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

Optimal. Leaf size=292 \[ \frac{x \left (13 a^3 f-7 a^2 b e+a b^2 d+5 b^3 c\right )}{18 a^2 b^3 \left (a+b x^3\right )}+\frac{x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a b^3 \left (a+b x^3\right )^2}-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-14 a^3 f+2 a^2 b e+a b^2 d+5 b^3 c\right )}{54 a^{8/3} b^{10/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-14 a^3 f+2 a^2 b e+a b^2 d+5 b^3 c\right )}{27 a^{8/3} b^{10/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-14 a^3 f+2 a^2 b e+a b^2 d+5 b^3 c\right )}{9 \sqrt{3} a^{8/3} b^{10/3}}+\frac{f x}{b^3} \]

[Out]

(f*x)/b^3 + ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(6*a*b^3*(a + b*x^3)^2) + ((
5*b^3*c + a*b^2*d - 7*a^2*b*e + 13*a^3*f)*x)/(18*a^2*b^3*(a + b*x^3)) - ((5*b^3*
c + a*b^2*d + 2*a^2*b*e - 14*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1
/3))])/(9*Sqrt[3]*a^(8/3)*b^(10/3)) + ((5*b^3*c + a*b^2*d + 2*a^2*b*e - 14*a^3*f
)*Log[a^(1/3) + b^(1/3)*x])/(27*a^(8/3)*b^(10/3)) - ((5*b^3*c + a*b^2*d + 2*a^2*
b*e - 14*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(8/3)*b^(1
0/3))

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Rubi [A]  time = 0.684743, antiderivative size = 292, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{x \left (13 a^3 f-7 a^2 b e+a b^2 d+5 b^3 c\right )}{18 a^2 b^3 \left (a+b x^3\right )}+\frac{x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a b^3 \left (a+b x^3\right )^2}-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-14 a^3 f+2 a^2 b e+a b^2 d+5 b^3 c\right )}{54 a^{8/3} b^{10/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-14 a^3 f+2 a^2 b e+a b^2 d+5 b^3 c\right )}{27 a^{8/3} b^{10/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-14 a^3 f+2 a^2 b e+a b^2 d+5 b^3 c\right )}{9 \sqrt{3} a^{8/3} b^{10/3}}+\frac{f x}{b^3} \]

Antiderivative was successfully verified.

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

[Out]

(f*x)/b^3 + ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(6*a*b^3*(a + b*x^3)^2) + ((
5*b^3*c + a*b^2*d - 7*a^2*b*e + 13*a^3*f)*x)/(18*a^2*b^3*(a + b*x^3)) - ((5*b^3*
c + a*b^2*d + 2*a^2*b*e - 14*a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1
/3))])/(9*Sqrt[3]*a^(8/3)*b^(10/3)) + ((5*b^3*c + a*b^2*d + 2*a^2*b*e - 14*a^3*f
)*Log[a^(1/3) + b^(1/3)*x])/(27*a^(8/3)*b^(10/3)) - ((5*b^3*c + a*b^2*d + 2*a^2*
b*e - 14*a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(8/3)*b^(1
0/3))

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Rubi in Sympy [A]  time = 116.6, size = 289, normalized size = 0.99 \[ \frac{f x}{b^{3}} - \frac{x \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{6 a b^{3} \left (a + b x^{3}\right )^{2}} + \frac{x \left (13 a^{3} f - 7 a^{2} b e + a b^{2} d + 5 b^{3} c\right )}{18 a^{2} b^{3} \left (a + b x^{3}\right )} - \frac{\left (14 a^{3} f - 2 a^{2} b e - a b^{2} d - 5 b^{3} c\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{27 a^{\frac{8}{3}} b^{\frac{10}{3}}} + \frac{\left (14 a^{3} f - 2 a^{2} b e - a b^{2} d - 5 b^{3} c\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{54 a^{\frac{8}{3}} b^{\frac{10}{3}}} + \frac{\sqrt{3} \left (14 a^{3} f - 2 a^{2} b e - a b^{2} d - 5 b^{3} c\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{27 a^{\frac{8}{3}} b^{\frac{10}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

f*x/b**3 - x*(a**3*f - a**2*b*e + a*b**2*d - b**3*c)/(6*a*b**3*(a + b*x**3)**2)
+ x*(13*a**3*f - 7*a**2*b*e + a*b**2*d + 5*b**3*c)/(18*a**2*b**3*(a + b*x**3)) -
(14*a**3*f - 2*a**2*b*e - a*b**2*d - 5*b**3*c)*log(a**(1/3) + b**(1/3)*x)/(27*a
**(8/3)*b**(10/3)) + (14*a**3*f - 2*a**2*b*e - a*b**2*d - 5*b**3*c)*log(a**(2/3)
- a**(1/3)*b**(1/3)*x + b**(2/3)*x**2)/(54*a**(8/3)*b**(10/3)) + sqrt(3)*(14*a*
*3*f - 2*a**2*b*e - a*b**2*d - 5*b**3*c)*atan(sqrt(3)*(a**(1/3)/3 - 2*b**(1/3)*x
/3)/a**(1/3))/(27*a**(8/3)*b**(10/3))

