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

Optimal. Leaf size=220 \[ -\frac{(A b-7 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{27 a^{2/3} b^{10/3}}+\frac{2 (A b-7 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{2/3} b^{10/3}}-\frac{2 (A b-7 a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{2/3} b^{10/3}}-\frac{2 x (A b-7 a B)}{9 a b^3}+\frac{x^4 (A b-7 a B)}{18 a b^2 \left (a+b x^3\right )}+\frac{x^7 (A b-a B)}{6 a b \left (a+b x^3\right )^2} \]

[Out]

(-2*(A*b - 7*a*B)*x)/(9*a*b^3) + ((A*b - a*B)*x^7)/(6*a*b*(a + b*x^3)^2) + ((A*b
- 7*a*B)*x^4)/(18*a*b^2*(a + b*x^3)) - (2*(A*b - 7*a*B)*ArcTan[(a^(1/3) - 2*b^(
1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(2/3)*b^(10/3)) + (2*(A*b - 7*a*B)*Log[
a^(1/3) + b^(1/3)*x])/(27*a^(2/3)*b^(10/3)) - ((A*b - 7*a*B)*Log[a^(2/3) - a^(1/
3)*b^(1/3)*x + b^(2/3)*x^2])/(27*a^(2/3)*b^(10/3))

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Rubi [A]  time = 0.356503, antiderivative size = 220, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.45 \[ -\frac{(A b-7 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{27 a^{2/3} b^{10/3}}+\frac{2 (A b-7 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{2/3} b^{10/3}}-\frac{2 (A b-7 a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{2/3} b^{10/3}}-\frac{2 x (A b-7 a B)}{9 a b^3}+\frac{x^4 (A b-7 a B)}{18 a b^2 \left (a+b x^3\right )}+\frac{x^7 (A b-a B)}{6 a b \left (a+b x^3\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(A*b - 7*a*B)*x)/(9*a*b^3) + ((A*b - a*B)*x^7)/(6*a*b*(a + b*x^3)^2) + ((A*b
- 7*a*B)*x^4)/(18*a*b^2*(a + b*x^3)) - (2*(A*b - 7*a*B)*ArcTan[(a^(1/3) - 2*b^(
1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(2/3)*b^(10/3)) + (2*(A*b - 7*a*B)*Log[
a^(1/3) + b^(1/3)*x])/(27*a^(2/3)*b^(10/3)) - ((A*b - 7*a*B)*Log[a^(2/3) - a^(1/
3)*b^(1/3)*x + b^(2/3)*x^2])/(27*a^(2/3)*b^(10/3))

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Rubi in Sympy [A]  time = 48.5323, size = 206, normalized size = 0.94 \[ \frac{x^{7} \left (A b - B a\right )}{6 a b \left (a + b x^{3}\right )^{2}} + \frac{x^{4} \left (A b - 7 B a\right )}{18 a b^{2} \left (a + b x^{3}\right )} - \frac{2 x \left (A b - 7 B a\right )}{9 a b^{3}} + \frac{2 \left (A b - 7 B a\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{27 a^{\frac{2}{3}} b^{\frac{10}{3}}} - \frac{\left (A b - 7 B a\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{27 a^{\frac{2}{3}} b^{\frac{10}{3}}} - \frac{2 \sqrt{3} \left (A b - 7 B a\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{27 a^{\frac{2}{3}} b^{\frac{10}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**7*(A*b - B*a)/(6*a*b*(a + b*x**3)**2) + x**4*(A*b - 7*B*a)/(18*a*b**2*(a + b*
x**3)) - 2*x*(A*b - 7*B*a)/(9*a*b**3) + 2*(A*b - 7*B*a)*log(a**(1/3) + b**(1/3)*
x)/(27*a**(2/3)*b**(10/3)) - (A*b - 7*B*a)*log(a**(2/3) - a**(1/3)*b**(1/3)*x +
b**(2/3)*x**2)/(27*a**(2/3)*b**(10/3)) - 2*sqrt(3)*(A*b - 7*B*a)*atan(sqrt(3)*(a
**(1/3)/3 - 2*b**(1/3)*x/3)/a**(1/3))/(27*a**(2/3)*b**(10/3))

