3.23 \(\int \frac{\left (c+d x^3\right )^2}{\left (a+b x^3\right )^2} \, dx\)

Optimal. Leaf size=203 \[ -\frac{(b c-a d) (2 a d+b c) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{9 a^{5/3} b^{7/3}}+\frac{2 (b c-a d) (2 a d+b c) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{7/3}}-\frac{2 (b c-a d) (2 a d+b c) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{5/3} b^{7/3}}+\frac{x (b c-a d)^2}{3 a b^2 \left (a+b x^3\right )}+\frac{d^2 x}{b^2} \]

[Out]

(d^2*x)/b^2 + ((b*c - a*d)^2*x)/(3*a*b^2*(a + b*x^3)) - (2*(b*c - a*d)*(b*c + 2*
a*d)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(5/3)*b^(7/
3)) + (2*(b*c - a*d)*(b*c + 2*a*d)*Log[a^(1/3) + b^(1/3)*x])/(9*a^(5/3)*b^(7/3))
- ((b*c - a*d)*(b*c + 2*a*d)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(9
*a^(5/3)*b^(7/3))

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Rubi [A]  time = 0.497695, antiderivative size = 203, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421 \[ -\frac{(b c-a d) (2 a d+b c) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{9 a^{5/3} b^{7/3}}+\frac{2 (b c-a d) (2 a d+b c) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{7/3}}-\frac{2 (b c-a d) (2 a d+b c) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{5/3} b^{7/3}}+\frac{x (b c-a d)^2}{3 a b^2 \left (a+b x^3\right )}+\frac{d^2 x}{b^2} \]

Antiderivative was successfully verified.

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

[Out]

(d^2*x)/b^2 + ((b*c - a*d)^2*x)/(3*a*b^2*(a + b*x^3)) - (2*(b*c - a*d)*(b*c + 2*
a*d)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(5/3)*b^(7/
3)) + (2*(b*c - a*d)*(b*c + 2*a*d)*Log[a^(1/3) + b^(1/3)*x])/(9*a^(5/3)*b^(7/3))
- ((b*c - a*d)*(b*c + 2*a*d)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(9
*a^(5/3)*b^(7/3))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ d^{2} \int \frac{1}{b^{2}}\, dx + \frac{x \left (a d - b c\right )^{2}}{3 a b^{2} \left (a + b x^{3}\right )} - \frac{2 \left (a d - b c\right ) \left (2 a d + b c\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{9 a^{\frac{5}{3}} b^{\frac{7}{3}}} + \frac{\left (a d - b c\right ) \left (2 a d + b c\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{9 a^{\frac{5}{3}} b^{\frac{7}{3}}} + \frac{2 \sqrt{3} \left (a d - b c\right ) \left (2 a d + b c\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{9 a^{\frac{5}{3}} b^{\frac{7}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x**3+c)**2/(b*x**3+a)**2,x)

[Out]

d**2*Integral(b**(-2), x) + x*(a*d - b*c)**2/(3*a*b**2*(a + b*x**3)) - 2*(a*d -
b*c)*(2*a*d + b*c)*log(a**(1/3) + b**(1/3)*x)/(9*a**(5/3)*b**(7/3)) + (a*d - b*c
)*(2*a*d + b*c)*log(a**(2/3) - a**(1/3)*b**(1/3)*x + b**(2/3)*x**2)/(9*a**(5/3)*
b**(7/3)) + 2*sqrt(3)*(a*d - b*c)*(2*a*d + b*c)*atan(sqrt(3)*(a**(1/3)/3 - 2*b**
(1/3)*x/3)/a**(1/3))/(9*a**(5/3)*b**(7/3))

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Mathematica [A]  time = 0.412179, size = 205, normalized size = 1.01 \[ \frac{\frac{2 \left (-2 a^2 d^2+a b c d+b^2 c^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{5/3}}-\frac{2 \sqrt{3} \left (-2 a^2 d^2+a b c d+b^2 c^2\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{a^{5/3}}-\frac{\left (-2 a^2 d^2+a b c d+b^2 c^2\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{5/3}}+\frac{3 \sqrt [3]{b} x (b c-a d)^2}{a \left (a+b x^3\right )}+9 \sqrt [3]{b} d^2 x}{9 b^{7/3}} \]

Antiderivative was successfully verified.

