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

Optimal. Leaf size=252 \[ -\frac{(b c-a d)^4 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} b^{13/3}}+\frac{(b c-a d)^4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{13/3}}-\frac{(b c-a d)^4 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} b^{13/3}}+\frac{d x (2 b c-a d) \left (a^2 d^2-2 a b c d+2 b^2 c^2\right )}{b^4}+\frac{d^2 x^4 \left (a^2 d^2-4 a b c d+6 b^2 c^2\right )}{4 b^3}+\frac{d^3 x^7 (4 b c-a d)}{7 b^2}+\frac{d^4 x^{10}}{10 b} \]

[Out]

(d*(2*b*c - a*d)*(2*b^2*c^2 - 2*a*b*c*d + a^2*d^2)*x)/b^4 + (d^2*(6*b^2*c^2 - 4*
a*b*c*d + a^2*d^2)*x^4)/(4*b^3) + (d^3*(4*b*c - a*d)*x^7)/(7*b^2) + (d^4*x^10)/(
10*b) - ((b*c - a*d)^4*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(Sqrt[
3]*a^(2/3)*b^(13/3)) + ((b*c - a*d)^4*Log[a^(1/3) + b^(1/3)*x])/(3*a^(2/3)*b^(13
/3)) - ((b*c - a*d)^4*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*a^(2/3)
*b^(13/3))

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Rubi [A]  time = 0.415438, antiderivative size = 252, 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 \[ -\frac{(b c-a d)^4 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} b^{13/3}}+\frac{(b c-a d)^4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{13/3}}-\frac{(b c-a d)^4 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} b^{13/3}}+\frac{d x (2 b c-a d) \left (a^2 d^2-2 a b c d+2 b^2 c^2\right )}{b^4}+\frac{d^2 x^4 \left (a^2 d^2-4 a b c d+6 b^2 c^2\right )}{4 b^3}+\frac{d^3 x^7 (4 b c-a d)}{7 b^2}+\frac{d^4 x^{10}}{10 b} \]

Antiderivative was successfully verified.

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

[Out]

(d*(2*b*c - a*d)*(2*b^2*c^2 - 2*a*b*c*d + a^2*d^2)*x)/b^4 + (d^2*(6*b^2*c^2 - 4*
a*b*c*d + a^2*d^2)*x^4)/(4*b^3) + (d^3*(4*b*c - a*d)*x^7)/(7*b^2) + (d^4*x^10)/(
10*b) - ((b*c - a*d)^4*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(Sqrt[
3]*a^(2/3)*b^(13/3)) + ((b*c - a*d)^4*Log[a^(1/3) + b^(1/3)*x])/(3*a^(2/3)*b^(13
/3)) - ((b*c - a*d)^4*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*a^(2/3)
*b^(13/3))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{d^{4} x^{10}}{10 b} - \frac{d^{3} x^{7} \left (a d - 4 b c\right )}{7 b^{2}} + \frac{d^{2} x^{4} \left (a^{2} d^{2} - 4 a b c d + 6 b^{2} c^{2}\right )}{4 b^{3}} - \frac{\left (a d - 2 b c\right ) \left (a^{2} d^{2} - 2 a b c d + 2 b^{2} c^{2}\right ) \int d\, dx}{b^{4}} + \frac{\left (a d - b c\right )^{4} \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 a^{\frac{2}{3}} b^{\frac{13}{3}}} - \frac{\left (a d - b c\right )^{4} \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 a^{\frac{2}{3}} b^{\frac{13}{3}}} - \frac{\sqrt{3} \left (a d - b c\right )^{4} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 a^{\frac{2}{3}} b^{\frac{13}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d**4*x**10/(10*b) - d**3*x**7*(a*d - 4*b*c)/(7*b**2) + d**2*x**4*(a**2*d**2 - 4*
a*b*c*d + 6*b**2*c**2)/(4*b**3) - (a*d - 2*b*c)*(a**2*d**2 - 2*a*b*c*d + 2*b**2*
c**2)*Integral(d, x)/b**4 + (a*d - b*c)**4*log(a**(1/3) + b**(1/3)*x)/(3*a**(2/3
)*b**(13/3)) - (a*d - b*c)**4*log(a**(2/3) - a**(1/3)*b**(1/3)*x + b**(2/3)*x**2
)/(6*a**(2/3)*b**(13/3)) - sqrt(3)*(a*d - b*c)**4*atan(sqrt(3)*(a**(1/3)/3 - 2*b
**(1/3)*x/3)/a**(1/3))/(3*a**(2/3)*b**(13/3))

