3.73 \(\int \frac{x^7 \left (A+B x^3\right )}{\left (a+b x^3\right )^2} \, dx\)

Optimal. Leaf size=215 \[ -\frac{a^{2/3} (5 A b-8 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 b^{11/3}}+\frac{a^{2/3} (5 A b-8 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 b^{11/3}}+\frac{a^{2/3} (5 A b-8 a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} b^{11/3}}+\frac{x^2 (5 A b-8 a B)}{6 b^3}-\frac{x^5 (5 A b-8 a B)}{15 a b^2}+\frac{x^8 (A b-a B)}{3 a b \left (a+b x^3\right )} \]

[Out]

((5*A*b - 8*a*B)*x^2)/(6*b^3) - ((5*A*b - 8*a*B)*x^5)/(15*a*b^2) + ((A*b - a*B)*
x^8)/(3*a*b*(a + b*x^3)) + (a^(2/3)*(5*A*b - 8*a*B)*ArcTan[(a^(1/3) - 2*b^(1/3)*
x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*b^(11/3)) + (a^(2/3)*(5*A*b - 8*a*B)*Log[a^(1/
3) + b^(1/3)*x])/(9*b^(11/3)) - (a^(2/3)*(5*A*b - 8*a*B)*Log[a^(2/3) - a^(1/3)*b
^(1/3)*x + b^(2/3)*x^2])/(18*b^(11/3))

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Rubi [A]  time = 0.380322, 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 \[ -\frac{a^{2/3} (5 A b-8 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 b^{11/3}}+\frac{a^{2/3} (5 A b-8 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 b^{11/3}}+\frac{a^{2/3} (5 A b-8 a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} b^{11/3}}+\frac{x^2 (5 A b-8 a B)}{6 b^3}-\frac{x^5 (5 A b-8 a B)}{15 a b^2}+\frac{x^8 (A b-a B)}{3 a b \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]  Int[(x^7*(A + B*x^3))/(a + b*x^3)^2,x]

[Out]

((5*A*b - 8*a*B)*x^2)/(6*b^3) - ((5*A*b - 8*a*B)*x^5)/(15*a*b^2) + ((A*b - a*B)*
x^8)/(3*a*b*(a + b*x^3)) + (a^(2/3)*(5*A*b - 8*a*B)*ArcTan[(a^(1/3) - 2*b^(1/3)*
x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*b^(11/3)) + (a^(2/3)*(5*A*b - 8*a*B)*Log[a^(1/
3) + b^(1/3)*x])/(9*b^(11/3)) - (a^(2/3)*(5*A*b - 8*a*B)*Log[a^(2/3) - a^(1/3)*b
^(1/3)*x + b^(2/3)*x^2])/(18*b^(11/3))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{a^{\frac{2}{3}} \left (5 A b - 8 B a\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{9 b^{\frac{11}{3}}} - \frac{a^{\frac{2}{3}} \left (5 A b - 8 B a\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{18 b^{\frac{11}{3}}} + \frac{\sqrt{3} a^{\frac{2}{3}} \left (5 A b - 8 B a\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{9 b^{\frac{11}{3}}} + \frac{\left (5 A b - 8 B a\right ) \int x\, dx}{3 b^{3}} + \frac{x^{8} \left (A b - B a\right )}{3 a b \left (a + b x^{3}\right )} - \frac{x^{5} \left (5 A b - 8 B a\right )}{15 a b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**7*(B*x**3+A)/(b*x**3+a)**2,x)

[Out]

a**(2/3)*(5*A*b - 8*B*a)*log(a**(1/3) + b**(1/3)*x)/(9*b**(11/3)) - a**(2/3)*(5*
A*b - 8*B*a)*log(a**(2/3) - a**(1/3)*b**(1/3)*x + b**(2/3)*x**2)/(18*b**(11/3))
+ sqrt(3)*a**(2/3)*(5*A*b - 8*B*a)*atan(sqrt(3)*(a**(1/3)/3 - 2*b**(1/3)*x/3)/a*
*(1/3))/(9*b**(11/3)) + (5*A*b - 8*B*a)*Integral(x, x)/(3*b**3) + x**8*(A*b - B*
a)/(3*a*b*(a + b*x**3)) - x**5*(5*A*b - 8*B*a)/(15*a*b**2)

