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

Optimal. Leaf size=234 \[ -\frac{(b c-a d)^2 (7 a d+2 b c) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{5/3} b^{10/3}}+\frac{(b c-a d)^2 (7 a d+2 b c) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{10/3}}-\frac{(b c-a d)^2 (7 a d+2 b c) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{5/3} b^{10/3}}+\frac{d^2 x (3 b c-2 a d)}{b^3}+\frac{x (b c-a d)^3}{3 a b^3 \left (a+b x^3\right )}+\frac{d^3 x^4}{4 b^2} \]

[Out]

(d^2*(3*b*c - 2*a*d)*x)/b^3 + (d^3*x^4)/(4*b^2) + ((b*c - a*d)^3*x)/(3*a*b^3*(a
+ b*x^3)) - ((b*c - a*d)^2*(2*b*c + 7*a*d)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[
3]*a^(1/3))])/(3*Sqrt[3]*a^(5/3)*b^(10/3)) + ((b*c - a*d)^2*(2*b*c + 7*a*d)*Log[
a^(1/3) + b^(1/3)*x])/(9*a^(5/3)*b^(10/3)) - ((b*c - a*d)^2*(2*b*c + 7*a*d)*Log[
a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*a^(5/3)*b^(10/3))

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Rubi [A]  time = 0.482122, antiderivative size = 234, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421 \[ -\frac{(b c-a d)^2 (7 a d+2 b c) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{5/3} b^{10/3}}+\frac{(b c-a d)^2 (7 a d+2 b c) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{10/3}}-\frac{(b c-a d)^2 (7 a d+2 b c) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{5/3} b^{10/3}}+\frac{d^2 x (3 b c-2 a d)}{b^3}+\frac{x (b c-a d)^3}{3 a b^3 \left (a+b x^3\right )}+\frac{d^3 x^4}{4 b^2} \]

Antiderivative was successfully verified.

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

[Out]

(d^2*(3*b*c - 2*a*d)*x)/b^3 + (d^3*x^4)/(4*b^2) + ((b*c - a*d)^3*x)/(3*a*b^3*(a
+ b*x^3)) - ((b*c - a*d)^2*(2*b*c + 7*a*d)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[
3]*a^(1/3))])/(3*Sqrt[3]*a^(5/3)*b^(10/3)) + ((b*c - a*d)^2*(2*b*c + 7*a*d)*Log[
a^(1/3) + b^(1/3)*x])/(9*a^(5/3)*b^(10/3)) - ((b*c - a*d)^2*(2*b*c + 7*a*d)*Log[
a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*a^(5/3)*b^(10/3))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - d^{2} \left (2 a d - 3 b c\right ) \int \frac{1}{b^{3}}\, dx + \frac{d^{3} x^{4}}{4 b^{2}} - \frac{x \left (a d - b c\right )^{3}}{3 a b^{3} \left (a + b x^{3}\right )} + \frac{\left (a d - b c\right )^{2} \left (7 a d + 2 b c\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{9 a^{\frac{5}{3}} b^{\frac{10}{3}}} - \frac{\left (a d - b c\right )^{2} \left (7 a d + 2 b c\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{18 a^{\frac{5}{3}} b^{\frac{10}{3}}} - \frac{\sqrt{3} \left (a d - b c\right )^{2} \left (7 a d + 2 b c\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{9 a^{\frac{5}{3}} b^{\frac{10}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-d**2*(2*a*d - 3*b*c)*Integral(b**(-3), x) + d**3*x**4/(4*b**2) - x*(a*d - b*c)*
*3/(3*a*b**3*(a + b*x**3)) + (a*d - b*c)**2*(7*a*d + 2*b*c)*log(a**(1/3) + b**(1
/3)*x)/(9*a**(5/3)*b**(10/3)) - (a*d - b*c)**2*(7*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**(10/3)) - sqrt(3)*(a*d - b*c)
**2*(7*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**(10/3))

