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

Optimal. Leaf size=222 \[ \frac{5 (A b-4 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 \sqrt [3]{a} b^{11/3}}-\frac{5 (A b-4 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 \sqrt [3]{a} b^{11/3}}-\frac{5 (A b-4 a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} \sqrt [3]{a} b^{11/3}}-\frac{5 x^2 (A b-4 a B)}{18 a b^3}+\frac{x^5 (A b-4 a B)}{9 a b^2 \left (a+b x^3\right )}+\frac{x^8 (A b-a B)}{6 a b \left (a+b x^3\right )^2} \]

[Out]

(-5*(A*b - 4*a*B)*x^2)/(18*a*b^3) + ((A*b - a*B)*x^8)/(6*a*b*(a + b*x^3)^2) + ((
A*b - 4*a*B)*x^5)/(9*a*b^2*(a + b*x^3)) - (5*(A*b - 4*a*B)*ArcTan[(a^(1/3) - 2*b
^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(1/3)*b^(11/3)) - (5*(A*b - 4*a*B)*Lo
g[a^(1/3) + b^(1/3)*x])/(27*a^(1/3)*b^(11/3)) + (5*(A*b - 4*a*B)*Log[a^(2/3) - a
^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(1/3)*b^(11/3))

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Rubi [A]  time = 0.39914, antiderivative size = 222, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.45 \[ \frac{5 (A b-4 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 \sqrt [3]{a} b^{11/3}}-\frac{5 (A b-4 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 \sqrt [3]{a} b^{11/3}}-\frac{5 (A b-4 a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} \sqrt [3]{a} b^{11/3}}-\frac{5 x^2 (A b-4 a B)}{18 a b^3}+\frac{x^5 (A b-4 a B)}{9 a b^2 \left (a+b x^3\right )}+\frac{x^8 (A b-a B)}{6 a b \left (a+b x^3\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

(-5*(A*b - 4*a*B)*x^2)/(18*a*b^3) + ((A*b - a*B)*x^8)/(6*a*b*(a + b*x^3)^2) + ((
A*b - 4*a*B)*x^5)/(9*a*b^2*(a + b*x^3)) - (5*(A*b - 4*a*B)*ArcTan[(a^(1/3) - 2*b
^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(1/3)*b^(11/3)) - (5*(A*b - 4*a*B)*Lo
g[a^(1/3) + b^(1/3)*x])/(27*a^(1/3)*b^(11/3)) + (5*(A*b - 4*a*B)*Log[a^(2/3) - a
^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(1/3)*b^(11/3))

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Rubi in Sympy [A]  time = 47.4581, size = 209, normalized size = 0.94 \[ \frac{x^{8} \left (A b - B a\right )}{6 a b \left (a + b x^{3}\right )^{2}} + \frac{x^{5} \left (A b - 4 B a\right )}{9 a b^{2} \left (a + b x^{3}\right )} - \frac{5 x^{2} \left (A b - 4 B a\right )}{18 a b^{3}} - \frac{5 \left (A b - 4 B a\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{27 \sqrt [3]{a} b^{\frac{11}{3}}} + \frac{5 \left (A b - 4 B a\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{54 \sqrt [3]{a} b^{\frac{11}{3}}} - \frac{5 \sqrt{3} \left (A b - 4 B a\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{27 \sqrt [3]{a} b^{\frac{11}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**8*(A*b - B*a)/(6*a*b*(a + b*x**3)**2) + x**5*(A*b - 4*B*a)/(9*a*b**2*(a + b*x
**3)) - 5*x**2*(A*b - 4*B*a)/(18*a*b**3) - 5*(A*b - 4*B*a)*log(a**(1/3) + b**(1/
3)*x)/(27*a**(1/3)*b**(11/3)) + 5*(A*b - 4*B*a)*log(a**(2/3) - a**(1/3)*b**(1/3)
*x + b**(2/3)*x**2)/(54*a**(1/3)*b**(11/3)) - 5*sqrt(3)*(A*b - 4*B*a)*atan(sqrt(
3)*(a**(1/3)/3 - 2*b**(1/3)*x/3)/a**(1/3))/(27*a**(1/3)*b**(11/3))

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Mathematica [A]  time = 0.342959, size = 194, normalized size = 0.87 \[ \frac{\frac{5 (A b-4 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{\sqrt [3]{a}}-\frac{6 b^{2/3} x^2 (4 A b-7 a B)}{a+b x^3}+\frac{9 a b^{2/3} x^2 (A b-a B)}{\left (a+b x^3\right )^2}+\frac{10 (4 a B-A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a}}+\frac{10 \sqrt{3} (4 a B-A b) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{\sqrt [3]{a}}+27 b^{2/3} B x^2}{54 b^{11/3}} \]

Antiderivative was successfully verified.

