3.111 \(\int \frac{x^4}{\left (a+b x^3\right ) \left (c+d x^3\right )} \, dx\)

Optimal. Leaf size=288 \[ -\frac{a^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{2/3} (b c-a d)}+\frac{a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{2/3} (b c-a d)}+\frac{a^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^{2/3} (b c-a d)}+\frac{c^{2/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 d^{2/3} (b c-a d)}-\frac{c^{2/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 d^{2/3} (b c-a d)}-\frac{c^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} d^{2/3} (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.39722, antiderivative size = 288, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318 \[ -\frac{a^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{2/3} (b c-a d)}+\frac{a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{2/3} (b c-a d)}+\frac{a^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^{2/3} (b c-a d)}+\frac{c^{2/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 d^{2/3} (b c-a d)}-\frac{c^{2/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 d^{2/3} (b c-a d)}-\frac{c^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} d^{2/3} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[x^4/((a + b*x^3)*(c + d*x^3)),x]

[Out]

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

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Rubi in Sympy [A]  time = 67.069, size = 260, normalized size = 0.9 \[ - \frac{a^{\frac{2}{3}} \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 b^{\frac{2}{3}} \left (a d - b c\right )} + \frac{a^{\frac{2}{3}} \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 b^{\frac{2}{3}} \left (a d - b c\right )} - \frac{\sqrt{3} a^{\frac{2}{3}} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 b^{\frac{2}{3}} \left (a d - b c\right )} + \frac{c^{\frac{2}{3}} \log{\left (\sqrt [3]{c} + \sqrt [3]{d} x \right )}}{3 d^{\frac{2}{3}} \left (a d - b c\right )} - \frac{c^{\frac{2}{3}} \log{\left (c^{\frac{2}{3}} - \sqrt [3]{c} \sqrt [3]{d} x + d^{\frac{2}{3}} x^{2} \right )}}{6 d^{\frac{2}{3}} \left (a d - b c\right )} + \frac{\sqrt{3} c^{\frac{2}{3}} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{c}}{3} - \frac{2 \sqrt [3]{d} x}{3}\right )}{\sqrt [3]{c}} \right )}}{3 d^{\frac{2}{3}} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4/(b*x**3+a)/(d*x**3+c),x)

[Out]

-a**(2/3)*log(a**(1/3) + b**(1/3)*x)/(3*b**(2/3)*(a*d - b*c)) + a**(2/3)*log(a**
(2/3) - a**(1/3)*b**(1/3)*x + b**(2/3)*x**2)/(6*b**(2/3)*(a*d - b*c)) - sqrt(3)*
a**(2/3)*atan(sqrt(3)*(a**(1/3)/3 - 2*b**(1/3)*x/3)/a**(1/3))/(3*b**(2/3)*(a*d -
b*c)) + c**(2/3)*log(c**(1/3) + d**(1/3)*x)/(3*d**(2/3)*(a*d - b*c)) - c**(2/3)
*log(c**(2/3) - c**(1/3)*d**(1/3)*x + d**(2/3)*x**2)/(6*d**(2/3)*(a*d - b*c)) +
sqrt(3)*c**(2/3)*atan(sqrt(3)*(c**(1/3)/3 - 2*d**(1/3)*x/3)/c**(1/3))/(3*d**(2/3
)*(a*d - b*c))

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Mathematica [A]  time = 0.194368, size = 224, normalized size = 0.78 \[ \frac{-\frac{a^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{b^{2/3}}+\frac{2 a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{b^{2/3}}+\frac{2 \sqrt{3} a^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{b^{2/3}}+\frac{c^{2/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{d^{2/3}}-\frac{2 c^{2/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{d^{2/3}}-\frac{2 \sqrt{3} c^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt{3}}\right )}{d^{2/3}}}{6 b c-6 a d} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/((a + b*x^3)*(c + d*x^3)),x]

[Out]

