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

Optimal. Leaf size=173 \[ -\frac{(b c-a d)^2 \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{2/3} d^{7/3}}+\frac{(b c-a d)^2 \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} d^{7/3}}-\frac{(b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} c^{2/3} d^{7/3}}-\frac{b x (b c-2 a d)}{d^2}+\frac{b^2 x^4}{4 d} \]

[Out]

-((b*(b*c - 2*a*d)*x)/d^2) + (b^2*x^4)/(4*d) - ((b*c - a*d)^2*ArcTan[(c^(1/3) -
2*d^(1/3)*x)/(Sqrt[3]*c^(1/3))])/(Sqrt[3]*c^(2/3)*d^(7/3)) + ((b*c - a*d)^2*Log[
c^(1/3) + d^(1/3)*x])/(3*c^(2/3)*d^(7/3)) - ((b*c - a*d)^2*Log[c^(2/3) - c^(1/3)
*d^(1/3)*x + d^(2/3)*x^2])/(6*c^(2/3)*d^(7/3))

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Rubi [A]  time = 0.27559, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368 \[ -\frac{(b c-a d)^2 \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{2/3} d^{7/3}}+\frac{(b c-a d)^2 \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} d^{7/3}}-\frac{(b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} c^{2/3} d^{7/3}}-\frac{b x (b c-2 a d)}{d^2}+\frac{b^2 x^4}{4 d} \]

Antiderivative was successfully verified.

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

[Out]

-((b*(b*c - 2*a*d)*x)/d^2) + (b^2*x^4)/(4*d) - ((b*c - a*d)^2*ArcTan[(c^(1/3) -
2*d^(1/3)*x)/(Sqrt[3]*c^(1/3))])/(Sqrt[3]*c^(2/3)*d^(7/3)) + ((b*c - a*d)^2*Log[
c^(1/3) + d^(1/3)*x])/(3*c^(2/3)*d^(7/3)) - ((b*c - a*d)^2*Log[c^(2/3) - c^(1/3)
*d^(1/3)*x + d^(2/3)*x^2])/(6*c^(2/3)*d^(7/3))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{b^{2} x^{4}}{4 d} + \frac{\left (2 a d - b c\right ) \int b\, dx}{d^{2}} + \frac{\left (a d - b c\right )^{2} \log{\left (\sqrt [3]{c} + \sqrt [3]{d} x \right )}}{3 c^{\frac{2}{3}} d^{\frac{7}{3}}} - \frac{\left (a d - b c\right )^{2} \log{\left (c^{\frac{2}{3}} - \sqrt [3]{c} \sqrt [3]{d} x + d^{\frac{2}{3}} x^{2} \right )}}{6 c^{\frac{2}{3}} d^{\frac{7}{3}}} - \frac{\sqrt{3} \left (a d - b c\right )^{2} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{c}}{3} - \frac{2 \sqrt [3]{d} x}{3}\right )}{\sqrt [3]{c}} \right )}}{3 c^{\frac{2}{3}} d^{\frac{7}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b**2*x**4/(4*d) + (2*a*d - b*c)*Integral(b, x)/d**2 + (a*d - b*c)**2*log(c**(1/3
) + d**(1/3)*x)/(3*c**(2/3)*d**(7/3)) - (a*d - b*c)**2*log(c**(2/3) - c**(1/3)*d
**(1/3)*x + d**(2/3)*x**2)/(6*c**(2/3)*d**(7/3)) - sqrt(3)*(a*d - b*c)**2*atan(s
qrt(3)*(c**(1/3)/3 - 2*d**(1/3)*x/3)/c**(1/3))/(3*c**(2/3)*d**(7/3))

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Mathematica [A]  time = 0.170788, size = 167, normalized size = 0.97 \[ \frac{-2 (b c-a d)^2 \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )-12 b c^{2/3} \sqrt [3]{d} x (b c-2 a d)+4 (b c-a d)^2 \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )+4 \sqrt{3} (b c-a d)^2 \tan ^{-1}\left (\frac{2 \sqrt [3]{d} x-\sqrt [3]{c}}{\sqrt{3} \sqrt [3]{c}}\right )+3 b^2 c^{2/3} d^{4/3} x^4}{12 c^{2/3} d^{7/3}} \]

Antiderivative was successfully verified.

