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

Optimal. Leaf size=201 \[ \frac{(a B+2 A b) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{7/3} b^{5/3}}-\frac{(a B+2 A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{7/3} b^{5/3}}-\frac{(a B+2 A b) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{7/3} b^{5/3}}+\frac{x^2 (a B+2 A b)}{9 a^2 b \left (a+b x^3\right )}+\frac{x^2 (A b-a B)}{6 a b \left (a+b x^3\right )^2} \]

[Out]

((A*b - a*B)*x^2)/(6*a*b*(a + b*x^3)^2) + ((2*A*b + a*B)*x^2)/(9*a^2*b*(a + b*x^
3)) - ((2*A*b + a*B)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[
3]*a^(7/3)*b^(5/3)) - ((2*A*b + a*B)*Log[a^(1/3) + b^(1/3)*x])/(27*a^(7/3)*b^(5/
3)) + ((2*A*b + a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(7/3)
*b^(5/3))

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Rubi [A]  time = 0.304704, antiderivative size = 201, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.444 \[ \frac{(a B+2 A b) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{7/3} b^{5/3}}-\frac{(a B+2 A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{7/3} b^{5/3}}-\frac{(a B+2 A b) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{7/3} b^{5/3}}+\frac{x^2 (a B+2 A b)}{9 a^2 b \left (a+b x^3\right )}+\frac{x^2 (A b-a B)}{6 a b \left (a+b x^3\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

((A*b - a*B)*x^2)/(6*a*b*(a + b*x^3)^2) + ((2*A*b + a*B)*x^2)/(9*a^2*b*(a + b*x^
3)) - ((2*A*b + a*B)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[
3]*a^(7/3)*b^(5/3)) - ((2*A*b + a*B)*Log[a^(1/3) + b^(1/3)*x])/(27*a^(7/3)*b^(5/
3)) + ((2*A*b + a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(7/3)
*b^(5/3))

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Rubi in Sympy [A]  time = 39.5477, size = 184, normalized size = 0.92 \[ \frac{x^{2} \left (A b - B a\right )}{6 a b \left (a + b x^{3}\right )^{2}} + \frac{x^{2} \left (2 A b + B a\right )}{9 a^{2} b \left (a + b x^{3}\right )} - \frac{\left (2 A b + B a\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{27 a^{\frac{7}{3}} b^{\frac{5}{3}}} + \frac{\left (2 A b + B a\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{54 a^{\frac{7}{3}} b^{\frac{5}{3}}} - \frac{\sqrt{3} \left (2 A b + B a\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{27 a^{\frac{7}{3}} b^{\frac{5}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**2*(A*b - B*a)/(6*a*b*(a + b*x**3)**2) + x**2*(2*A*b + B*a)/(9*a**2*b*(a + b*x
**3)) - (2*A*b + B*a)*log(a**(1/3) + b**(1/3)*x)/(27*a**(7/3)*b**(5/3)) + (2*A*b
+ B*a)*log(a**(2/3) - a**(1/3)*b**(1/3)*x + b**(2/3)*x**2)/(54*a**(7/3)*b**(5/3
)) - sqrt(3)*(2*A*b + B*a)*atan(sqrt(3)*(a**(1/3)/3 - 2*b**(1/3)*x/3)/a**(1/3))/
(27*a**(7/3)*b**(5/3))

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Mathematica [A]  time = 0.290272, size = 178, normalized size = 0.89 \[ \frac{(a B+2 A b) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-\frac{9 a^{4/3} b^{2/3} x^2 (a B-A b)}{\left (a+b x^3\right )^2}+\frac{6 \sqrt [3]{a} b^{2/3} x^2 (a B+2 A b)}{a+b x^3}-2 (a B+2 A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-2 \sqrt{3} (a B+2 A b) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{54 a^{7/3} b^{5/3}} \]

Antiderivative was successfully verified.