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

[Out]

(9*b^(1/3)*d^2*x + (3*b^(1/3)*(b*c - a*d)^2*x)/(a*(a + b*x^3)) - (2*Sqrt[3]*(b^2
*c^2 + a*b*c*d - 2*a^2*d^2)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(5/3)
+ (2*(b^2*c^2 + a*b*c*d - 2*a^2*d^2)*Log[a^(1/3) + b^(1/3)*x])/a^(5/3) - ((b^2*
c^2 + a*b*c*d - 2*a^2*d^2)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(5/
3))/(9*b^(7/3))

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Maple [B]  time = 0.013, size = 367, normalized size = 1.8 \[{\frac{{d}^{2}x}{{b}^{2}}}+{\frac{ax{d}^{2}}{3\,{b}^{2} \left ( b{x}^{3}+a \right ) }}-{\frac{2\,cxd}{3\,b \left ( b{x}^{3}+a \right ) }}+{\frac{x{c}^{2}}{3\,a \left ( b{x}^{3}+a \right ) }}-{\frac{4\,a{d}^{2}}{9\,{b}^{3}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{2\,cd}{9\,{b}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{2\,{c}^{2}}{9\,ab}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{2\,a{d}^{2}}{9\,{b}^{3}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{cd}{9\,{b}^{2}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{{c}^{2}}{9\,ab}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{4\,a\sqrt{3}{d}^{2}}{9\,{b}^{3}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{2\,\sqrt{3}cd}{9\,{b}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{2\,\sqrt{3}{c}^{2}}{9\,ab}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x^3+c)^2/(b*x^3+a)^2,x)

[Out]

d^2*x/b^2+1/3/b^2*x*a/(b*x^3+a)*d^2-2/3/b*x/(b*x^3+a)*c*d+1/3*x/a/(b*x^3+a)*c^2-
4/9/b^3*a/(a/b)^(2/3)*ln(x+(a/b)^(1/3))*d^2+2/9/b^2/(a/b)^(2/3)*ln(x+(a/b)^(1/3)
)*c*d+2/9/b/a/(a/b)^(2/3)*ln(x+(a/b)^(1/3))*c^2+2/9/b^3*a/(a/b)^(2/3)*ln(x^2-x*(
a/b)^(1/3)+(a/b)^(2/3))*d^2-1/9/b^2/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3)
)*c*d-1/9/b/a/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*c^2-4/9/b^3*a/(a/b)^
(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*d^2+2/9/b^2/(a/b)^(2/3)*3^
(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*c*d+2/9/b/a/(a/b)^(2/3)*3^(1/2)*ar
ctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*c^2

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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((d*x^3 + c)^2/(b*x^3 + a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.218815, size = 427, normalized size = 2.1 \[ \frac{\sqrt{3}{\left (\sqrt{3}{\left (a b^{2} c^{2} + a^{2} b c d - 2 \, a^{3} d^{2} +{\left (b^{3} c^{2} + a b^{2} c d - 2 \, a^{2} b d^{2}\right )} x^{3}\right )} \log \left (\left (-a^{2} b\right )^{\frac{2}{3}} x^{2} + \left (-a^{2} b\right )^{\frac{1}{3}} a x + a^{2}\right ) - 2 \, \sqrt{3}{\left (a b^{2} c^{2} + a^{2} b c d - 2 \, a^{3} d^{2} +{\left (b^{3} c^{2} + a b^{2} c d - 2 \, a^{2} b d^{2}\right )} x^{3}\right )} \log \left (\left (-a^{2} b\right )^{\frac{1}{3}} x - a\right ) + 6 \,{\left (a b^{2} c^{2} + a^{2} b c d - 2 \, a^{3} d^{2} +{\left (b^{3} c^{2} + a b^{2} c d - 2 \, a^{2} b d^{2}\right )} x^{3}\right )} \arctan \left (\frac{2 \, \sqrt{3} \left (-a^{2} b\right )^{\frac{1}{3}} x + \sqrt{3} a}{3 \, a}\right ) + 3 \, \sqrt{3}{\left (3 \, a b d^{2} x^{4} +{\left (b^{2} c^{2} - 2 \, a b c d + 4 \, a^{2} d^{2}\right )} x\right )} \left (-a^{2} b\right )^{\frac{1}{3}}\right )}}{27 \,{\left (a b^{3} x^{3} + a^{2} b^{2}\right )} \left (-a^{2} b\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/27*sqrt(3)*(sqrt(3)*(a*b^2*c^2 + a^2*b*c*d - 2*a^3*d^2 + (b^3*c^2 + a*b^2*c*d
- 2*a^2*b*d^2)*x^3)*log((-a^2*b)^(2/3)*x^2 + (-a^2*b)^(1/3)*a*x + a^2) - 2*sqrt(
3)*(a*b^2*c^2 + a^2*b*c*d - 2*a^3*d^2 + (b^3*c^2 + a*b^2*c*d - 2*a^2*b*d^2)*x^3)
*log((-a^2*b)^(1/3)*x - a) + 6*(a*b^2*c^2 + a^2*b*c*d - 2*a^3*d^2 + (b^3*c^2 + a
*b^2*c*d - 2*a^2*b*d^2)*x^3)*arctan(1/3*(2*sqrt(3)*(-a^2*b)^(1/3)*x + sqrt(3)*a)
/a) + 3*sqrt(3)*(3*a*b*d^2*x^4 + (b^2*c^2 - 2*a*b*c*d + 4*a^2*d^2)*x)*(-a^2*b)^(
1/3))/((a*b^3*x^3 + a^2*b^2)*(-a^2*b)^(1/3))

