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

Optimal. Leaf size=307 \[ \frac{x \left (-19 a^3 f+13 a^2 b e-7 a b^2 d+b^3 c\right )}{18 a b^4 \left (a+b x^3\right )}-\frac{x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^4 \left (a+b x^3\right )^2}-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (35 a^3 f-14 a^2 b e+2 a b^2 d+b^3 c\right )}{54 a^{5/3} b^{13/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (35 a^3 f-14 a^2 b e+2 a b^2 d+b^3 c\right )}{27 a^{5/3} b^{13/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (35 a^3 f-14 a^2 b e+2 a b^2 d+b^3 c\right )}{9 \sqrt{3} a^{5/3} b^{13/3}}+\frac{x (b e-3 a f)}{b^4}+\frac{f x^4}{4 b^3} \]

[Out]

((b*e - 3*a*f)*x)/b^4 + (f*x^4)/(4*b^3) - ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x
)/(6*b^4*(a + b*x^3)^2) + ((b^3*c - 7*a*b^2*d + 13*a^2*b*e - 19*a^3*f)*x)/(18*a*
b^4*(a + b*x^3)) - ((b^3*c + 2*a*b^2*d - 14*a^2*b*e + 35*a^3*f)*ArcTan[(a^(1/3)
- 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(5/3)*b^(13/3)) + ((b^3*c + 2*a*
b^2*d - 14*a^2*b*e + 35*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(27*a^(5/3)*b^(13/3)) -
((b^3*c + 2*a*b^2*d - 14*a^2*b*e + 35*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x +
b^(2/3)*x^2])/(54*a^(5/3)*b^(13/3))

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Rubi [A]  time = 0.87972, antiderivative size = 307, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 10, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{x \left (-19 a^3 f+13 a^2 b e-7 a b^2 d+b^3 c\right )}{18 a b^4 \left (a+b x^3\right )}-\frac{x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^4 \left (a+b x^3\right )^2}-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (35 a^3 f-14 a^2 b e+2 a b^2 d+b^3 c\right )}{54 a^{5/3} b^{13/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (35 a^3 f-14 a^2 b e+2 a b^2 d+b^3 c\right )}{27 a^{5/3} b^{13/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (35 a^3 f-14 a^2 b e+2 a b^2 d+b^3 c\right )}{9 \sqrt{3} a^{5/3} b^{13/3}}+\frac{x (b e-3 a f)}{b^4}+\frac{f x^4}{4 b^3} \]

Antiderivative was successfully verified.

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

[Out]

((b*e - 3*a*f)*x)/b^4 + (f*x^4)/(4*b^3) - ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x
)/(6*b^4*(a + b*x^3)^2) + ((b^3*c - 7*a*b^2*d + 13*a^2*b*e - 19*a^3*f)*x)/(18*a*
b^4*(a + b*x^3)) - ((b^3*c + 2*a*b^2*d - 14*a^2*b*e + 35*a^3*f)*ArcTan[(a^(1/3)
- 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(5/3)*b^(13/3)) + ((b^3*c + 2*a*
b^2*d - 14*a^2*b*e + 35*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(27*a^(5/3)*b^(13/3)) -
((b^3*c + 2*a*b^2*d - 14*a^2*b*e + 35*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x +
b^(2/3)*x^2])/(54*a^(5/3)*b^(13/3))

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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**3*(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 = 0.373461, size = 294, normalized size = 0.96 \[ \frac{\frac{6 \sqrt [3]{b} x \left (-19 a^3 f+13 a^2 b e-7 a b^2 d+b^3 c\right )}{a \left (a+b x^3\right )}-\frac{18 \sqrt [3]{b} x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{\left (a+b x^3\right )^2}+\frac{4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (35 a^3 f-14 a^2 b e+2 a b^2 d+b^3 c\right )}{a^{5/3}}-\frac{4 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (35 a^3 f-14 a^2 b e+2 a b^2 d+b^3 c\right )}{a^{5/3}}-\frac{2 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (35 a^3 f-14 a^2 b e+2 a b^2 d+b^3 c\right )}{a^{5/3}}+108 \sqrt [3]{b} x (b e-3 a f)+27 b^{4/3} f x^4}{108 b^{13/3}} \]

Antiderivative was successfully verified.

