3.76 \(\int \frac{x^4 \left (A+B x^3\right )}{\left (a+b x^3\right )^2} \, dx\)

Optimal. Leaf size=196 \[ \frac{(2 A b-5 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 \sqrt [3]{a} b^{8/3}}-\frac{(2 A b-5 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 \sqrt [3]{a} b^{8/3}}-\frac{(2 A b-5 a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} \sqrt [3]{a} b^{8/3}}-\frac{x^2 (2 A b-5 a B)}{6 a b^2}+\frac{x^5 (A b-a B)}{3 a b \left (a+b x^3\right )} \]

[Out]

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

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Rubi [A]  time = 0.326517, antiderivative size = 196, 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 \[ \frac{(2 A b-5 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 \sqrt [3]{a} b^{8/3}}-\frac{(2 A b-5 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 \sqrt [3]{a} b^{8/3}}-\frac{(2 A b-5 a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} \sqrt [3]{a} b^{8/3}}-\frac{x^2 (2 A b-5 a B)}{6 a b^2}+\frac{x^5 (A b-a B)}{3 a b \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]  Int[(x^4*(A + B*x^3))/(a + b*x^3)^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 40.4904, size = 182, normalized size = 0.93 \[ \frac{x^{5} \left (A b - B a\right )}{3 a b \left (a + b x^{3}\right )} - \frac{x^{2} \left (2 A b - 5 B a\right )}{6 a b^{2}} - \frac{\left (2 A b - 5 B a\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{9 \sqrt [3]{a} b^{\frac{8}{3}}} + \frac{\left (2 A b - 5 B a\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{18 \sqrt [3]{a} b^{\frac{8}{3}}} - \frac{\sqrt{3} \left (2 A b - 5 B a\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{9 \sqrt [3]{a} b^{\frac{8}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4*(B*x**3+A)/(b*x**3+a)**2,x)

[Out]

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

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Mathematica [A]  time = 0.251768, size = 165, normalized size = 0.84 \[ \frac{\frac{(2 A b-5 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{\sqrt [3]{a}}-\frac{6 b^{2/3} x^2 (A b-a B)}{a+b x^3}+\frac{2 (5 a B-2 A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a}}+\frac{2 \sqrt{3} (5 a B-2 A b) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{\sqrt [3]{a}}+9 b^{2/3} B x^2}{18 b^{8/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^4*(A + B*x^3))/(a + b*x^3)^2,x]

[Out]

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

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Maple [A]  time = 0.013, size = 235, normalized size = 1.2 \[{\frac{B{x}^{2}}{2\,{b}^{2}}}-{\frac{{x}^{2}A}{3\,b \left ( b{x}^{3}+a \right ) }}+{\frac{B{x}^{2}a}{3\,{b}^{2} \left ( b{x}^{3}+a \right ) }}+{\frac{5\,Ba}{9\,{b}^{3}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{5\,Ba}{18\,{b}^{3}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{5\,Ba\sqrt{3}}{9\,{b}^{3}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{2\,A}{9\,{b}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{A}{9\,{b}^{2}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{2\,A\sqrt{3}}{9\,{b}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*(B*x^3+A)/(b*x^3+a)^2,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.241338, size = 313, normalized size = 1.6 \[ \frac{\sqrt{3}{\left (\sqrt{3}{\left ({\left (5 \, B a b - 2 \, A b^{2}\right )} x^{3} + 5 \, B a^{2} - 2 \, A a b\right )} \log \left (\left (-a b^{2}\right )^{\frac{1}{3}} b x^{2} - a b + \left (-a b^{2}\right )^{\frac{2}{3}} x\right ) - 2 \, \sqrt{3}{\left ({\left (5 \, B a b - 2 \, A b^{2}\right )} x^{3} + 5 \, B a^{2} - 2 \, A a b\right )} \log \left (a b + \left (-a b^{2}\right )^{\frac{2}{3}} x\right ) + 6 \,{\left ({\left (5 \, B a b - 2 \, A b^{2}\right )} x^{3} + 5 \, B a^{2} - 2 \, A a b\right )} \arctan \left (-\frac{\sqrt{3} a b - 2 \, \sqrt{3} \left (-a b^{2}\right )^{\frac{2}{3}} x}{3 \, a b}\right ) + 3 \, \sqrt{3}{\left (3 \, B b x^{5} +{\left (5 \, B a - 2 \, A b\right )} x^{2}\right )} \left (-a b^{2}\right )^{\frac{1}{3}}\right )}}{54 \,{\left (b^{3} x^{3} + a b^{2}\right )} \left (-a b^{2}\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*x^4/(b*x^3 + a)^2,x, algorithm="fricas")