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Mathematica [A]  time = 0.355843, size = 279, normalized size = 0.96 \[ \frac{\frac{3 \sqrt [3]{b} x \left (13 a^3 f-7 a^2 b e+a b^2 d+5 b^3 c\right )}{a^2 \left (a+b x^3\right )}+\frac{9 \sqrt [3]{b} x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{a \left (a+b x^3\right )^2}+\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-14 a^3 f+2 a^2 b e+a b^2 d+5 b^3 c\right )}{a^{8/3}}-\frac{2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (-14 a^3 f+2 a^2 b e+a b^2 d+5 b^3 c\right )}{a^{8/3}}-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-14 a^3 f+2 a^2 b e+a b^2 d+5 b^3 c\right )}{a^{8/3}}+54 \sqrt [3]{b} f x}{54 b^{10/3}} \]

Antiderivative was successfully verified.

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

[Out]

(54*b^(1/3)*f*x + (9*b^(1/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(a*(a + b*x^
3)^2) + (3*b^(1/3)*(5*b^3*c + a*b^2*d - 7*a^2*b*e + 13*a^3*f)*x)/(a^2*(a + b*x^3
)) - (2*Sqrt[3]*(5*b^3*c + a*b^2*d + 2*a^2*b*e - 14*a^3*f)*ArcTan[(1 - (2*b^(1/3
)*x)/a^(1/3))/Sqrt[3]])/a^(8/3) + (2*(5*b^3*c + a*b^2*d + 2*a^2*b*e - 14*a^3*f)*
Log[a^(1/3) + b^(1/3)*x])/a^(8/3) - ((5*b^3*c + a*b^2*d + 2*a^2*b*e - 14*a^3*f)*
Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(8/3))/(54*b^(10/3))

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Maple [B]  time = 0.017, size = 539, normalized size = 1.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

f*x/b^3+13/18/b^2/(b*x^3+a)^2*x^4*a*f-7/18/b/(b*x^3+a)^2*x^4*e+1/18/(b*x^3+a)^2/
a*x^4*d+5/18*b/(b*x^3+a)^2/a^2*x^4*c+5/9/b^3/(b*x^3+a)^2*a^2*f*x-2/9/b^2/(b*x^3+
a)^2*a*e*x-1/9/b/(b*x^3+a)^2*d*x+4/9*c/a*x/(b*x^3+a)^2-14/27/b^4*a/(a/b)^(2/3)*l
n(x+(a/b)^(1/3))*f+2/27/b^3/(a/b)^(2/3)*ln(x+(a/b)^(1/3))*e+1/27/b^2/a/(a/b)^(2/
3)*ln(x+(a/b)^(1/3))*d+5/27*c/a^2/b/(a/b)^(2/3)*ln(x+(a/b)^(1/3))+7/27/b^4*a/(a/
b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*f-1/27/b^3/(a/b)^(2/3)*ln(x^2-x*(a/b)
^(1/3)+(a/b)^(2/3))*e-1/54/b^2/a/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*d
-5/54*c/a^2/b/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))-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))*f+2/27/b^3/(a/b)^(2/3)*3^(1/
2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*e+1/27/b^2/a/(a/b)^(2/3)*3^(1/2)*arct
an(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*d+5/27*c/a^2/b/(a/b)^(2/3)*3^(1/2)*arctan(1/
3*3^(1/2)*(2/(a/b)^(1/3)*x-1))