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Mathematica [A]  time = 0.351913, size = 188, normalized size = 0.85 \[ \frac{\frac{2 (7 a B-A b) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{2/3}}+\frac{4 (A b-7 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{2/3}}+\frac{4 \sqrt{3} (7 a B-A b) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{a^{2/3}}-\frac{3 \sqrt [3]{b} x (7 A b-13 a B)}{a+b x^3}+\frac{9 a \sqrt [3]{b} x (A b-a B)}{\left (a+b x^3\right )^2}+54 \sqrt [3]{b} B x}{54 b^{10/3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.017, size = 268, normalized size = 1.2 \[{\frac{Bx}{{b}^{3}}}-{\frac{7\,A{x}^{4}}{18\,b \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{13\,B{x}^{4}a}{18\,{b}^{2} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{2\,aAx}{9\,{b}^{2} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{5\,Bx{a}^{2}}{9\,{b}^{3} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{2\,A}{27\,{b}^{3}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{A}{27\,{b}^{3}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{2\,A\sqrt{3}}{27\,{b}^{3}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{14\,Ba}{27\,{b}^{4}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{7\,Ba}{27\,{b}^{4}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{14\,Ba\sqrt{3}}{27\,{b}^{4}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*x/b^3-7/18/b/(b*x^3+a)^2*A*x^4+13/18/b^2/(b*x^3+a)^2*B*x^4*a-2/9/b^2/(b*x^3+a)
^2*A*x*a+5/9/b^3/(b*x^3+a)^2*B*x*a^2+2/27/b^3*A/(a/b)^(2/3)*ln(x+(a/b)^(1/3))-1/
27/b^3*A/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))+2/27/b^3*A/(a/b)^(2/3)*3^
(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))-14/27/b^4*B*a/(a/b)^(2/3)*ln(x+(a/
b)^(1/3))+7/27/b^4*B*a/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))-14/27/b^4*B
*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^6/(b*x^3 + a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.238091, size = 440, normalized size = 2. \[ \frac{\sqrt{3}{\left (2 \, \sqrt{3}{\left ({\left (7 \, B a b^{2} - A b^{3}\right )} x^{6} + 7 \, B a^{3} - A a^{2} b + 2 \,{\left (7 \, B a^{2} b - A a b^{2}\right )} x^{3}\right )} \log \left (\left (a^{2} b\right )^{\frac{2}{3}} x^{2} - \left (a^{2} b\right )^{\frac{1}{3}} a x + a^{2}\right ) - 4 \, \sqrt{3}{\left ({\left (7 \, B a b^{2} - A b^{3}\right )} x^{6} + 7 \, B a^{3} - A a^{2} b + 2 \,{\left (7 \, B a^{2} b - A a b^{2}\right )} x^{3}\right )} \log \left (\left (a^{2} b\right )^{\frac{1}{3}} x + a\right ) - 12 \,{\left ({\left (7 \, B a b^{2} - A b^{3}\right )} x^{6} + 7 \, B a^{3} - A a^{2} b + 2 \,{\left (7 \, B a^{2} b - A a b^{2}\right )} x^{3}\right )} \arctan \left (\frac{2 \, \sqrt{3} \left (a^{2} b\right )^{\frac{1}{3}} x - \sqrt{3} a}{3 \, a}\right ) + 3 \, \sqrt{3}{\left (18 \, B b^{2} x^{7} + 7 \,{\left (7 \, B a b - A b^{2}\right )} x^{4} + 4 \,{\left (7 \, B a^{2} - A a b\right )} x\right )} \left (a^{2} b\right )^{\frac{1}{3}}\right )}}{162 \,{\left (b^{5} x^{6} + 2 \, a b^{4} x^{3} + a^{2} b^{3}\right )} \left (a^{2} b\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/162*sqrt(3)*(2*sqrt(3)*((7*B*a*b^2 - A*b^3)*x^6 + 7*B*a^3 - A*a^2*b + 2*(7*B*a
^2*b - A*a*b^2)*x^3)*log((a^2*b)^(2/3)*x^2 - (a^2*b)^(1/3)*a*x + a^2) - 4*sqrt(3
)*((7*B*a*b^2 - A*b^3)*x^6 + 7*B*a^3 - A*a^2*b + 2*(7*B*a^2*b - A*a*b^2)*x^3)*lo
g((a^2*b)^(1/3)*x + a) - 12*((7*B*a*b^2 - A*b^3)*x^6 + 7*B*a^3 - A*a^2*b + 2*(7*
B*a^2*b - A*a*b^2)*x^3)*arctan(1/3*(2*sqrt(3)*(a^2*b)^(1/3)*x - sqrt(3)*a)/a) +
3*sqrt(3)*(18*B*b^2*x^7 + 7*(7*B*a*b - A*b^2)*x^4 + 4*(7*B*a^2 - A*a*b)*x)*(a^2*
b)^(1/3))/((b^5*x^6 + 2*a*b^4*x^3 + a^2*b^3)*(a^2*b)^(1/3))