[Out]

1/54*sqrt(3)*(sqrt(3)*((5*B*a*b - 2*A*b^2)*x^3 + 5*B*a^2 - 2*A*a*b)*log((-a*b^2)
^(1/3)*b*x^2 - a*b + (-a*b^2)^(2/3)*x) - 2*sqrt(3)*((5*B*a*b - 2*A*b^2)*x^3 + 5*
B*a^2 - 2*A*a*b)*log(a*b + (-a*b^2)^(2/3)*x) + 6*((5*B*a*b - 2*A*b^2)*x^3 + 5*B*
a^2 - 2*A*a*b)*arctan(-1/3*(sqrt(3)*a*b - 2*sqrt(3)*(-a*b^2)^(2/3)*x)/(a*b)) + 3
*sqrt(3)*(3*B*b*x^5 + (5*B*a - 2*A*b)*x^2)*(-a*b^2)^(1/3))/((b^3*x^3 + a*b^2)*(-
a*b^2)^(1/3))

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Sympy [A]  time = 4.20981, size = 126, normalized size = 0.64 \[ \frac{B x^{2}}{2 b^{2}} + \frac{x^{2} \left (- A b + B a\right )}{3 a b^{2} + 3 b^{3} x^{3}} + \operatorname{RootSum}{\left (729 t^{3} a b^{8} + 8 A^{3} b^{3} - 60 A^{2} B a b^{2} + 150 A B^{2} a^{2} b - 125 B^{3} a^{3}, \left ( t \mapsto t \log{\left (\frac{81 t^{2} a b^{5}}{4 A^{2} b^{2} - 20 A B a b + 25 B^{2} a^{2}} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4*(B*x**3+A)/(b*x**3+a)**2,x)

[Out]

B*x**2/(2*b**2) + x**2*(-A*b + B*a)/(3*a*b**2 + 3*b**3*x**3) + RootSum(729*_t**3
*a*b**8 + 8*A**3*b**3 - 60*A**2*B*a*b**2 + 150*A*B**2*a**2*b - 125*B**3*a**3, La
mbda(_t, _t*log(81*_t**2*a*b**5/(4*A**2*b**2 - 20*A*B*a*b + 25*B**2*a**2) + x)))

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GIAC/XCAS [A]  time = 0.220965, size = 285, normalized size = 1.45 \[ \frac{B x^{2}}{2 \, b^{2}} + \frac{{\left (5 \, B a \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 2 \, A b \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a b^{2}} + \frac{B a x^{2} - A b x^{2}}{3 \,{\left (b x^{3} + a\right )} b^{2}} + \frac{\sqrt{3}{\left (5 \, \left (-a b^{2}\right )^{\frac{2}{3}} B a - 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} A b\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{9 \, a b^{4}} - \frac{{\left (5 \, \left (-a b^{2}\right )^{\frac{2}{3}} B a - 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} A b\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{18 \, a b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*x^4/(b*x^3 + a)^2,x, algorithm="giac")

[Out]

1/2*B*x^2/b^2 + 1/9*(5*B*a*(-a/b)^(1/3) - 2*A*b*(-a/b)^(1/3))*(-a/b)^(1/3)*ln(ab
s(x - (-a/b)^(1/3)))/(a*b^2) + 1/3*(B*a*x^2 - A*b*x^2)/((b*x^3 + a)*b^2) + 1/9*s
qrt(3)*(5*(-a*b^2)^(2/3)*B*a - 2*(-a*b^2)^(2/3)*A*b)*arctan(1/3*sqrt(3)*(2*x + (
-a/b)^(1/3))/(-a/b)^(1/3))/(a*b^4) - 1/18*(5*(-a*b^2)^(2/3)*B*a - 2*(-a*b^2)^(2/
3)*A*b)*ln(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a*b^4)