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

Optimal. Leaf size=258 \[ -\frac{\left (5 a^2 d^2+2 a b c d+2 b^2 c^2\right ) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{54 c^{8/3} d^{7/3}}+\frac{\left (5 a^2 d^2+2 a b c d+2 b^2 c^2\right ) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{27 c^{8/3} d^{7/3}}-\frac{\left (5 a^2 d^2+2 a b c d+2 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{9 \sqrt{3} c^{8/3} d^{7/3}}-\frac{x (b c-a d) (5 a d+4 b c)}{18 c^2 d^2 \left (c+d x^3\right )}-\frac{x \left (a+b x^3\right ) (b c-a d)}{6 c d \left (c+d x^3\right )^2} \]

[Out]

-((b*c - a*d)*x*(a + b*x^3))/(6*c*d*(c + d*x^3)^2) - ((b*c - a*d)*(4*b*c + 5*a*d
)*x)/(18*c^2*d^2*(c + d*x^3)) - ((2*b^2*c^2 + 2*a*b*c*d + 5*a^2*d^2)*ArcTan[(c^(
1/3) - 2*d^(1/3)*x)/(Sqrt[3]*c^(1/3))])/(9*Sqrt[3]*c^(8/3)*d^(7/3)) + ((2*b^2*c^
2 + 2*a*b*c*d + 5*a^2*d^2)*Log[c^(1/3) + d^(1/3)*x])/(27*c^(8/3)*d^(7/3)) - ((2*
b^2*c^2 + 2*a*b*c*d + 5*a^2*d^2)*Log[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2])
/(54*c^(8/3)*d^(7/3))

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Rubi [A]  time = 0.477267, antiderivative size = 258, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421 \[ -\frac{\left (5 a^2 d^2+2 a b c d+2 b^2 c^2\right ) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{54 c^{8/3} d^{7/3}}+\frac{\left (5 a^2 d^2+2 a b c d+2 b^2 c^2\right ) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{27 c^{8/3} d^{7/3}}-\frac{\left (5 a^2 d^2+2 a b c d+2 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{9 \sqrt{3} c^{8/3} d^{7/3}}-\frac{x (b c-a d) (5 a d+4 b c)}{18 c^2 d^2 \left (c+d x^3\right )}-\frac{x \left (a+b x^3\right ) (b c-a d)}{6 c d \left (c+d x^3\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

-((b*c - a*d)*x*(a + b*x^3))/(6*c*d*(c + d*x^3)^2) - ((b*c - a*d)*(4*b*c + 5*a*d
)*x)/(18*c^2*d^2*(c + d*x^3)) - ((2*b^2*c^2 + 2*a*b*c*d + 5*a^2*d^2)*ArcTan[(c^(
1/3) - 2*d^(1/3)*x)/(Sqrt[3]*c^(1/3))])/(9*Sqrt[3]*c^(8/3)*d^(7/3)) + ((2*b^2*c^
2 + 2*a*b*c*d + 5*a^2*d^2)*Log[c^(1/3) + d^(1/3)*x])/(27*c^(8/3)*d^(7/3)) - ((2*
b^2*c^2 + 2*a*b*c*d + 5*a^2*d^2)*Log[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2])
/(54*c^(8/3)*d^(7/3))

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Rubi in Sympy [A]  time = 53.106, size = 243, normalized size = 0.94 \[ \frac{x \left (a + b x^{3}\right ) \left (a d - b c\right )}{6 c d \left (c + d x^{3}\right )^{2}} + \frac{x \left (a d - b c\right ) \left (5 a d + 4 b c\right )}{18 c^{2} d^{2} \left (c + d x^{3}\right )} + \frac{\left (a d \left (5 a d + b c\right ) + b c \left (a d + 2 b c\right )\right ) \log{\left (\sqrt [3]{c} + \sqrt [3]{d} x \right )}}{27 c^{\frac{8}{3}} d^{\frac{7}{3}}} - \frac{\left (a d \left (5 a d + b c\right ) + b c \left (a d + 2 b c\right )\right ) \log{\left (c^{\frac{2}{3}} - \sqrt [3]{c} \sqrt [3]{d} x + d^{\frac{2}{3}} x^{2} \right )}}{54 c^{\frac{8}{3}} d^{\frac{7}{3}}} - \frac{\sqrt{3} \left (a d \left (5 a d + b c\right ) + b c \left (a d + 2 b c\right )\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{c}}{3} - \frac{2 \sqrt [3]{d} x}{3}\right )}{\sqrt [3]{c}} \right )}}{27 c^{\frac{8}{3}} d^{\frac{7}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(a + b*x**3)*(a*d - b*c)/(6*c*d*(c + d*x**3)**2) + x*(a*d - b*c)*(5*a*d + 4*b*
c)/(18*c**2*d**2*(c + d*x**3)) + (a*d*(5*a*d + b*c) + b*c*(a*d + 2*b*c))*log(c**
(1/3) + d**(1/3)*x)/(27*c**(8/3)*d**(7/3)) - (a*d*(5*a*d + b*c) + b*c*(a*d + 2*b
*c))*log(c**(2/3) - c**(1/3)*d**(1/3)*x + d**(2/3)*x**2)/(54*c**(8/3)*d**(7/3))
- sqrt(3)*(a*d*(5*a*d + b*c) + b*c*(a*d + 2*b*c))*atan(sqrt(3)*(c**(1/3)/3 - 2*d
**(1/3)*x/3)/c**(1/3))/(27*c**(8/3)*d**(7/3))

