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

Optimal. Leaf size=208 \[ -\frac{(b c-a d)^3 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} b^{10/3}}+\frac{(b c-a d)^3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{10/3}}-\frac{(b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} b^{10/3}}+\frac{d x \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{b^3}+\frac{d^2 x^4 (3 b c-a d)}{4 b^2}+\frac{d^3 x^7}{7 b} \]

[Out]

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

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Rubi [A]  time = 0.31675, antiderivative size = 208, 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 \[ -\frac{(b c-a d)^3 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} b^{10/3}}+\frac{(b c-a d)^3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{10/3}}-\frac{(b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} b^{10/3}}+\frac{d x \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{b^3}+\frac{d^2 x^4 (3 b c-a d)}{4 b^2}+\frac{d^3 x^7}{7 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{d^{3} x^{7}}{7 b} - \frac{d^{2} x^{4} \left (a d - 3 b c\right )}{4 b^{2}} + \frac{\left (a^{2} d^{2} - 3 a b c d + 3 b^{2} c^{2}\right ) \int d\, dx}{b^{3}} - \frac{\left (a d - b c\right )^{3} \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 a^{\frac{2}{3}} b^{\frac{10}{3}}} + \frac{\left (a d - b c\right )^{3} \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 a^{\frac{2}{3}} b^{\frac{10}{3}}} + \frac{\sqrt{3} \left (a d - b c\right )^{3} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 a^{\frac{2}{3}} b^{\frac{10}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d**3*x**7/(7*b) - d**2*x**4*(a*d - 3*b*c)/(4*b**2) + (a**2*d**2 - 3*a*b*c*d + 3*
b**2*c**2)*Integral(d, x)/b**3 - (a*d - b*c)**3*log(a**(1/3) + b**(1/3)*x)/(3*a*
*(2/3)*b**(10/3)) + (a*d - b*c)**3*log(a**(2/3) - a**(1/3)*b**(1/3)*x + b**(2/3)
*x**2)/(6*a**(2/3)*b**(10/3)) + sqrt(3)*(a*d - b*c)**3*atan(sqrt(3)*(a**(1/3)/3
- 2*b**(1/3)*x/3)/a**(1/3))/(3*a**(2/3)*b**(10/3))

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Mathematica [A]  time = 0.155733, size = 203, normalized size = 0.98 \[ \frac{\frac{14 (a d-b c)^3 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{2/3}}+\frac{28 (b c-a d)^3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{2/3}}+\frac{28 \sqrt{3} (b c-a d)^3 \tan ^{-1}\left (\frac{2 \sqrt [3]{b} x-\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{a^{2/3}}+84 \sqrt [3]{b} d x \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )+21 b^{4/3} d^2 x^4 (3 b c-a d)+12 b^{7/3} d^3 x^7}{84 b^{10/3}} \]

Antiderivative was successfully verified.

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

[Out]

(84*b^(1/3)*d*(3*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*x + 21*b^(4/3)*d^2*(3*b*c - a*d)
*x^4 + 12*b^(7/3)*d^3*x^7 + (28*Sqrt[3]*(b*c - a*d)^3*ArcTan[(-a^(1/3) + 2*b^(1/
3)*x)/(Sqrt[3]*a^(1/3))])/a^(2/3) + (28*(b*c - a*d)^3*Log[a^(1/3) + b^(1/3)*x])/
a^(2/3) + (14*(-(b*c) + a*d)^3*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a
^(2/3))/(84*b^(10/3))

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Maple [B]  time = 0.004, size = 486, 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),x)

[Out]

1/7*d^3*x^7/b-1/4*d^3/b^2*x^4*a+3/4*d^2/b*x^4*c+d^3/b^3*a^2*x-3*d^2/b^2*a*c*x+3*
d/b*c^2*x-1/3/b^4/(a/b)^(2/3)*ln(x+(a/b)^(1/3))*a^3*d^3+1/b^3/(a/b)^(2/3)*ln(x+(
a/b)^(1/3))*a^2*c*d^2-1/b^2/(a/b)^(2/3)*ln(x+(a/b)^(1/3))*a*c^2*d+1/3/b/(a/b)^(2
/3)*ln(x+(a/b)^(1/3))*c^3+1/6/b^4/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*
a^3*d^3-1/2/b^3/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*a^2*c*d^2+1/2/b^2/
(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*a*c^2*d-1/6/b/(a/b)^(2/3)*ln(x^2-x
*(a/b)^(1/3)+(a/b)^(2/3))*c^3-1/3/b^4/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/
(a/b)^(1/3)*x-1))*a^3*d^3+1/b^3/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^
(1/3)*x-1))*a^2*c*d^2-1/b^2/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3
)*x-1))*a*c^2*d+1/3/b/(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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.215725, size = 379, normalized size = 1.82 \[ \frac{\sqrt{3}{\left (14 \, \sqrt{3}{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (\left (-a^{2} b\right )^{\frac{2}{3}} x^{2} + \left (-a^{2} b\right )^{\frac{1}{3}} a x + a^{2}\right ) - 28 \, \sqrt{3}{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (\left (-a^{2} b\right )^{\frac{1}{3}} x - a\right ) + 84 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \arctan \left (\frac{2 \, \sqrt{3} \left (-a^{2} b\right )^{\frac{1}{3}} x + \sqrt{3} a}{3 \, a}\right ) + 3 \, \sqrt{3}{\left (4 \, b^{2} d^{3} x^{7} + 7 \,{\left (3 \, b^{2} c d^{2} - a b d^{3}\right )} x^{4} + 28 \,{\left (3 \, b^{2} c^{2} d - 3 \, a b c d^{2} + a^{2} d^{3}\right )} x\right )} \left (-a^{2} b\right )^{\frac{1}{3}}\right )}}{252 \, \left (-a^{2} b\right )^{\frac{1}{3}} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/252*sqrt(3)*(14*sqrt(3)*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*lo
g((-a^2*b)^(2/3)*x^2 + (-a^2*b)^(1/3)*a*x + a^2) - 28*sqrt(3)*(b^3*c^3 - 3*a*b^2
*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*log((-a^2*b)^(1/3)*x - a) + 84*(b^3*c^3 - 3*a*
b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*arctan(1/3*(2*sqrt(3)*(-a^2*b)^(1/3)*x + sq
rt(3)*a)/a) + 3*sqrt(3)*(4*b^2*d^3*x^7 + 7*(3*b^2*c*d^2 - a*b*d^3)*x^4 + 28*(3*b
^2*c^2*d - 3*a*b*c*d^2 + a^2*d^3)*x)*(-a^2*b)^(1/3))/((-a^2*b)^(1/3)*b^3)