((2*Sqrt[3]*a^(2/3)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(2/3) - (2*Sq
rt[3]*c^(2/3)*ArcTan[(1 - (2*d^(1/3)*x)/c^(1/3))/Sqrt[3]])/d^(2/3) + (2*a^(2/3)*
Log[a^(1/3) + b^(1/3)*x])/b^(2/3) - (2*c^(2/3)*Log[c^(1/3) + d^(1/3)*x])/d^(2/3)
- (a^(2/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(2/3) + (c^(2/3)*L
og[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2])/d^(2/3))/(6*b*c - 6*a*d)

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Maple [A]  time = 0.01, size = 246, normalized size = 0.9 \[ -{\frac{a}{ \left ( 3\,ad-3\,bc \right ) b}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{a}{ \left ( 6\,ad-6\,bc \right ) b}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{a\sqrt{3}}{ \left ( 3\,ad-3\,bc \right ) b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{c}{ \left ( 3\,ad-3\,bc \right ) d}\ln \left ( x+\sqrt [3]{{\frac{c}{d}}} \right ){\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}-{\frac{c}{ \left ( 6\,ad-6\,bc \right ) d}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{c}{d}}}+ \left ({\frac{c}{d}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}-{\frac{c\sqrt{3}}{ \left ( 3\,ad-3\,bc \right ) d}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(b*x^3+a)/(d*x^3+c),x)

[Out]

-1/3*a/(a*d-b*c)/b/(a/b)^(1/3)*ln(x+(a/b)^(1/3))+1/6*a/(a*d-b*c)/b/(a/b)^(1/3)*l
n(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))+1/3*a/(a*d-b*c)*3^(1/2)/b/(a/b)^(1/3)*arctan(1/
3*3^(1/2)*(2/(a/b)^(1/3)*x-1))+1/3*c/(a*d-b*c)/d/(c/d)^(1/3)*ln(x+(c/d)^(1/3))-1
/6*c/(a*d-b*c)/d/(c/d)^(1/3)*ln(x^2-x*(c/d)^(1/3)+(c/d)^(2/3))-1/3*c/(a*d-b*c)*3
^(1/2)/d/(c/d)^(1/3)*arctan(1/3*3^(1/2)*(2/(c/d)^(1/3)*x-1))

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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(x^4/((b*x^3 + a)*(d*x^3 + c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.273086, size = 366, normalized size = 1.27 \[ \frac{\sqrt{3}{\left (\sqrt{3} \left (-\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \log \left (a x^{2} - b x \left (-\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}} - a \left (-\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}}\right ) + \sqrt{3} \left (\frac{c^{2}}{d^{2}}\right )^{\frac{1}{3}} \log \left (c x^{2} - d x \left (\frac{c^{2}}{d^{2}}\right )^{\frac{2}{3}} + c \left (\frac{c^{2}}{d^{2}}\right )^{\frac{1}{3}}\right ) - 2 \, \sqrt{3} \left (-\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \log \left (a x + b \left (-\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}}\right ) - 2 \, \sqrt{3} \left (\frac{c^{2}}{d^{2}}\right )^{\frac{1}{3}} \log \left (c x + d \left (\frac{c^{2}}{d^{2}}\right )^{\frac{2}{3}}\right ) - 6 \, \left (-\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \arctan \left (-\frac{2 \, \sqrt{3} a x - \sqrt{3} b \left (-\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}}}{3 \, b \left (-\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}}}\right ) - 6 \, \left (\frac{c^{2}}{d^{2}}\right )^{\frac{1}{3}} \arctan \left (-\frac{2 \, \sqrt{3} c x - \sqrt{3} d \left (\frac{c^{2}}{d^{2}}\right )^{\frac{2}{3}}}{3 \, d \left (\frac{c^{2}}{d^{2}}\right )^{\frac{2}{3}}}\right )\right )}}{18 \,{\left (b c - a d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/((b*x^3 + a)*(d*x^3 + c)),x, algorithm="fricas")

[Out]

