3.291 \(\int \frac{x^4 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx\)

Optimal. Leaf size=316 \[ \frac{x^2 \left (-10 a^3 f+7 a^2 b e-4 a b^2 d+b^3 c\right )}{9 a b^4 \left (a+b x^3\right )}-\frac{x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^4 \left (a+b x^3\right )^2}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (44 a^3 f-20 a^2 b e+5 a b^2 d+b^3 c\right )}{54 a^{4/3} b^{14/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (44 a^3 f-20 a^2 b e+5 a b^2 d+b^3 c\right )}{27 a^{4/3} b^{14/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (44 a^3 f-20 a^2 b e+5 a b^2 d+b^3 c\right )}{9 \sqrt{3} a^{4/3} b^{14/3}}+\frac{x^2 (b e-3 a f)}{2 b^4}+\frac{f x^5}{5 b^3} \]

[Out]

((b*e - 3*a*f)*x^2)/(2*b^4) + (f*x^5)/(5*b^3) - ((b^3*c - a*b^2*d + a^2*b*e - a^
3*f)*x^2)/(6*b^4*(a + b*x^3)^2) + ((b^3*c - 4*a*b^2*d + 7*a^2*b*e - 10*a^3*f)*x^
2)/(9*a*b^4*(a + b*x^3)) - ((b^3*c + 5*a*b^2*d - 20*a^2*b*e + 44*a^3*f)*ArcTan[(
a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(4/3)*b^(14/3)) - ((b^3*
c + 5*a*b^2*d - 20*a^2*b*e + 44*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(27*a^(4/3)*b^(
14/3)) + ((b^3*c + 5*a*b^2*d - 20*a^2*b*e + 44*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1
/3)*x + b^(2/3)*x^2])/(54*a^(4/3)*b^(14/3))

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Rubi [A]  time = 1.01798, antiderivative size = 316, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 10, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{x^2 \left (-10 a^3 f+7 a^2 b e-4 a b^2 d+b^3 c\right )}{9 a b^4 \left (a+b x^3\right )}-\frac{x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^4 \left (a+b x^3\right )^2}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (44 a^3 f-20 a^2 b e+5 a b^2 d+b^3 c\right )}{54 a^{4/3} b^{14/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (44 a^3 f-20 a^2 b e+5 a b^2 d+b^3 c\right )}{27 a^{4/3} b^{14/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (44 a^3 f-20 a^2 b e+5 a b^2 d+b^3 c\right )}{9 \sqrt{3} a^{4/3} b^{14/3}}+\frac{x^2 (b e-3 a f)}{2 b^4}+\frac{f x^5}{5 b^3} \]

Antiderivative was successfully verified.

[In]  Int[(x^4*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^3,x]

[Out]

((b*e - 3*a*f)*x^2)/(2*b^4) + (f*x^5)/(5*b^3) - ((b^3*c - a*b^2*d + a^2*b*e - a^
3*f)*x^2)/(6*b^4*(a + b*x^3)^2) + ((b^3*c - 4*a*b^2*d + 7*a^2*b*e - 10*a^3*f)*x^
2)/(9*a*b^4*(a + b*x^3)) - ((b^3*c + 5*a*b^2*d - 20*a^2*b*e + 44*a^3*f)*ArcTan[(
a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(4/3)*b^(14/3)) - ((b^3*
c + 5*a*b^2*d - 20*a^2*b*e + 44*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(27*a^(4/3)*b^(
14/3)) + ((b^3*c + 5*a*b^2*d - 20*a^2*b*e + 44*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1
/3)*x + b^(2/3)*x^2])/(54*a^(4/3)*b^(14/3))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4*(f*x**9+e*x**6+d*x**3+c)/(b*x**3+a)**3,x)

[Out]

Timed out

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Mathematica [A]  time = 0.394382, size = 300, normalized size = 0.95 \[ \frac{\frac{30 b^{2/3} x^2 \left (-10 a^3 f+7 a^2 b e-4 a b^2 d+b^3 c\right )}{a \left (a+b x^3\right )}-\frac{45 b^{2/3} x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{\left (a+b x^3\right )^2}-\frac{10 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (44 a^3 f-20 a^2 b e+5 a b^2 d+b^3 c\right )}{a^{4/3}}-\frac{10 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (44 a^3 f-20 a^2 b e+5 a b^2 d+b^3 c\right )}{a^{4/3}}+\frac{5 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (44 a^3 f-20 a^2 b e+5 a b^2 d+b^3 c\right )}{a^{4/3}}+135 b^{2/3} x^2 (b e-3 a f)+54 b^{5/3} f x^5}{270 b^{14/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^4*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^3,x]

[Out]