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Mathematica [A]  time = 0.199474, size = 253, normalized size = 1. \[ \frac{-\frac{70 (b c-a d)^4 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{2/3}}+\frac{140 (b c-a d)^4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{2/3}}+\frac{140 \sqrt{3} (b c-a d)^4 \tan ^{-1}\left (\frac{2 \sqrt [3]{b} x-\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{a^{2/3}}+105 b^{4/3} d^2 x^4 \left (a^2 d^2-4 a b c d+6 b^2 c^2\right )+420 \sqrt [3]{b} d x \left (-a^3 d^3+4 a^2 b c d^2-6 a b^2 c^2 d+4 b^3 c^3\right )+60 b^{7/3} d^3 x^7 (4 b c-a d)+42 b^{10/3} d^4 x^{10}}{420 b^{13/3}} \]

Antiderivative was successfully verified.

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

[Out]

(420*b^(1/3)*d*(4*b^3*c^3 - 6*a*b^2*c^2*d + 4*a^2*b*c*d^2 - a^3*d^3)*x + 105*b^(
4/3)*d^2*(6*b^2*c^2 - 4*a*b*c*d + a^2*d^2)*x^4 + 60*b^(7/3)*d^3*(4*b*c - a*d)*x^
7 + 42*b^(10/3)*d^4*x^10 + (140*Sqrt[3]*(b*c - a*d)^4*ArcTan[(-a^(1/3) + 2*b^(1/
3)*x)/(Sqrt[3]*a^(1/3))])/a^(2/3) + (140*(b*c - a*d)^4*Log[a^(1/3) + b^(1/3)*x])
/a^(2/3) - (70*(b*c - a*d)^4*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(
2/3))/(420*b^(13/3))

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Maple [B]  time = 0.006, size = 661, normalized size = 2.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x^3+c)^4/(b*x^3+a),x)

[Out]

-4/3/b^2/(a/b)^(2/3)*ln(x+(a/b)^(1/3))*a*c^3*d+2/3/b^4/(a/b)^(2/3)*ln(x^2-x*(a/b
)^(1/3)+(a/b)^(2/3))*a^3*c*d^3+2/3/b^2/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2
/3))*a*c^3*d+1/3/b^5/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))
*a^4*d^4-1/b^3/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*a^2*c^2*d^2-4/3/b^4
/(a/b)^(2/3)*ln(x+(a/b)^(1/3))*a^3*c*d^3+2/b^3/(a/b)^(2/3)*ln(x+(a/b)^(1/3))*a^2
*c^2*d^2+1/3/b^5/(a/b)^(2/3)*ln(x+(a/b)^(1/3))*a^4*d^4-d^3/b^2*x^4*a*c+4*d^3/b^3
*a^2*c*x-6*d^2/b^2*a*c^2*x-4/3/b^4/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/
b)^(1/3)*x-1))*a^3*c*d^3+2/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^2*d^2-1/6/b^5/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*a^4
*d^4+1/3/b/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*c^4+4/7*d
^3/b*x^7*c-1/7*d^4/b^2*x^7*a+3/2*d^2/b*x^4*c^2-d^4/b^4*a^3*x+4*d/b*c^3*x+1/3/b/(
a/b)^(2/3)*ln(x+(a/b)^(1/3))*c^4-1/6/b/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2
/3))*c^4+1/4*d^4/b^3*x^4*a^2-4/3/b^2/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(
a/b)^(1/3)*x-1))*a*c^3*d+1/10*d^4*x^10/b

