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

Optimal. Leaf size=244 \[ \frac{7 \sqrt [3]{a} (2 A b-5 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 b^{13/3}}-\frac{7 \sqrt [3]{a} (2 A b-5 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 b^{13/3}}+\frac{7 \sqrt [3]{a} (2 A b-5 a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} b^{13/3}}+\frac{7 x (2 A b-5 a B)}{9 b^4}-\frac{7 x^4 (2 A b-5 a B)}{36 a b^3}+\frac{x^7 (2 A b-5 a B)}{9 a b^2 \left (a+b x^3\right )}+\frac{x^{10} (A b-a B)}{6 a b \left (a+b x^3\right )^2} \]

[Out]

(7*(2*A*b - 5*a*B)*x)/(9*b^4) - (7*(2*A*b - 5*a*B)*x^4)/(36*a*b^3) + ((A*b - a*B
)*x^10)/(6*a*b*(a + b*x^3)^2) + ((2*A*b - 5*a*B)*x^7)/(9*a*b^2*(a + b*x^3)) + (7
*a^(1/3)*(2*A*b - 5*a*B)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*S
qrt[3]*b^(13/3)) - (7*a^(1/3)*(2*A*b - 5*a*B)*Log[a^(1/3) + b^(1/3)*x])/(27*b^(1
3/3)) + (7*a^(1/3)*(2*A*b - 5*a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2
])/(54*b^(13/3))

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Rubi [A]  time = 0.44012, antiderivative size = 244, 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 \[ \frac{7 \sqrt [3]{a} (2 A b-5 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 b^{13/3}}-\frac{7 \sqrt [3]{a} (2 A b-5 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 b^{13/3}}+\frac{7 \sqrt [3]{a} (2 A b-5 a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} b^{13/3}}+\frac{7 x (2 A b-5 a B)}{9 b^4}-\frac{7 x^4 (2 A b-5 a B)}{36 a b^3}+\frac{x^7 (2 A b-5 a B)}{9 a b^2 \left (a+b x^3\right )}+\frac{x^{10} (A b-a B)}{6 a b \left (a+b x^3\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

(7*(2*A*b - 5*a*B)*x)/(9*b^4) - (7*(2*A*b - 5*a*B)*x^4)/(36*a*b^3) + ((A*b - a*B
)*x^10)/(6*a*b*(a + b*x^3)^2) + ((2*A*b - 5*a*B)*x^7)/(9*a*b^2*(a + b*x^3)) + (7
*a^(1/3)*(2*A*b - 5*a*B)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*S
qrt[3]*b^(13/3)) - (7*a^(1/3)*(2*A*b - 5*a*B)*Log[a^(1/3) + b^(1/3)*x])/(27*b^(1
3/3)) + (7*a^(1/3)*(2*A*b - 5*a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2
])/(54*b^(13/3))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{7 \sqrt [3]{a} \left (2 A b - 5 B a\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{27 b^{\frac{13}{3}}} + \frac{7 \sqrt [3]{a} \left (2 A b - 5 B a\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{54 b^{\frac{13}{3}}} + \frac{7 \sqrt{3} \sqrt [3]{a} \left (2 A b - 5 B a\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{27 b^{\frac{13}{3}}} + \frac{x^{10} \left (A b - B a\right )}{6 a b \left (a + b x^{3}\right )^{2}} + \frac{x^{7} \left (2 A b - 5 B a\right )}{9 a b^{2} \left (a + b x^{3}\right )} - \frac{7 x^{4} \left (2 A b - 5 B a\right )}{36 a b^{3}} + \frac{7 \left (2 A b - 5 B a\right ) \int a\, dx}{9 a b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-7*a**(1/3)*(2*A*b - 5*B*a)*log(a**(1/3) + b**(1/3)*x)/(27*b**(13/3)) + 7*a**(1/
3)*(2*A*b - 5*B*a)*log(a**(2/3) - a**(1/3)*b**(1/3)*x + b**(2/3)*x**2)/(54*b**(1
3/3)) + 7*sqrt(3)*a**(1/3)*(2*A*b - 5*B*a)*atan(sqrt(3)*(a**(1/3)/3 - 2*b**(1/3)
*x/3)/a**(1/3))/(27*b**(13/3)) + x**10*(A*b - B*a)/(6*a*b*(a + b*x**3)**2) + x**
7*(2*A*b - 5*B*a)/(9*a*b**2*(a + b*x**3)) - 7*x**4*(2*A*b - 5*B*a)/(36*a*b**3) +
7*(2*A*b - 5*B*a)*Integral(a, x)/(9*a*b**4)