(135*b^(2/3)*(b*e - 3*a*f)*x^2 + 54*b^(5/3)*f*x^5 - (45*b^(2/3)*(b^3*c - a*b^2*d
+ a^2*b*e - a^3*f)*x^2)/(a + b*x^3)^2 + (30*b^(2/3)*(b^3*c - 4*a*b^2*d + 7*a^2*
b*e - 10*a^3*f)*x^2)/(a*(a + b*x^3)) - (10*Sqrt[3]*(b^3*c + 5*a*b^2*d - 20*a^2*b
*e + 44*a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(4/3) - (10*(b^3*c
+ 5*a*b^2*d - 20*a^2*b*e + 44*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/a^(4/3) + (5*(b^
3*c + 5*a*b^2*d - 20*a^2*b*e + 44*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/
3)*x^2])/a^(4/3))/(270*b^(14/3))

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Maple [B]  time = 0.017, size = 574, normalized size = 1.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^3,x)

[Out]

1/5*f*x^5/b^3-3/2/b^4*x^2*a*f+1/2/b^3*x^2*e-10/9/b^3/(b*x^3+a)^2*a^2*x^5*f+7/9/b
^2/(b*x^3+a)^2*a*x^5*e-4/9/b/(b*x^3+a)^2*x^5*d+1/9/(b*x^3+a)^2/a*x^5*c-17/18/b^4
/(b*x^3+a)^2*x^2*a^3*f+11/18/b^3/(b*x^3+a)^2*x^2*a^2*e-5/18/b^2/(b*x^3+a)^2*x^2*
a*d-1/18/b/(b*x^3+a)^2*x^2*c-44/27/b^5*a^2/(a/b)^(1/3)*ln(x+(a/b)^(1/3))*f+20/27
/b^4*a/(a/b)^(1/3)*ln(x+(a/b)^(1/3))*e-5/27/b^3/(a/b)^(1/3)*ln(x+(a/b)^(1/3))*d-
1/27/b^2/a/(a/b)^(1/3)*ln(x+(a/b)^(1/3))*c+22/27/b^5*a^2/(a/b)^(1/3)*ln(x^2-x*(a
/b)^(1/3)+(a/b)^(2/3))*f-10/27/b^4*a/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3
))*e+5/54/b^3/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*d+1/54/b^2/a/(a/b)^(
1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*c+44/27/b^5*a^2*3^(1/2)/(a/b)^(1/3)*arcta
n(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*f-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))*e+5/27/b^3*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(
2/(a/b)^(1/3)*x-1))*d+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))*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^4/(b*x^3 + a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.236581, size = 747, normalized size = 2.36 \[ -\frac{\sqrt{3}{\left (5 \, \sqrt{3}{\left ({\left (b^{5} c + 5 \, a b^{4} d - 20 \, a^{2} b^{3} e + 44 \, a^{3} b^{2} f\right )} x^{6} + a^{2} b^{3} c + 5 \, a^{3} b^{2} d - 20 \, a^{4} b e + 44 \, a^{5} f + 2 \,{\left (a b^{4} c + 5 \, a^{2} b^{3} d - 20 \, a^{3} b^{2} e + 44 \, a^{4} b f\right )} x^{3}\right )} \log \left (\left (-a b^{2}\right )^{\frac{1}{3}} b x^{2} - a b + \left (-a b^{2}\right )^{\frac{2}{3}} x\right ) - 10 \, \sqrt{3}{\left ({\left (b^{5} c + 5 \, a b^{4} d - 20 \, a^{2} b^{3} e + 44 \, a^{3} b^{2} f\right )} x^{6} + a^{2} b^{3} c + 5 \, a^{3} b^{2} d - 20 \, a^{4} b e + 44 \, a^{5} f + 2 \,{\left (a b^{4} c + 5 \, a^{2} b^{3} d - 20 \, a^{3} b^{2} e + 44 \, a^{4} b f\right )} x^{3}\right )} \log \left (a b + \left (-a b^{2}\right )^{\frac{2}{3}} x\right ) + 30 \,{\left ({\left (b^{5} c + 5 \, a b^{4} d - 20 \, a^{2} b^{3} e + 44 \, a^{3} b^{2} f\right )} x^{6} + a^{2} b^{3} c + 5 \, a^{3} b^{2} d - 20 \, a^{4} b e + 44 \, a^{5} f + 2 \,{\left (a b^{4} c + 5 \, a^{2} b^{3} d - 20 \, a^{3} b^{2} e + 44 \, a^{4} b f\right )} x^{3}\right )} \arctan \left (-\frac{\sqrt{3} a b - 2 \, \sqrt{3} \left (-a b^{2}\right )^{\frac{2}{3}} x}{3 \, a b}\right ) - 3 \, \sqrt{3}{\left (18 \, a b^{3} f x^{11} + 9 \,{\left (5 \, a b^{3} e - 11 \, a^{2} b^{2} f\right )} x^{8} + 2 \,{\left (5 \, b^{4} c - 20 \, a b^{3} d + 80 \, a^{2} b^{2} e - 176 \, a^{3} b f\right )} x^{5} - 5 \,{\left (a b^{3} c + 5 \, a^{2} b^{2} d - 20 \, a^{3} b e + 44 \, a^{4} f\right )} x^{2}\right )} \left (-a b^{2}\right )^{\frac{1}{3}}\right )}}{810 \,{\left (a b^{6} x^{6} + 2 \, a^{2} b^{5} x^{3} + a^{3} b^{4}\right )} \left (-a b^{2}\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^9 + e*x^6 + d*x^3 + c)*x^4/(b*x^3 + a)^3,x, algorithm="fricas")

