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

Optimal. Leaf size=190 \[ -\frac{(A b-4 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{2/3} b^{7/3}}+\frac{(A b-4 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{2/3} b^{7/3}}-\frac{(A b-4 a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{2/3} b^{7/3}}-\frac{x (A b-4 a B)}{3 a b^2}+\frac{x^4 (A b-a B)}{3 a b \left (a+b x^3\right )} \]

[Out]

-((A*b - 4*a*B)*x)/(3*a*b^2) + ((A*b - a*B)*x^4)/(3*a*b*(a + b*x^3)) - ((A*b - 4
*a*B)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(2/3)*b^(7
/3)) + ((A*b - 4*a*B)*Log[a^(1/3) + b^(1/3)*x])/(9*a^(2/3)*b^(7/3)) - ((A*b - 4*
a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*a^(2/3)*b^(7/3))

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Rubi [A]  time = 0.30186, antiderivative size = 190, 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 \[ -\frac{(A b-4 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{2/3} b^{7/3}}+\frac{(A b-4 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{2/3} b^{7/3}}-\frac{(A b-4 a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{2/3} b^{7/3}}-\frac{x (A b-4 a B)}{3 a b^2}+\frac{x^4 (A b-a B)}{3 a b \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

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

[Out]

-((A*b - 4*a*B)*x)/(3*a*b^2) + ((A*b - a*B)*x^4)/(3*a*b*(a + b*x^3)) - ((A*b - 4
*a*B)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(2/3)*b^(7
/3)) + ((A*b - 4*a*B)*Log[a^(1/3) + b^(1/3)*x])/(9*a^(2/3)*b^(7/3)) - ((A*b - 4*
a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*a^(2/3)*b^(7/3))

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Rubi in Sympy [A]  time = 43.0382, size = 173, normalized size = 0.91 \[ \frac{x^{4} \left (A b - B a\right )}{3 a b \left (a + b x^{3}\right )} - \frac{x \left (A b - 4 B a\right )}{3 a b^{2}} + \frac{\left (A b - 4 B a\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{9 a^{\frac{2}{3}} b^{\frac{7}{3}}} - \frac{\left (A b - 4 B a\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{18 a^{\frac{2}{3}} b^{\frac{7}{3}}} - \frac{\sqrt{3} \left (A b - 4 B a\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{9 a^{\frac{2}{3}} b^{\frac{7}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**4*(A*b - B*a)/(3*a*b*(a + b*x**3)) - x*(A*b - 4*B*a)/(3*a*b**2) + (A*b - 4*B*
a)*log(a**(1/3) + b**(1/3)*x)/(9*a**(2/3)*b**(7/3)) - (A*b - 4*B*a)*log(a**(2/3)
- a**(1/3)*b**(1/3)*x + b**(2/3)*x**2)/(18*a**(2/3)*b**(7/3)) - sqrt(3)*(A*b -
4*B*a)*atan(sqrt(3)*(a**(1/3)/3 - 2*b**(1/3)*x/3)/a**(1/3))/(9*a**(2/3)*b**(7/3)
)

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Mathematica [A]  time = 0.240734, size = 160, normalized size = 0.84 \[ \frac{\frac{(4 a B-A b) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{2/3}}+\frac{2 (A b-4 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{2/3}}+\frac{2 \sqrt{3} (4 a B-A b) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{a^{2/3}}-\frac{6 \sqrt [3]{b} x (A b-a B)}{a+b x^3}+18 \sqrt [3]{b} B x}{18 b^{7/3}} \]

Antiderivative was successfully verified.

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

[Out]

(18*b^(1/3)*B*x - (6*b^(1/3)*(A*b - a*B)*x)/(a + b*x^3) + (2*Sqrt[3]*(-(A*b) + 4
*a*B)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(2/3) + (2*(A*b - 4*a*B)*Lo
g[a^(1/3) + b^(1/3)*x])/a^(2/3) + ((-(A*b) + 4*a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3
)*x + b^(2/3)*x^2])/a^(2/3))/(18*b^(7/3))