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

[Out]

((-9*a^(4/3)*b^(2/3)*(-(A*b) + a*B)*x^2)/(a + b*x^3)^2 + (6*a^(1/3)*b^(2/3)*(2*A
*b + a*B)*x^2)/(a + b*x^3) - 2*Sqrt[3]*(2*A*b + a*B)*ArcTan[(1 - (2*b^(1/3)*x)/a
^(1/3))/Sqrt[3]] - 2*(2*A*b + a*B)*Log[a^(1/3) + b^(1/3)*x] + (2*A*b + a*B)*Log[
a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(7/3)*b^(5/3))

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Maple [A]  time = 0.013, size = 251, normalized size = 1.3 \[{\frac{1}{ \left ( b{x}^{3}+a \right ) ^{2}} \left ({\frac{ \left ( 2\,Ab+Ba \right ){x}^{5}}{9\,{a}^{2}}}+{\frac{ \left ( 7\,Ab-Ba \right ){x}^{2}}{18\,ab}} \right ) }-{\frac{2\,A}{27\,{a}^{2}b}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{B}{27\,a{b}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{A}{27\,{a}^{2}b}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{B}{54\,a{b}^{2}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{2\,\sqrt{3}A}{27\,{a}^{2}b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{\sqrt{3}B}{27\,a{b}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(B*x^3+A)/(b*x^3+a)^3,x)

[Out]

(1/9*(2*A*b+B*a)/a^2*x^5+1/18*(7*A*b-B*a)/a/b*x^2)/(b*x^3+a)^2-2/27/a^2/b/(a/b)^
(1/3)*ln(x+(a/b)^(1/3))*A-1/27/a/b^2/(a/b)^(1/3)*ln(x+(a/b)^(1/3))*B+1/27/a^2/b/
(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*A+1/54/a/b^2/(a/b)^(1/3)*ln(x^2-x*
(a/b)^(1/3)+(a/b)^(2/3))*B+2/27/a^2/b*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/
(a/b)^(1/3)*x-1))*A+1/27/a/b^2*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/(b*x^3 + a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.235225, size = 433, normalized size = 2.15 \[ -\frac{\sqrt{3}{\left (\sqrt{3}{\left ({\left (B a b^{2} + 2 \, A b^{3}\right )} x^{6} + B a^{3} + 2 \, A a^{2} b + 2 \,{\left (B a^{2} b + 2 \, A a b^{2}\right )} x^{3}\right )} \log \left (\left (-a b^{2}\right )^{\frac{1}{3}} b x^{2} - a b + \left (-a b^{2}\right )^{\frac{2}{3}} x\right ) - 2 \, \sqrt{3}{\left ({\left (B a b^{2} + 2 \, A b^{3}\right )} x^{6} + B a^{3} + 2 \, A a^{2} b + 2 \,{\left (B a^{2} b + 2 \, A a b^{2}\right )} x^{3}\right )} \log \left (a b + \left (-a b^{2}\right )^{\frac{2}{3}} x\right ) + 6 \,{\left ({\left (B a b^{2} + 2 \, A b^{3}\right )} x^{6} + B a^{3} + 2 \, A a^{2} b + 2 \,{\left (B a^{2} b + 2 \, A a b^{2}\right )} x^{3}\right )} \arctan \left (-\frac{\sqrt{3} a b - 2 \, \sqrt{3} \left (-a b^{2}\right )^{\frac{2}{3}} x}{3 \, a b}\right ) - 3 \, \sqrt{3}{\left (2 \,{\left (B a b + 2 \, A b^{2}\right )} x^{5} -{\left (B a^{2} - 7 \, A a b\right )} x^{2}\right )} \left (-a b^{2}\right )^{\frac{1}{3}}\right )}}{162 \,{\left (a^{2} b^{3} x^{6} + 2 \, a^{3} b^{2} x^{3} + a^{4} b\right )} \left (-a b^{2}\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/162*sqrt(3)*(sqrt(3)*((B*a*b^2 + 2*A*b^3)*x^6 + B*a^3 + 2*A*a^2*b + 2*(B*a^2*
b + 2*A*a*b^2)*x^3)*log((-a*b^2)^(1/3)*b*x^2 - a*b + (-a*b^2)^(2/3)*x) - 2*sqrt(
3)*((B*a*b^2 + 2*A*b^3)*x^6 + B*a^3 + 2*A*a^2*b + 2*(B*a^2*b + 2*A*a*b^2)*x^3)*l
og(a*b + (-a*b^2)^(2/3)*x) + 6*((B*a*b^2 + 2*A*b^3)*x^6 + B*a^3 + 2*A*a^2*b + 2*
(B*a^2*b + 2*A*a*b^2)*x^3)*arctan(-1/3*(sqrt(3)*a*b - 2*sqrt(3)*(-a*b^2)^(2/3)*x
)/(a*b)) - 3*sqrt(3)*(2*(B*a*b + 2*A*b^2)*x^5 - (B*a^2 - 7*A*a*b)*x^2)*(-a*b^2)^
(1/3))/((a^2*b^3*x^6 + 2*a^3*b^2*x^3 + a^4*b)*(-a*b^2)^(1/3))

