3.416 \(\int \frac{c+d x+e x^2+f x^3+g x^4+h x^5}{x^3 \left (a+b x^3\right )^3} \, dx\)

Optimal. Leaf size=360 \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (5 \sqrt [3]{b} (4 b c-a f)-2 \sqrt [3]{a} (7 b d-a g)\right )}{54 a^{11/3} b^{2/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (5 \sqrt [3]{b} (4 b c-a f)-2 \sqrt [3]{a} (7 b d-a g)\right )}{27 a^{11/3} b^{2/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-2 a^{4/3} g+14 \sqrt [3]{a} b d-5 a \sqrt [3]{b} f+20 b^{4/3} c\right )}{9 \sqrt{3} a^{11/3} b^{2/3}}-\frac{x \left (2 x (5 b d-2 a g)+3 x^2 (3 b e-a h)-5 a f+11 b c\right )}{18 a^3 \left (a+b x^3\right )}-\frac{e \log \left (a+b x^3\right )}{3 a^3}-\frac{c}{2 a^3 x^2}-\frac{d}{a^3 x}+\frac{e \log (x)}{a^3}-\frac{x \left (x (b d-a g)+x^2 (b e-a h)-a f+b c\right )}{6 a^2 \left (a+b x^3\right )^2} \]

[Out]

-c/(2*a^3*x^2) - d/(a^3*x) - (x*(b*c - a*f + (b*d - a*g)*x + (b*e - a*h)*x^2))/(
6*a^2*(a + b*x^3)^2) - (x*(11*b*c - 5*a*f + 2*(5*b*d - 2*a*g)*x + 3*(3*b*e - a*h
)*x^2))/(18*a^3*(a + b*x^3)) + ((20*b^(4/3)*c + 14*a^(1/3)*b*d - 5*a*b^(1/3)*f -
2*a^(4/3)*g)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(1
1/3)*b^(2/3)) + (e*Log[x])/a^3 - ((5*b^(1/3)*(4*b*c - a*f) - 2*a^(1/3)*(7*b*d -
a*g))*Log[a^(1/3) + b^(1/3)*x])/(27*a^(11/3)*b^(2/3)) + ((5*b^(1/3)*(4*b*c - a*f
) - 2*a^(1/3)*(7*b*d - a*g))*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54
*a^(11/3)*b^(2/3)) - (e*Log[a + b*x^3])/(3*a^3)

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Rubi [A]  time = 1.56156, antiderivative size = 357, normalized size of antiderivative = 0.99, number of steps used = 12, number of rules used = 9, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.237 \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-\frac{2 \sqrt [3]{a} (7 b d-a g)}{\sqrt [3]{b}}-5 a f+20 b c\right )}{54 a^{11/3} \sqrt [3]{b}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (5 \sqrt [3]{b} (4 b c-a f)-2 \sqrt [3]{a} (7 b d-a g)\right )}{27 a^{11/3} b^{2/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-2 a^{4/3} g+14 \sqrt [3]{a} b d-5 a \sqrt [3]{b} f+20 b^{4/3} c\right )}{9 \sqrt{3} a^{11/3} b^{2/3}}-\frac{x \left (2 x (5 b d-2 a g)+3 x^2 (3 b e-a h)-5 a f+11 b c\right )}{18 a^3 \left (a+b x^3\right )}-\frac{e \log \left (a+b x^3\right )}{3 a^3}-\frac{c}{2 a^3 x^2}-\frac{d}{a^3 x}+\frac{e \log (x)}{a^3}-\frac{x \left (x (b d-a g)+x^2 (b e-a h)-a f+b c\right )}{6 a^2 \left (a+b x^3\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5)/(x^3*(a + b*x^3)^3),x]

[Out]

-c/(2*a^3*x^2) - d/(a^3*x) - (x*(b*c - a*f + (b*d - a*g)*x + (b*e - a*h)*x^2))/(
6*a^2*(a + b*x^3)^2) - (x*(11*b*c - 5*a*f + 2*(5*b*d - 2*a*g)*x + 3*(3*b*e - a*h
)*x^2))/(18*a^3*(a + b*x^3)) + ((20*b^(4/3)*c + 14*a^(1/3)*b*d - 5*a*b^(1/3)*f -
2*a^(4/3)*g)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(1
1/3)*b^(2/3)) + (e*Log[x])/a^3 - ((5*b^(1/3)*(4*b*c - a*f) - 2*a^(1/3)*(7*b*d -
a*g))*Log[a^(1/3) + b^(1/3)*x])/(27*a^(11/3)*b^(2/3)) + ((20*b*c - 5*a*f - (2*a^
(1/3)*(7*b*d - a*g))/b^(1/3))*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(5
4*a^(11/3)*b^(1/3)) - (e*Log[a + b*x^3])/(3*a^3)