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Mathematica [A]  time = 0.261119, size = 227, normalized size = 0.97 \[ \frac{-\frac{2 (b c-a d)^2 (7 a d+2 b c) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{5/3}}+\frac{4 (b c-a d)^2 (7 a d+2 b c) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{5/3}}+\frac{4 \sqrt{3} (b c-a d)^2 (7 a d+2 b c) \tan ^{-1}\left (\frac{2 \sqrt [3]{b} x-\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{a^{5/3}}+36 \sqrt [3]{b} d^2 x (3 b c-2 a d)+\frac{12 \sqrt [3]{b} x (b c-a d)^3}{a \left (a+b x^3\right )}+9 b^{4/3} d^3 x^4}{36 b^{10/3}} \]

Antiderivative was successfully verified.

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

[Out]

(36*b^(1/3)*d^2*(3*b*c - 2*a*d)*x + 9*b^(4/3)*d^3*x^4 + (12*b^(1/3)*(b*c - a*d)^
3*x)/(a*(a + b*x^3)) + (4*Sqrt[3]*(b*c - a*d)^2*(2*b*c + 7*a*d)*ArcTan[(-a^(1/3)
+ 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/a^(5/3) + (4*(b*c - a*d)^2*(2*b*c + 7*a*d)*L
og[a^(1/3) + b^(1/3)*x])/a^(5/3) - (2*(b*c - a*d)^2*(2*b*c + 7*a*d)*Log[a^(2/3)
- a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(5/3))/(36*b^(10/3))

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Maple [B]  time = 0.013, size = 529, normalized size = 2.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x^3+c)^3/(b*x^3+a)^2,x)

[Out]

1/4*d^3*x^4/b^2-2*d^3/b^3*a*x+3*d^2/b^2*x*c-1/3/b^3*x*a^2/(b*x^3+a)*d^3+1/b^2*x*
a/(b*x^3+a)*c*d^2-1/b*x/(b*x^3+a)*c^2*d+1/3*x/a/(b*x^3+a)*c^3+7/9/b^4*a^2/(a/b)^
(2/3)*ln(x+(a/b)^(1/3))*d^3-4/3/b^3*a/(a/b)^(2/3)*ln(x+(a/b)^(1/3))*c*d^2+1/3/b^
2/(a/b)^(2/3)*ln(x+(a/b)^(1/3))*c^2*d+2/9/b/a/(a/b)^(2/3)*ln(x+(a/b)^(1/3))*c^3-
7/18/b^4*a^2/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*d^3+2/3/b^3*a/(a/b)^(
2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*c*d^2-1/6/b^2/(a/b)^(2/3)*ln(x^2-x*(a/b)^
(1/3)+(a/b)^(2/3))*c^2*d-1/9/b/a/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*c
^3+7/9/b^4*a^2/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*d^3-4
/3/b^3*a/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*c*d^2+1/3/b
^2/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*c^2*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^3