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.216267, size = 494, normalized size = 1.96 \[ -\frac{\sqrt{3}{\left (70 \, \sqrt{3}{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \log \left (\left (a^{2} b\right )^{\frac{2}{3}} x^{2} - \left (a^{2} b\right )^{\frac{1}{3}} a x + a^{2}\right ) - 140 \, \sqrt{3}{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \log \left (\left (a^{2} b\right )^{\frac{1}{3}} x + a\right ) - 420 \,{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \arctan \left (\frac{2 \, \sqrt{3} \left (a^{2} b\right )^{\frac{1}{3}} x - \sqrt{3} a}{3 \, a}\right ) - 3 \, \sqrt{3}{\left (14 \, b^{3} d^{4} x^{10} + 20 \,{\left (4 \, b^{3} c d^{3} - a b^{2} d^{4}\right )} x^{7} + 35 \,{\left (6 \, b^{3} c^{2} d^{2} - 4 \, a b^{2} c d^{3} + a^{2} b d^{4}\right )} x^{4} + 140 \,{\left (4 \, b^{3} c^{3} d - 6 \, a b^{2} c^{2} d^{2} + 4 \, a^{2} b c d^{3} - a^{3} d^{4}\right )} x\right )} \left (a^{2} b\right )^{\frac{1}{3}}\right )}}{1260 \, \left (a^{2} b\right )^{\frac{1}{3}} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/1260*sqrt(3)*(70*sqrt(3)*(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3
*b*c*d^3 + a^4*d^4)*log((a^2*b)^(2/3)*x^2 - (a^2*b)^(1/3)*a*x + a^2) - 140*sqrt(
3)*(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*log((
a^2*b)^(1/3)*x + a) - 420*(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b
*c*d^3 + a^4*d^4)*arctan(1/3*(2*sqrt(3)*(a^2*b)^(1/3)*x - sqrt(3)*a)/a) - 3*sqrt
(3)*(14*b^3*d^4*x^10 + 20*(4*b^3*c*d^3 - a*b^2*d^4)*x^7 + 35*(6*b^3*c^2*d^2 - 4*
a*b^2*c*d^3 + a^2*b*d^4)*x^4 + 140*(4*b^3*c^3*d - 6*a*b^2*c^2*d^2 + 4*a^2*b*c*d^
3 - a^3*d^4)*x)*(a^2*b)^(1/3))/((a^2*b)^(1/3)*b^4)

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Sympy [A]  time = 5.40425, size = 369, normalized size = 1.46 \[ \operatorname{RootSum}{\left (27 t^{3} a^{2} b^{13} - a^{12} d^{12} + 12 a^{11} b c d^{11} - 66 a^{10} b^{2} c^{2} d^{10} + 220 a^{9} b^{3} c^{3} d^{9} - 495 a^{8} b^{4} c^{4} d^{8} + 792 a^{7} b^{5} c^{5} d^{7} - 924 a^{6} b^{6} c^{6} d^{6} + 792 a^{5} b^{7} c^{7} d^{5} - 495 a^{4} b^{8} c^{8} d^{4} + 220 a^{3} b^{9} c^{9} d^{3} - 66 a^{2} b^{10} c^{10} d^{2} + 12 a b^{11} c^{11} d - b^{12} c^{12}, \left ( t \mapsto t \log{\left (\frac{3 t a b^{4}}{a^{4} d^{4} - 4 a^{3} b c d^{3} + 6 a^{2} b^{2} c^{2} d^{2} - 4 a b^{3} c^{3} d + b^{4} c^{4}} + x \right )} \right )\right )} + \frac{d^{4} x^{10}}{10 b} - \frac{x^{7} \left (a d^{4} - 4 b c d^{3}\right )}{7 b^{2}} + \frac{x^{4} \left (a^{2} d^{4} - 4 a b c d^{3} + 6 b^{2} c^{2} d^{2}\right )}{4 b^{3}} - \frac{x \left (a^{3} d^{4} - 4 a^{2} b c d^{3} + 6 a b^{2} c^{2} d^{2} - 4 b^{3} c^{3} d\right )}{b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

