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

Optimal. Leaf size=199 \[ -\frac{(2 a B+A b) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{5/3} b^{7/3}}+\frac{(2 a B+A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{5/3} b^{7/3}}-\frac{(2 a B+A b) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{5/3} b^{7/3}}-\frac{x (2 a B+A b)}{9 a b^2 \left (a+b x^3\right )}+\frac{x^4 (A b-a B)}{6 a b \left (a+b x^3\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.317874, antiderivative size = 199, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ -\frac{(2 a B+A b) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{5/3} b^{7/3}}+\frac{(2 a B+A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{5/3} b^{7/3}}-\frac{(2 a B+A b) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{5/3} b^{7/3}}-\frac{x (2 a B+A b)}{9 a b^2 \left (a+b x^3\right )}+\frac{x^4 (A b-a B)}{6 a b \left (a+b x^3\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 42.4002, size = 182, normalized size = 0.91 \[ \frac{x^{4} \left (A b - B a\right )}{6 a b \left (a + b x^{3}\right )^{2}} - \frac{x \left (A b + 2 B a\right )}{9 a b^{2} \left (a + b x^{3}\right )} + \frac{\left (A b + 2 B a\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{27 a^{\frac{5}{3}} b^{\frac{7}{3}}} - \frac{\left (A b + 2 B a\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{54 a^{\frac{5}{3}} b^{\frac{7}{3}}} - \frac{\sqrt{3} \left (A b + 2 B a\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{27 a^{\frac{5}{3}} b^{\frac{7}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.345429, size = 178, normalized size = 0.89 \[ \frac{-\frac{(2 a B+A b) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{5/3}}+\frac{2 (2 a B+A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{5/3}}-\frac{2 \sqrt{3} (2 a B+A b) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{a^{5/3}}+\frac{3 \sqrt [3]{b} x (A b-7 a B)}{a \left (a+b x^3\right )}-\frac{9 \sqrt [3]{b} x (A b-a B)}{\left (a+b x^3\right )^2}}{54 b^{7/3}} \]

Antiderivative was successfully verified.

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

[Out]

((-9*b^(1/3)*(A*b - a*B)*x)/(a + b*x^3)^2 + (3*b^(1/3)*(A*b - 7*a*B)*x)/(a*(a +
b*x^3)) - (2*Sqrt[3]*(A*b + 2*a*B)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/
a^(5/3) + (2*(A*b + 2*a*B)*Log[a^(1/3) + b^(1/3)*x])/a^(5/3) - ((A*b + 2*a*B)*Lo
g[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(5/3))/(54*b^(7/3))

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Maple [A]  time = 0.015, size = 239, normalized size = 1.2 \[{\frac{1}{ \left ( b{x}^{3}+a \right ) ^{2}} \left ({\frac{ \left ( Ab-7\,Ba \right ){x}^{4}}{18\,ab}}-{\frac{ \left ( Ab+2\,Ba \right ) x}{9\,{b}^{2}}} \right ) }+{\frac{A}{27\,a{b}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{2\,B}{27\,{b}^{3}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{A}{54\,a{b}^{2}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{B}{27\,{b}^{3}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{\sqrt{3}A}{27\,a{b}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{2\,\sqrt{3}B}{27\,{b}^{3}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(1/18*(A*b-7*B*a)/a/b*x^4-1/9*(A*b+2*B*a)/b^2*x)/(b*x^3+a)^2+1/27/b^2/a/(a/b)^(2
/3)*ln(x+(a/b)^(1/3))*A+2/27/b^3/(a/b)^(2/3)*ln(x+(a/b)^(1/3))*B-1/54/b^2/a/(a/b
)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*A-1/27/b^3/(a/b)^(2/3)*ln(x^2-x*(a/b)^
(1/3)+(a/b)^(2/3))*B+1/27/b^2/a/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^
(1/3)*x-1))*A+2/27/b^3/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1
))*B