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

[Out]

(-12*b*c^(2/3)*d^(1/3)*(b*c - 2*a*d)*x + 3*b^2*c^(2/3)*d^(4/3)*x^4 + 4*Sqrt[3]*(
b*c - a*d)^2*ArcTan[(-c^(1/3) + 2*d^(1/3)*x)/(Sqrt[3]*c^(1/3))] + 4*(b*c - a*d)^
2*Log[c^(1/3) + d^(1/3)*x] - 2*(b*c - a*d)^2*Log[c^(2/3) - c^(1/3)*d^(1/3)*x + d
^(2/3)*x^2])/(12*c^(2/3)*d^(7/3))

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Maple [B]  time = 0.004, size = 334, normalized size = 1.9 \[{\frac{{b}^{2}{x}^{4}}{4\,d}}+2\,{\frac{abx}{d}}-{\frac{{b}^{2}xc}{{d}^{2}}}+{\frac{{a}^{2}}{3\,d}\ln \left ( x+\sqrt [3]{{\frac{c}{d}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}-{\frac{2\,abc}{3\,{d}^{2}}\ln \left ( x+\sqrt [3]{{\frac{c}{d}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}+{\frac{{b}^{2}{c}^{2}}{3\,{d}^{3}}\ln \left ( x+\sqrt [3]{{\frac{c}{d}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}-{\frac{{a}^{2}}{6\,d}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{c}{d}}}+ \left ({\frac{c}{d}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}+{\frac{abc}{3\,{d}^{2}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{c}{d}}}+ \left ({\frac{c}{d}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}-{\frac{{b}^{2}{c}^{2}}{6\,{d}^{3}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{c}{d}}}+ \left ({\frac{c}{d}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}+{\frac{\sqrt{3}{a}^{2}}{3\,d}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}-1 \right ) } \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}-{\frac{2\,\sqrt{3}cab}{3\,{d}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}-1 \right ) } \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}+{\frac{\sqrt{3}{b}^{2}{c}^{2}}{3\,{d}^{3}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}-1 \right ) } \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^2/(d*x^3+c),x)

[Out]

1/4*b^2*x^4/d+2*b/d*a*x-b^2/d^2*x*c+1/3/d/(c/d)^(2/3)*ln(x+(c/d)^(1/3))*a^2-2/3/
d^2/(c/d)^(2/3)*ln(x+(c/d)^(1/3))*c*a*b+1/3/d^3/(c/d)^(2/3)*ln(x+(c/d)^(1/3))*b^
2*c^2-1/6/d/(c/d)^(2/3)*ln(x^2-x*(c/d)^(1/3)+(c/d)^(2/3))*a^2+1/3/d^2/(c/d)^(2/3
)*ln(x^2-x*(c/d)^(1/3)+(c/d)^(2/3))*c*a*b-1/6/d^3/(c/d)^(2/3)*ln(x^2-x*(c/d)^(1/
3)+(c/d)^(2/3))*b^2*c^2+1/3/d/(c/d)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(c/d)^(1
/3)*x-1))*a^2-2/3/d^2/(c/d)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(c/d)^(1/3)*x-1)
)*c*a*b+1/3/d^3/(c/d)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(c/d)^(1/3)*x-1))*b^2*
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((b*x^3 + a)^2/(d*x^3 + c),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.212159, size = 258, normalized size = 1.49 \[ -\frac{\sqrt{3}{\left (2 \, \sqrt{3}{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (\left (c^{2} d\right )^{\frac{2}{3}} x^{2} - \left (c^{2} d\right )^{\frac{1}{3}} c x + c^{2}\right ) - 4 \, \sqrt{3}{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (\left (c^{2} d\right )^{\frac{1}{3}} x + c\right ) - 12 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac{2 \, \sqrt{3} \left (c^{2} d\right )^{\frac{1}{3}} x - \sqrt{3} c}{3 \, c}\right ) - 3 \, \sqrt{3}{\left (b^{2} d x^{4} - 4 \,{\left (b^{2} c - 2 \, a b d\right )} x\right )} \left (c^{2} d\right )^{\frac{1}{3}}\right )}}{36 \, \left (c^{2} d\right )^{\frac{1}{3}} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/36*sqrt(3)*(2*sqrt(3)*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*log((c^2*d)^(2/3)*x^2 -
(c^2*d)^(1/3)*c*x + c^2) - 4*sqrt(3)*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*log((c^2*d
)^(1/3)*x + c) - 12*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*arctan(1/3*(2*sqrt(3)*(c^2*d
)^(1/3)*x - sqrt(3)*c)/c) - 3*sqrt(3)*(b^2*d*x^4 - 4*(b^2*c - 2*a*b*d)*x)*(c^2*d
)^(1/3))/((c^2*d)^(1/3)*d^2)

