3.74 \(\int \frac{x^6 \left (A+B x^3\right )}{\left (a+b x^3\right )^2} \, dx\)

Optimal. Leaf size=213 \[ \frac{\sqrt [3]{a} (4 A b-7 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 b^{10/3}}-\frac{\sqrt [3]{a} (4 A b-7 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 b^{10/3}}+\frac{\sqrt [3]{a} (4 A b-7 a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} b^{10/3}}+\frac{x (4 A b-7 a B)}{3 b^3}-\frac{x^4 (4 A b-7 a B)}{12 a b^2}+\frac{x^7 (A b-a B)}{3 a b \left (a+b x^3\right )} \]

[Out]

((4*A*b - 7*a*B)*x)/(3*b^3) - ((4*A*b - 7*a*B)*x^4)/(12*a*b^2) + ((A*b - a*B)*x^
7)/(3*a*b*(a + b*x^3)) + (a^(1/3)*(4*A*b - 7*a*B)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)
/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*b^(10/3)) - (a^(1/3)*(4*A*b - 7*a*B)*Log[a^(1/3)
+ b^(1/3)*x])/(9*b^(10/3)) + (a^(1/3)*(4*A*b - 7*a*B)*Log[a^(2/3) - a^(1/3)*b^(
1/3)*x + b^(2/3)*x^2])/(18*b^(10/3))

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Rubi [A]  time = 0.351085, antiderivative size = 213, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ \frac{\sqrt [3]{a} (4 A b-7 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 b^{10/3}}-\frac{\sqrt [3]{a} (4 A b-7 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 b^{10/3}}+\frac{\sqrt [3]{a} (4 A b-7 a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} b^{10/3}}+\frac{x (4 A b-7 a B)}{3 b^3}-\frac{x^4 (4 A b-7 a B)}{12 a b^2}+\frac{x^7 (A b-a B)}{3 a b \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]  Int[(x^6*(A + B*x^3))/(a + b*x^3)^2,x]

[Out]

((4*A*b - 7*a*B)*x)/(3*b^3) - ((4*A*b - 7*a*B)*x^4)/(12*a*b^2) + ((A*b - a*B)*x^
7)/(3*a*b*(a + b*x^3)) + (a^(1/3)*(4*A*b - 7*a*B)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)
/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*b^(10/3)) - (a^(1/3)*(4*A*b - 7*a*B)*Log[a^(1/3)
+ b^(1/3)*x])/(9*b^(10/3)) + (a^(1/3)*(4*A*b - 7*a*B)*Log[a^(2/3) - a^(1/3)*b^(
1/3)*x + b^(2/3)*x^2])/(18*b^(10/3))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{\sqrt [3]{a} \left (4 A b - 7 B a\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{9 b^{\frac{10}{3}}} + \frac{\sqrt [3]{a} \left (4 A b - 7 B a\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{18 b^{\frac{10}{3}}} + \frac{\sqrt{3} \sqrt [3]{a} \left (4 A b - 7 B a\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{9 b^{\frac{10}{3}}} + \frac{x^{7} \left (A b - B a\right )}{3 a b \left (a + b x^{3}\right )} - \frac{x^{4} \left (4 A b - 7 B a\right )}{12 a b^{2}} + \frac{\left (4 A b - 7 B a\right ) \int a\, dx}{3 a b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**6*(B*x**3+A)/(b*x**3+a)**2,x)

[Out]

-a**(1/3)*(4*A*b - 7*B*a)*log(a**(1/3) + b**(1/3)*x)/(9*b**(10/3)) + a**(1/3)*(4
*A*b - 7*B*a)*log(a**(2/3) - a**(1/3)*b**(1/3)*x + b**(2/3)*x**2)/(18*b**(10/3))
+ sqrt(3)*a**(1/3)*(4*A*b - 7*B*a)*atan(sqrt(3)*(a**(1/3)/3 - 2*b**(1/3)*x/3)/a
**(1/3))/(9*b**(10/3)) + x**7*(A*b - B*a)/(3*a*b*(a + b*x**3)) - x**4*(4*A*b - 7
*B*a)/(12*a*b**2) + (4*A*b - 7*B*a)*Integral(a, x)/(3*a*b**3)