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

[Out]

(27*b^(2/3)*B*x^2 + (9*a*b^(2/3)*(A*b - a*B)*x^2)/(a + b*x^3)^2 - (6*b^(2/3)*(4*
A*b - 7*a*B)*x^2)/(a + b*x^3) + (10*Sqrt[3]*(-(A*b) + 4*a*B)*ArcTan[(1 - (2*b^(1
/3)*x)/a^(1/3))/Sqrt[3]])/a^(1/3) + (10*(-(A*b) + 4*a*B)*Log[a^(1/3) + b^(1/3)*x
])/a^(1/3) + (5*(A*b - 4*a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^
(1/3))/(54*b^(11/3))

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Maple [A]  time = 0.014, size = 275, normalized size = 1.2 \[{\frac{B{x}^{2}}{2\,{b}^{3}}}-{\frac{4\,A{x}^{5}}{9\,b \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{7\,B{x}^{5}a}{9\,{b}^{2} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{5\,aA{x}^{2}}{18\,{b}^{2} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{11\,B{x}^{2}{a}^{2}}{18\,{b}^{3} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{5\,A}{27\,{b}^{3}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{5\,A}{54\,{b}^{3}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{5\,A\sqrt{3}}{27\,{b}^{3}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{20\,Ba}{27\,{b}^{4}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{10\,Ba}{27\,{b}^{4}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{20\,Ba\sqrt{3}}{27\,{b}^{4}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*B*x^2/b^3-4/9/b/(b*x^3+a)^2*A*x^5+7/9/b^2/(b*x^3+a)^2*B*x^5*a-5/18/b^2/(b*x^
3+a)^2*A*x^2*a+11/18/b^3/(b*x^3+a)^2*B*x^2*a^2-5/27/b^3*A/(a/b)^(1/3)*ln(x+(a/b)
^(1/3))+5/54/b^3*A/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))+5/27/b^3*A*3^(1
/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))+20/27/b^4*B*a/(a/b)^(1/3
)*ln(x+(a/b)^(1/3))-10/27/b^4*B*a/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))-
20/27/b^4*B*a*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)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.234363, size = 458, normalized size = 2.06 \[ \frac{\sqrt{3}{\left (5 \, \sqrt{3}{\left ({\left (4 \, B a b^{2} - A b^{3}\right )} x^{6} + 4 \, B a^{3} - A a^{2} b + 2 \,{\left (4 \, B a^{2} b - A a b^{2}\right )} x^{3}\right )} \log \left (\left (-a b^{2}\right )^{\frac{1}{3}} b x^{2} - a b + \left (-a b^{2}\right )^{\frac{2}{3}} x\right ) - 10 \, \sqrt{3}{\left ({\left (4 \, B a b^{2} - A b^{3}\right )} x^{6} + 4 \, B a^{3} - A a^{2} b + 2 \,{\left (4 \, B a^{2} b - A a b^{2}\right )} x^{3}\right )} \log \left (a b + \left (-a b^{2}\right )^{\frac{2}{3}} x\right ) + 30 \,{\left ({\left (4 \, B a b^{2} - A b^{3}\right )} x^{6} + 4 \, B a^{3} - A a^{2} b + 2 \,{\left (4 \, B a^{2} b - A a b^{2}\right )} x^{3}\right )} \arctan \left (-\frac{\sqrt{3} a b - 2 \, \sqrt{3} \left (-a b^{2}\right )^{\frac{2}{3}} x}{3 \, a b}\right ) + 3 \, \sqrt{3}{\left (9 \, B b^{2} x^{8} + 8 \,{\left (4 \, B a b - A b^{2}\right )} x^{5} + 5 \,{\left (4 \, B a^{2} - A a b\right )} x^{2}\right )} \left (-a b^{2}\right )^{\frac{1}{3}}\right )}}{162 \,{\left (b^{5} x^{6} + 2 \, a b^{4} x^{3} + a^{2} b^{3}\right )} \left (-a b^{2}\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/162*sqrt(3)*(5*sqrt(3)*((4*B*a*b^2 - A*b^3)*x^6 + 4*B*a^3 - A*a^2*b + 2*(4*B*a
^2*b - A*a*b^2)*x^3)*log((-a*b^2)^(1/3)*b*x^2 - a*b + (-a*b^2)^(2/3)*x) - 10*sqr
t(3)*((4*B*a*b^2 - A*b^3)*x^6 + 4*B*a^3 - A*a^2*b + 2*(4*B*a^2*b - A*a*b^2)*x^3)
*log(a*b + (-a*b^2)^(2/3)*x) + 30*((4*B*a*b^2 - A*b^3)*x^6 + 4*B*a^3 - A*a^2*b +
2*(4*B*a^2*b - A*a*b^2)*x^3)*arctan(-1/3*(sqrt(3)*a*b - 2*sqrt(3)*(-a*b^2)^(2/3
)*x)/(a*b)) + 3*sqrt(3)*(9*B*b^2*x^8 + 8*(4*B*a*b - A*b^2)*x^5 + 5*(4*B*a^2 - A*
a*b)*x^2)*(-a*b^2)^(1/3))/((b^5*x^6 + 2*a*b^4*x^3 + a^2*b^3)*(-a*b^2)^(1/3))