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Mathematica [A]  time = 0.258326, size = 185, normalized size = 0.86 \[ \frac{5 a^{2/3} (8 a B-5 A b) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-10 a^{2/3} (8 a B-5 A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-10 \sqrt{3} a^{2/3} (8 a B-5 A b) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )+45 b^{2/3} x^2 (A b-2 a B)+\frac{30 a b^{2/3} x^2 (A b-a B)}{a+b x^3}+18 b^{5/3} B x^5}{90 b^{11/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^7*(A + B*x^3))/(a + b*x^3)^2,x]

[Out]

(45*b^(2/3)*(A*b - 2*a*B)*x^2 + 18*b^(5/3)*B*x^5 + (30*a*b^(2/3)*(A*b - a*B)*x^2
)/(a + b*x^3) - 10*Sqrt[3]*a^(2/3)*(-5*A*b + 8*a*B)*ArcTan[(1 - (2*b^(1/3)*x)/a^
(1/3))/Sqrt[3]] - 10*a^(2/3)*(-5*A*b + 8*a*B)*Log[a^(1/3) + b^(1/3)*x] + 5*a^(2/
3)*(-5*A*b + 8*a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(90*b^(11/3)
)

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Maple [A]  time = 0.013, size = 266, normalized size = 1.2 \[{\frac{B{x}^{5}}{5\,{b}^{2}}}+{\frac{A{x}^{2}}{2\,{b}^{2}}}-{\frac{B{x}^{2}a}{{b}^{3}}}+{\frac{aA{x}^{2}}{3\,{b}^{2} \left ( b{x}^{3}+a \right ) }}-{\frac{B{x}^{2}{a}^{2}}{3\,{b}^{3} \left ( b{x}^{3}+a \right ) }}+{\frac{5\,aA}{9\,{b}^{3}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{5\,aA}{18\,{b}^{3}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{5\,aA\sqrt{3}}{9\,{b}^{3}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{8\,{a}^{2}B}{9\,{b}^{4}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{4\,{a}^{2}B}{9\,{b}^{4}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{8\,{a}^{2}B\sqrt{3}}{9\,{b}^{4}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7*(B*x^3+A)/(b*x^3+a)^2,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.233097, size = 375, normalized size = 1.74 \[ \frac{\sqrt{3}{\left (5 \, \sqrt{3}{\left ({\left (8 \, B a b - 5 \, A b^{2}\right )} x^{3} + 8 \, B a^{2} - 5 \, A a b\right )} \left (\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \log \left (a x^{2} - b x \left (\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}} + a \left (\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}}\right ) - 10 \, \sqrt{3}{\left ({\left (8 \, B a b - 5 \, A b^{2}\right )} x^{3} + 8 \, B a^{2} - 5 \, A a b\right )} \left (\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \log \left (a x + b \left (\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}}\right ) - 30 \,{\left ({\left (8 \, B a b - 5 \, A b^{2}\right )} x^{3} + 8 \, B a^{2} - 5 \, A a b\right )} \left (\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \arctan \left (-\frac{2 \, \sqrt{3} a x - \sqrt{3} b \left (\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}}}{3 \, b \left (\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}}}\right ) + 3 \, \sqrt{3}{\left (6 \, B b^{2} x^{8} - 3 \,{\left (8 \, B a b - 5 \, A b^{2}\right )} x^{5} - 5 \,{\left (8 \, B a^{2} - 5 \, A a b\right )} x^{2}\right )}\right )}}{270 \,{\left (b^{4} x^{3} + a b^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*x^7/(b*x^3 + a)^2,x, algorithm="fricas")

