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

Optimal. Leaf size=301 \[ \frac{x^2 \left (7 a^3 f-4 a^2 b e+a b^2 d+2 b^3 c\right )}{9 a^2 b^3 \left (a+b x^3\right )}+\frac{x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a b^3 \left (a+b x^3\right )^2}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-20 a^3 f+5 a^2 b e+a b^2 d+2 b^3 c\right )}{54 a^{7/3} b^{11/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-20 a^3 f+5 a^2 b e+a b^2 d+2 b^3 c\right )}{27 a^{7/3} b^{11/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-20 a^3 f+5 a^2 b e+a b^2 d+2 b^3 c\right )}{9 \sqrt{3} a^{7/3} b^{11/3}}+\frac{f x^2}{2 b^3} \]

[Out]

(f*x^2)/(2*b^3) + ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^2)/(6*a*b^3*(a + b*x^3)
^2) + ((2*b^3*c + a*b^2*d - 4*a^2*b*e + 7*a^3*f)*x^2)/(9*a^2*b^3*(a + b*x^3)) -
((2*b^3*c + a*b^2*d + 5*a^2*b*e - 20*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt
[3]*a^(1/3))])/(9*Sqrt[3]*a^(7/3)*b^(11/3)) - ((2*b^3*c + a*b^2*d + 5*a^2*b*e -
20*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(27*a^(7/3)*b^(11/3)) + ((2*b^3*c + a*b^2*d
+ 5*a^2*b*e - 20*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(7
/3)*b^(11/3))

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Rubi [A]  time = 0.805486, antiderivative size = 301, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 10, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357 \[ \frac{x^2 \left (7 a^3 f-4 a^2 b e+a b^2 d+2 b^3 c\right )}{9 a^2 b^3 \left (a+b x^3\right )}+\frac{x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a b^3 \left (a+b x^3\right )^2}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-20 a^3 f+5 a^2 b e+a b^2 d+2 b^3 c\right )}{54 a^{7/3} b^{11/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-20 a^3 f+5 a^2 b e+a b^2 d+2 b^3 c\right )}{27 a^{7/3} b^{11/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-20 a^3 f+5 a^2 b e+a b^2 d+2 b^3 c\right )}{9 \sqrt{3} a^{7/3} b^{11/3}}+\frac{f x^2}{2 b^3} \]

Antiderivative was successfully verified.

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

[Out]

(f*x^2)/(2*b^3) + ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^2)/(6*a*b^3*(a + b*x^3)
^2) + ((2*b^3*c + a*b^2*d - 4*a^2*b*e + 7*a^3*f)*x^2)/(9*a^2*b^3*(a + b*x^3)) -
((2*b^3*c + a*b^2*d + 5*a^2*b*e - 20*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt
[3]*a^(1/3))])/(9*Sqrt[3]*a^(7/3)*b^(11/3)) - ((2*b^3*c + a*b^2*d + 5*a^2*b*e -
20*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(27*a^(7/3)*b^(11/3)) + ((2*b^3*c + a*b^2*d
+ 5*a^2*b*e - 20*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(7
/3)*b^(11/3))

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Rubi in Sympy [A]  time = 128.405, size = 291, normalized size = 0.97 \[ \frac{f x^{2}}{2 b^{3}} - \frac{x^{2} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{6 a b^{3} \left (a + b x^{3}\right )^{2}} + \frac{x^{2} \left (7 a^{3} f - 4 a^{2} b e + a b^{2} d + 2 b^{3} c\right )}{9 a^{2} b^{3} \left (a + b x^{3}\right )} + \frac{\left (a \left (20 a^{2} f - 5 a b e - b^{2} d\right ) - 2 b^{3} c\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{27 a^{\frac{7}{3}} b^{\frac{11}{3}}} - \frac{\left (a \left (20 a^{2} f - 5 a b e - b^{2} d\right ) - 2 b^{3} c\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{54 a^{\frac{7}{3}} b^{\frac{11}{3}}} + \frac{\sqrt{3} \left (a \left (20 a^{2} f - 5 a b e - b^{2} d\right ) - 2 b^{3} c\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{27 a^{\frac{7}{3}} b^{\frac{11}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