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

[Out]

(108*b^(1/3)*(b*e - 3*a*f)*x + 27*b^(4/3)*f*x^4 - (18*b^(1/3)*(b^3*c - a*b^2*d +
a^2*b*e - a^3*f)*x)/(a + b*x^3)^2 + (6*b^(1/3)*(b^3*c - 7*a*b^2*d + 13*a^2*b*e
- 19*a^3*f)*x)/(a*(a + b*x^3)) - (4*Sqrt[3]*(b^3*c + 2*a*b^2*d - 14*a^2*b*e + 35
*a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(5/3) + (4*(b^3*c + 2*a*b
^2*d - 14*a^2*b*e + 35*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/a^(5/3) - (2*(b^3*c + 2*
a*b^2*d - 14*a^2*b*e + 35*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])
/a^(5/3))/(108*b^(13/3))

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Maple [B]  time = 0.017, size = 561, normalized size = 1.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*f*x^4/b^3-3/b^4*a*f*x+1/b^3*e*x-19/18/b^3/(b*x^3+a)^2*x^4*a^2*f+13/18/b^2/(b
*x^3+a)^2*x^4*a*e-7/18/b/(b*x^3+a)^2*x^4*d+1/18/(b*x^3+a)^2/a*x^4*c-8/9/b^4/(b*x
^3+a)^2*a^3*f*x+5/9/b^3/(b*x^3+a)^2*a^2*e*x-2/9/b^2/(b*x^3+a)^2*a*d*x-1/9/b/(b*x
^3+a)^2*c*x+35/27/b^5*a^2/(a/b)^(2/3)*ln(x+(a/b)^(1/3))*f-14/27/b^4*a/(a/b)^(2/3
)*ln(x+(a/b)^(1/3))*e+2/27/b^3/(a/b)^(2/3)*ln(x+(a/b)^(1/3))*d+1/27/b^2/a/(a/b)^
(2/3)*ln(x+(a/b)^(1/3))*c-35/54/b^5*a^2/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(
2/3))*f+7/27/b^4*a/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*e-1/27/b^3/(a/b
)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*d-1/54/b^2/a/(a/b)^(2/3)*ln(x^2-x*(a/b
)^(1/3)+(a/b)^(2/3))*c+35/27/b^5*a^2/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(
a/b)^(1/3)*x-1))*f-14/27/b^4*a/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(
1/3)*x-1))*e+2/27/b^3/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1)
)*d+1/27/b^2/a/(a/b)^(2/3)*3^(1/2)*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^3/(b*x^3 + a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.230494, size = 728, normalized size = 2.37 \[ -\frac{\sqrt{3}{\left (2 \, \sqrt{3}{\left ({\left (b^{5} c + 2 \, a b^{4} d - 14 \, a^{2} b^{3} e + 35 \, a^{3} b^{2} f\right )} x^{6} + a^{2} b^{3} c + 2 \, a^{3} b^{2} d - 14 \, a^{4} b e + 35 \, a^{5} f + 2 \,{\left (a b^{4} c + 2 \, a^{2} b^{3} d - 14 \, a^{3} b^{2} e + 35 \, a^{4} b f\right )} x^{3}\right )} \log \left (\left (a^{2} b\right )^{\frac{2}{3}} x^{2} - \left (a^{2} b\right )^{\frac{1}{3}} a x + a^{2}\right ) - 4 \, \sqrt{3}{\left ({\left (b^{5} c + 2 \, a b^{4} d - 14 \, a^{2} b^{3} e + 35 \, a^{3} b^{2} f\right )} x^{6} + a^{2} b^{3} c + 2 \, a^{3} b^{2} d - 14 \, a^{4} b e + 35 \, a^{5} f + 2 \,{\left (a b^{4} c + 2 \, a^{2} b^{3} d - 14 \, a^{3} b^{2} e + 35 \, a^{4} b f\right )} x^{3}\right )} \log \left (\left (a^{2} b\right )^{\frac{1}{3}} x + a\right ) - 12 \,{\left ({\left (b^{5} c + 2 \, a b^{4} d - 14 \, a^{2} b^{3} e + 35 \, a^{3} b^{2} f\right )} x^{6} + a^{2} b^{3} c + 2 \, a^{3} b^{2} d - 14 \, a^{4} b e + 35 \, a^{5} f + 2 \,{\left (a b^{4} c + 2 \, a^{2} b^{3} d - 14 \, a^{3} b^{2} e + 35 \, a^{4} b f\right )} x^{3}\right )} \arctan \left (\frac{2 \, \sqrt{3} \left (a^{2} b\right )^{\frac{1}{3}} x - \sqrt{3} a}{3 \, a}\right ) - 3 \, \sqrt{3}{\left (9 \, a b^{3} f x^{10} + 18 \,{\left (2 \, a b^{3} e - 5 \, a^{2} b^{2} f\right )} x^{7} +{\left (2 \, b^{4} c - 14 \, a b^{3} d + 98 \, a^{2} b^{2} e - 245 \, a^{3} b f\right )} x^{4} - 4 \,{\left (a b^{3} c + 2 \, a^{2} b^{2} d - 14 \, a^{3} b e + 35 \, a^{4} f\right )} x\right )} \left (a^{2} b\right )^{\frac{1}{3}}\right )}}{324 \,{\left (a b^{6} x^{6} + 2 \, a^{2} b^{5} x^{3} + a^{3} b^{4}\right )} \left (a^{2} b\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/324*sqrt(3)*(2*sqrt(3)*((b^5*c + 2*a*b^4*d - 14*a^2*b^3*e + 35*a^3*b^2*f)*x^6
+ a^2*b^3*c + 2*a^3*b^2*d - 14*a^4*b*e + 35*a^5*f + 2*(a*b^4*c + 2*a^2*b^3*d -
14*a^3*b^2*e + 35*a^4*b*f)*x^3)*log((a^2*b)^(2/3)*x^2 - (a^2*b)^(1/3)*a*x + a^2)
- 4*sqrt(3)*((b^5*c + 2*a*b^4*d - 14*a^2*b^3*e + 35*a^3*b^2*f)*x^6 + a^2*b^3*c
+ 2*a^3*b^2*d - 14*a^4*b*e + 35*a^5*f + 2*(a*b^4*c + 2*a^2*b^3*d - 14*a^3*b^2*e
+ 35*a^4*b*f)*x^3)*log((a^2*b)^(1/3)*x + a) - 12*((b^5*c + 2*a*b^4*d - 14*a^2*b^
3*e + 35*a^3*b^2*f)*x^6 + a^2*b^3*c + 2*a^3*b^2*d - 14*a^4*b*e + 35*a^5*f + 2*(a
*b^4*c + 2*a^2*b^3*d - 14*a^3*b^2*e + 35*a^4*b*f)*x^3)*arctan(1/3*(2*sqrt(3)*(a^
2*b)^(1/3)*x - sqrt(3)*a)/a) - 3*sqrt(3)*(9*a*b^3*f*x^10 + 18*(2*a*b^3*e - 5*a^2
*b^2*f)*x^7 + (2*b^4*c - 14*a*b^3*d + 98*a^2*b^2*e - 245*a^3*b*f)*x^4 - 4*(a*b^3
*c + 2*a^2*b^2*d - 14*a^3*b*e + 35*a^4*f)*x)*(a^2*b)^(1/3))/((a*b^6*x^6 + 2*a^2*
b^5*x^3 + a^3*b^4)*(a^2*b)^(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**3*(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 = 0.219125, size = 495, normalized size = 1.61 \[ -\frac{{\left (b^{3} c + 2 \, a b^{2} d + 35 \, a^{3} f - 14 \, a^{2} b e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{27 \, a^{2} b^{4}} + \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c + 2 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d + 35 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f - 14 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{27 \, a^{2} b^{5}} + \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c + 2 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d + 35 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f - 14 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{54 \, a^{2} b^{5}} + \frac{b^{4} c x^{4} - 7 \, a b^{3} d x^{4} - 19 \, a^{3} b f x^{4} + 13 \, a^{2} b^{2} x^{4} e - 2 \, a b^{3} c x - 4 \, a^{2} b^{2} d x - 16 \, a^{4} f x + 10 \, a^{3} b x e}{18 \,{\left (b x^{3} + a\right )}^{2} a b^{4}} + \frac{b^{9} f x^{4} - 12 \, a b^{8} f x + 4 \, b^{9} x e}{4 \, b^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/27*(b^3*c + 2*a*b^2*d + 35*a^3*f - 14*a^2*b*e)*(-a/b)^(1/3)*ln(abs(x - (-a/b)
^(1/3)))/(a^2*b^4) + 1/27*sqrt(3)*((-a*b^2)^(1/3)*b^3*c + 2*(-a*b^2)^(1/3)*a*b^2
*d + 35*(-a*b^2)^(1/3)*a^3*f - 14*(-a*b^2)^(1/3)*a^2*b*e)*arctan(1/3*sqrt(3)*(2*
x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a^2*b^5) + 1/54*((-a*b^2)^(1/3)*b^3*c + 2*(-a*b
^2)^(1/3)*a*b^2*d + 35*(-a*b^2)^(1/3)*a^3*f - 14*(-a*b^2)^(1/3)*a^2*b*e)*ln(x^2
+ x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^2*b^5) + 1/18*(b^4*c*x^4 - 7*a*b^3*d*x^4 - 1
9*a^3*b*f*x^4 + 13*a^2*b^2*x^4*e - 2*a*b^3*c*x - 4*a^2*b^2*d*x - 16*a^4*f*x + 10
*a^3*b*x*e)/((b*x^3 + a)^2*a*b^4) + 1/4*(b^9*f*x^4 - 12*a*b^8*f*x + 4*b^9*x*e)/b
^12