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Rubi in Sympy [A]  time = 51.3888, size = 83, normalized size = 0.23 \[ \frac{x \left (\frac{6 f}{x^{3}} + \frac{6 g}{x^{2}} + \frac{6 h}{x}\right )}{18 a b \left (a + b x^{3}\right )} - \frac{x \left (\frac{a f}{x^{3}} + \frac{a g}{x^{2}} + \frac{a h}{x} - \frac{b c}{x^{3}} - \frac{b d}{x^{2}} - \frac{b e}{x}\right )}{6 a b \left (a + b x^{3}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((h*x**5+g*x**4+f*x**3+e*x**2+d*x+c)/x**3/(b*x**3+a)**3,x)

[Out]

x*(6*f/x**3 + 6*g/x**2 + 6*h/x)/(18*a*b*(a + b*x**3)) - x*(a*f/x**3 + a*g/x**2 +
a*h/x - b*c/x**3 - b*d/x**2 - b*e/x)/(6*a*b*(a + b*x**3)**2)

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Mathematica [A]  time = 1.29957, size = 337, normalized size = 0.94 \[ -\frac{-\frac{\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (2 a^{4/3} g-14 \sqrt [3]{a} b d-5 a \sqrt [3]{b} f+20 b^{4/3} c\right )}{b^{2/3}}+\frac{2 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (2 a^{4/3} g-14 \sqrt [3]{a} b d-5 a \sqrt [3]{b} f+20 b^{4/3} c\right )}{b^{2/3}}+\frac{2 \sqrt{3} \sqrt [3]{a} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (2 a^{4/3} g-14 \sqrt [3]{a} b d+5 a \sqrt [3]{b} f-20 b^{4/3} c\right )}{b^{2/3}}+\frac{9 a^2 \left (a^2 h-a b (e+x (f+g x))+b^2 x (c+d x)\right )}{b \left (a+b x^3\right )^2}-\frac{3 a (6 a e+a x (5 f+4 g x)-b x (11 c+10 d x))}{a+b x^3}+18 a e \log \left (a+b x^3\right )+\frac{27 a c}{x^2}+\frac{54 a d}{x}-54 a e \log (x)}{54 a^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5)/(x^3*(a + b*x^3)^3),x]

[Out]

-((27*a*c)/x^2 + (54*a*d)/x - (3*a*(6*a*e - b*x*(11*c + 10*d*x) + a*x*(5*f + 4*g
*x)))/(a + b*x^3) + (9*a^2*(a^2*h + b^2*x*(c + d*x) - a*b*(e + x*(f + g*x))))/(b
*(a + b*x^3)^2) + (2*Sqrt[3]*a^(1/3)*(-20*b^(4/3)*c - 14*a^(1/3)*b*d + 5*a*b^(1/
3)*f + 2*a^(4/3)*g)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(2/3) - 54*a*
e*Log[x] + (2*a^(1/3)*(20*b^(4/3)*c - 14*a^(1/3)*b*d - 5*a*b^(1/3)*f + 2*a^(4/3)
*g)*Log[a^(1/3) + b^(1/3)*x])/b^(2/3) - (a^(1/3)*(20*b^(4/3)*c - 14*a^(1/3)*b*d
- 5*a*b^(1/3)*f + 2*a^(4/3)*g)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b
^(2/3) + 18*a*e*Log[a + b*x^3])/(54*a^4)

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Maple [B]  time = 0.027, size = 626, normalized size = 1.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/x^3/(b*x^3+a)^3,x)

[Out]