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Sympy [A]  time = 4.63911, size = 153, normalized size = 0.76 \[ \frac{x^{5} \left (4 A b^{2} + 2 B a b\right ) + x^{2} \left (7 A a b - B a^{2}\right )}{18 a^{4} b + 36 a^{3} b^{2} x^{3} + 18 a^{2} b^{3} x^{6}} + \operatorname{RootSum}{\left (19683 t^{3} a^{7} b^{5} + 8 A^{3} b^{3} + 12 A^{2} B a b^{2} + 6 A B^{2} a^{2} b + B^{3} a^{3}, \left ( t \mapsto t \log{\left (\frac{729 t^{2} a^{5} b^{3}}{4 A^{2} b^{2} + 4 A B a b + B^{2} a^{2}} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(x**5*(4*A*b**2 + 2*B*a*b) + x**2*(7*A*a*b - B*a**2))/(18*a**4*b + 36*a**3*b**2*
x**3 + 18*a**2*b**3*x**6) + RootSum(19683*_t**3*a**7*b**5 + 8*A**3*b**3 + 12*A**
2*B*a*b**2 + 6*A*B**2*a**2*b + B**3*a**3, Lambda(_t, _t*log(729*_t**2*a**5*b**3/
(4*A**2*b**2 + 4*A*B*a*b + B**2*a**2) + x)))

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GIAC/XCAS [A]  time = 0.221142, size = 301, normalized size = 1.5 \[ -\frac{{\left (B a \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 2 \, A b \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{27 \, a^{3} b} - \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{2}{3}} B a + 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} A b\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{27 \, a^{3} b^{3}} + \frac{2 \, B a b x^{5} + 4 \, A b^{2} x^{5} - B a^{2} x^{2} + 7 \, A a b x^{2}}{18 \,{\left (b x^{3} + a\right )}^{2} a^{2} b} + \frac{{\left (\left (-a b^{2}\right )^{\frac{2}{3}} B a + 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} A b\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{54 \, a^{3} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/27*(B*a*(-a/b)^(1/3) + 2*A*b*(-a/b)^(1/3))*(-a/b)^(1/3)*ln(abs(x - (-a/b)^(1/
3)))/(a^3*b) - 1/27*sqrt(3)*((-a*b^2)^(2/3)*B*a + 2*(-a*b^2)^(2/3)*A*b)*arctan(1
/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a^3*b^3) + 1/18*(2*B*a*b*x^5 + 4*
A*b^2*x^5 - B*a^2*x^2 + 7*A*a*b*x^2)/((b*x^3 + a)^2*a^2*b) + 1/54*((-a*b^2)^(2/3
)*B*a + 2*(-a*b^2)^(2/3)*A*b)*ln(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^3*b^3)