3.235 \(\int \frac{x^6 \left (c+d x^3+e x^6+f x^9\right )}{a+b x^3} \, dx\)

Optimal. Leaf size=312 \[ \frac{x^7 \left (a^2 f-a b e+b^2 d\right )}{7 b^3}-\frac{a x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{b^5}+\frac{x^4 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{4 b^4}-\frac{a^{4/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^{16/3}}+\frac{a^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 b^{16/3}}-\frac{a^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{\sqrt{3} b^{16/3}}+\frac{x^{10} (b e-a f)}{10 b^2}+\frac{f x^{13}}{13 b} \]

[Out]

-((a*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/b^5) + ((b^3*c - a*b^2*d + a^2*b*e -
a^3*f)*x^4)/(4*b^4) + ((b^2*d - a*b*e + a^2*f)*x^7)/(7*b^3) + ((b*e - a*f)*x^10
)/(10*b^2) + (f*x^13)/(13*b) - (a^(4/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*ArcT
an[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*b^(16/3)) + (a^(4/3)*(b^
3*c - a*b^2*d + a^2*b*e - a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(3*b^(16/3)) - (a^(4/
3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)
*x^2])/(6*b^(16/3))

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Rubi [A]  time = 0.663587, antiderivative size = 312, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{x^7 \left (a^2 f-a b e+b^2 d\right )}{7 b^3}-\frac{a x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{b^5}+\frac{x^4 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{4 b^4}-\frac{a^{4/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^{16/3}}+\frac{a^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 b^{16/3}}-\frac{a^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{\sqrt{3} b^{16/3}}+\frac{x^{10} (b e-a f)}{10 b^2}+\frac{f x^{13}}{13 b} \]

Antiderivative was successfully verified.

[In]  Int[(x^6*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3),x]

[Out]

-((a*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/b^5) + ((b^3*c - a*b^2*d + a^2*b*e -
a^3*f)*x^4)/(4*b^4) + ((b^2*d - a*b*e + a^2*f)*x^7)/(7*b^3) + ((b*e - a*f)*x^10
)/(10*b^2) + (f*x^13)/(13*b) - (a^(4/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*ArcT
an[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*b^(16/3)) + (a^(4/3)*(b^
3*c - a*b^2*d + a^2*b*e - a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(3*b^(16/3)) - (a^(4/
3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)
*x^2])/(6*b^(16/3))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{\frac{4}{3}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 b^{\frac{16}{3}}} + \frac{a^{\frac{4}{3}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 b^{\frac{16}{3}}} + \frac{\sqrt{3} a^{\frac{4}{3}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 b^{\frac{16}{3}}} + \frac{f x^{13}}{13 b} - \frac{x^{10} \left (a f - b e\right )}{10 b^{2}} + \frac{x^{7} \left (a^{2} f - a b e + b^{2} d\right )}{7 b^{3}} - \frac{x^{4} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{4 b^{4}} + \frac{\left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \int a\, dx}{b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**6*(f*x**9+e*x**6+d*x**3+c)/(b*x**3+a),x)

[Out]

-a**(4/3)*(a**3*f - a**2*b*e + a*b**2*d - b**3*c)*log(a**(1/3) + b**(1/3)*x)/(3*
b**(16/3)) + a**(4/3)*(a**3*f - a**2*b*e + a*b**2*d - b**3*c)*log(a**(2/3) - a**
(1/3)*b**(1/3)*x + b**(2/3)*x**2)/(6*b**(16/3)) + sqrt(3)*a**(4/3)*(a**3*f - a**
2*b*e + a*b**2*d - b**3*c)*atan(sqrt(3)*(a**(1/3)/3 - 2*b**(1/3)*x/3)/a**(1/3))/
(3*b**(16/3)) + f*x**13/(13*b) - x**10*(a*f - b*e)/(10*b**2) + x**7*(a**2*f - a*
b*e + b**2*d)/(7*b**3) - x**4*(a**3*f - a**2*b*e + a*b**2*d - b**3*c)/(4*b**4) +
(a**3*f - a**2*b*e + a*b**2*d - b**3*c)*Integral(a, x)/b**5

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Mathematica [A]  time = 0.171048, size = 306, normalized size = 0.98 \[ \frac{x^7 \left (a^2 f-a b e+b^2 d\right )}{7 b^3}+\frac{a x \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{b^5}+\frac{x^4 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{4 b^4}+\frac{a^{4/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{6 b^{16/3}}-\frac{a^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{3 b^{16/3}}+\frac{a^{4/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{\sqrt{3} b^{16/3}}+\frac{x^{10} (b e-a f)}{10 b^2}+\frac{f x^{13}}{13 b} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^6*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3),x]

[Out]

(a*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x)/b^5 + ((b^3*c - a*b^2*d + a^2*b*e -
a^3*f)*x^4)/(4*b^4) + ((b^2*d - a*b*e + a^2*f)*x^7)/(7*b^3) + ((b*e - a*f)*x^10
)/(10*b^2) + (f*x^13)/(13*b) + (a^(4/3)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*A
rcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/(Sqrt[3]*b^(16/3)) - (a^(4/3)*(-(b^3
*c) + a*b^2*d - a^2*b*e + a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(3*b^(16/3)) + (a^(4/
3)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2
/3)*x^2])/(6*b^(16/3))