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Mathematica [A]  time = 0.31591, size = 210, normalized size = 0.86 \[ \frac{-14 \sqrt [3]{a} (5 a B-2 A b) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+\frac{18 a^2 \sqrt [3]{b} x (a B-A b)}{\left (a+b x^3\right )^2}+\frac{6 a \sqrt [3]{b} x (13 A b-19 a B)}{a+b x^3}+108 \sqrt [3]{b} x (A b-3 a B)+28 \sqrt [3]{a} (5 a B-2 A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-28 \sqrt{3} \sqrt [3]{a} (5 a B-2 A b) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )+27 b^{4/3} B x^4}{108 b^{13/3}} \]

Antiderivative was successfully verified.

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

[Out]

(108*b^(1/3)*(A*b - 3*a*B)*x + 27*b^(4/3)*B*x^4 + (18*a^2*b^(1/3)*(-(A*b) + a*B)
*x)/(a + b*x^3)^2 + (6*a*b^(1/3)*(13*A*b - 19*a*B)*x)/(a + b*x^3) - 28*Sqrt[3]*a
^(1/3)*(-2*A*b + 5*a*B)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 28*a^(1/3)
*(-2*A*b + 5*a*B)*Log[a^(1/3) + b^(1/3)*x] - 14*a^(1/3)*(-2*A*b + 5*a*B)*Log[a^(
2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(108*b^(13/3))

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Maple [A]  time = 0.018, size = 299, normalized size = 1.2 \[{\frac{B{x}^{4}}{4\,{b}^{3}}}+{\frac{Ax}{{b}^{3}}}-3\,{\frac{Bxa}{{b}^{4}}}+{\frac{13\,Aa{x}^{4}}{18\,{b}^{2} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{19\,{a}^{2}B{x}^{4}}{18\,{b}^{3} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{5\,{a}^{2}Ax}{9\,{b}^{3} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{8\,B{a}^{3}x}{9\,{b}^{4} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{14\,Aa}{27\,{b}^{4}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{7\,Aa}{27\,{b}^{4}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{14\,Aa\sqrt{3}}{27\,{b}^{4}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{35\,{a}^{2}B}{27\,{b}^{5}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{35\,{a}^{2}B}{54\,{b}^{5}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{35\,{a}^{2}B\sqrt{3}}{27\,{b}^{5}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.241562, size = 487, normalized size = 2. \[ \frac{\sqrt{3}{\left (14 \, \sqrt{3}{\left ({\left (5 \, B a b^{2} - 2 \, A b^{3}\right )} x^{6} + 5 \, B a^{3} - 2 \, A a^{2} b + 2 \,{\left (5 \, B a^{2} b - 2 \, A a b^{2}\right )} x^{3}\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right ) - 28 \, \sqrt{3}{\left ({\left (5 \, B a b^{2} - 2 \, A b^{3}\right )} x^{6} + 5 \, B a^{3} - 2 \, A a^{2} b + 2 \,{\left (5 \, B a^{2} b - 2 \, A a b^{2}\right )} x^{3}\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left (x - \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right ) + 84 \,{\left ({\left (5 \, B a b^{2} - 2 \, A b^{3}\right )} x^{6} + 5 \, B a^{3} - 2 \, A a^{2} b + 2 \,{\left (5 \, B a^{2} b - 2 \, A a b^{2}\right )} x^{3}\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3} x + \sqrt{3} \left (-\frac{a}{b}\right )^{\frac{1}{3}}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right ) + 3 \, \sqrt{3}{\left (9 \, B b^{3} x^{10} - 18 \,{\left (5 \, B a b^{2} - 2 \, A b^{3}\right )} x^{7} - 49 \,{\left (5 \, B a^{2} b - 2 \, A a b^{2}\right )} x^{4} - 28 \,{\left (5 \, B a^{3} - 2 \, A a^{2} b\right )} x\right )}\right )}}{324 \,{\left (b^{6} x^{6} + 2 \, a b^{5} x^{3} + a^{2} b^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 9.29287, size = 162, normalized size = 0.66 \[ \frac{B x^{4}}{4 b^{3}} - \frac{x^{4} \left (- 13 A a b^{2} + 19 B a^{2} b\right ) + x \left (- 10 A a^{2} b + 16 B a^{3}\right )}{18 a^{2} b^{4} + 36 a b^{5} x^{3} + 18 b^{6} x^{6}} + \operatorname{RootSum}{\left (19683 t^{3} b^{13} + 2744 A^{3} a b^{3} - 20580 A^{2} B a^{2} b^{2} + 51450 A B^{2} a^{3} b - 42875 B^{3} a^{4}, \left ( t \mapsto t \log{\left (\frac{27 t b^{4}}{- 14 A b + 35 B a} + x \right )} \right )\right )} - \frac{x \left (- A b + 3 B a\right )}{b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*x**4/(4*b**3) - (x**4*(-13*A*a*b**2 + 19*B*a**2*b) + x*(-10*A*a**2*b + 16*B*a*
*3))/(18*a**2*b**4 + 36*a*b**5*x**3 + 18*b**6*x**6) + RootSum(19683*_t**3*b**13
+ 2744*A**3*a*b**3 - 20580*A**2*B*a**2*b**2 + 51450*A*B**2*a**3*b - 42875*B**3*a
**4, Lambda(_t, _t*log(27*_t*b**4/(-14*A*b + 35*B*a) + x))) - x*(-A*b + 3*B*a)/b
**4

