt(
3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/b^5 + 1/9*(10*B*a^3 - 7*A*a^2*b)*(-a/b)^(1
/3)*ln(abs(x - (-a/b)^(1/3)))/(a*b^4) - 1/18*(10*(-a*b^2)^(1/3)*B*a^2 - 7*(-a*b^
2)^(1/3)*A*a*b)*ln(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/b^5 + 1/3*(B*a^3*x - A*a
^2*b*x)/((b*x^3 + a)*b^4) + 1/28*(4*B*b^12*x^7 - 14*B*a*b^11*x^4 + 7*A*b^12*x^4
+ 84*B*a^2*b^10*x - 56*A*a*b^11*x)/b^14