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Maple [B]  time = 0.007, size = 544, normalized size = 1.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^6*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a),x)

[Out]

-1/3*a^3/b^4/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*d-1/6*a
^2/b^3/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*c+1/6*a^3/b^4/(a/b)^(2/3)*l
n(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*d+1/13*f*x^13/b+1/4/b^3*x^4*a^2*e-1/4/b^2*x^4*a
*d+1/b^5*a^4*f*x-1/b^4*a^3*e*x+1/b^3*a^2*d*x-1/b^2*a*c*x-1/10/b^2*x^10*a*f+1/7/b
^3*x^7*a^2*f-1/7/b^2*x^7*a*e-1/4/b^4*x^4*a^3*f-1/3*a^5/b^6/(a/b)^(2/3)*ln(x+(a/b
)^(1/3))*f+1/10/b*x^10*e+1/3*a^4/b^5/(a/b)^(2/3)*ln(x+(a/b)^(1/3))*e-1/3*a^3/b^4
/(a/b)^(2/3)*ln(x+(a/b)^(1/3))*d+1/3*a^2/b^3/(a/b)^(2/3)*ln(x+(a/b)^(1/3))*c+1/6
*a^5/b^6/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*f-1/6*a^4/b^5/(a/b)^(2/3)
*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*e-1/3*a^5/b^6/(a/b)^(2/3)*3^(1/2)*arctan(1/3*
3^(1/2)*(2/(a/b)^(1/3)*x-1))*f+1/3*a^4/b^5/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2
)*(2/(a/b)^(1/3)*x-1))*e+1/3*a^2/b^3/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(
a/b)^(1/3)*x-1))*c+1/4/b*x^4*c+1/7/b*x^7*d

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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((f*x^9 + e*x^6 + d*x^3 + c)*x^6/(b*x^3 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.219294, size = 429, normalized size = 1.38 \[ \frac{\sqrt{3}{\left (910 \, \sqrt{3}{\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right ) - 1820 \, \sqrt{3}{\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left (x - \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right ) + 5460 \,{\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3} x + \sqrt{3} \left (-\frac{a}{b}\right )^{\frac{1}{3}}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right ) + 3 \, \sqrt{3}{\left (140 \, b^{4} f x^{13} + 182 \,{\left (b^{4} e - a b^{3} f\right )} x^{10} + 260 \,{\left (b^{4} d - a b^{3} e + a^{2} b^{2} f\right )} x^{7} + 455 \,{\left (b^{4} c - a b^{3} d + a^{2} b^{2} e - a^{3} b f\right )} x^{4} - 1820 \,{\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} x\right )}\right )}}{16380 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^9 + e*x^6 + d*x^3 + c)*x^6/(b*x^3 + a),x, algorithm="fricas")

[Out]

1/16380*sqrt(3)*(910*sqrt(3)*(a*b^3*c - a^2*b^2*d + a^3*b*e - a^4*f)*(-a/b)^(1/3
)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3)) - 1820*sqrt(3)*(a*b^3*c - a^2*b^2*d +
a^3*b*e - a^4*f)*(-a/b)^(1/3)*log(x - (-a/b)^(1/3)) + 5460*(a*b^3*c - a^2*b^2*d
+ a^3*b*e - a^4*f)*(-a/b)^(1/3)*arctan(1/3*(2*sqrt(3)*x + sqrt(3)*(-a/b)^(1/3))
/(-a/b)^(1/3)) + 3*sqrt(3)*(140*b^4*f*x^13 + 182*(b^4*e - a*b^3*f)*x^10 + 260*(b
^4*d - a*b^3*e + a^2*b^2*f)*x^7 + 455*(b^4*c - a*b^3*d + a^2*b^2*e - a^3*b*f)*x^
4 - 1820*(a*b^3*c - a^2*b^2*d + a^3*b*e - a^4*f)*x))/b^5

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Sympy [A]  time = 3.67061, size = 411, normalized size = 1.32 \[ \operatorname{RootSum}{\left (27 t^{3} b^{16} + a^{13} f^{3} - 3 a^{12} b e f^{2} + 3 a^{11} b^{2} d f^{2} + 3 a^{11} b^{2} e^{2} f - 3 a^{10} b^{3} c f^{2} - 6 a^{10} b^{3} d e f - a^{10} b^{3} e^{3} + 6 a^{9} b^{4} c e f + 3 a^{9} b^{4} d^{2} f + 3 a^{9} b^{4} d e^{2} - 6 a^{8} b^{5} c d f - 3 a^{8} b^{5} c e^{2} - 3 a^{8} b^{5} d^{2} e + 3 a^{7} b^{6} c^{2} f + 6 a^{7} b^{6} c d e + a^{7} b^{6} d^{3} - 3 a^{6} b^{7} c^{2} e - 3 a^{6} b^{7} c d^{2} + 3 a^{5} b^{8} c^{2} d - a^{4} b^{9} c^{3}, \left ( t \mapsto t \log{\left (- \frac{3 t b^{5}}{a^{4} f - a^{3} b e + a^{2} b^{2} d - a b^{3} c} + x \right )} \right )\right )} + \frac{f x^{13}}{13 b} - \frac{x^{10} \left (a f - b e\right )}{10 b^{2}} + \frac{x^{7} \left (a^{2} f - a b e + b^{2} d\right )}{7 b^{3}} - \frac{x^{4} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{4 b^{4}} + \frac{x \left (a^{4} f - a^{3} b e + a^{2} b^{2} d - a b^{3} c\right )}{b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**6*(f*x**9+e*x**6+d*x**3+c)/(b*x**3+a),x)