f*x**2/(2*b**3) - x**2*(a**3*f - a**2*b*e + a*b**2*d - b**3*c)/(6*a*b**3*(a + b*
x**3)**2) + x**2*(7*a**3*f - 4*a**2*b*e + a*b**2*d + 2*b**3*c)/(9*a**2*b**3*(a +
b*x**3)) + (a*(20*a**2*f - 5*a*b*e - b**2*d) - 2*b**3*c)*log(a**(1/3) + b**(1/3
)*x)/(27*a**(7/3)*b**(11/3)) - (a*(20*a**2*f - 5*a*b*e - b**2*d) - 2*b**3*c)*log
(a**(2/3) - a**(1/3)*b**(1/3)*x + b**(2/3)*x**2)/(54*a**(7/3)*b**(11/3)) + sqrt(
3)*(a*(20*a**2*f - 5*a*b*e - b**2*d) - 2*b**3*c)*atan(sqrt(3)*(a**(1/3)/3 - 2*b*
*(1/3)*x/3)/a**(1/3))/(27*a**(7/3)*b**(11/3))

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Mathematica [A]  time = 0.379933, size = 284, normalized size = 0.94 \[ \frac{\frac{6 b^{2/3} x^2 \left (7 a^3 f-4 a^2 b e+a b^2 d+2 b^3 c\right )}{a^2 \left (a+b x^3\right )}+\frac{9 b^{2/3} x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{a \left (a+b x^3\right )^2}-\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-20 a^3 f+5 a^2 b e+a b^2 d+2 b^3 c\right )}{a^{7/3}}-\frac{2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (-20 a^3 f+5 a^2 b e+a b^2 d+2 b^3 c\right )}{a^{7/3}}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-20 a^3 f+5 a^2 b e+a b^2 d+2 b^3 c\right )}{a^{7/3}}+27 b^{2/3} f x^2}{54 b^{11/3}} \]

Antiderivative was successfully verified.

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

[Out]

(27*b^(2/3)*f*x^2 + (9*b^(2/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^2)/(a*(a +
b*x^3)^2) + (6*b^(2/3)*(2*b^3*c + a*b^2*d - 4*a^2*b*e + 7*a^3*f)*x^2)/(a^2*(a +
b*x^3)) - (2*Sqrt[3]*(2*b^3*c + a*b^2*d + 5*a^2*b*e - 20*a^3*f)*ArcTan[(1 - (2*b
^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(7/3) - (2*(2*b^3*c + a*b^2*d + 5*a^2*b*e - 20*a^
3*f)*Log[a^(1/3) + b^(1/3)*x])/a^(7/3) + ((2*b^3*c + a*b^2*d + 5*a^2*b*e - 20*a^
3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(7/3))/(54*b^(11/3))