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Mathematica [A]  time = 0.27194, size = 181, normalized size = 0.85 \[ \frac{-2 \sqrt [3]{a} (7 a B-4 A b) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+\frac{12 a \sqrt [3]{b} x (A b-a B)}{a+b x^3}+36 \sqrt [3]{b} x (A b-2 a B)+4 \sqrt [3]{a} (7 a B-4 A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-4 \sqrt{3} \sqrt [3]{a} (7 a B-4 A b) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )+9 b^{4/3} B x^4}{36 b^{10/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^6*(A + B*x^3))/(a + b*x^3)^2,x]

[Out]

(36*b^(1/3)*(A*b - 2*a*B)*x + 9*b^(4/3)*B*x^4 + (12*a*b^(1/3)*(A*b - a*B)*x)/(a
+ b*x^3) - 4*Sqrt[3]*a^(1/3)*(-4*A*b + 7*a*B)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))
/Sqrt[3]] + 4*a^(1/3)*(-4*A*b + 7*a*B)*Log[a^(1/3) + b^(1/3)*x] - 2*a^(1/3)*(-4*
A*b + 7*a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(36*b^(10/3))

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Maple [A]  time = 0.013, size = 257, normalized size = 1.2 \[{\frac{B{x}^{4}}{4\,{b}^{2}}}+{\frac{Ax}{{b}^{2}}}-2\,{\frac{Bxa}{{b}^{3}}}+{\frac{aAx}{3\,{b}^{2} \left ( b{x}^{3}+a \right ) }}-{\frac{Bx{a}^{2}}{3\,{b}^{3} \left ( b{x}^{3}+a \right ) }}-{\frac{4\,Aa}{9\,{b}^{3}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{2\,Aa}{9\,{b}^{3}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{4\,Aa\sqrt{3}}{9\,{b}^{3}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{7\,{a}^{2}B}{9\,{b}^{4}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{7\,{a}^{2}B}{18\,{b}^{4}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{7\,{a}^{2}B\sqrt{3}}{9\,{b}^{4}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^6*(B*x^3+A)/(b*x^3+a)^2,x)

[Out]

1/4/b^2*B*x^4+1/b^2*A*x-2/b^3*B*x*a+1/3*a/b^2*x/(b*x^3+a)*A-1/3*a^2/b^3*x/(b*x^3
+a)*B-4/9*a/b^3*A/(a/b)^(2/3)*ln(x+(a/b)^(1/3))+2/9*a/b^3*A/(a/b)^(2/3)*ln(x^2-x
*(a/b)^(1/3)+(a/b)^(2/3))-4/9*a/b^3*A/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/
(a/b)^(1/3)*x-1))+7/9*a^2/b^4*B/(a/b)^(2/3)*ln(x+(a/b)^(1/3))-7/18*a^2/b^4*B/(a/
b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))+7/9*a^2/b^4*B/(a/b)^(2/3)*3^(1/2)*arc
tan(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^6/(b*x^3 + a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.241183, size = 343, normalized size = 1.61 \[ \frac{\sqrt{3}{\left (2 \, \sqrt{3}{\left ({\left (7 \, B a b - 4 \, A b^{2}\right )} x^{3} + 7 \, B a^{2} - 4 \, A a b\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right ) - 4 \, \sqrt{3}{\left ({\left (7 \, B a b - 4 \, A b^{2}\right )} x^{3} + 7 \, B a^{2} - 4 \, A a b\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left (x - \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right ) + 12 \,{\left ({\left (7 \, B a b - 4 \, A b^{2}\right )} x^{3} + 7 \, B a^{2} - 4 \, A a b\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3} x + \sqrt{3} \left (-\frac{a}{b}\right )^{\frac{1}{3}}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right ) + 3 \, \sqrt{3}{\left (3 \, B b^{2} x^{7} - 3 \,{\left (7 \, B a b - 4 \, A b^{2}\right )} x^{4} - 4 \,{\left (7 \, B a^{2} - 4 \, A a b\right )} x\right )}\right )}}{108 \,{\left (b^{4} x^{3} + a b^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*x^6/(b*x^3 + a)^2,x, algorithm="fricas")