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Sympy [A]  time = 5.08011, size = 189, normalized size = 0.93 \[ \frac{x \left (a^{2} d^{2} - 2 a b c d + b^{2} c^{2}\right )}{3 a^{2} b^{2} + 3 a b^{3} x^{3}} + \operatorname{RootSum}{\left (729 t^{3} a^{5} b^{7} + 64 a^{6} d^{6} - 96 a^{5} b c d^{5} - 48 a^{4} b^{2} c^{2} d^{4} + 88 a^{3} b^{3} c^{3} d^{3} + 24 a^{2} b^{4} c^{4} d^{2} - 24 a b^{5} c^{5} d - 8 b^{6} c^{6}, \left ( t \mapsto t \log{\left (- \frac{9 t a^{2} b^{2}}{4 a^{2} d^{2} - 2 a b c d - 2 b^{2} c^{2}} + x \right )} \right )\right )} + \frac{d^{2} x}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x**3+c)**2/(b*x**3+a)**2,x)

[Out]

x*(a**2*d**2 - 2*a*b*c*d + b**2*c**2)/(3*a**2*b**2 + 3*a*b**3*x**3) + RootSum(72
9*_t**3*a**5*b**7 + 64*a**6*d**6 - 96*a**5*b*c*d**5 - 48*a**4*b**2*c**2*d**4 + 8
8*a**3*b**3*c**3*d**3 + 24*a**2*b**4*c**4*d**2 - 24*a*b**5*c**5*d - 8*b**6*c**6,
Lambda(_t, _t*log(-9*_t*a**2*b**2/(4*a**2*d**2 - 2*a*b*c*d - 2*b**2*c**2) + x))
) + d**2*x/b**2

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GIAC/XCAS [A]  time = 0.218488, size = 350, normalized size = 1.72 \[ \frac{d^{2} x}{b^{2}} - \frac{2 \,{\left (b^{2} c^{2} + a b c d - 2 \, a^{2} d^{2}\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a^{2} b^{2}} + \frac{2 \, \sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{2} c^{2} + \left (-a b^{2}\right )^{\frac{1}{3}} a b c d - 2 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{9 \, a^{2} b^{3}} + \frac{b^{2} c^{2} x - 2 \, a b c d x + a^{2} d^{2} x}{3 \,{\left (b x^{3} + a\right )} a b^{2}} + \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{2} c^{2} + \left (-a b^{2}\right )^{\frac{1}{3}} a b c d - 2 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} d^{2}\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{9 \, a^{2} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d^2*x/b^2 - 2/9*(b^2*c^2 + a*b*c*d - 2*a^2*d^2)*(-a/b)^(1/3)*ln(abs(x - (-a/b)^(
1/3)))/(a^2*b^2) + 2/9*sqrt(3)*((-a*b^2)^(1/3)*b^2*c^2 + (-a*b^2)^(1/3)*a*b*c*d
- 2*(-a*b^2)^(1/3)*a^2*d^2)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3)
)/(a^2*b^3) + 1/3*(b^2*c^2*x - 2*a*b*c*d*x + a^2*d^2*x)/((b*x^3 + a)*a*b^2) + 1/
9*((-a*b^2)^(1/3)*b^2*c^2 + (-a*b^2)^(1/3)*a*b*c*d - 2*(-a*b^2)^(1/3)*a^2*d^2)*l
n(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^2*b^3)