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

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

[Out]

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

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Rubi [A]  time = 0.495065, 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 \[ \frac{(b c-a d) (a d+2 b c) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{9 c^{5/3} d^{7/3}}-\frac{2 (b c-a d) (a d+2 b c) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{9 c^{5/3} d^{7/3}}+\frac{2 (b c-a d) (a d+2 b c) \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{3 \sqrt{3} c^{5/3} d^{7/3}}+\frac{x (b c-a d)^2}{3 c d^2 \left (c+d x^3\right )}+\frac{b^2 x}{d^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.379356, size = 210, normalized size = 1.03 \[ \frac{-\frac{2 \left (-a^2 d^2-a b c d+2 b^2 c^2\right ) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{c^{5/3}}+\frac{2 \sqrt{3} \left (-a^2 d^2-a b c d+2 b^2 c^2\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt{3}}\right )}{c^{5/3}}+\frac{\left (-a^2 d^2-a b c d+2 b^2 c^2\right ) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{c^{5/3}}+\frac{3 \sqrt [3]{d} x (b c-a d)^2}{c \left (c+d x^3\right )}+9 b^2 \sqrt [3]{d} x}{9 d^{7/3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.014, size = 367, normalized size = 1.8 \[{\frac{{b}^{2}x}{{d}^{2}}}+{\frac{x{a}^{2}}{3\,c \left ( d{x}^{3}+c \right ) }}-{\frac{2\,axb}{3\,d \left ( d{x}^{3}+c \right ) }}+{\frac{cx{b}^{2}}{3\,{d}^{2} \left ( d{x}^{3}+c \right ) }}+{\frac{2\,{a}^{2}}{9\,cd}\ln \left ( x+\sqrt [3]{{\frac{c}{d}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}+{\frac{2\,ab}{9\,{d}^{2}}\ln \left ( x+\sqrt [3]{{\frac{c}{d}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}-{\frac{4\,{b}^{2}c}{9\,{d}^{3}}\ln \left ( x+\sqrt [3]{{\frac{c}{d}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}-{\frac{{a}^{2}}{9\,cd}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{c}{d}}}+ \left ({\frac{c}{d}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}-{\frac{ab}{9\,{d}^{2}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{c}{d}}}+ \left ({\frac{c}{d}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}+{\frac{2\,{b}^{2}c}{9\,{d}^{3}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{c}{d}}}+ \left ({\frac{c}{d}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}+{\frac{2\,\sqrt{3}{a}^{2}}{9\,cd}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}-1 \right ) } \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}+{\frac{2\,\sqrt{3}ab}{9\,{d}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}-1 \right ) } \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}-{\frac{4\,c\sqrt{3}{b}^{2}}{9\,{d}^{3}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}-1 \right ) } \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^2/(d*x^3+c)^2,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.218018, size = 435, normalized size = 2.14 \[ \frac{\sqrt{3}{\left (\sqrt{3}{\left (2 \, b^{2} c^{3} - a b c^{2} d - a^{2} c d^{2} +{\left (2 \, b^{2} c^{2} d - a b c d^{2} - a^{2} d^{3}\right )} x^{3}\right )} \log \left (\left (c^{2} d\right )^{\frac{2}{3}} x^{2} - \left (c^{2} d\right )^{\frac{1}{3}} c x + c^{2}\right ) - 2 \, \sqrt{3}{\left (2 \, b^{2} c^{3} - a b c^{2} d - a^{2} c d^{2} +{\left (2 \, b^{2} c^{2} d - a b c d^{2} - a^{2} d^{3}\right )} x^{3}\right )} \log \left (\left (c^{2} d\right )^{\frac{1}{3}} x + c\right ) - 6 \,{\left (2 \, b^{2} c^{3} - a b c^{2} d - a^{2} c d^{2} +{\left (2 \, b^{2} c^{2} d - a b c d^{2} - a^{2} d^{3}\right )} x^{3}\right )} \arctan \left (\frac{2 \, \sqrt{3} \left (c^{2} d\right )^{\frac{1}{3}} x - \sqrt{3} c}{3 \, c}\right ) + 3 \, \sqrt{3}{\left (3 \, b^{2} c d x^{4} +{\left (4 \, b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x\right )} \left (c^{2} d\right )^{\frac{1}{3}}\right )}}{27 \,{\left (c d^{3} x^{3} + c^{2} d^{2}\right )} \left (c^{2} d\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 4.8321, size = 189, normalized size = 0.93 \[ \frac{b^{2} x}{d^{2}} + \frac{x \left (a^{2} d^{2} - 2 a b c d + b^{2} c^{2}\right )}{3 c^{2} d^{2} + 3 c d^{3} x^{3}} + \operatorname{RootSum}{\left (729 t^{3} c^{5} d^{7} - 8 a^{6} d^{6} - 24 a^{5} b c d^{5} + 24 a^{4} b^{2} c^{2} d^{4} + 88 a^{3} b^{3} c^{3} d^{3} - 48 a^{2} b^{4} c^{4} d^{2} - 96 a b^{5} c^{5} d + 64 b^{6} c^{6}, \left ( t \mapsto t \log{\left (\frac{9 t c^{2} d^{2}}{2 a^{2} d^{2} + 2 a b c d - 4 b^{2} c^{2}} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b**2*x/d**2 + x*(a**2*d**2 - 2*a*b*c*d + b**2*c**2)/(3*c**2*d**2 + 3*c*d**3*x**3
) + RootSum(729*_t**3*c**5*d**7 - 8*a**6*d**6 - 24*a**5*b*c*d**5 + 24*a**4*b**2*
c**2*d**4 + 88*a**3*b**3*c**3*d**3 - 48*a**2*b**4*c**4*d**2 - 96*a*b**5*c**5*d +
64*b**6*c**6, Lambda(_t, _t*log(9*_t*c**2*d**2/(2*a**2*d**2 + 2*a*b*c*d - 4*b**
2*c**2) + x)))

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GIAC/XCAS [A]  time = 0.219868, size = 358, normalized size = 1.76 \[ \frac{b^{2} x}{d^{2}} + \frac{2 \,{\left (2 \, b^{2} c^{2} - a b c d - a^{2} d^{2}\right )} \left (-\frac{c}{d}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{c}{d}\right )^{\frac{1}{3}} \right |}\right )}{9 \, c^{2} d^{2}} - \frac{2 \, \sqrt{3}{\left (2 \, \left (-c d^{2}\right )^{\frac{1}{3}} b^{2} c^{2} - \left (-c d^{2}\right )^{\frac{1}{3}} a b c d - \left (-c d^{2}\right )^{\frac{1}{3}} a^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{c}{d}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{c}{d}\right )^{\frac{1}{3}}}\right )}{9 \, c^{2} d^{3}} + \frac{b^{2} c^{2} x - 2 \, a b c d x + a^{2} d^{2} x}{3 \,{\left (d x^{3} + c\right )} c d^{2}} - \frac{{\left (2 \, \left (-c d^{2}\right )^{\frac{1}{3}} b^{2} c^{2} - \left (-c d^{2}\right )^{\frac{1}{3}} a b c d - \left (-c d^{2}\right )^{\frac{1}{3}} a^{2} d^{2}\right )}{\rm ln}\left (x^{2} + x \left (-\frac{c}{d}\right )^{\frac{1}{3}} + \left (-\frac{c}{d}\right )^{\frac{2}{3}}\right )}{9 \, c^{2} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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