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

Optimal. Leaf size=197 \[ -\frac{(a B+5 A b) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{8/3} b^{4/3}}+\frac{(a B+5 A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{8/3} b^{4/3}}-\frac{(a B+5 A b) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{8/3} b^{4/3}}+\frac{x (a B+5 A b)}{18 a^2 b \left (a+b x^3\right )}+\frac{x (A b-a B)}{6 a b \left (a+b x^3\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.260993, antiderivative size = 197, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.471 \[ -\frac{(a B+5 A b) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{8/3} b^{4/3}}+\frac{(a B+5 A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{8/3} b^{4/3}}-\frac{(a B+5 A b) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{8/3} b^{4/3}}+\frac{x (a B+5 A b)}{18 a^2 b \left (a+b x^3\right )}+\frac{x (A b-a B)}{6 a b \left (a+b x^3\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 38.1519, size = 180, normalized size = 0.91 \[ \frac{x \left (A b - B a\right )}{6 a b \left (a + b x^{3}\right )^{2}} + \frac{x \left (5 A b + B a\right )}{18 a^{2} b \left (a + b x^{3}\right )} + \frac{\left (5 A b + B a\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{27 a^{\frac{8}{3}} b^{\frac{4}{3}}} - \frac{\left (5 A b + B a\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{54 a^{\frac{8}{3}} b^{\frac{4}{3}}} - \frac{\sqrt{3} \left (5 A b + B a\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{27 a^{\frac{8}{3}} b^{\frac{4}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.257669, size = 175, normalized size = 0.89 \[ \frac{-(a B+5 A b) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-\frac{9 a^{5/3} \sqrt [3]{b} x (a B-A b)}{\left (a+b x^3\right )^2}+\frac{3 a^{2/3} \sqrt [3]{b} x (a B+5 A b)}{a+b x^3}+2 (a B+5 A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-2 \sqrt{3} (a B+5 A b) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{54 a^{8/3} b^{4/3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.014, size = 249, normalized size = 1.3 \[{\frac{1}{ \left ( b{x}^{3}+a \right ) ^{2}} \left ({\frac{ \left ( 5\,Ab+Ba \right ){x}^{4}}{18\,{a}^{2}}}+{\frac{ \left ( 4\,Ab-Ba \right ) x}{9\,ab}} \right ) }+{\frac{5\,A}{27\,{a}^{2}b}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{B}{27\,a{b}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{5\,A}{54\,{a}^{2}b}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{B}{54\,a{b}^{2}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{5\,\sqrt{3}A}{27\,{a}^{2}b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{\sqrt{3}B}{27\,a{b}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^3+A)/(b*x^3+a)^3,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.240919, size = 414, normalized size = 2.1 \[ -\frac{\sqrt{3}{\left (\sqrt{3}{\left ({\left (B a b^{2} + 5 \, A b^{3}\right )} x^{6} + B a^{3} + 5 \, A a^{2} b + 2 \,{\left (B a^{2} b + 5 \, A a b^{2}\right )} x^{3}\right )} \log \left (\left (a^{2} b\right )^{\frac{2}{3}} x^{2} - \left (a^{2} b\right )^{\frac{1}{3}} a x + a^{2}\right ) - 2 \, \sqrt{3}{\left ({\left (B a b^{2} + 5 \, A b^{3}\right )} x^{6} + B a^{3} + 5 \, A a^{2} b + 2 \,{\left (B a^{2} b + 5 \, A a b^{2}\right )} x^{3}\right )} \log \left (\left (a^{2} b\right )^{\frac{1}{3}} x + a\right ) - 6 \,{\left ({\left (B a b^{2} + 5 \, A b^{3}\right )} x^{6} + B a^{3} + 5 \, A a^{2} b + 2 \,{\left (B a^{2} b + 5 \, A a b^{2}\right )} x^{3}\right )} \arctan \left (\frac{2 \, \sqrt{3} \left (a^{2} b\right )^{\frac{1}{3}} x - \sqrt{3} a}{3 \, a}\right ) - 3 \, \sqrt{3}{\left ({\left (B a b + 5 \, A b^{2}\right )} x^{4} - 2 \,{\left (B a^{2} - 4 \, A a b\right )} x\right )} \left (a^{2} b\right )^{\frac{1}{3}}\right )}}{162 \,{\left (a^{2} b^{3} x^{6} + 2 \, a^{3} b^{2} x^{3} + a^{4} b\right )} \left (a^{2} b\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/162*sqrt(3)*(sqrt(3)*((B*a*b^2 + 5*A*b^3)*x^6 + B*a^3 + 5*A*a^2*b + 2*(B*a^2*
b + 5*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)
*((B*a*b^2 + 5*A*b^3)*x^6 + B*a^3 + 5*A*a^2*b + 2*(B*a^2*b + 5*A*a*b^2)*x^3)*log
((a^2*b)^(1/3)*x + a) - 6*((B*a*b^2 + 5*A*b^3)*x^6 + B*a^3 + 5*A*a^2*b + 2*(B*a^
2*b + 5*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)*((B*a*b + 5*A*b^2)*x^4 - 2*(B*a^2 - 4*A*a*b)*x)*(a^2*b)^(1/3))/((a^2*b^3
*x^6 + 2*a^3*b^2*x^3 + a^4*b)*(a^2*b)^(1/3))

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Sympy [A]  time = 3.85969, size = 133, normalized size = 0.68 \[ \frac{x^{4} \left (5 A b^{2} + B a b\right ) + x \left (8 A a b - 2 B a^{2}\right )}{18 a^{4} b + 36 a^{3} b^{2} x^{3} + 18 a^{2} b^{3} x^{6}} + \operatorname{RootSum}{\left (19683 t^{3} a^{8} b^{4} - 125 A^{3} b^{3} - 75 A^{2} B a b^{2} - 15 A B^{2} a^{2} b - B^{3} a^{3}, \left ( t \mapsto t \log{\left (\frac{27 t a^{3} b}{5 A b + B a} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(x**4*(5*A*b**2 + B*a*b) + x*(8*A*a*b - 2*B*a**2))/(18*a**4*b + 36*a**3*b**2*x**
3 + 18*a**2*b**3*x**6) + RootSum(19683*_t**3*a**8*b**4 - 125*A**3*b**3 - 75*A**2
*B*a*b**2 - 15*A*B**2*a**2*b - B**3*a**3, Lambda(_t, _t*log(27*_t*a**3*b/(5*A*b
+ B*a) + x)))

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GIAC/XCAS [A]  time = 0.220958, size = 273, normalized size = 1.39 \[ -\frac{{\left (B a + 5 \, A b\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{27 \, a^{3} b} + \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} B a + 5 \, \left (-a b^{2}\right )^{\frac{1}{3}} A b\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{27 \, a^{3} b^{2}} + \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} B a + 5 \, \left (-a b^{2}\right )^{\frac{1}{3}} A b\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{54 \, a^{3} b^{2}} + \frac{B a b x^{4} + 5 \, A b^{2} x^{4} - 2 \, B a^{2} x + 8 \, A a b x}{18 \,{\left (b x^{3} + a\right )}^{2} a^{2} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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