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Sympy [A]  time = 11.4758, size = 162, normalized size = 0.73 \[ \frac{B x^{2}}{2 b^{3}} + \frac{x^{5} \left (- 8 A b^{2} + 14 B a b\right ) + x^{2} \left (- 5 A a b + 11 B a^{2}\right )}{18 a^{2} b^{3} + 36 a b^{4} x^{3} + 18 b^{5} x^{6}} + \operatorname{RootSum}{\left (19683 t^{3} a b^{11} + 125 A^{3} b^{3} - 1500 A^{2} B a b^{2} + 6000 A B^{2} a^{2} b - 8000 B^{3} a^{3}, \left ( t \mapsto t \log{\left (\frac{729 t^{2} a b^{7}}{25 A^{2} b^{2} - 200 A B a b + 400 B^{2} a^{2}} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*x**2/(2*b**3) + (x**5*(-8*A*b**2 + 14*B*a*b) + x**2*(-5*A*a*b + 11*B*a**2))/(1
8*a**2*b**3 + 36*a*b**4*x**3 + 18*b**5*x**6) + RootSum(19683*_t**3*a*b**11 + 125
*A**3*b**3 - 1500*A**2*B*a*b**2 + 6000*A*B**2*a**2*b - 8000*B**3*a**3, Lambda(_t
, _t*log(729*_t**2*a*b**7/(25*A**2*b**2 - 200*A*B*a*b + 400*B**2*a**2) + x)))

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GIAC/XCAS [A]  time = 0.222259, size = 313, normalized size = 1.41 \[ \frac{B x^{2}}{2 \, b^{3}} + \frac{5 \,{\left (4 \, B a \left (-\frac{a}{b}\right )^{\frac{1}{3}} - A b \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{27 \, a b^{3}} + \frac{5 \, \sqrt{3}{\left (4 \, \left (-a b^{2}\right )^{\frac{2}{3}} B a - \left (-a b^{2}\right )^{\frac{2}{3}} A b\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{27 \, a b^{5}} + \frac{14 \, B a b x^{5} - 8 \, A b^{2} x^{5} + 11 \, B a^{2} x^{2} - 5 \, A a b x^{2}}{18 \,{\left (b x^{3} + a\right )}^{2} b^{3}} - \frac{5 \,{\left (4 \, \left (-a b^{2}\right )^{\frac{2}{3}} B a - \left (-a b^{2}\right )^{\frac{2}{3}} A b\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{54 \, a b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*B*x^2/b^3 + 5/27*(4*B*a*(-a/b)^(1/3) - A*b*(-a/b)^(1/3))*(-a/b)^(1/3)*ln(abs
(x - (-a/b)^(1/3)))/(a*b^3) + 5/27*sqrt(3)*(4*(-a*b^2)^(2/3)*B*a - (-a*b^2)^(2/3
)*A*b)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a*b^5) + 1/18*(14*
B*a*b*x^5 - 8*A*b^2*x^5 + 11*B*a^2*x^2 - 5*A*a*b*x^2)/((b*x^3 + a)^2*b^3) - 5/54
*(4*(-a*b^2)^(2/3)*B*a - (-a*b^2)^(2/3)*A*b)*ln(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2
/3))/(a*b^5)