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Sympy [A]  time = 4.16369, size = 255, normalized size = 1.23 \[ \operatorname{RootSum}{\left (27 t^{3} a^{2} b^{10} + a^{9} d^{9} - 9 a^{8} b c d^{8} + 36 a^{7} b^{2} c^{2} d^{7} - 84 a^{6} b^{3} c^{3} d^{6} + 126 a^{5} b^{4} c^{4} d^{5} - 126 a^{4} b^{5} c^{5} d^{4} + 84 a^{3} b^{6} c^{6} d^{3} - 36 a^{2} b^{7} c^{7} d^{2} + 9 a b^{8} c^{8} d - b^{9} c^{9}, \left ( t \mapsto t \log{\left (- \frac{3 t a b^{3}}{a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}} + x \right )} \right )\right )} + \frac{d^{3} x^{7}}{7 b} - \frac{x^{4} \left (a d^{3} - 3 b c d^{2}\right )}{4 b^{2}} + \frac{x \left (a^{2} d^{3} - 3 a b c d^{2} + 3 b^{2} c^{2} d\right )}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

RootSum(27*_t**3*a**2*b**10 + a**9*d**9 - 9*a**8*b*c*d**8 + 36*a**7*b**2*c**2*d*
*7 - 84*a**6*b**3*c**3*d**6 + 126*a**5*b**4*c**4*d**5 - 126*a**4*b**5*c**5*d**4
+ 84*a**3*b**6*c**6*d**3 - 36*a**2*b**7*c**7*d**2 + 9*a*b**8*c**8*d - b**9*c**9,
Lambda(_t, _t*log(-3*_t*a*b**3/(a**3*d**3 - 3*a**2*b*c*d**2 + 3*a*b**2*c**2*d -
b**3*c**3) + x))) + d**3*x**7/(7*b) - x**4*(a*d**3 - 3*b*c*d**2)/(4*b**2) + x*(
a**2*d**3 - 3*a*b*c*d**2 + 3*b**2*c**2*d)/b**3

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GIAC/XCAS [A]  time = 0.21963, size = 473, normalized size = 2.27 \[ \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c^{3} - 3 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} c^{2} d + 3 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b c d^{2} - \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} d^{3}\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, a b^{4}} + \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c^{3} - 3 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} c^{2} d + 3 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b c d^{2} - \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} d^{3}\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, a b^{4}} - \frac{{\left (b^{7} c^{3} - 3 \, a b^{6} c^{2} d + 3 \, a^{2} b^{5} c d^{2} - a^{3} b^{4} d^{3}\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a b^{7}} + \frac{4 \, b^{6} d^{3} x^{7} + 21 \, b^{6} c d^{2} x^{4} - 7 \, a b^{5} d^{3} x^{4} + 84 \, b^{6} c^{2} d x - 84 \, a b^{5} c d^{2} x + 28 \, a^{2} b^{4} d^{3} x}{28 \, b^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*sqrt(3)*((-a*b^2)^(1/3)*b^3*c^3 - 3*(-a*b^2)^(1/3)*a*b^2*c^2*d + 3*(-a*b^2)^
(1/3)*a^2*b*c*d^2 - (-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*b^4) + 1/6*((-a*b^2)^(1/3)*b^3*c^3 - 3*(-a*b^2)^(1/3)*a*b^2
*c^2*d + 3*(-a*b^2)^(1/3)*a^2*b*c*d^2 - (-a*b^2)^(1/3)*a^3*d^3)*ln(x^2 + x*(-a/b
)^(1/3) + (-a/b)^(2/3))/(a*b^4) - 1/3*(b^7*c^3 - 3*a*b^6*c^2*d + 3*a^2*b^5*c*d^2
- a^3*b^4*d^3)*(-a/b)^(1/3)*ln(abs(x - (-a/b)^(1/3)))/(a*b^7) + 1/28*(4*b^6*d^3
*x^7 + 21*b^6*c*d^2*x^4 - 7*a*b^5*d^3*x^4 + 84*b^6*c^2*d*x - 84*a*b^5*c*d^2*x +
28*a^2*b^4*d^3*x)/b^7