3.246 \(\int \frac{c+d x^3+e x^6+f x^9}{x^{11} \left (a+b x^3\right )} \, dx\)

Optimal. Leaf size=277 \[ \frac{b c-a d}{7 a^2 x^7}-\frac{a^2 e-a b d+b^2 c}{4 a^3 x^4}-\frac{\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^{13/3}}-\frac{\sqrt [3]{b} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{\sqrt{3} a^{13/3}}+\frac{\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^{13/3}}+\frac{a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{a^4 x}-\frac{c}{10 a x^{10}} \]

[Out]

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

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Rubi [A]  time = 0.501596, antiderivative size = 277, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.233 \[ \frac{b c-a d}{7 a^2 x^7}-\frac{a^2 e-a b d+b^2 c}{4 a^3 x^4}-\frac{\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^{13/3}}-\frac{\sqrt [3]{b} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{\sqrt{3} a^{13/3}}+\frac{\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^{13/3}}+\frac{a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{a^4 x}-\frac{c}{10 a x^{10}} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x^3 + e*x^6 + f*x^9)/(x^11*(a + b*x^3)),x]

[Out]

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

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Rubi in Sympy [A]  time = 78.6698, size = 257, normalized size = 0.93 \[ - \frac{c}{10 a x^{10}} - \frac{a d - b c}{7 a^{2} x^{7}} - \frac{a^{2} e - a b d + b^{2} c}{4 a^{3} x^{4}} - \frac{a^{3} f - a^{2} b e + a b^{2} d - b^{3} c}{a^{4} x} + \frac{\sqrt [3]{b} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 a^{\frac{13}{3}}} - \frac{\sqrt [3]{b} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 a^{\frac{13}{3}}} + \frac{\sqrt{3} \sqrt [3]{b} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 a^{\frac{13}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((f*x**9+e*x**6+d*x**3+c)/x**11/(b*x**3+a),x)

[Out]

-c/(10*a*x**10) - (a*d - b*c)/(7*a**2*x**7) - (a**2*e - a*b*d + b**2*c)/(4*a**3*
x**4) - (a**3*f - a**2*b*e + a*b**2*d - b**3*c)/(a**4*x) + b**(1/3)*(a**3*f - a*
*2*b*e + a*b**2*d - b**3*c)*log(a**(1/3) + b**(1/3)*x)/(3*a**(13/3)) - b**(1/3)*
(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*a**(13/3)) + sqrt(3)*b**(1/3)*(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*a**(13/3))

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Mathematica [A]  time = 0.205059, size = 266, normalized size = 0.96 \[ \frac{\frac{60 a^{7/3} (b c-a d)}{x^7}-\frac{42 a^{10/3} c}{x^{10}}-\frac{105 a^{4/3} \left (a^2 e-a b d+b^2 c\right )}{x^4}+\frac{420 \sqrt [3]{a} \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{x}+140 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )-140 \sqrt{3} \sqrt [3]{b} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )+70 \sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{420 a^{13/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^11*(a + b*x^3)),x]

[Out]

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

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Maple [B]  time = 0.013, size = 491, normalized size = 1.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((f*x^9+e*x^6+d*x^3+c)/x^11/(b*x^3+a),x)

[Out]

-1/10*c/a/x^10-1/7/a/x^7*d+1/7/a^2/x^7*b*c-1/4/a/x^4*e+1/4/a^2/x^4*b*d-1/4/a^3/x
^4*b^2*c-1/a/x*f+1/a^2/x*b*e-1/a^3/x*b^2*d+1/a^4/x*b^3*c+1/3/a/(a/b)^(1/3)*ln(x+
(a/b)^(1/3))*f-1/3*b/a^2/(a/b)^(1/3)*ln(x+(a/b)^(1/3))*e+1/3*b^2/a^3/(a/b)^(1/3)
*ln(x+(a/b)^(1/3))*d-1/3*b^3/a^4/(a/b)^(1/3)*ln(x+(a/b)^(1/3))*c-1/6/a/(a/b)^(1/
3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*f+1/6*b/a^2/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3
)+(a/b)^(2/3))*e-1/6*b^2/a^3/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*d+1/6
*b^3/a^4/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*c-1/3/a*3^(1/2)/(a/b)^(1/
3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*f+1/3*b/a^2*3^(1/2)/(a/b)^(1/3)*arcta
n(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*e-1/3*b^2/a^3*3^(1/2)/(a/b)^(1/3)*arctan(1/3*
3^(1/2)*(2/(a/b)^(1/3)*x-1))*d+1/3*b^3/a^4*3^(1/2)/(a/b)^(1/3)*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((f*x^9 + e*x^6 + d*x^3 + c)/((b*x^3 + a)*x^11),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.226273, size = 389, normalized size = 1.4 \[ \frac{\sqrt{3}{\left (70 \, \sqrt{3}{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{10} \left (\frac{b}{a}\right )^{\frac{1}{3}} \log \left (b x^{2} - a x \left (\frac{b}{a}\right )^{\frac{2}{3}} + a \left (\frac{b}{a}\right )^{\frac{1}{3}}\right ) - 140 \, \sqrt{3}{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{10} \left (\frac{b}{a}\right )^{\frac{1}{3}} \log \left (b x + a \left (\frac{b}{a}\right )^{\frac{2}{3}}\right ) - 420 \,{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{10} \left (\frac{b}{a}\right )^{\frac{1}{3}} \arctan \left (-\frac{2 \, \sqrt{3} b x - \sqrt{3} a \left (\frac{b}{a}\right )^{\frac{2}{3}}}{3 \, a \left (\frac{b}{a}\right )^{\frac{2}{3}}}\right ) + 3 \, \sqrt{3}{\left (140 \,{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{9} - 35 \,{\left (a b^{2} c - a^{2} b d + a^{3} e\right )} x^{6} - 14 \, a^{3} c + 20 \,{\left (a^{2} b c - a^{3} d\right )} x^{3}\right )}\right )}}{1260 \, a^{4} x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^9 + e*x^6 + d*x^3 + c)/((b*x^3 + a)*x^11),x, algorithm="fricas")