[Out]

-1/810*sqrt(3)*(5*sqrt(3)*((b^5*c + 5*a*b^4*d - 20*a^2*b^3*e + 44*a^3*b^2*f)*x^6
+ a^2*b^3*c + 5*a^3*b^2*d - 20*a^4*b*e + 44*a^5*f + 2*(a*b^4*c + 5*a^2*b^3*d -
20*a^3*b^2*e + 44*a^4*b*f)*x^3)*log((-a*b^2)^(1/3)*b*x^2 - a*b + (-a*b^2)^(2/3)*
x) - 10*sqrt(3)*((b^5*c + 5*a*b^4*d - 20*a^2*b^3*e + 44*a^3*b^2*f)*x^6 + a^2*b^3
*c + 5*a^3*b^2*d - 20*a^4*b*e + 44*a^5*f + 2*(a*b^4*c + 5*a^2*b^3*d - 20*a^3*b^2
*e + 44*a^4*b*f)*x^3)*log(a*b + (-a*b^2)^(2/3)*x) + 30*((b^5*c + 5*a*b^4*d - 20*
a^2*b^3*e + 44*a^3*b^2*f)*x^6 + a^2*b^3*c + 5*a^3*b^2*d - 20*a^4*b*e + 44*a^5*f
+ 2*(a*b^4*c + 5*a^2*b^3*d - 20*a^3*b^2*e + 44*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)*(18*a*b^3*f*x^11 + 9*(5*a
*b^3*e - 11*a^2*b^2*f)*x^8 + 2*(5*b^4*c - 20*a*b^3*d + 80*a^2*b^2*e - 176*a^3*b*
f)*x^5 - 5*(a*b^3*c + 5*a^2*b^2*d - 20*a^3*b*e + 44*a^4*f)*x^2)*(-a*b^2)^(1/3))/
((a*b^6*x^6 + 2*a^2*b^5*x^3 + a^3*b^4)*(-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**4*(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 = 0.221729, size = 558, normalized size = 1.77 \[ -\frac{{\left (b^{3} c \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 5 \, a b^{2} d \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 44 \, a^{3} f \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 20 \, a^{2} b \left (-\frac{a}{b}\right )^{\frac{1}{3}} e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{27 \, a^{2} b^{4}} - \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c + 5 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d + 44 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f - 20 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{2} b e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{27 \, a^{2} b^{6}} + \frac{2 \, b^{4} c x^{5} - 8 \, a b^{3} d x^{5} - 20 \, a^{3} b f x^{5} + 14 \, a^{2} b^{2} x^{5} e - a b^{3} c x^{2} - 5 \, a^{2} b^{2} d x^{2} - 17 \, a^{4} f x^{2} + 11 \, a^{3} b x^{2} e}{18 \,{\left (b x^{3} + a\right )}^{2} a b^{4}} + \frac{{\left (\left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c + 5 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d + 44 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f - 20 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{2} b e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{54 \, a^{2} b^{6}} + \frac{2 \, b^{12} f x^{5} - 15 \, a b^{11} f x^{2} + 5 \, b^{12} x^{2} e}{10 \, b^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^9 + e*x^6 + d*x^3 + c)*x^4/(b*x^3 + a)^3,x, algorithm="giac")

[Out]

-1/27*(b^3*c*(-a/b)^(1/3) + 5*a*b^2*d*(-a/b)^(1/3) + 44*a^3*f*(-a/b)^(1/3) - 20*
a^2*b*(-a/b)^(1/3)*e)*(-a/b)^(1/3)*ln(abs(x - (-a/b)^(1/3)))/(a^2*b^4) - 1/27*sq
rt(3)*((-a*b^2)^(2/3)*b^3*c + 5*(-a*b^2)^(2/3)*a*b^2*d + 44*(-a*b^2)^(2/3)*a^3*f
- 20*(-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^2*b^6) + 1/18*(2*b^4*c*x^5 - 8*a*b^3*d*x^5 - 20*a^3*b*f*x^5 + 14*a^2*b^2*
x^5*e - a*b^3*c*x^2 - 5*a^2*b^2*d*x^2 - 17*a^4*f*x^2 + 11*a^3*b*x^2*e)/((b*x^3 +
a)^2*a*b^4) + 1/54*((-a*b^2)^(2/3)*b^3*c + 5*(-a*b^2)^(2/3)*a*b^2*d + 44*(-a*b^
2)^(2/3)*a^3*f - 20*(-a*b^2)^(2/3)*a^2*b*e)*ln(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/
3))/(a^2*b^6) + 1/10*(2*b^12*f*x^5 - 15*a*b^11*f*x^2 + 5*b^12*x^2*e)/b^15