1/27/a^2*g/b/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))+5/27/a^2*f/b/(a/b)^(2
/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))+4/9/a/(b*x^3+a)^2*f*x-13/18/
a^2/(b*x^3+a)^2*x^2*b*d+5/27/a^2*f/b/(a/b)^(2/3)*ln(x+(a/b)^(1/3))-5/54/a^2*f/b/
(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))-1/2*c/a^3/x^2-20/27/a^3/(a/b)^(2/3
)*ln(x+(a/b)^(1/3))*c+10/27/a^3/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*c+
14/27/a^3/(a/b)^(1/3)*ln(x+(a/b)^(1/3))*d-7/27/a^3/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1
/3)+(a/b)^(2/3))*d-1/6/(b*x^3+a)^2/b*h-d/a^3/x+2/9/a^2/(b*x^3+a)^2*x^5*b*g-7/9/a
^2/(b*x^3+a)^2*x*b*c-20/27/a^3/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(
1/3)*x-1))*c-2/27/a^2*g/b/(a/b)^(1/3)*ln(x+(a/b)^(1/3))+2/27/a^2*g*3^(1/2)/b/(a/
b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))-11/18/a^3/(b*x^3+a)^2*x^4*b^2*c
-14/27/a^3*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*d+7/18/a/
(b*x^3+a)^2*x^2*g-5/9/a^3/(b*x^3+a)^2*x^5*b^2*d+5/18/a^2/(b*x^3+a)^2*x^4*b*f+1/2
/a/(b*x^3+a)^2*e+1/3*b/a^2/(b*x^3+a)^2*e*x^3+e*ln(x)/a^3-1/3*e*ln(b*x^3+a)/a^3

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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((h*x^5 + g*x^4 + f*x^3 + e*x^2 + d*x + c)/((b*x^3 + a)^3*x^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((h*x^5 + g*x^4 + f*x^3 + e*x^2 + d*x + c)/((b*x^3 + a)^3*x^3),x, algorithm="fricas")

[Out]

Exception raised: NotImplementedError

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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((h*x**5+g*x**4+f*x**3+e*x**2+d*x+c)/x**3/(b*x**3+a)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.228878, size = 568, normalized size = 1.58 \[ -\frac{e{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{3}} + \frac{e{\rm ln}\left ({\left | x \right |}\right )}{a^{3}} - \frac{\sqrt{3}{\left (20 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{2} c - 5 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b f - 14 \, \left (-a b^{2}\right )^{\frac{2}{3}} b d + 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} a g\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{27 \, a^{4} b^{2}} - \frac{{\left (20 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{2} c - 5 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b f + 14 \, \left (-a b^{2}\right )^{\frac{2}{3}} b d - 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} a g\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{54 \, a^{4} b^{2}} - \frac{28 \, b^{3} d x^{7} - 4 \, a b^{2} g x^{7} + 20 \, b^{3} c x^{6} - 5 \, a b^{2} f x^{6} - 6 \, a b^{2} x^{5} e + 49 \, a b^{2} d x^{4} - 7 \, a^{2} b g x^{4} + 32 \, a b^{2} c x^{3} - 8 \, a^{2} b f x^{3} + 3 \, a^{3} h x^{2} - 9 \, a^{2} b x^{2} e + 18 \, a^{2} b d x + 9 \, a^{2} b c}{18 \,{\left (b x^{4} + a x\right )}^{2} a^{3} b} + \frac{{\left (14 \, a^{3} b^{2} d \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 2 \, a^{4} b g \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 20 \, a^{3} b^{2} c - 5 \, a^{4} b f\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{27 \, a^{7} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((h*x^5 + g*x^4 + f*x^3 + e*x^2 + d*x + c)/((b*x^3 + a)^3*x^3),x, algorithm="giac")

[Out]

-1/3*e*ln(abs(b*x^3 + a))/a^3 + e*ln(abs(x))/a^3 - 1/27*sqrt(3)*(20*(-a*b^2)^(1/
3)*b^2*c - 5*(-a*b^2)^(1/3)*a*b*f - 14*(-a*b^2)^(2/3)*b*d + 2*(-a*b^2)^(2/3)*a*g
)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a^4*b^2) - 1/54*(20*(-a
*b^2)^(1/3)*b^2*c - 5*(-a*b^2)^(1/3)*a*b*f + 14*(-a*b^2)^(2/3)*b*d - 2*(-a*b^2)^
(2/3)*a*g)*ln(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^4*b^2) - 1/18*(28*b^3*d*x^
7 - 4*a*b^2*g*x^7 + 20*b^3*c*x^6 - 5*a*b^2*f*x^6 - 6*a*b^2*x^5*e + 49*a*b^2*d*x^
4 - 7*a^2*b*g*x^4 + 32*a*b^2*c*x^3 - 8*a^2*b*f*x^3 + 3*a^3*h*x^2 - 9*a^2*b*x^2*e
+ 18*a^2*b*d*x + 9*a^2*b*c)/((b*x^4 + a*x)^2*a^3*b) + 1/27*(14*a^3*b^2*d*(-a/b)
^(1/3) - 2*a^4*b*g*(-a/b)^(1/3) + 20*a^3*b^2*c - 5*a^4*b*f)*(-a/b)^(1/3)*ln(abs(
x - (-a/b)^(1/3)))/(a^7*b)