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.219973, size = 606, normalized size = 2.59 \[ -\frac{\sqrt{3}{\left (2 \, \sqrt{3}{\left (2 \, a b^{3} c^{3} + 3 \, a^{2} b^{2} c^{2} d - 12 \, a^{3} b c d^{2} + 7 \, a^{4} d^{3} +{\left (2 \, b^{4} c^{3} + 3 \, a b^{3} c^{2} d - 12 \, a^{2} b^{2} c d^{2} + 7 \, a^{3} b d^{3}\right )} x^{3}\right )} \log \left (\left (a^{2} b\right )^{\frac{2}{3}} x^{2} - \left (a^{2} b\right )^{\frac{1}{3}} a x + a^{2}\right ) - 4 \, \sqrt{3}{\left (2 \, a b^{3} c^{3} + 3 \, a^{2} b^{2} c^{2} d - 12 \, a^{3} b c d^{2} + 7 \, a^{4} d^{3} +{\left (2 \, b^{4} c^{3} + 3 \, a b^{3} c^{2} d - 12 \, a^{2} b^{2} c d^{2} + 7 \, a^{3} b d^{3}\right )} x^{3}\right )} \log \left (\left (a^{2} b\right )^{\frac{1}{3}} x + a\right ) - 12 \,{\left (2 \, a b^{3} c^{3} + 3 \, a^{2} b^{2} c^{2} d - 12 \, a^{3} b c d^{2} + 7 \, a^{4} d^{3} +{\left (2 \, b^{4} c^{3} + 3 \, a b^{3} c^{2} d - 12 \, a^{2} b^{2} c d^{2} + 7 \, a^{3} b d^{3}\right )} x^{3}\right )} \arctan \left (\frac{2 \, \sqrt{3} \left (a^{2} b\right )^{\frac{1}{3}} x - \sqrt{3} a}{3 \, a}\right ) - 3 \, \sqrt{3}{\left (3 \, a b^{2} d^{3} x^{7} + 3 \,{\left (12 \, a b^{2} c d^{2} - 7 \, a^{2} b d^{3}\right )} x^{4} + 4 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 12 \, a^{2} b c d^{2} - 7 \, a^{3} d^{3}\right )} x\right )} \left (a^{2} b\right )^{\frac{1}{3}}\right )}}{108 \,{\left (a b^{4} x^{3} + a^{2} b^{3}\right )} \left (a^{2} b\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/108*sqrt(3)*(2*sqrt(3)*(2*a*b^3*c^3 + 3*a^2*b^2*c^2*d - 12*a^3*b*c*d^2 + 7*a^
4*d^3 + (2*b^4*c^3 + 3*a*b^3*c^2*d - 12*a^2*b^2*c*d^2 + 7*a^3*b*d^3)*x^3)*log((a
^2*b)^(2/3)*x^2 - (a^2*b)^(1/3)*a*x + a^2) - 4*sqrt(3)*(2*a*b^3*c^3 + 3*a^2*b^2*
c^2*d - 12*a^3*b*c*d^2 + 7*a^4*d^3 + (2*b^4*c^3 + 3*a*b^3*c^2*d - 12*a^2*b^2*c*d
^2 + 7*a^3*b*d^3)*x^3)*log((a^2*b)^(1/3)*x + a) - 12*(2*a*b^3*c^3 + 3*a^2*b^2*c^
2*d - 12*a^3*b*c*d^2 + 7*a^4*d^3 + (2*b^4*c^3 + 3*a*b^3*c^2*d - 12*a^2*b^2*c*d^2
+ 7*a^3*b*d^3)*x^3)*arctan(1/3*(2*sqrt(3)*(a^2*b)^(1/3)*x - sqrt(3)*a)/a) - 3*s
qrt(3)*(3*a*b^2*d^3*x^7 + 3*(12*a*b^2*c*d^2 - 7*a^2*b*d^3)*x^4 + 4*(b^3*c^3 - 3*
a*b^2*c^2*d + 12*a^2*b*c*d^2 - 7*a^3*d^3)*x)*(a^2*b)^(1/3))/((a*b^4*x^3 + a^2*b^
3)*(a^2*b)^(1/3))