[Out]

1/270*sqrt(3)*(5*sqrt(3)*((8*B*a*b - 5*A*b^2)*x^3 + 8*B*a^2 - 5*A*a*b)*(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)) - 10*sqrt(3)*((8*B*a
*b - 5*A*b^2)*x^3 + 8*B*a^2 - 5*A*a*b)*(a^2/b^2)^(1/3)*log(a*x + b*(a^2/b^2)^(2/
3)) - 30*((8*B*a*b - 5*A*b^2)*x^3 + 8*B*a^2 - 5*A*a*b)*(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)*
(6*B*b^2*x^8 - 3*(8*B*a*b - 5*A*b^2)*x^5 - 5*(8*B*a^2 - 5*A*a*b)*x^2))/(b^4*x^3
+ a*b^3)

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Sympy [A]  time = 4.64935, size = 151, normalized size = 0.7 \[ \frac{B x^{5}}{5 b^{2}} - \frac{x^{2} \left (- A a b + B a^{2}\right )}{3 a b^{3} + 3 b^{4} x^{3}} + \operatorname{RootSum}{\left (729 t^{3} b^{11} - 125 A^{3} a^{2} b^{3} + 600 A^{2} B a^{3} b^{2} - 960 A B^{2} a^{4} b + 512 B^{3} a^{5}, \left ( t \mapsto t \log{\left (\frac{81 t^{2} b^{7}}{25 A^{2} a b^{2} - 80 A B a^{2} b + 64 B^{2} a^{3}} + x \right )} \right )\right )} - \frac{x^{2} \left (- A b + 2 B a\right )}{2 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**7*(B*x**3+A)/(b*x**3+a)**2,x)

[Out]

B*x**5/(5*b**2) - x**2*(-A*a*b + B*a**2)/(3*a*b**3 + 3*b**4*x**3) + RootSum(729*
_t**3*b**11 - 125*A**3*a**2*b**3 + 600*A**2*B*a**3*b**2 - 960*A*B**2*a**4*b + 51
2*B**3*a**5, Lambda(_t, _t*log(81*_t**2*b**7/(25*A**2*a*b**2 - 80*A*B*a**2*b + 6
4*B**2*a**3) + x))) - x**2*(-A*b + 2*B*a)/(2*b**3)

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GIAC/XCAS [A]  time = 0.218887, size = 319, normalized size = 1.48 \[ -\frac{{\left (8 \, B a^{2} \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 5 \, A a b \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a b^{3}} - \frac{\sqrt{3}{\left (8 \, \left (-a b^{2}\right )^{\frac{2}{3}} B a - 5 \, \left (-a b^{2}\right )^{\frac{2}{3}} A b\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{9 \, b^{5}} - \frac{B a^{2} x^{2} - A a b x^{2}}{3 \,{\left (b x^{3} + a\right )} b^{3}} + \frac{{\left (8 \, \left (-a b^{2}\right )^{\frac{2}{3}} B a - 5 \, \left (-a b^{2}\right )^{\frac{2}{3}} A b\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{18 \, b^{5}} + \frac{2 \, B b^{8} x^{5} - 10 \, B a b^{7} x^{2} + 5 \, A b^{8} x^{2}}{10 \, b^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*x^7/(b*x^3 + a)^2,x, algorithm="giac")

[Out]

-1/9*(8*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^3) - 1/9*sqrt(3)*(8*(-a*b^2)^(2/3)*B*a - 5*(-a*b^2)^(2/3)*A*b)*ar
ctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/b^5 - 1/3*(B*a^2*x^2 - A*a*b
*x^2)/((b*x^3 + a)*b^3) + 1/18*(8*(-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^5 + 1/10*(2*B*b^8*x^5 - 10*B*a*b^7*x^2 +
5*A*b^8*x^2)/b^10