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Maple [A]  time = 0.011, size = 228, normalized size = 1.2 \[{\frac{Bx}{{b}^{2}}}-{\frac{xA}{3\,b \left ( b{x}^{3}+a \right ) }}+{\frac{Bxa}{3\,{b}^{2} \left ( b{x}^{3}+a \right ) }}-{\frac{4\,Ba}{9\,{b}^{3}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{2\,Ba}{9\,{b}^{3}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{4\,Ba\sqrt{3}}{9\,{b}^{3}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{A}{9\,{b}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{A}{18\,{b}^{2}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{A\sqrt{3}}{9\,{b}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.237483, size = 296, normalized size = 1.56 \[ \frac{\sqrt{3}{\left (\sqrt{3}{\left ({\left (4 \, B a b - A b^{2}\right )} x^{3} + 4 \, B a^{2} - A a b\right )} \log \left (\left (a^{2} b\right )^{\frac{2}{3}} x^{2} - \left (a^{2} b\right )^{\frac{1}{3}} a x + a^{2}\right ) - 2 \, \sqrt{3}{\left ({\left (4 \, B a b - A b^{2}\right )} x^{3} + 4 \, B a^{2} - A a b\right )} \log \left (\left (a^{2} b\right )^{\frac{1}{3}} x + a\right ) - 6 \,{\left ({\left (4 \, B a b - A b^{2}\right )} x^{3} + 4 \, B a^{2} - A a b\right )} \arctan \left (\frac{2 \, \sqrt{3} \left (a^{2} b\right )^{\frac{1}{3}} x - \sqrt{3} a}{3 \, a}\right ) + 6 \, \sqrt{3}{\left (3 \, B b x^{4} +{\left (4 \, B a - A b\right )} x\right )} \left (a^{2} b\right )^{\frac{1}{3}}\right )}}{54 \,{\left (b^{3} x^{3} + a b^{2}\right )} \left (a^{2} b\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/54*sqrt(3)*(sqrt(3)*((4*B*a*b - A*b^2)*x^3 + 4*B*a^2 - A*a*b)*log((a^2*b)^(2/3
)*x^2 - (a^2*b)^(1/3)*a*x + a^2) - 2*sqrt(3)*((4*B*a*b - A*b^2)*x^3 + 4*B*a^2 -
A*a*b)*log((a^2*b)^(1/3)*x + a) - 6*((4*B*a*b - A*b^2)*x^3 + 4*B*a^2 - A*a*b)*ar
ctan(1/3*(2*sqrt(3)*(a^2*b)^(1/3)*x - sqrt(3)*a)/a) + 6*sqrt(3)*(3*B*b*x^4 + (4*
B*a - A*b)*x)*(a^2*b)^(1/3))/((b^3*x^3 + a*b^2)*(a^2*b)^(1/3))

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Sympy [A]  time = 3.59246, size = 102, normalized size = 0.54 \[ \frac{B x}{b^{2}} + \frac{x \left (- A b + B a\right )}{3 a b^{2} + 3 b^{3} x^{3}} + \operatorname{RootSum}{\left (729 t^{3} a^{2} b^{7} - A^{3} b^{3} + 12 A^{2} B a b^{2} - 48 A B^{2} a^{2} b + 64 B^{3} a^{3}, \left ( t \mapsto t \log{\left (- \frac{9 t a b^{2}}{- A b + 4 B a} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*x/b**2 + x*(-A*b + B*a)/(3*a*b**2 + 3*b**3*x**3) + RootSum(729*_t**3*a**2*b**7
- A**3*b**3 + 12*A**2*B*a*b**2 - 48*A*B**2*a**2*b + 64*B**3*a**3, Lambda(_t, _t
*log(-9*_t*a*b**2/(-A*b + 4*B*a) + x)))

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GIAC/XCAS [A]  time = 0.219189, size = 254, normalized size = 1.34 \[ \frac{B x}{b^{2}} + \frac{{\left (4 \, B a - A b\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a b^{2}} - \frac{\sqrt{3}{\left (4 \, \left (-a b^{2}\right )^{\frac{1}{3}} B a - \left (-a b^{2}\right )^{\frac{1}{3}} A b\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{9 \, a b^{3}} + \frac{B a x - A b x}{3 \,{\left (b x^{3} + a\right )} b^{2}} - \frac{{\left (4 \, \left (-a b^{2}\right )^{\frac{1}{3}} B a - \left (-a b^{2}\right )^{\frac{1}{3}} A b\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{18 \, a b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*x/b^2 + 1/9*(4*B*a - A*b)*(-a/b)^(1/3)*ln(abs(x - (-a/b)^(1/3)))/(a*b^2) - 1/9
*sqrt(3)*(4*(-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))/(a*b^3) + 1/3*(B*a*x - A*b*x)/((b*x^3 + a)*b^2) - 1/1
8*(4*(-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))/(a*b^3)