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Sympy [A]  time = 8.05575, size = 289, normalized size = 1.24 \[ - \frac{x \left (a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}\right )}{3 a^{2} b^{3} + 3 a b^{4} x^{3}} + \operatorname{RootSum}{\left (729 t^{3} a^{5} b^{10} - 343 a^{9} d^{9} + 1764 a^{8} b c d^{8} - 3465 a^{7} b^{2} c^{2} d^{7} + 2946 a^{6} b^{3} c^{3} d^{6} - 477 a^{5} b^{4} c^{4} d^{5} - 792 a^{4} b^{5} c^{5} d^{4} + 321 a^{3} b^{6} c^{6} d^{3} + 90 a^{2} b^{7} c^{7} d^{2} - 36 a b^{8} c^{8} d - 8 b^{9} c^{9}, \left ( t \mapsto t \log{\left (\frac{9 t a^{2} b^{3}}{7 a^{3} d^{3} - 12 a^{2} b c d^{2} + 3 a b^{2} c^{2} d + 2 b^{3} c^{3}} + x \right )} \right )\right )} + \frac{d^{3} x^{4}}{4 b^{2}} - \frac{x \left (2 a d^{3} - 3 b c d^{2}\right )}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x*(a**3*d**3 - 3*a**2*b*c*d**2 + 3*a*b**2*c**2*d - b**3*c**3)/(3*a**2*b**3 + 3*
a*b**4*x**3) + RootSum(729*_t**3*a**5*b**10 - 343*a**9*d**9 + 1764*a**8*b*c*d**8
- 3465*a**7*b**2*c**2*d**7 + 2946*a**6*b**3*c**3*d**6 - 477*a**5*b**4*c**4*d**5
- 792*a**4*b**5*c**5*d**4 + 321*a**3*b**6*c**6*d**3 + 90*a**2*b**7*c**7*d**2 -
36*a*b**8*c**8*d - 8*b**9*c**9, Lambda(_t, _t*log(9*_t*a**2*b**3/(7*a**3*d**3 -
12*a**2*b*c*d**2 + 3*a*b**2*c**2*d + 2*b**3*c**3) + x))) + d**3*x**4/(4*b**2) -
x*(2*a*d**3 - 3*b*c*d**2)/b**3

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GIAC/XCAS [A]  time = 0.220806, size = 495, normalized size = 2.12 \[ -\frac{{\left (2 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d - 12 \, a^{2} b c d^{2} + 7 \, a^{3} d^{3}\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a^{2} b^{3}} + \frac{\sqrt{3}{\left (2 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c^{3} + 3 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} c^{2} d - 12 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b c d^{2} + 7 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} d^{3}\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{9 \, a^{2} b^{4}} + \frac{b^{3} c^{3} x - 3 \, a b^{2} c^{2} d x + 3 \, a^{2} b c d^{2} x - a^{3} d^{3} x}{3 \,{\left (b x^{3} + a\right )} a b^{3}} + \frac{{\left (2 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c^{3} + 3 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} c^{2} d - 12 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b c d^{2} + 7 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} d^{3}\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{18 \, a^{2} b^{4}} + \frac{b^{6} d^{3} x^{4} + 12 \, b^{6} c d^{2} x - 8 \, a b^{5} d^{3} x}{4 \, b^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/9*(2*b^3*c^3 + 3*a*b^2*c^2*d - 12*a^2*b*c*d^2 + 7*a^3*d^3)*(-a/b)^(1/3)*ln(ab
s(x - (-a/b)^(1/3)))/(a^2*b^3) + 1/9*sqrt(3)*(2*(-a*b^2)^(1/3)*b^3*c^3 + 3*(-a*b
^2)^(1/3)*a*b^2*c^2*d - 12*(-a*b^2)^(1/3)*a^2*b*c*d^2 + 7*(-a*b^2)^(1/3)*a^3*d^3
)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a^2*b^4) + 1/3*(b^3*c^3
*x - 3*a*b^2*c^2*d*x + 3*a^2*b*c*d^2*x - a^3*d^3*x)/((b*x^3 + a)*a*b^3) + 1/18*(
2*(-a*b^2)^(1/3)*b^3*c^3 + 3*(-a*b^2)^(1/3)*a*b^2*c^2*d - 12*(-a*b^2)^(1/3)*a^2*
b*c*d^2 + 7*(-a*b^2)^(1/3)*a^3*d^3)*ln(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^2
*b^4) + 1/4*(b^6*d^3*x^4 + 12*b^6*c*d^2*x - 8*a*b^5*d^3*x)/b^8