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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^3/(b*x^3 + a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.238126, size = 416, normalized size = 2.09 \[ -\frac{\sqrt{3}{\left (\sqrt{3}{\left ({\left (2 \, B a b^{2} + A b^{3}\right )} x^{6} + 2 \, B a^{3} + A a^{2} b + 2 \,{\left (2 \, B a^{2} b + A a b^{2}\right )} x^{3}\right )} \log \left (\left (a^{2} b\right )^{\frac{2}{3}} x^{2} - \left (a^{2} b\right )^{\frac{1}{3}} a x + a^{2}\right ) - 2 \, \sqrt{3}{\left ({\left (2 \, B a b^{2} + A b^{3}\right )} x^{6} + 2 \, B a^{3} + A a^{2} b + 2 \,{\left (2 \, B a^{2} b + A a b^{2}\right )} x^{3}\right )} \log \left (\left (a^{2} b\right )^{\frac{1}{3}} x + a\right ) - 6 \,{\left ({\left (2 \, B a b^{2} + A b^{3}\right )} x^{6} + 2 \, B a^{3} + A a^{2} b + 2 \,{\left (2 \, B a^{2} b + A a b^{2}\right )} x^{3}\right )} \arctan \left (\frac{2 \, \sqrt{3} \left (a^{2} b\right )^{\frac{1}{3}} x - \sqrt{3} a}{3 \, a}\right ) + 3 \, \sqrt{3}{\left ({\left (7 \, B a b - A b^{2}\right )} x^{4} + 2 \,{\left (2 \, B a^{2} + A a b\right )} x\right )} \left (a^{2} b\right )^{\frac{1}{3}}\right )}}{162 \,{\left (a b^{4} x^{6} + 2 \, a^{2} b^{3} x^{3} + a^{3} b^{2}\right )} \left (a^{2} b\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/162*sqrt(3)*(sqrt(3)*((2*B*a*b^2 + A*b^3)*x^6 + 2*B*a^3 + A*a^2*b + 2*(2*B*a^
2*b + A*a*b^2)*x^3)*log((a^2*b)^(2/3)*x^2 - (a^2*b)^(1/3)*a*x + a^2) - 2*sqrt(3)
*((2*B*a*b^2 + A*b^3)*x^6 + 2*B*a^3 + A*a^2*b + 2*(2*B*a^2*b + A*a*b^2)*x^3)*log
((a^2*b)^(1/3)*x + a) - 6*((2*B*a*b^2 + A*b^3)*x^6 + 2*B*a^3 + A*a^2*b + 2*(2*B*
a^2*b + A*a*b^2)*x^3)*arctan(1/3*(2*sqrt(3)*(a^2*b)^(1/3)*x - sqrt(3)*a)/a) + 3*
sqrt(3)*((7*B*a*b - A*b^2)*x^4 + 2*(2*B*a^2 + A*a*b)*x)*(a^2*b)^(1/3))/((a*b^4*x
^6 + 2*a^2*b^3*x^3 + a^3*b^2)*(a^2*b)^(1/3))

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Sympy [A]  time = 5.46619, size = 134, normalized size = 0.67 \[ - \frac{x^{4} \left (- A b^{2} + 7 B a b\right ) + x \left (2 A a b + 4 B a^{2}\right )}{18 a^{3} b^{2} + 36 a^{2} b^{3} x^{3} + 18 a b^{4} x^{6}} + \operatorname{RootSum}{\left (19683 t^{3} a^{5} b^{7} - A^{3} b^{3} - 6 A^{2} B a b^{2} - 12 A B^{2} a^{2} b - 8 B^{3} a^{3}, \left ( t \mapsto t \log{\left (\frac{27 t a^{2} b^{2}}{A b + 2 B a} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(x**4*(-A*b**2 + 7*B*a*b) + x*(2*A*a*b + 4*B*a**2))/(18*a**3*b**2 + 36*a**2*b**
3*x**3 + 18*a*b**4*x**6) + RootSum(19683*_t**3*a**5*b**7 - A**3*b**3 - 6*A**2*B*
a*b**2 - 12*A*B**2*a**2*b - 8*B**3*a**3, Lambda(_t, _t*log(27*_t*a**2*b**2/(A*b
+ 2*B*a) + x)))

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GIAC/XCAS [A]  time = 0.222972, size = 274, normalized size = 1.38 \[ -\frac{{\left (2 \, B a + A b\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{27 \, a^{2} b^{2}} + \frac{\sqrt{3}{\left (2 \, \left (-a b^{2}\right )^{\frac{1}{3}} B a + \left (-a b^{2}\right )^{\frac{1}{3}} A b\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{27 \, a^{2} b^{3}} + \frac{{\left (2 \, \left (-a b^{2}\right )^{\frac{1}{3}} B a + \left (-a b^{2}\right )^{\frac{1}{3}} A b\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{54 \, a^{2} b^{3}} - \frac{7 \, B a b x^{4} - A b^{2} x^{4} + 4 \, B a^{2} x + 2 \, A a b x}{18 \,{\left (b x^{3} + a\right )}^{2} a b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/27*(2*B*a + A*b)*(-a/b)^(1/3)*ln(abs(x - (-a/b)^(1/3)))/(a^2*b^2) + 1/27*sqrt
(3)*(2*(-a*b^2)^(1/3)*B*a + (-a*b^2)^(1/3)*A*b)*arctan(1/3*sqrt(3)*(2*x + (-a/b)
^(1/3))/(-a/b)^(1/3))/(a^2*b^3) + 1/54*(2*(-a*b^2)^(1/3)*B*a + (-a*b^2)^(1/3)*A*
b)*ln(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^2*b^3) - 1/18*(7*B*a*b*x^4 - A*b^2
*x^4 + 4*B*a^2*x + 2*A*a*b*x)/((b*x^3 + a)^2*a*b^2)