1/18*sqrt(3)*(sqrt(3)*(-a^2/b^2)^(1/3)*log(a*x^2 - b*x*(-a^2/b^2)^(2/3) - a*(-a^
2/b^2)^(1/3)) + sqrt(3)*(c^2/d^2)^(1/3)*log(c*x^2 - d*x*(c^2/d^2)^(2/3) + c*(c^2
/d^2)^(1/3)) - 2*sqrt(3)*(-a^2/b^2)^(1/3)*log(a*x + b*(-a^2/b^2)^(2/3)) - 2*sqrt
(3)*(c^2/d^2)^(1/3)*log(c*x + d*(c^2/d^2)^(2/3)) - 6*(-a^2/b^2)^(1/3)*arctan(-1/
3*(2*sqrt(3)*a*x - sqrt(3)*b*(-a^2/b^2)^(2/3))/(b*(-a^2/b^2)^(2/3))) - 6*(c^2/d^
2)^(1/3)*arctan(-1/3*(2*sqrt(3)*c*x - sqrt(3)*d*(c^2/d^2)^(2/3))/(d*(c^2/d^2)^(2
/3))))/(b*c - a*d)

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Sympy [A]  time = 29.973, size = 573, normalized size = 1.99 \[ \operatorname{RootSum}{\left (t^{3} \left (27 a^{3} d^{5} - 81 a^{2} b c d^{4} + 81 a b^{2} c^{2} d^{3} - 27 b^{3} c^{3} d^{2}\right ) - c^{2}, \left ( t \mapsto t \log{\left (x + \frac{243 t^{5} a^{6} b^{2} d^{8} - 1458 t^{5} a^{5} b^{3} c d^{7} + 3645 t^{5} a^{4} b^{4} c^{2} d^{6} - 4860 t^{5} a^{3} b^{5} c^{3} d^{5} + 3645 t^{5} a^{2} b^{6} c^{4} d^{4} - 1458 t^{5} a b^{7} c^{5} d^{3} + 243 t^{5} b^{8} c^{6} d^{2} + 9 t^{2} a^{5} d^{5} - 18 t^{2} a^{4} b c d^{4} + 9 t^{2} a^{3} b^{2} c^{2} d^{3} + 9 t^{2} a^{2} b^{3} c^{3} d^{2} - 18 t^{2} a b^{4} c^{4} d + 9 t^{2} b^{5} c^{5}}{a^{3} c d^{2} + a b^{2} c^{3}} \right )} \right )\right )} + \operatorname{RootSum}{\left (t^{3} \left (27 a^{3} b^{2} d^{3} - 81 a^{2} b^{3} c d^{2} + 81 a b^{4} c^{2} d - 27 b^{5} c^{3}\right ) + a^{2}, \left ( t \mapsto t \log{\left (x + \frac{243 t^{5} a^{6} b^{2} d^{8} - 1458 t^{5} a^{5} b^{3} c d^{7} + 3645 t^{5} a^{4} b^{4} c^{2} d^{6} - 4860 t^{5} a^{3} b^{5} c^{3} d^{5} + 3645 t^{5} a^{2} b^{6} c^{4} d^{4} - 1458 t^{5} a b^{7} c^{5} d^{3} + 243 t^{5} b^{8} c^{6} d^{2} + 9 t^{2} a^{5} d^{5} - 18 t^{2} a^{4} b c d^{4} + 9 t^{2} a^{3} b^{2} c^{2} d^{3} + 9 t^{2} a^{2} b^{3} c^{3} d^{2} - 18 t^{2} a b^{4} c^{4} d + 9 t^{2} b^{5} c^{5}}{a^{3} c d^{2} + a b^{2} c^{3}} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(b*x**3+a)/(d*x**3+c),x)

[Out]