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GIAC/XCAS [A]  time = 0.222218, size = 316, normalized size = 1.3 \[ \frac{7 \, \sqrt{3}{\left (5 \, \left (-a b^{2}\right )^{\frac{1}{3}} B a - 2 \, \left (-a b^{2}\right )^{\frac{1}{3}} A b\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{27 \, b^{5}} - \frac{7 \,{\left (5 \, B a^{2} - 2 \, A a b\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{27 \, a b^{4}} + \frac{7 \,{\left (5 \, \left (-a b^{2}\right )^{\frac{1}{3}} B a - 2 \, \left (-a b^{2}\right )^{\frac{1}{3}} A b\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{54 \, b^{5}} - \frac{19 \, B a^{2} b x^{4} - 13 \, A a b^{2} x^{4} + 16 \, B a^{3} x - 10 \, A a^{2} b x}{18 \,{\left (b x^{3} + a\right )}^{2} b^{4}} + \frac{B b^{9} x^{4} - 12 \, B a b^{8} x + 4 \, A b^{9} x}{4 \, b^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

7/27*sqrt(3)*(5*(-a*b^2)^(1/3)*B*a - 2*(-a*b^2)^(1/3)*A*b)*arctan(1/3*sqrt(3)*(2
*x + (-a/b)^(1/3))/(-a/b)^(1/3))/b^5 - 7/27*(5*B*a^2 - 2*A*a*b)*(-a/b)^(1/3)*ln(
abs(x - (-a/b)^(1/3)))/(a*b^4) + 7/54*(5*(-a*b^2)^(1/3)*B*a - 2*(-a*b^2)^(1/3)*A
*b)*ln(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/b^5 - 1/18*(19*B*a^2*b*x^4 - 13*A*a*
b^2*x^4 + 16*B*a^3*x - 10*A*a^2*b*x)/((b*x^3 + a)^2*b^4) + 1/4*(B*b^9*x^4 - 12*B
*a*b^8*x + 4*A*b^9*x)/b^12