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Sympy [A]  time = 7.83489, size = 141, normalized size = 0.64 \[ \frac{B x}{b^{3}} + \frac{x^{4} \left (- 7 A b^{2} + 13 B a b\right ) + x \left (- 4 A a b + 10 B a^{2}\right )}{18 a^{2} b^{3} + 36 a b^{4} x^{3} + 18 b^{5} x^{6}} + \operatorname{RootSum}{\left (19683 t^{3} a^{2} b^{10} - 8 A^{3} b^{3} + 168 A^{2} B a b^{2} - 1176 A B^{2} a^{2} b + 2744 B^{3} a^{3}, \left ( t \mapsto t \log{\left (- \frac{27 t a b^{3}}{- 2 A b + 14 B a} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*x/b**3 + (x**4*(-7*A*b**2 + 13*B*a*b) + x*(-4*A*a*b + 10*B*a**2))/(18*a**2*b**
3 + 36*a*b**4*x**3 + 18*b**5*x**6) + RootSum(19683*_t**3*a**2*b**10 - 8*A**3*b**
3 + 168*A**2*B*a*b**2 - 1176*A*B**2*a**2*b + 2744*B**3*a**3, Lambda(_t, _t*log(-
27*_t*a*b**3/(-2*A*b + 14*B*a) + x)))

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GIAC/XCAS [A]  time = 0.223261, size = 282, normalized size = 1.28 \[ \frac{B x}{b^{3}} + \frac{2 \,{\left (7 \, B a - A b\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{27 \, a b^{3}} - \frac{2 \, \sqrt{3}{\left (7 \, \left (-a b^{2}\right )^{\frac{1}{3}} B a - \left (-a b^{2}\right )^{\frac{1}{3}} A b\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{27 \, a b^{4}} - \frac{{\left (7 \, \left (-a b^{2}\right )^{\frac{1}{3}} B a - \left (-a b^{2}\right )^{\frac{1}{3}} A b\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{27 \, a b^{4}} + \frac{13 \, B a b x^{4} - 7 \, A b^{2} x^{4} + 10 \, B a^{2} x - 4 \, A a b x}{18 \,{\left (b x^{3} + a\right )}^{2} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*x/b^3 + 2/27*(7*B*a - A*b)*(-a/b)^(1/3)*ln(abs(x - (-a/b)^(1/3)))/(a*b^3) - 2/
27*sqrt(3)*(7*(-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^4) - 1/27*(7*(-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^4) + 1/18*(13*B*a*b*x^4 - 7
*A*b^2*x^4 + 10*B*a^2*x - 4*A*a*b*x)/((b*x^3 + a)^2*b^3)