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

Optimal. Leaf size=169 \[ -\frac{(a d+2 b c) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{5/3} b^{4/3}}+\frac{(a d+2 b c) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{4/3}}-\frac{(a d+2 b c) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{5/3} b^{4/3}}+\frac{x (b c-a d)}{3 a b \left (a+b x^3\right )} \]

[Out]

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

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Rubi [A]  time = 0.193477, antiderivative size = 169, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.412 \[ -\frac{(a d+2 b c) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{5/3} b^{4/3}}+\frac{(a d+2 b c) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{4/3}}-\frac{(a d+2 b c) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{5/3} b^{4/3}}+\frac{x (b c-a d)}{3 a b \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 34.112, size = 155, normalized size = 0.92 \[ - \frac{x \left (a d - b c\right )}{3 a b \left (a + b x^{3}\right )} + \frac{\left (a d + 2 b c\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{9 a^{\frac{5}{3}} b^{\frac{4}{3}}} - \frac{\left (a d + 2 b c\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{18 a^{\frac{5}{3}} b^{\frac{4}{3}}} - \frac{\sqrt{3} \left (a d + 2 b c\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{9 a^{\frac{5}{3}} b^{\frac{4}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x*(a*d - b*c)/(3*a*b*(a + b*x**3)) + (a*d + 2*b*c)*log(a**(1/3) + b**(1/3)*x)/(
9*a**(5/3)*b**(4/3)) - (a*d + 2*b*c)*log(a**(2/3) - a**(1/3)*b**(1/3)*x + b**(2/
3)*x**2)/(18*a**(5/3)*b**(4/3)) - sqrt(3)*(a*d + 2*b*c)*atan(sqrt(3)*(a**(1/3)/3
- 2*b**(1/3)*x/3)/a**(1/3))/(9*a**(5/3)*b**(4/3))

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Mathematica [A]  time = 0.173095, size = 145, normalized size = 0.86 \[ \frac{-(a d+2 b c) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-\frac{6 a^{2/3} \sqrt [3]{b} x (a d-b c)}{a+b x^3}+2 (a d+2 b c) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-2 \sqrt{3} (a d+2 b c) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{18 a^{5/3} b^{4/3}} \]

Antiderivative was successfully verified.

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

[Out]

((-6*a^(2/3)*b^(1/3)*(-(b*c) + a*d)*x)/(a + b*x^3) - 2*Sqrt[3]*(2*b*c + a*d)*Arc
Tan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 2*(2*b*c + a*d)*Log[a^(1/3) + b^(1/3)
*x] - (2*b*c + a*d)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*a^(5/3)*
b^(4/3))

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Maple [A]  time = 0.012, size = 221, normalized size = 1.3 \[ -{\frac{ \left ( ad-bc \right ) x}{3\,ab \left ( b{x}^{3}+a \right ) }}+{\frac{d}{9\,{b}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{2\,c}{9\,ab}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{d}{18\,{b}^{2}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{c}{9\,ab}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{\sqrt{3}d}{9\,{b}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{2\,\sqrt{3}c}{9\,ab}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x^3+c)/(b*x^3+a)^2,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.216441, size = 277, normalized size = 1.64 \[ \frac{\sqrt{3}{\left (6 \, \sqrt{3} \left (a^{2} b\right )^{\frac{1}{3}}{\left (b c - a d\right )} x - \sqrt{3}{\left ({\left (2 \, b^{2} c + a b d\right )} x^{3} + 2 \, a b c + a^{2} d\right )} \log \left (\left (a^{2} b\right )^{\frac{2}{3}} x^{2} - \left (a^{2} b\right )^{\frac{1}{3}} a x + a^{2}\right ) + 2 \, \sqrt{3}{\left ({\left (2 \, b^{2} c + a b d\right )} x^{3} + 2 \, a b c + a^{2} d\right )} \log \left (\left (a^{2} b\right )^{\frac{1}{3}} x + a\right ) + 6 \,{\left ({\left (2 \, b^{2} c + a b d\right )} x^{3} + 2 \, a b c + a^{2} d\right )} \arctan \left (\frac{2 \, \sqrt{3} \left (a^{2} b\right )^{\frac{1}{3}} x - \sqrt{3} a}{3 \, a}\right )\right )}}{54 \,{\left (a b^{2} x^{3} + a^{2} b\right )} \left (a^{2} b\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/54*sqrt(3)*(6*sqrt(3)*(a^2*b)^(1/3)*(b*c - a*d)*x - sqrt(3)*((2*b^2*c + a*b*d)
*x^3 + 2*a*b*c + a^2*d)*log((a^2*b)^(2/3)*x^2 - (a^2*b)^(1/3)*a*x + a^2) + 2*sqr
t(3)*((2*b^2*c + a*b*d)*x^3 + 2*a*b*c + a^2*d)*log((a^2*b)^(1/3)*x + a) + 6*((2*
b^2*c + a*b*d)*x^3 + 2*a*b*c + a^2*d)*arctan(1/3*(2*sqrt(3)*(a^2*b)^(1/3)*x - sq
rt(3)*a)/a))/((a*b^2*x^3 + a^2*b)*(a^2*b)^(1/3))

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Sympy [A]  time = 2.68576, size = 97, normalized size = 0.57 \[ - \frac{x \left (a d - b c\right )}{3 a^{2} b + 3 a b^{2} x^{3}} + \operatorname{RootSum}{\left (729 t^{3} a^{5} b^{4} - a^{3} d^{3} - 6 a^{2} b c d^{2} - 12 a b^{2} c^{2} d - 8 b^{3} c^{3}, \left ( t \mapsto t \log{\left (\frac{9 t a^{2} b}{a d + 2 b c} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x*(a*d - b*c)/(3*a**2*b + 3*a*b**2*x**3) + RootSum(729*_t**3*a**5*b**4 - a**3*d
**3 - 6*a**2*b*c*d**2 - 12*a*b**2*c**2*d - 8*b**3*c**3, Lambda(_t, _t*log(9*_t*a
**2*b/(a*d + 2*b*c) + x)))

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GIAC/XCAS [A]  time = 0.217503, size = 246, normalized size = 1.46 \[ -\frac{{\left (2 \, b c + a d\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a^{2} b} + \frac{\sqrt{3}{\left (2 \, \left (-a b^{2}\right )^{\frac{1}{3}} b c + \left (-a b^{2}\right )^{\frac{1}{3}} a d\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{9 \, a^{2} b^{2}} + \frac{b c x - a d x}{3 \,{\left (b x^{3} + a\right )} a b} + \frac{{\left (2 \, \left (-a b^{2}\right )^{\frac{1}{3}} b c + \left (-a b^{2}\right )^{\frac{1}{3}} a d\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{18 \, a^{2} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/9*(2*b*c + a*d)*(-a/b)^(1/3)*ln(abs(x - (-a/b)^(1/3)))/(a^2*b) + 1/9*sqrt(3)*
(2*(-a*b^2)^(1/3)*b*c + (-a*b^2)^(1/3)*a*d)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/
3))/(-a/b)^(1/3))/(a^2*b^2) + 1/3*(b*c*x - a*d*x)/((b*x^3 + a)*a*b) + 1/18*(2*(-
a*b^2)^(1/3)*b*c + (-a*b^2)^(1/3)*a*d)*ln(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(
a^2*b^2)