[Out]

1/108*sqrt(3)*(2*sqrt(3)*((7*B*a*b - 4*A*b^2)*x^3 + 7*B*a^2 - 4*A*a*b)*(-a/b)^(1
/3)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3)) - 4*sqrt(3)*((7*B*a*b - 4*A*b^2)*x^
3 + 7*B*a^2 - 4*A*a*b)*(-a/b)^(1/3)*log(x - (-a/b)^(1/3)) + 12*((7*B*a*b - 4*A*b
^2)*x^3 + 7*B*a^2 - 4*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)*(3*B*b^2*x^7 - 3*(7*B*a*b - 4*A*b^2)*x^4 - 4
*(7*B*a^2 - 4*A*a*b)*x))/(b^4*x^3 + a*b^3)

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Sympy [A]  time = 4.1506, size = 124, normalized size = 0.58 \[ \frac{B x^{4}}{4 b^{2}} - \frac{x \left (- A a b + B a^{2}\right )}{3 a b^{3} + 3 b^{4} x^{3}} + \operatorname{RootSum}{\left (729 t^{3} b^{10} + 64 A^{3} a b^{3} - 336 A^{2} B a^{2} b^{2} + 588 A B^{2} a^{3} b - 343 B^{3} a^{4}, \left ( t \mapsto t \log{\left (\frac{9 t b^{3}}{- 4 A b + 7 B a} + x \right )} \right )\right )} - \frac{x \left (- A b + 2 B a\right )}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**6*(B*x**3+A)/(b*x**3+a)**2,x)

[Out]

B*x**4/(4*b**2) - x*(-A*a*b + B*a**2)/(3*a*b**3 + 3*b**4*x**3) + RootSum(729*_t*
*3*b**10 + 64*A**3*a*b**3 - 336*A**2*B*a**2*b**2 + 588*A*B**2*a**3*b - 343*B**3*
a**4, Lambda(_t, _t*log(9*_t*b**3/(-4*A*b + 7*B*a) + x))) - x*(-A*b + 2*B*a)/b**
3

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GIAC/XCAS [A]  time = 0.218601, size = 285, normalized size = 1.34 \[ \frac{\sqrt{3}{\left (7 \, \left (-a b^{2}\right )^{\frac{1}{3}} B a - 4 \, \left (-a b^{2}\right )^{\frac{1}{3}} A b\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{9 \, b^{4}} - \frac{{\left (7 \, B a^{2} - 4 \, A a b\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a b^{3}} + \frac{{\left (7 \, \left (-a b^{2}\right )^{\frac{1}{3}} B a - 4 \, \left (-a b^{2}\right )^{\frac{1}{3}} A b\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{18 \, b^{4}} - \frac{B a^{2} x - A a b x}{3 \,{\left (b x^{3} + a\right )} b^{3}} + \frac{B b^{6} x^{4} - 8 \, B a b^{5} x + 4 \, A b^{6} x}{4 \, b^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*x^6/(b*x^3 + a)^2,x, algorithm="giac")

[Out]

1/9*sqrt(3)*(7*(-a*b^2)^(1/3)*B*a - 4*(-a*b^2)^(1/3)*A*b)*arctan(1/3*sqrt(3)*(2*
x + (-a/b)^(1/3))/(-a/b)^(1/3))/b^4 - 1/9*(7*B*a^2 - 4*A*a*b)*(-a/b)^(1/3)*ln(ab
s(x - (-a/b)^(1/3)))/(a*b^3) + 1/18*(7*(-a*b^2)^(1/3)*B*a - 4*(-a*b^2)^(1/3)*A*b
)*ln(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/b^4 - 1/3*(B*a^2*x - A*a*b*x)/((b*x^3
+ a)*b^3) + 1/4*(B*b^6*x^4 - 8*B*a*b^5*x + 4*A*b^6*x)/b^8