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

Optimal. Leaf size=201 \[ \frac{(5 a B+A b) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{4/3} b^{8/3}}-\frac{(5 a B+A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{4/3} b^{8/3}}-\frac{(5 a B+A b) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{4/3} b^{8/3}}-\frac{x^2 (5 a B+A b)}{18 a b^2 \left (a+b x^3\right )}+\frac{x^5 (A b-a B)}{6 a b \left (a+b x^3\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.30396, antiderivative size = 201, 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 \[ \frac{(5 a B+A b) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{4/3} b^{8/3}}-\frac{(5 a B+A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{4/3} b^{8/3}}-\frac{(5 a B+A b) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{4/3} b^{8/3}}-\frac{x^2 (5 a B+A b)}{18 a b^2 \left (a+b x^3\right )}+\frac{x^5 (A b-a B)}{6 a b \left (a+b x^3\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 40.8409, size = 184, normalized size = 0.92 \[ \frac{x^{5} \left (A b - B a\right )}{6 a b \left (a + b x^{3}\right )^{2}} - \frac{x^{2} \left (A b + 5 B a\right )}{18 a b^{2} \left (a + b x^{3}\right )} - \frac{\left (A b + 5 B a\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{27 a^{\frac{4}{3}} b^{\frac{8}{3}}} + \frac{\left (A b + 5 B a\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{54 a^{\frac{4}{3}} b^{\frac{8}{3}}} - \frac{\sqrt{3} \left (A b + 5 B a\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{27 a^{\frac{4}{3}} b^{\frac{8}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.338836, size = 181, normalized size = 0.9 \[ \frac{\frac{(5 a B+A b) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{4/3}}-\frac{2 (5 a B+A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{4/3}}-\frac{2 \sqrt{3} (5 a B+A b) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{a^{4/3}}+\frac{6 b^{2/3} x^2 (A b-4 a B)}{a \left (a+b x^3\right )}-\frac{9 b^{2/3} x^2 (A b-a B)}{\left (a+b x^3\right )^2}}{54 b^{8/3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.014, size = 241, normalized size = 1.2 \[{\frac{1}{ \left ( b{x}^{3}+a \right ) ^{2}} \left ({\frac{ \left ( Ab-4\,Ba \right ){x}^{5}}{9\,ab}}-{\frac{ \left ( Ab+5\,Ba \right ){x}^{2}}{18\,{b}^{2}}} \right ) }-{\frac{A}{27\,a{b}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{5\,B}{27\,{b}^{3}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{A}{54\,a{b}^{2}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{5\,B}{54\,{b}^{3}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{\sqrt{3}A}{27\,a{b}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{5\,\sqrt{3}B}{27\,{b}^{3}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.238096, size = 433, normalized size = 2.15 \[ -\frac{\sqrt{3}{\left (\sqrt{3}{\left ({\left (5 \, B a b^{2} + A b^{3}\right )} x^{6} + 5 \, B a^{3} + A a^{2} b + 2 \,{\left (5 \, B a^{2} b + A a b^{2}\right )} x^{3}\right )} \log \left (\left (-a b^{2}\right )^{\frac{1}{3}} b x^{2} - a b + \left (-a b^{2}\right )^{\frac{2}{3}} x\right ) - 2 \, \sqrt{3}{\left ({\left (5 \, B a b^{2} + A b^{3}\right )} x^{6} + 5 \, B a^{3} + A a^{2} b + 2 \,{\left (5 \, B a^{2} b + A a b^{2}\right )} x^{3}\right )} \log \left (a b + \left (-a b^{2}\right )^{\frac{2}{3}} x\right ) + 6 \,{\left ({\left (5 \, B a b^{2} + A b^{3}\right )} x^{6} + 5 \, B a^{3} + A a^{2} b + 2 \,{\left (5 \, B a^{2} b + A a b^{2}\right )} x^{3}\right )} \arctan \left (-\frac{\sqrt{3} a b - 2 \, \sqrt{3} \left (-a b^{2}\right )^{\frac{2}{3}} x}{3 \, a b}\right ) + 3 \, \sqrt{3}{\left (2 \,{\left (4 \, B a b - A b^{2}\right )} x^{5} +{\left (5 \, B a^{2} + A a b\right )} x^{2}\right )} \left (-a b^{2}\right )^{\frac{1}{3}}\right )}}{162 \,{\left (a b^{4} x^{6} + 2 \, a^{2} b^{3} x^{3} + a^{3} b^{2}\right )} \left (-a b^{2}\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 8.02935, size = 153, normalized size = 0.76 \[ - \frac{x^{5} \left (- 2 A b^{2} + 8 B a b\right ) + x^{2} \left (A a b + 5 B a^{2}\right )}{18 a^{3} b^{2} + 36 a^{2} b^{3} x^{3} + 18 a b^{4} x^{6}} + \operatorname{RootSum}{\left (19683 t^{3} a^{4} b^{8} + A^{3} b^{3} + 15 A^{2} B a b^{2} + 75 A B^{2} a^{2} b + 125 B^{3} a^{3}, \left ( t \mapsto t \log{\left (\frac{729 t^{2} a^{3} b^{5}}{A^{2} b^{2} + 10 A B a b + 25 B^{2} a^{2}} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(x**5*(-2*A*b**2 + 8*B*a*b) + x**2*(A*a*b + 5*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**4*b**8 + A**3*b**3 + 15*A**
2*B*a*b**2 + 75*A*B**2*a**2*b + 125*B**3*a**3, Lambda(_t, _t*log(729*_t**2*a**3*
b**5/(A**2*b**2 + 10*A*B*a*b + 25*B**2*a**2) + x)))

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GIAC/XCAS [A]  time = 0.224988, size = 300, normalized size = 1.49 \[ -\frac{{\left (5 \, B a \left (-\frac{a}{b}\right )^{\frac{1}{3}} + A b \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{27 \, a^{2} b^{2}} - \frac{\sqrt{3}{\left (5 \, \left (-a b^{2}\right )^{\frac{2}{3}} B a + \left (-a b^{2}\right )^{\frac{2}{3}} A b\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{27 \, a^{2} b^{4}} - \frac{8 \, B a b x^{5} - 2 \, A b^{2} x^{5} + 5 \, B a^{2} x^{2} + A a b x^{2}}{18 \,{\left (b x^{3} + a\right )}^{2} a b^{2}} + \frac{{\left (5 \, \left (-a b^{2}\right )^{\frac{2}{3}} B a + \left (-a b^{2}\right )^{\frac{2}{3}} A b\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{54 \, a^{2} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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