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

Optimal. Leaf size=162 \[ \frac{\sqrt [3]{a} (A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{7/3}}-\frac{\sqrt [3]{a} (A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{7/3}}+\frac{\sqrt [3]{a} (A b-a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^{7/3}}+\frac{x (A b-a B)}{b^2}+\frac{B x^4}{4 b} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.289126, antiderivative size = 162, 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 \[ \frac{\sqrt [3]{a} (A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{7/3}}-\frac{\sqrt [3]{a} (A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{7/3}}+\frac{\sqrt [3]{a} (A b-a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^{7/3}}+\frac{x (A b-a B)}{b^2}+\frac{B x^4}{4 b} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 40.1321, size = 150, normalized size = 0.93 \[ \frac{B x^{4}}{4 b} - \frac{\sqrt [3]{a} \left (A b - B a\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 b^{\frac{7}{3}}} + \frac{\sqrt [3]{a} \left (A b - B a\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 b^{\frac{7}{3}}} + \frac{\sqrt{3} \sqrt [3]{a} \left (A b - B a\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 b^{\frac{7}{3}}} + \frac{x \left (A b - B a\right )}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.150298, size = 152, normalized size = 0.94 \[ \frac{-2 \sqrt [3]{a} (a B-A b) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+12 \sqrt [3]{b} x (A b-a B)+4 \sqrt [3]{a} (a B-A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-4 \sqrt{3} \sqrt [3]{a} (a B-A b) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )+3 b^{4/3} B x^4}{12 b^{7/3}} \]

Antiderivative was successfully verified.

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

[Out]

(12*b^(1/3)*(A*b - a*B)*x + 3*b^(4/3)*B*x^4 - 4*Sqrt[3]*a^(1/3)*(-(A*b) + a*B)*A
rcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 4*a^(1/3)*(-(A*b) + a*B)*Log[a^(1/3
) + b^(1/3)*x] - 2*a^(1/3)*(-(A*b) + a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2
/3)*x^2])/(12*b^(7/3))

_______________________________________________________________________________________

Maple [A]  time = 0.003, size = 221, normalized size = 1.4 \[{\frac{B{x}^{4}}{4\,b}}+{\frac{Ax}{b}}-{\frac{Bxa}{{b}^{2}}}-{\frac{aA}{3\,{b}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{{a}^{2}B}{3\,{b}^{3}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{aA}{6\,{b}^{2}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{{a}^{2}B}{6\,{b}^{3}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{a\sqrt{3}A}{3\,{b}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{{a}^{2}\sqrt{3}B}{3\,{b}^{3}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(B*x^3+A)/(b*x^3+a),x)

[Out]

1/4*B*x^4/b+1/b*A*x-1/b^2*B*x*a-1/3*a/b^2/(a/b)^(2/3)*ln(x+(a/b)^(1/3))*A+1/3*a^
2/b^3/(a/b)^(2/3)*ln(x+(a/b)^(1/3))*B+1/6*a/b^2/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)
+(a/b)^(2/3))*A-1/6*a^2/b^3/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*B-1/3*
a/b^2/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*A+1/3*a^2/b^3/
(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*B

_______________________________________________________________________________________

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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.236106, size = 213, normalized size = 1.31 \[ \frac{\sqrt{3}{\left (2 \, \sqrt{3}{\left (B a - A b\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right ) - 4 \, \sqrt{3}{\left (B a - A b\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left (x - \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right ) + 12 \,{\left (B a - A b\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3} x + \sqrt{3} \left (-\frac{a}{b}\right )^{\frac{1}{3}}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right ) + 3 \, \sqrt{3}{\left (B b x^{4} - 4 \,{\left (B a - A b\right )} x\right )}\right )}}{36 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/36*sqrt(3)*(2*sqrt(3)*(B*a - A*b)*(-a/b)^(1/3)*log(x^2 + x*(-a/b)^(1/3) + (-a/
b)^(2/3)) - 4*sqrt(3)*(B*a - A*b)*(-a/b)^(1/3)*log(x - (-a/b)^(1/3)) + 12*(B*a -
A*b)*(-a/b)^(1/3)*arctan(1/3*(2*sqrt(3)*x + sqrt(3)*(-a/b)^(1/3))/(-a/b)^(1/3))
+ 3*sqrt(3)*(B*b*x^4 - 4*(B*a - A*b)*x))/b^2

_______________________________________________________________________________________

Sympy [A]  time = 2.26818, size = 87, normalized size = 0.54 \[ \frac{B x^{4}}{4 b} + \operatorname{RootSum}{\left (27 t^{3} b^{7} + A^{3} a b^{3} - 3 A^{2} B a^{2} b^{2} + 3 A B^{2} a^{3} b - B^{3} a^{4}, \left ( t \mapsto t \log{\left (\frac{3 t b^{2}}{- A b + B a} + x \right )} \right )\right )} - \frac{x \left (- A b + B a\right )}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*x**4/(4*b) + RootSum(27*_t**3*b**7 + A**3*a*b**3 - 3*A**2*B*a**2*b**2 + 3*A*B*
*2*a**3*b - B**3*a**4, Lambda(_t, _t*log(3*_t*b**2/(-A*b + B*a) + x))) - x*(-A*b
+ B*a)/b**2

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.216273, size = 251, normalized size = 1.55 \[ \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} B a - \left (-a b^{2}\right )^{\frac{1}{3}} A b\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, b^{3}} + \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} B a - \left (-a b^{2}\right )^{\frac{1}{3}} A b\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, b^{3}} - \frac{{\left (B a^{2} b^{2} - A a b^{3}\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a b^{4}} + \frac{B b^{3} x^{4} - 4 \, B a b^{2} x + 4 \, A b^{3} x}{4 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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