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

Optimal. Leaf size=264 \[ -\frac{\sqrt [3]{a} d \left (\sqrt [3]{a} d+\sqrt [3]{b} c\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{2/3} \left (b c^3-a d^3\right )}-\frac{\sqrt [3]{a} d \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^{2/3} \left (a^{2/3} d^2+\sqrt [3]{a} \sqrt [3]{b} c d+b^{2/3} c^2\right )}+\frac{\sqrt [3]{a} d \left (\sqrt [3]{a} d+\sqrt [3]{b} c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{2/3} \left (b c^3-a d^3\right )}+\frac{c^2 \log \left (a+b x^3\right )}{3 \left (b c^3-a d^3\right )}-\frac{c^2 \log (c+d x)}{b c^3-a d^3} \]

[Out]

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

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Rubi [A]  time = 0.960649, antiderivative size = 264, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ -\frac{\sqrt [3]{a} d \left (\sqrt [3]{a} d+\sqrt [3]{b} c\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{2/3} \left (b c^3-a d^3\right )}-\frac{\sqrt [3]{a} d \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^{2/3} \left (a^{2/3} d^2+\sqrt [3]{a} \sqrt [3]{b} c d+b^{2/3} c^2\right )}+\frac{\sqrt [3]{a} d \left (\sqrt [3]{a} d+\sqrt [3]{b} c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{2/3} \left (b c^3-a d^3\right )}+\frac{c^2 \log \left (a+b x^3\right )}{3 \left (b c^3-a d^3\right )}-\frac{c^2 \log (c+d x)}{b c^3-a d^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 113.591, size = 231, normalized size = 0.88 \[ - \frac{\sqrt{3} \sqrt [3]{a} d \left (\sqrt [3]{a} d - \sqrt [3]{b} c\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 b^{\frac{2}{3}} \left (a d^{3} - b c^{3}\right )} - \frac{\sqrt [3]{a} d \left (\sqrt [3]{a} d + \sqrt [3]{b} c\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 b^{\frac{2}{3}} \left (a d^{3} - b c^{3}\right )} + \frac{\sqrt [3]{a} d \left (\sqrt [3]{a} d + \sqrt [3]{b} c\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 b^{\frac{2}{3}} \left (a d^{3} - b c^{3}\right )} - \frac{c^{2} \log{\left (a + b x^{3} \right )}}{3 \left (a d^{3} - b c^{3}\right )} + \frac{c^{2} \log{\left (c + d x \right )}}{a d^{3} - b c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.169607, size = 228, normalized size = 0.86 \[ \frac{-\sqrt [3]{a} \sqrt [3]{b} c d \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-a^{2/3} d^2 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+2 b^{2/3} c^2 \log \left (a+b x^3\right )+2 \sqrt [3]{a} d \left (\sqrt [3]{a} d+\sqrt [3]{b} c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+2 \sqrt{3} \sqrt [3]{a} d \left (\sqrt [3]{a} d-\sqrt [3]{b} c\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )-6 b^{2/3} c^2 \log (c+d x)}{6 b^{2/3} \left (b c^3-a d^3\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.008, size = 336, normalized size = 1.3 \[ -{\frac{acd}{ \left ( 3\,a{d}^{3}-3\,b{c}^{3} \right ) b}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{acd}{ \left ( 6\,a{d}^{3}-6\,b{c}^{3} \right ) b}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{acd\sqrt{3}}{ \left ( 3\,a{d}^{3}-3\,b{c}^{3} \right ) b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{a{d}^{2}}{ \left ( 3\,a{d}^{3}-3\,b{c}^{3} \right ) b}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{a{d}^{2}}{ \left ( 6\,a{d}^{3}-6\,b{c}^{3} \right ) b}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{a{d}^{2}\sqrt{3}}{ \left ( 3\,a{d}^{3}-3\,b{c}^{3} \right ) b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{{c}^{2}\ln \left ( b{x}^{3}+a \right ) }{3\,a{d}^{3}-3\,b{c}^{3}}}+{\frac{{c}^{2}\ln \left ( dx+c \right ) }{a{d}^{3}-b{c}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(d*x+c)/(b*x^3+a),x)

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: NotImplementedError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.270792, size = 432, normalized size = 1.64 \[ -\frac{c^{2} d{\rm ln}\left ({\left | d x + c \right |}\right )}{b c^{3} d - a d^{4}} + \frac{c^{2}{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \,{\left (b c^{3} - a d^{3}\right )}} + \frac{\left (-a b^{2}\right )^{\frac{1}{3}} d \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{\sqrt{3} b^{2} c^{2} - \sqrt{3} \left (-a b^{2}\right )^{\frac{1}{3}} b c d + \sqrt{3} \left (-a b^{2}\right )^{\frac{2}{3}} d^{2}} + \frac{{\left (a b^{2} c^{3} d^{2} \left (-\frac{a}{b}\right )^{\frac{1}{3}} - a^{2} b d^{5} \left (-\frac{a}{b}\right )^{\frac{1}{3}} - a b^{2} c^{4} d + a^{2} b c d^{4}\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \,{\left (a b^{3} c^{6} - 2 \, a^{2} b^{2} c^{3} d^{3} + a^{3} b d^{6}\right )}} + \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b c d - \left (-a b^{2}\right )^{\frac{2}{3}} d^{2}\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \,{\left (b^{3} c^{3} - a b^{2} d^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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