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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)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.219346, size = 710, normalized size = 2.43 \[ \frac{\sqrt{3}{\left (\sqrt{3}{\left ({\left (5 \, b^{5} c + a b^{4} d + 2 \, a^{2} b^{3} e - 14 \, a^{3} b^{2} f\right )} x^{6} + 5 \, a^{2} b^{3} c + a^{3} b^{2} d + 2 \, a^{4} b e - 14 \, a^{5} f + 2 \,{\left (5 \, a b^{4} c + a^{2} b^{3} d + 2 \, a^{3} b^{2} e - 14 \, a^{4} b f\right )} x^{3}\right )} \log \left (\left (-a^{2} b\right )^{\frac{2}{3}} x^{2} + \left (-a^{2} b\right )^{\frac{1}{3}} a x + a^{2}\right ) - 2 \, \sqrt{3}{\left ({\left (5 \, b^{5} c + a b^{4} d + 2 \, a^{2} b^{3} e - 14 \, a^{3} b^{2} f\right )} x^{6} + 5 \, a^{2} b^{3} c + a^{3} b^{2} d + 2 \, a^{4} b e - 14 \, a^{5} f + 2 \,{\left (5 \, a b^{4} c + a^{2} b^{3} d + 2 \, a^{3} b^{2} e - 14 \, a^{4} b f\right )} x^{3}\right )} \log \left (\left (-a^{2} b\right )^{\frac{1}{3}} x - a\right ) + 6 \,{\left ({\left (5 \, b^{5} c + a b^{4} d + 2 \, a^{2} b^{3} e - 14 \, a^{3} b^{2} f\right )} x^{6} + 5 \, a^{2} b^{3} c + a^{3} b^{2} d + 2 \, a^{4} b e - 14 \, a^{5} f + 2 \,{\left (5 \, a b^{4} c + a^{2} b^{3} d + 2 \, a^{3} b^{2} e - 14 \, a^{4} b f\right )} x^{3}\right )} \arctan \left (\frac{2 \, \sqrt{3} \left (-a^{2} b\right )^{\frac{1}{3}} x + \sqrt{3} a}{3 \, a}\right ) + 3 \, \sqrt{3}{\left (18 \, a^{2} b^{2} f x^{7} +{\left (5 \, b^{4} c + a b^{3} d - 7 \, a^{2} b^{2} e + 49 \, a^{3} b f\right )} x^{4} + 2 \,{\left (4 \, a b^{3} c - a^{2} b^{2} d - 2 \, a^{3} b e + 14 \, a^{4} f\right )} x\right )} \left (-a^{2} b\right )^{\frac{1}{3}}\right )}}{162 \,{\left (a^{2} b^{5} x^{6} + 2 \, a^{3} b^{4} x^{3} + a^{4} b^{3}\right )} \left (-a^{2} b\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/162*sqrt(3)*(sqrt(3)*((5*b^5*c + a*b^4*d + 2*a^2*b^3*e - 14*a^3*b^2*f)*x^6 + 5
*a^2*b^3*c + a^3*b^2*d + 2*a^4*b*e - 14*a^5*f + 2*(5*a*b^4*c + a^2*b^3*d + 2*a^3
*b^2*e - 14*a^4*b*f)*x^3)*log((-a^2*b)^(2/3)*x^2 + (-a^2*b)^(1/3)*a*x + a^2) - 2
*sqrt(3)*((5*b^5*c + a*b^4*d + 2*a^2*b^3*e - 14*a^3*b^2*f)*x^6 + 5*a^2*b^3*c + a
^3*b^2*d + 2*a^4*b*e - 14*a^5*f + 2*(5*a*b^4*c + a^2*b^3*d + 2*a^3*b^2*e - 14*a^
4*b*f)*x^3)*log((-a^2*b)^(1/3)*x - a) + 6*((5*b^5*c + a*b^4*d + 2*a^2*b^3*e - 14
*a^3*b^2*f)*x^6 + 5*a^2*b^3*c + a^3*b^2*d + 2*a^4*b*e - 14*a^5*f + 2*(5*a*b^4*c
+ a^2*b^3*d + 2*a^3*b^2*e - 14*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)*(18*a^2*b^2*f*x^7 + (5*b^4*c + a*b^3*d - 7*a^2*b
^2*e + 49*a^3*b*f)*x^4 + 2*(4*a*b^3*c - a^2*b^2*d - 2*a^3*b*e + 14*a^4*f)*x)*(-a
^2*b)^(1/3))/((a^2*b^5*x^6 + 2*a^3*b^4*x^3 + a^4*b^3)*(-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((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 = 0.219098, size = 463, normalized size = 1.59 \[ \frac{f x}{b^{3}} - \frac{{\left (5 \, b^{3} c + a b^{2} d - 14 \, a^{3} f + 2 \, a^{2} b e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{27 \, a^{3} b^{3}} + \frac{\sqrt{3}{\left (5 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c + \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d - 14 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f + 2 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{27 \, a^{3} b^{4}} + \frac{{\left (5 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c + \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d - 14 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f + 2 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{54 \, a^{3} b^{4}} + \frac{5 \, b^{4} c x^{4} + a b^{3} d x^{4} + 13 \, a^{3} b f x^{4} - 7 \, a^{2} b^{2} x^{4} e + 8 \, a b^{3} c x - 2 \, a^{2} b^{2} d x + 10 \, a^{4} f x - 4 \, a^{3} b x e}{18 \,{\left (b x^{3} + a\right )}^{2} a^{2} b^{3}} \]

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)^3,x, algorithm="giac")

[Out]

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