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

Optimal. Leaf size=167 \[ -\frac{a^{2/3} (A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{8/3}}+\frac{a^{2/3} (A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{8/3}}+\frac{a^{2/3} (A b-a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^{8/3}}+\frac{x^2 (A b-a B)}{2 b^2}+\frac{B x^5}{5 b} \]

[Out]

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

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Rubi [A]  time = 0.316466, antiderivative size = 167, 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 \[ -\frac{a^{2/3} (A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{8/3}}+\frac{a^{2/3} (A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{8/3}}+\frac{a^{2/3} (A b-a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^{8/3}}+\frac{x^2 (A b-a B)}{2 b^2}+\frac{B x^5}{5 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 37.7182, size = 153, normalized size = 0.92 \[ \frac{B x^{5}}{5 b} + \frac{a^{\frac{2}{3}} \left (A b - B a\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 b^{\frac{8}{3}}} - \frac{a^{\frac{2}{3}} \left (A b - B a\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 b^{\frac{8}{3}}} + \frac{\sqrt{3} a^{\frac{2}{3}} \left (A b - B a\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 b^{\frac{8}{3}}} + \frac{x^{2} \left (A b - B a\right )}{2 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.168146, size = 154, normalized size = 0.92 \[ \frac{5 a^{2/3} (a B-A b) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-10 a^{2/3} (a B-A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-10 \sqrt{3} a^{2/3} (a B-A b) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )+15 b^{2/3} x^2 (A b-a B)+6 b^{5/3} B x^5}{30 b^{8/3}} \]

Antiderivative was successfully verified.

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

[Out]

(15*b^(2/3)*(A*b - a*B)*x^2 + 6*b^(5/3)*B*x^5 - 10*Sqrt[3]*a^(2/3)*(-(A*b) + a*B
)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] - 10*a^(2/3)*(-(A*b) + a*B)*Log[a^
(1/3) + b^(1/3)*x] + 5*a^(2/3)*(-(A*b) + a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x +
b^(2/3)*x^2])/(30*b^(8/3))

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Maple [A]  time = 0.006, size = 226, normalized size = 1.4 \[{\frac{B{x}^{5}}{5\,b}}+{\frac{A{x}^{2}}{2\,b}}-{\frac{B{x}^{2}a}{2\,{b}^{2}}}+{\frac{aA}{3\,{b}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{{a}^{2}B}{3\,{b}^{3}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{aA}{6\,{b}^{2}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{{a}^{2}B}{6\,{b}^{3}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{a\sqrt{3}A}{3\,{b}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{{a}^{2}\sqrt{3}B}{3\,{b}^{3}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*(B*x^3+A)/(b*x^3+a),x)

[Out]

1/5*B*x^5/b+1/2/b*A*x^2-1/2/b^2*B*x^2*a+1/3*a/b^2/(a/b)^(1/3)*ln(x+(a/b)^(1/3))*
A-1/3*a^2/b^3/(a/b)^(1/3)*ln(x+(a/b)^(1/3))*B-1/6*a/b^2/(a/b)^(1/3)*ln(x^2-x*(a/
b)^(1/3)+(a/b)^(2/3))*A+1/6*a^2/b^3/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3)
)*B-1/3*a/b^2*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*A+1/3*
a^2/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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.235575, size = 247, normalized size = 1.48 \[ \frac{\sqrt{3}{\left (5 \, \sqrt{3}{\left (B a - A b\right )} \left (\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \log \left (a x^{2} - b x \left (\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}} + a \left (\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}}\right ) - 10 \, \sqrt{3}{\left (B a - A b\right )} \left (\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \log \left (a x + b \left (\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}}\right ) - 30 \,{\left (B a - A b\right )} \left (\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \arctan \left (-\frac{2 \, \sqrt{3} a x - \sqrt{3} b \left (\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}}}{3 \, b \left (\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}}}\right ) + 3 \, \sqrt{3}{\left (2 \, B b x^{5} - 5 \,{\left (B a - A b\right )} x^{2}\right )}\right )}}{90 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/90*sqrt(3)*(5*sqrt(3)*(B*a - A*b)*(a^2/b^2)^(1/3)*log(a*x^2 - b*x*(a^2/b^2)^(2
/3) + a*(a^2/b^2)^(1/3)) - 10*sqrt(3)*(B*a - A*b)*(a^2/b^2)^(1/3)*log(a*x + b*(a
^2/b^2)^(2/3)) - 30*(B*a - A*b)*(a^2/b^2)^(1/3)*arctan(-1/3*(2*sqrt(3)*a*x - sqr
t(3)*b*(a^2/b^2)^(2/3))/(b*(a^2/b^2)^(2/3))) + 3*sqrt(3)*(2*B*b*x^5 - 5*(B*a - A
*b)*x^2))/b^2

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Sympy [A]  time = 2.12063, size = 112, normalized size = 0.67 \[ \frac{B x^{5}}{5 b} + \operatorname{RootSum}{\left (27 t^{3} b^{8} - A^{3} a^{2} b^{3} + 3 A^{2} B a^{3} b^{2} - 3 A B^{2} a^{4} b + B^{3} a^{5}, \left ( t \mapsto t \log{\left (\frac{9 t^{2} b^{5}}{A^{2} a b^{2} - 2 A B a^{2} b + B^{2} a^{3}} + x \right )} \right )\right )} - \frac{x^{2} \left (- A b + B a\right )}{2 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*x**5/(5*b) + RootSum(27*_t**3*b**8 - A**3*a**2*b**3 + 3*A**2*B*a**3*b**2 - 3*A
*B**2*a**4*b + B**3*a**5, Lambda(_t, _t*log(9*_t**2*b**5/(A**2*a*b**2 - 2*A*B*a*
*2*b + B**2*a**3) + x))) - x**2*(-A*b + B*a)/(2*b**2)

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GIAC/XCAS [A]  time = 0.219821, size = 279, normalized size = 1.67 \[ -\frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{2}{3}} B a - \left (-a b^{2}\right )^{\frac{2}{3}} A b\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, b^{4}} + \frac{{\left (\left (-a b^{2}\right )^{\frac{2}{3}} B a - \left (-a b^{2}\right )^{\frac{2}{3}} A b\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, b^{4}} - \frac{{\left (B a^{2} b^{3} \left (-\frac{a}{b}\right )^{\frac{1}{3}} - A a b^{4} \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a b^{5}} + \frac{2 \, B b^{4} x^{5} - 5 \, B a b^{3} x^{2} + 5 \, A b^{4} x^{2}}{10 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*sqrt(3)*((-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))/b^4 + 1/6*((-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))/b^4 - 1/3*(B*a^2*b^3*(-a/b)^(1/3) - A*a
*b^4*(-a/b)^(1/3))*(-a/b)^(1/3)*ln(abs(x - (-a/b)^(1/3)))/(a*b^5) + 1/10*(2*B*b^
4*x^5 - 5*B*a*b^3*x^2 + 5*A*b^4*x^2)/b^5