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Sympy [A]  time = 2.86124, size = 156, normalized size = 0.9 \[ \frac{b^{2} x^{4}}{4 d} + \operatorname{RootSum}{\left (27 t^{3} c^{2} d^{7} - a^{6} d^{6} + 6 a^{5} b c d^{5} - 15 a^{4} b^{2} c^{2} d^{4} + 20 a^{3} b^{3} c^{3} d^{3} - 15 a^{2} b^{4} c^{4} d^{2} + 6 a b^{5} c^{5} d - b^{6} c^{6}, \left ( t \mapsto t \log{\left (\frac{3 t c d^{2}}{a^{2} d^{2} - 2 a b c d + b^{2} c^{2}} + x \right )} \right )\right )} + \frac{x \left (2 a b d - b^{2} c\right )}{d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b**2*x**4/(4*d) + RootSum(27*_t**3*c**2*d**7 - a**6*d**6 + 6*a**5*b*c*d**5 - 15*
a**4*b**2*c**2*d**4 + 20*a**3*b**3*c**3*d**3 - 15*a**2*b**4*c**4*d**2 + 6*a*b**5
*c**5*d - b**6*c**6, Lambda(_t, _t*log(3*_t*c*d**2/(a**2*d**2 - 2*a*b*c*d + b**2
*c**2) + x))) + x*(2*a*b*d - b**2*c)/d**2

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GIAC/XCAS [A]  time = 0.218936, size = 336, normalized size = 1.94 \[ \frac{\sqrt{3}{\left (\left (-c d^{2}\right )^{\frac{1}{3}} b^{2} c^{2} - 2 \, \left (-c d^{2}\right )^{\frac{1}{3}} a b c d + \left (-c d^{2}\right )^{\frac{1}{3}} a^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{c}{d}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{c}{d}\right )^{\frac{1}{3}}}\right )}{3 \, c d^{3}} + \frac{{\left (\left (-c d^{2}\right )^{\frac{1}{3}} b^{2} c^{2} - 2 \, \left (-c d^{2}\right )^{\frac{1}{3}} a b c d + \left (-c d^{2}\right )^{\frac{1}{3}} a^{2} d^{2}\right )}{\rm ln}\left (x^{2} + x \left (-\frac{c}{d}\right )^{\frac{1}{3}} + \left (-\frac{c}{d}\right )^{\frac{2}{3}}\right )}{6 \, c d^{3}} - \frac{{\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} \left (-\frac{c}{d}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{c}{d}\right )^{\frac{1}{3}} \right |}\right )}{3 \, c d^{4}} + \frac{b^{2} d^{3} x^{4} - 4 \, b^{2} c d^{2} x + 8 \, a b d^{3} x}{4 \, d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*sqrt(3)*((-c*d^2)^(1/3)*b^2*c^2 - 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*d^3) + 1/6*((-
c*d^2)^(1/3)*b^2*c^2 - 2*(-c*d^2)^(1/3)*a*b*c*d + (-c*d^2)^(1/3)*a^2*d^2)*ln(x^2
+ x*(-c/d)^(1/3) + (-c/d)^(2/3))/(c*d^3) - 1/3*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2
*d^4)*(-c/d)^(1/3)*ln(abs(x - (-c/d)^(1/3)))/(c*d^4) + 1/4*(b^2*d^3*x^4 - 4*b^2*
c*d^2*x + 8*a*b*d^3*x)/d^4