RootSum(_t**3*(27*a**3*d**5 - 81*a**2*b*c*d**4 + 81*a*b**2*c**2*d**3 - 27*b**3*c
**3*d**2) - c**2, Lambda(_t, _t*log(x + (243*_t**5*a**6*b**2*d**8 - 1458*_t**5*a
**5*b**3*c*d**7 + 3645*_t**5*a**4*b**4*c**2*d**6 - 4860*_t**5*a**3*b**5*c**3*d**
5 + 3645*_t**5*a**2*b**6*c**4*d**4 - 1458*_t**5*a*b**7*c**5*d**3 + 243*_t**5*b**
8*c**6*d**2 + 9*_t**2*a**5*d**5 - 18*_t**2*a**4*b*c*d**4 + 9*_t**2*a**3*b**2*c**
2*d**3 + 9*_t**2*a**2*b**3*c**3*d**2 - 18*_t**2*a*b**4*c**4*d + 9*_t**2*b**5*c**
5)/(a**3*c*d**2 + a*b**2*c**3)))) + RootSum(_t**3*(27*a**3*b**2*d**3 - 81*a**2*b
**3*c*d**2 + 81*a*b**4*c**2*d - 27*b**5*c**3) + a**2, Lambda(_t, _t*log(x + (243
*_t**5*a**6*b**2*d**8 - 1458*_t**5*a**5*b**3*c*d**7 + 3645*_t**5*a**4*b**4*c**2*
d**6 - 4860*_t**5*a**3*b**5*c**3*d**5 + 3645*_t**5*a**2*b**6*c**4*d**4 - 1458*_t
**5*a*b**7*c**5*d**3 + 243*_t**5*b**8*c**6*d**2 + 9*_t**2*a**5*d**5 - 18*_t**2*a
**4*b*c*d**4 + 9*_t**2*a**3*b**2*c**2*d**3 + 9*_t**2*a**2*b**3*c**3*d**2 - 18*_t
**2*a*b**4*c**4*d + 9*_t**2*b**5*c**5)/(a**3*c*d**2 + a*b**2*c**3))))

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GIAC/XCAS [A]  time = 0.230233, size = 386, normalized size = 1.34 \[ \frac{a \left (-\frac{a}{b}\right )^{\frac{2}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \,{\left (a b c - a^{2} d\right )}} - \frac{c \left (-\frac{c}{d}\right )^{\frac{2}{3}}{\rm ln}\left ({\left | x - \left (-\frac{c}{d}\right )^{\frac{1}{3}} \right |}\right )}{3 \,{\left (b c^{2} - a c d\right )}} + \frac{\left (-a b^{2}\right )^{\frac{2}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{\sqrt{3} b^{3} c - \sqrt{3} a b^{2} d} - \frac{\left (-c d^{2}\right )^{\frac{2}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{c}{d}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{c}{d}\right )^{\frac{1}{3}}}\right )}{\sqrt{3} b c d^{2} - \sqrt{3} a d^{3}} - \frac{\left (-a b^{2}\right )^{\frac{2}{3}}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \,{\left (b^{3} c - a b^{2} d\right )}} + \frac{\left (-c d^{2}\right )^{\frac{2}{3}}{\rm ln}\left (x^{2} + x \left (-\frac{c}{d}\right )^{\frac{1}{3}} + \left (-\frac{c}{d}\right )^{\frac{2}{3}}\right )}{6 \,{\left (b c d^{2} - a d^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/((b*x^3 + a)*(d*x^3 + c)),x, algorithm="giac")

[Out]

1/3*a*(-a/b)^(2/3)*ln(abs(x - (-a/b)^(1/3)))/(a*b*c - a^2*d) - 1/3*c*(-c/d)^(2/3
)*ln(abs(x - (-c/d)^(1/3)))/(b*c^2 - a*c*d) + (-a*b^2)^(2/3)*arctan(1/3*sqrt(3)*
(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(sqrt(3)*b^3*c - sqrt(3)*a*b^2*d) - (-c*d^2)^
(2/3)*arctan(1/3*sqrt(3)*(2*x + (-c/d)^(1/3))/(-c/d)^(1/3))/(sqrt(3)*b*c*d^2 - s
qrt(3)*a*d^3) - 1/6*(-a*b^2)^(2/3)*ln(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(b^3*
c - a*b^2*d) + 1/6*(-c*d^2)^(2/3)*ln(x^2 + x*(-c/d)^(1/3) + (-c/d)^(2/3))/(b*c*d
^2 - a*d^3)