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Maple [B]  time = 0.017, size = 550, normalized size = 1.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*f*x^2/b^3+7/9/b^2/(b*x^3+a)^2*a*x^5*f-4/9/b/(b*x^3+a)^2*x^5*e+1/9/(b*x^3+a)^
2/a*x^5*d+2/9*b/(b*x^3+a)^2/a^2*x^5*c+11/18/b^3/(b*x^3+a)^2*a^2*x^2*f-5/18/b^2/(
b*x^3+a)^2*a*x^2*e-1/18/b/(b*x^3+a)^2*x^2*d+7/18/(b*x^3+a)^2/a*x^2*c+20/27/b^4*a
/(a/b)^(1/3)*ln(x+(a/b)^(1/3))*f-5/27/b^3/(a/b)^(1/3)*ln(x+(a/b)^(1/3))*e-1/27/b
^2/a/(a/b)^(1/3)*ln(x+(a/b)^(1/3))*d-2/27/b/a^2/(a/b)^(1/3)*ln(x+(a/b)^(1/3))*c-
10/27/b^4*a/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*f+5/54/b^3/(a/b)^(1/3)
*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*e+1/54/b^2/a/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)
+(a/b)^(2/3))*d+1/27/b/a^2/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*c-20/27
/b^4*a*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*f+5/27/b^3*3^
(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*e+1/27/b^2/a*3^(1/2)/(
a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*d+2/27/b/a^2*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)*x/(b*x^3 + a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.222284, size = 711, normalized size = 2.36 \[ \frac{\sqrt{3}{\left (\sqrt{3}{\left ({\left (2 \, b^{5} c + a b^{4} d + 5 \, a^{2} b^{3} e - 20 \, a^{3} b^{2} f\right )} x^{6} + 2 \, a^{2} b^{3} c + a^{3} b^{2} d + 5 \, a^{4} b e - 20 \, a^{5} f + 2 \,{\left (2 \, a b^{4} c + a^{2} b^{3} d + 5 \, a^{3} b^{2} e - 20 \, a^{4} b f\right )} x^{3}\right )} \log \left (\left (a b^{2}\right )^{\frac{1}{3}} b x^{2} + a b - \left (a b^{2}\right )^{\frac{2}{3}} x\right ) - 2 \, \sqrt{3}{\left ({\left (2 \, b^{5} c + a b^{4} d + 5 \, a^{2} b^{3} e - 20 \, a^{3} b^{2} f\right )} x^{6} + 2 \, a^{2} b^{3} c + a^{3} b^{2} d + 5 \, a^{4} b e - 20 \, a^{5} f + 2 \,{\left (2 \, a b^{4} c + a^{2} b^{3} d + 5 \, a^{3} b^{2} e - 20 \, a^{4} b f\right )} x^{3}\right )} \log \left (a b + \left (a b^{2}\right )^{\frac{2}{3}} x\right ) + 6 \,{\left ({\left (2 \, b^{5} c + a b^{4} d + 5 \, a^{2} b^{3} e - 20 \, a^{3} b^{2} f\right )} x^{6} + 2 \, a^{2} b^{3} c + a^{3} b^{2} d + 5 \, a^{4} b e - 20 \, a^{5} f + 2 \,{\left (2 \, a b^{4} c + a^{2} b^{3} d + 5 \, a^{3} b^{2} e - 20 \, a^{4} b f\right )} x^{3}\right )} \arctan \left (-\frac{\sqrt{3} a b - 2 \, \sqrt{3} \left (a b^{2}\right )^{\frac{2}{3}} x}{3 \, a b}\right ) + 3 \, \sqrt{3}{\left (9 \, a^{2} b^{2} f x^{8} + 2 \,{\left (2 \, b^{4} c + a b^{3} d - 4 \, a^{2} b^{2} e + 16 \, a^{3} b f\right )} x^{5} +{\left (7 \, a b^{3} c - a^{2} b^{2} d - 5 \, a^{3} b e + 20 \, a^{4} f\right )} x^{2}\right )} \left (a b^{2}\right )^{\frac{1}{3}}\right )}}{162 \,{\left (a^{2} b^{5} x^{6} + 2 \, a^{3} b^{4} x^{3} + a^{4} b^{3}\right )} \left (a b^{2}\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/162*sqrt(3)*(sqrt(3)*((2*b^5*c + a*b^4*d + 5*a^2*b^3*e - 20*a^3*b^2*f)*x^6 + 2
*a^2*b^3*c + a^3*b^2*d + 5*a^4*b*e - 20*a^5*f + 2*(2*a*b^4*c + a^2*b^3*d + 5*a^3
*b^2*e - 20*a^4*b*f)*x^3)*log((a*b^2)^(1/3)*b*x^2 + a*b - (a*b^2)^(2/3)*x) - 2*s
qrt(3)*((2*b^5*c + a*b^4*d + 5*a^2*b^3*e - 20*a^3*b^2*f)*x^6 + 2*a^2*b^3*c + a^3
*b^2*d + 5*a^4*b*e - 20*a^5*f + 2*(2*a*b^4*c + a^2*b^3*d + 5*a^3*b^2*e - 20*a^4*
b*f)*x^3)*log(a*b + (a*b^2)^(2/3)*x) + 6*((2*b^5*c + a*b^4*d + 5*a^2*b^3*e - 20*
a^3*b^2*f)*x^6 + 2*a^2*b^3*c + a^3*b^2*d + 5*a^4*b*e - 20*a^5*f + 2*(2*a*b^4*c +
a^2*b^3*d + 5*a^3*b^2*e - 20*a^4*b*f)*x^3)*arctan(-1/3*(sqrt(3)*a*b - 2*sqrt(3)
*(a*b^2)^(2/3)*x)/(a*b)) + 3*sqrt(3)*(9*a^2*b^2*f*x^8 + 2*(2*b^4*c + a*b^3*d - 4
*a^2*b^2*e + 16*a^3*b*f)*x^5 + (7*a*b^3*c - a^2*b^2*d - 5*a^3*b*e + 20*a^4*f)*x^
2)*(a*b^2)^(1/3))/((a^2*b^5*x^6 + 2*a^3*b^4*x^3 + a^4*b^3)*(a*b^2)^(1/3))