[Out]

1/1260*sqrt(3)*(70*sqrt(3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^10*(b/a)^(1/3)*
log(b*x^2 - a*x*(b/a)^(2/3) + a*(b/a)^(1/3)) - 140*sqrt(3)*(b^3*c - a*b^2*d + a^
2*b*e - a^3*f)*x^10*(b/a)^(1/3)*log(b*x + a*(b/a)^(2/3)) - 420*(b^3*c - a*b^2*d
+ a^2*b*e - a^3*f)*x^10*(b/a)^(1/3)*arctan(-1/3*(2*sqrt(3)*b*x - sqrt(3)*a*(b/a)
^(2/3))/(a*(b/a)^(2/3))) + 3*sqrt(3)*(140*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^
9 - 35*(a*b^2*c - a^2*b*d + a^3*e)*x^6 - 14*a^3*c + 20*(a^2*b*c - a^3*d)*x^3))/(
a^4*x^10)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x**9+e*x**6+d*x**3+c)/x**11/(b*x**3+a),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.220606, size = 508, normalized size = 1.83 \[ -\frac{{\left (b^{4} c \left (-\frac{a}{b}\right )^{\frac{1}{3}} - a b^{3} d \left (-\frac{a}{b}\right )^{\frac{1}{3}} - a^{3} b f \left (-\frac{a}{b}\right )^{\frac{1}{3}} + a^{2} b^{2} \left (-\frac{a}{b}\right )^{\frac{1}{3}} e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a^{5}} - \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d - \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + \left (-a b^{2}\right )^{\frac{2}{3}} a^{2} b e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, a^{5} b} + \frac{{\left (\left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d - \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + \left (-a b^{2}\right )^{\frac{2}{3}} a^{2} b e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, a^{5} b} + \frac{140 \, b^{3} c x^{9} - 140 \, a b^{2} d x^{9} - 140 \, a^{3} f x^{9} + 140 \, a^{2} b x^{9} e - 35 \, a b^{2} c x^{6} + 35 \, a^{2} b d x^{6} - 35 \, a^{3} x^{6} e + 20 \, a^{2} b c x^{3} - 20 \, a^{3} d x^{3} - 14 \, a^{3} c}{140 \, a^{4} x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^9 + e*x^6 + d*x^3 + c)/((b*x^3 + a)*x^11),x, algorithm="giac")

[Out]

-1/3*(b^4*c*(-a/b)^(1/3) - a*b^3*d*(-a/b)^(1/3) - a^3*b*f*(-a/b)^(1/3) + a^2*b^2
*(-a/b)^(1/3)*e)*(-a/b)^(1/3)*ln(abs(x - (-a/b)^(1/3)))/a^5 - 1/3*sqrt(3)*((-a*b
^2)^(2/3)*b^3*c - (-a*b^2)^(2/3)*a*b^2*d - (-a*b^2)^(2/3)*a^3*f + (-a*b^2)^(2/3)
*a^2*b*e)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a^5*b) + 1/6*((
-a*b^2)^(2/3)*b^3*c - (-a*b^2)^(2/3)*a*b^2*d - (-a*b^2)^(2/3)*a^3*f + (-a*b^2)^(
2/3)*a^2*b*e)*ln(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^5*b) + 1/140*(140*b^3*c
*x^9 - 140*a*b^2*d*x^9 - 140*a^3*f*x^9 + 140*a^2*b*x^9*e - 35*a*b^2*c*x^6 + 35*a
^2*b*d*x^6 - 35*a^3*x^6*e + 20*a^2*b*c*x^3 - 20*a^3*d*x^3 - 14*a^3*c)/(a^4*x^10)