RootSum(27*_t**3*a**2*b**13 - a**12*d**12 + 12*a**11*b*c*d**11 - 66*a**10*b**2*c
**2*d**10 + 220*a**9*b**3*c**3*d**9 - 495*a**8*b**4*c**4*d**8 + 792*a**7*b**5*c*
*5*d**7 - 924*a**6*b**6*c**6*d**6 + 792*a**5*b**7*c**7*d**5 - 495*a**4*b**8*c**8
*d**4 + 220*a**3*b**9*c**9*d**3 - 66*a**2*b**10*c**10*d**2 + 12*a*b**11*c**11*d
- b**12*c**12, Lambda(_t, _t*log(3*_t*a*b**4/(a**4*d**4 - 4*a**3*b*c*d**3 + 6*a*
*2*b**2*c**2*d**2 - 4*a*b**3*c**3*d + b**4*c**4) + x))) + d**4*x**10/(10*b) - x*
*7*(a*d**4 - 4*b*c*d**3)/(7*b**2) + x**4*(a**2*d**4 - 4*a*b*c*d**3 + 6*b**2*c**2
*d**2)/(4*b**3) - x*(a**3*d**4 - 4*a**2*b*c*d**3 + 6*a*b**2*c**2*d**2 - 4*b**3*c
**3*d)/b**4

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GIAC/XCAS [A]  time = 0.220205, size = 622, normalized size = 2.47 \[ \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{4} c^{4} - 4 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{3} c^{3} d + 6 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b^{2} c^{2} d^{2} - 4 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} b c d^{3} + \left (-a b^{2}\right )^{\frac{1}{3}} a^{4} d^{4}\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, a b^{5}} + \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{4} c^{4} - 4 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{3} c^{3} d + 6 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b^{2} c^{2} d^{2} - 4 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} b c d^{3} + \left (-a b^{2}\right )^{\frac{1}{3}} a^{4} d^{4}\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, a b^{5}} - \frac{{\left (b^{10} c^{4} - 4 \, a b^{9} c^{3} d + 6 \, a^{2} b^{8} c^{2} d^{2} - 4 \, a^{3} b^{7} c d^{3} + a^{4} b^{6} d^{4}\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a b^{10}} + \frac{14 \, b^{9} d^{4} x^{10} + 80 \, b^{9} c d^{3} x^{7} - 20 \, a b^{8} d^{4} x^{7} + 210 \, b^{9} c^{2} d^{2} x^{4} - 140 \, a b^{8} c d^{3} x^{4} + 35 \, a^{2} b^{7} d^{4} x^{4} + 560 \, b^{9} c^{3} d x - 840 \, a b^{8} c^{2} d^{2} x + 560 \, a^{2} b^{7} c d^{3} x - 140 \, a^{3} b^{6} d^{4} x}{140 \, b^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*sqrt(3)*((-a*b^2)^(1/3)*b^4*c^4 - 4*(-a*b^2)^(1/3)*a*b^3*c^3*d + 6*(-a*b^2)^
(1/3)*a^2*b^2*c^2*d^2 - 4*(-a*b^2)^(1/3)*a^3*b*c*d^3 + (-a*b^2)^(1/3)*a^4*d^4)*a
rctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a*b^5) + 1/6*((-a*b^2)^(1/
3)*b^4*c^4 - 4*(-a*b^2)^(1/3)*a*b^3*c^3*d + 6*(-a*b^2)^(1/3)*a^2*b^2*c^2*d^2 - 4
*(-a*b^2)^(1/3)*a^3*b*c*d^3 + (-a*b^2)^(1/3)*a^4*d^4)*ln(x^2 + x*(-a/b)^(1/3) +
(-a/b)^(2/3))/(a*b^5) - 1/3*(b^10*c^4 - 4*a*b^9*c^3*d + 6*a^2*b^8*c^2*d^2 - 4*a^
3*b^7*c*d^3 + a^4*b^6*d^4)*(-a/b)^(1/3)*ln(abs(x - (-a/b)^(1/3)))/(a*b^10) + 1/1
40*(14*b^9*d^4*x^10 + 80*b^9*c*d^3*x^7 - 20*a*b^8*d^4*x^7 + 210*b^9*c^2*d^2*x^4
- 140*a*b^8*c*d^3*x^4 + 35*a^2*b^7*d^4*x^4 + 560*b^9*c^3*d*x - 840*a*b^8*c^2*d^2
*x + 560*a^2*b^7*c*d^3*x - 140*a^3*b^6*d^4*x)/b^10