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.220603, size = 522, normalized size = 1.73 \[ \frac{f x^{2}}{2 \, b^{3}} - \frac{{\left (2 \, b^{3} c \left (-\frac{a}{b}\right )^{\frac{1}{3}} + a b^{2} d \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 20 \, a^{3} f \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 5 \, a^{2} b \left (-\frac{a}{b}\right )^{\frac{1}{3}} e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{27 \, a^{3} b^{3}} - \frac{\sqrt{3}{\left (2 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c + \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d - 20 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + 5 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{2} b e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{27 \, a^{3} b^{5}} + \frac{4 \, b^{4} c x^{5} + 2 \, a b^{3} d x^{5} + 14 \, a^{3} b f x^{5} - 8 \, a^{2} b^{2} x^{5} e + 7 \, a b^{3} c x^{2} - a^{2} b^{2} d x^{2} + 11 \, a^{4} f x^{2} - 5 \, a^{3} b x^{2} e}{18 \,{\left (b x^{3} + a\right )}^{2} a^{2} b^{3}} + \frac{{\left (2 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c + \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d - 20 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + 5 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{2} b e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{54 \, a^{3} b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*f*x^2/b^3 - 1/27*(2*b^3*c*(-a/b)^(1/3) + a*b^2*d*(-a/b)^(1/3) - 20*a^3*f*(-a
/b)^(1/3) + 5*a^2*b*(-a/b)^(1/3)*e)*(-a/b)^(1/3)*ln(abs(x - (-a/b)^(1/3)))/(a^3*
b^3) - 1/27*sqrt(3)*(2*(-a*b^2)^(2/3)*b^3*c + (-a*b^2)^(2/3)*a*b^2*d - 20*(-a*b^
2)^(2/3)*a^3*f + 5*(-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^3*b^5) + 1/18*(4*b^4*c*x^5 + 2*a*b^3*d*x^5 + 14*a^3*b*f*x^5
- 8*a^2*b^2*x^5*e + 7*a*b^3*c*x^2 - a^2*b^2*d*x^2 + 11*a^4*f*x^2 - 5*a^3*b*x^2*e
)/((b*x^3 + a)^2*a^2*b^3) + 1/54*(2*(-a*b^2)^(2/3)*b^3*c + (-a*b^2)^(2/3)*a*b^2*
d - 20*(-a*b^2)^(2/3)*a^3*f + 5*(-a*b^2)^(2/3)*a^2*b*e)*ln(x^2 + x*(-a/b)^(1/3)
+ (-a/b)^(2/3))/(a^3*b^5)