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Mathematica [A]  time = 0.465576, size = 234, normalized size = 0.91 \[ \frac{2 \left (5 a^2 d^2+2 a b c d+2 b^2 c^2\right ) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )-2 \sqrt{3} \left (5 a^2 d^2+2 a b c d+2 b^2 c^2\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt{3}}\right )-\frac{3 c^{2/3} \sqrt [3]{d} x \left (-a^2 d^2 \left (8 c+5 d x^3\right )+2 a b c d \left (2 c-d x^3\right )+b^2 c^2 \left (4 c+7 d x^3\right )\right )}{\left (c+d x^3\right )^2}-\left (5 a^2 d^2+2 a b c d+2 b^2 c^2\right ) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{54 c^{8/3} d^{7/3}} \]

Antiderivative was successfully verified.

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

[Out]

((-3*c^(2/3)*d^(1/3)*x*(2*a*b*c*d*(2*c - d*x^3) - a^2*d^2*(8*c + 5*d*x^3) + b^2*
c^2*(4*c + 7*d*x^3)))/(c + d*x^3)^2 - 2*Sqrt[3]*(2*b^2*c^2 + 2*a*b*c*d + 5*a^2*d
^2)*ArcTan[(1 - (2*d^(1/3)*x)/c^(1/3))/Sqrt[3]] + 2*(2*b^2*c^2 + 2*a*b*c*d + 5*a
^2*d^2)*Log[c^(1/3) + d^(1/3)*x] - (2*b^2*c^2 + 2*a*b*c*d + 5*a^2*d^2)*Log[c^(2/
3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2])/(54*c^(8/3)*d^(7/3))

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Maple [A]  time = 0.015, size = 388, normalized size = 1.5 \[{\frac{1}{ \left ( d{x}^{3}+c \right ) ^{2}} \left ({\frac{ \left ( 5\,{a}^{2}{d}^{2}+2\,cabd-7\,{b}^{2}{c}^{2} \right ){x}^{4}}{18\,{c}^{2}d}}+{\frac{ \left ( 4\,{a}^{2}{d}^{2}-2\,cabd-2\,{b}^{2}{c}^{2} \right ) x}{9\,{d}^{2}c}} \right ) }+{\frac{5\,{a}^{2}}{27\,{c}^{2}d}\ln \left ( x+\sqrt [3]{{\frac{c}{d}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}+{\frac{2\,ab}{27\,{d}^{2}c}\ln \left ( x+\sqrt [3]{{\frac{c}{d}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}+{\frac{2\,{b}^{2}}{27\,{d}^{3}}\ln \left ( x+\sqrt [3]{{\frac{c}{d}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}-{\frac{5\,{a}^{2}}{54\,{c}^{2}d}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{c}{d}}}+ \left ({\frac{c}{d}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}-{\frac{ab}{27\,{d}^{2}c}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{c}{d}}}+ \left ({\frac{c}{d}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}-{\frac{{b}^{2}}{27\,{d}^{3}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{c}{d}}}+ \left ({\frac{c}{d}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}+{\frac{5\,\sqrt{3}{a}^{2}}{27\,{c}^{2}d}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}-1 \right ) } \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}+{\frac{2\,\sqrt{3}ab}{27\,{d}^{2}c}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}-1 \right ) } \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}+{\frac{2\,\sqrt{3}{b}^{2}}{27\,{d}^{3}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}-1 \right ) } \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^2/(d*x^3+c)^3,x)

[Out]