[Out]

RootSum(27*_t**3*b**16 + a**13*f**3 - 3*a**12*b*e*f**2 + 3*a**11*b**2*d*f**2 + 3
*a**11*b**2*e**2*f - 3*a**10*b**3*c*f**2 - 6*a**10*b**3*d*e*f - a**10*b**3*e**3
+ 6*a**9*b**4*c*e*f + 3*a**9*b**4*d**2*f + 3*a**9*b**4*d*e**2 - 6*a**8*b**5*c*d*
f - 3*a**8*b**5*c*e**2 - 3*a**8*b**5*d**2*e + 3*a**7*b**6*c**2*f + 6*a**7*b**6*c
*d*e + a**7*b**6*d**3 - 3*a**6*b**7*c**2*e - 3*a**6*b**7*c*d**2 + 3*a**5*b**8*c*
*2*d - a**4*b**9*c**3, Lambda(_t, _t*log(-3*_t*b**5/(a**4*f - a**3*b*e + a**2*b*
*2*d - a*b**3*c) + x))) + f*x**13/(13*b) - x**10*(a*f - b*e)/(10*b**2) + x**7*(a
**2*f - a*b*e + b**2*d)/(7*b**3) - x**4*(a**3*f - a**2*b*e + a*b**2*d - b**3*c)/
(4*b**4) + x*(a**4*f - a**3*b*e + a**2*b**2*d - a*b**3*c)/b**5

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GIAC/XCAS [A]  time = 0.215241, size = 541, normalized size = 1.73 \[ \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} a b^{3} c - \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b^{2} d - \left (-a b^{2}\right )^{\frac{1}{3}} a^{4} f + \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} b e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, b^{6}} + \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} a b^{3} c - \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b^{2} d - \left (-a b^{2}\right )^{\frac{1}{3}} a^{4} f + \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} b e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, b^{6}} - \frac{{\left (a^{2} b^{11} c - a^{3} b^{10} d - a^{5} b^{8} f + a^{4} b^{9} e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a b^{13}} + \frac{140 \, b^{12} f x^{13} - 182 \, a b^{11} f x^{10} + 182 \, b^{12} x^{10} e + 260 \, b^{12} d x^{7} + 260 \, a^{2} b^{10} f x^{7} - 260 \, a b^{11} x^{7} e + 455 \, b^{12} c x^{4} - 455 \, a b^{11} d x^{4} - 455 \, a^{3} b^{9} f x^{4} + 455 \, a^{2} b^{10} x^{4} e - 1820 \, a b^{11} c x + 1820 \, a^{2} b^{10} d x + 1820 \, a^{4} b^{8} f x - 1820 \, a^{3} b^{9} x e}{1820 \, b^{13}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^9 + e*x^6 + d*x^3 + c)*x^6/(b*x^3 + a),x, algorithm="giac")

[Out]

1/3*sqrt(3)*((-a*b^2)^(1/3)*a*b^3*c - (-a*b^2)^(1/3)*a^2*b^2*d - (-a*b^2)^(1/3)*
a^4*f + (-a*b^2)^(1/3)*a^3*b*e)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(
1/3))/b^6 + 1/6*((-a*b^2)^(1/3)*a*b^3*c - (-a*b^2)^(1/3)*a^2*b^2*d - (-a*b^2)^(1
/3)*a^4*f + (-a*b^2)^(1/3)*a^3*b*e)*ln(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/b^6
- 1/3*(a^2*b^11*c - a^3*b^10*d - a^5*b^8*f + a^4*b^9*e)*(-a/b)^(1/3)*ln(abs(x -
(-a/b)^(1/3)))/(a*b^13) + 1/1820*(140*b^12*f*x^13 - 182*a*b^11*f*x^10 + 182*b^12
*x^10*e + 260*b^12*d*x^7 + 260*a^2*b^10*f*x^7 - 260*a*b^11*x^7*e + 455*b^12*c*x^
4 - 455*a*b^11*d*x^4 - 455*a^3*b^9*f*x^4 + 455*a^2*b^10*x^4*e - 1820*a*b^11*c*x
+ 1820*a^2*b^10*d*x + 1820*a^4*b^8*f*x - 1820*a^3*b^9*x*e)/b^13