(1/18*(5*a^2*d^2+2*a*b*c*d-7*b^2*c^2)/c^2/d*x^4+2/9*(2*a^2*d^2-a*b*c*d-b^2*c^2)/
d^2/c*x)/(d*x^3+c)^2+5/27/c^2/d/(c/d)^(2/3)*ln(x+(c/d)^(1/3))*a^2+2/27/c/d^2/(c/
d)^(2/3)*ln(x+(c/d)^(1/3))*a*b+2/27/d^3/(c/d)^(2/3)*ln(x+(c/d)^(1/3))*b^2-5/54/c
^2/d/(c/d)^(2/3)*ln(x^2-x*(c/d)^(1/3)+(c/d)^(2/3))*a^2-1/27/c/d^2/(c/d)^(2/3)*ln
(x^2-x*(c/d)^(1/3)+(c/d)^(2/3))*a*b-1/27/d^3/(c/d)^(2/3)*ln(x^2-x*(c/d)^(1/3)+(c
/d)^(2/3))*b^2+5/27/c^2/d/(c/d)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(c/d)^(1/3)*
x-1))*a^2+2/27/c/d^2/(c/d)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(c/d)^(1/3)*x-1))
*a*b+2/27/d^3/(c/d)^(2/3)*3^(1/2)*arctan(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)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.218697, size = 637, normalized size = 2.47 \[ -\frac{\sqrt{3}{\left (\sqrt{3}{\left ({\left (2 \, b^{2} c^{2} d^{2} + 2 \, a b c d^{3} + 5 \, a^{2} d^{4}\right )} x^{6} + 2 \, b^{2} c^{4} + 2 \, a b c^{3} d + 5 \, a^{2} c^{2} d^{2} + 2 \,{\left (2 \, b^{2} c^{3} d + 2 \, a b c^{2} d^{2} + 5 \, a^{2} c d^{3}\right )} x^{3}\right )} \log \left (\left (c^{2} d\right )^{\frac{2}{3}} x^{2} - \left (c^{2} d\right )^{\frac{1}{3}} c x + c^{2}\right ) - 2 \, \sqrt{3}{\left ({\left (2 \, b^{2} c^{2} d^{2} + 2 \, a b c d^{3} + 5 \, a^{2} d^{4}\right )} x^{6} + 2 \, b^{2} c^{4} + 2 \, a b c^{3} d + 5 \, a^{2} c^{2} d^{2} + 2 \,{\left (2 \, b^{2} c^{3} d + 2 \, a b c^{2} d^{2} + 5 \, a^{2} c d^{3}\right )} x^{3}\right )} \log \left (\left (c^{2} d\right )^{\frac{1}{3}} x + c\right ) - 6 \,{\left ({\left (2 \, b^{2} c^{2} d^{2} + 2 \, a b c d^{3} + 5 \, a^{2} d^{4}\right )} x^{6} + 2 \, b^{2} c^{4} + 2 \, a b c^{3} d + 5 \, a^{2} c^{2} d^{2} + 2 \,{\left (2 \, b^{2} c^{3} d + 2 \, a b c^{2} d^{2} + 5 \, a^{2} c d^{3}\right )} x^{3}\right )} \arctan \left (\frac{2 \, \sqrt{3} \left (c^{2} d\right )^{\frac{1}{3}} x - \sqrt{3} c}{3 \, c}\right ) + 3 \, \sqrt{3}{\left ({\left (7 \, b^{2} c^{2} d - 2 \, a b c d^{2} - 5 \, a^{2} d^{3}\right )} x^{4} + 4 \,{\left (b^{2} c^{3} + a b c^{2} d - 2 \, a^{2} c d^{2}\right )} x\right )} \left (c^{2} d\right )^{\frac{1}{3}}\right )}}{162 \,{\left (c^{2} d^{4} x^{6} + 2 \, c^{3} d^{3} x^{3} + c^{4} d^{2}\right )} \left (c^{2} d\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/162*sqrt(3)*(sqrt(3)*((2*b^2*c^2*d^2 + 2*a*b*c*d^3 + 5*a^2*d^4)*x^6 + 2*b^2*c
^4 + 2*a*b*c^3*d + 5*a^2*c^2*d^2 + 2*(2*b^2*c^3*d + 2*a*b*c^2*d^2 + 5*a^2*c*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^2*d
^2 + 2*a*b*c*d^3 + 5*a^2*d^4)*x^6 + 2*b^2*c^4 + 2*a*b*c^3*d + 5*a^2*c^2*d^2 + 2*
(2*b^2*c^3*d + 2*a*b*c^2*d^2 + 5*a^2*c*d^3)*x^3)*log((c^2*d)^(1/3)*x + c) - 6*((
2*b^2*c^2*d^2 + 2*a*b*c*d^3 + 5*a^2*d^4)*x^6 + 2*b^2*c^4 + 2*a*b*c^3*d + 5*a^2*c
^2*d^2 + 2*(2*b^2*c^3*d + 2*a*b*c^2*d^2 + 5*a^2*c*d^3)*x^3)*arctan(1/3*(2*sqrt(3
)*(c^2*d)^(1/3)*x - sqrt(3)*c)/c) + 3*sqrt(3)*((7*b^2*c^2*d - 2*a*b*c*d^2 - 5*a^
2*d^3)*x^4 + 4*(b^2*c^3 + a*b*c^2*d - 2*a^2*c*d^2)*x)*(c^2*d)^(1/3))/((c^2*d^4*x
^6 + 2*c^3*d^3*x^3 + c^4*d^2)*(c^2*d)^(1/3))

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Sympy [A]  time = 7.43143, size = 233, normalized size = 0.9 \[ \frac{x^{4} \left (5 a^{2} d^{3} + 2 a b c d^{2} - 7 b^{2} c^{2} d\right ) + x \left (8 a^{2} c d^{2} - 4 a b c^{2} d - 4 b^{2} c^{3}\right )}{18 c^{4} d^{2} + 36 c^{3} d^{3} x^{3} + 18 c^{2} d^{4} x^{6}} + \operatorname{RootSum}{\left (19683 t^{3} c^{8} d^{7} - 125 a^{6} d^{6} - 150 a^{5} b c d^{5} - 210 a^{4} b^{2} c^{2} d^{4} - 128 a^{3} b^{3} c^{3} d^{3} - 84 a^{2} b^{4} c^{4} d^{2} - 24 a b^{5} c^{5} d - 8 b^{6} c^{6}, \left ( t \mapsto t \log{\left (\frac{27 t c^{3} d^{2}}{5 a^{2} d^{2} + 2 a b c d + 2 b^{2} c^{2}} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(x**4*(5*a**2*d**3 + 2*a*b*c*d**2 - 7*b**2*c**2*d) + x*(8*a**2*c*d**2 - 4*a*b*c*
*2*d - 4*b**2*c**3))/(18*c**4*d**2 + 36*c**3*d**3*x**3 + 18*c**2*d**4*x**6) + Ro
otSum(19683*_t**3*c**8*d**7 - 125*a**6*d**6 - 150*a**5*b*c*d**5 - 210*a**4*b**2*
c**2*d**4 - 128*a**3*b**3*c**3*d**3 - 84*a**2*b**4*c**4*d**2 - 24*a*b**5*c**5*d
- 8*b**6*c**6, Lambda(_t, _t*log(27*_t*c**3*d**2/(5*a**2*d**2 + 2*a*b*c*d + 2*b*
*2*c**2) + x)))

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GIAC/XCAS [A]  time = 0.222457, size = 400, normalized size = 1.55 \[ -\frac{{\left (2 \, b^{2} c^{2} + 2 \, a b c d + 5 \, a^{2} d^{2}\right )} \left (-\frac{c}{d}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{c}{d}\right )^{\frac{1}{3}} \right |}\right )}{27 \, c^{3} d^{2}} + \frac{\sqrt{3}{\left (2 \, \left (-c d^{2}\right )^{\frac{1}{3}} b^{2} c^{2} + 2 \, \left (-c d^{2}\right )^{\frac{1}{3}} a b c d + 5 \, \left (-c d^{2}\right )^{\frac{1}{3}} a^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{c}{d}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{c}{d}\right )^{\frac{1}{3}}}\right )}{27 \, c^{3} d^{3}} + \frac{{\left (2 \, \left (-c d^{2}\right )^{\frac{1}{3}} b^{2} c^{2} + 2 \, \left (-c d^{2}\right )^{\frac{1}{3}} a b c d + 5 \, \left (-c d^{2}\right )^{\frac{1}{3}} a^{2} d^{2}\right )}{\rm ln}\left (x^{2} + x \left (-\frac{c}{d}\right )^{\frac{1}{3}} + \left (-\frac{c}{d}\right )^{\frac{2}{3}}\right )}{54 \, c^{3} d^{3}} - \frac{7 \, b^{2} c^{2} d x^{4} - 2 \, a b c d^{2} x^{4} - 5 \, a^{2} d^{3} x^{4} + 4 \, b^{2} c^{3} x + 4 \, a b c^{2} d x - 8 \, a^{2} c d^{2} x}{18 \,{\left (d x^{3} + c\right )}^{2} c^{2} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/27*(2*b^2*c^2 + 2*a*b*c*d + 5*a^2*d^2)*(-c/d)^(1/3)*ln(abs(x - (-c/d)^(1/3)))
/(c^3*d^2) + 1/27*sqrt(3)*(2*(-c*d^2)^(1/3)*b^2*c^2 + 2*(-c*d^2)^(1/3)*a*b*c*d +
5*(-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^3*d^3) + 1/54*(2*(-c*d^2)^(1/3)*b^2*c^2 + 2*(-c*d^2)^(1/3)*a*b*c*d + 5*(-c*d
^2)^(1/3)*a^2*d^2)*ln(x^2 + x*(-c/d)^(1/3) + (-c/d)^(2/3))/(c^3*d^3) - 1/18*(7*b
^2*c^2*d*x^4 - 2*a*b*c*d^2*x^4 - 5*a^2*d^3*x^4 + 4*b^2*c^3*x + 4*a*b*c^2*d*x - 8
*a^2*c*d^2*x)/((d*x^3 + c)^2*c^2*d^2)