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

Optimal. Leaf size=395 \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (5 \sqrt [3]{b} (4 b d-a g)-2 \sqrt [3]{a} (7 b e-a h)\right )}{54 a^{11/3} b^{2/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (5 \sqrt [3]{b} (4 b d-a g)-2 \sqrt [3]{a} (7 b e-a h)\right )}{27 a^{11/3} b^{2/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-2 a^{4/3} h+14 \sqrt [3]{a} b e-5 a \sqrt [3]{b} g+20 b^{4/3} d\right )}{9 \sqrt{3} a^{11/3} b^{2/3}}+\frac{(3 b c-a f) \log \left (a+b x^3\right )}{3 a^4}-\frac{\log (x) (3 b c-a f)}{a^4}-\frac{x \left (-3 b x^2 \left (\frac{5 b c}{a}-3 f\right )+2 x (5 b e-2 a h)-5 a g+11 b d\right )}{18 a^3 \left (a+b x^3\right )}-\frac{c}{3 a^3 x^3}-\frac{d}{2 a^3 x^2}-\frac{e}{a^3 x}-\frac{x \left (-b x^2 \left (\frac{b c}{a}-f\right )+x (b e-a h)-a g+b d\right )}{6 a^2 \left (a+b x^3\right )^2} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 1.97627, antiderivative size = 392, 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 \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-\frac{2 \sqrt [3]{a} (7 b e-a h)}{\sqrt [3]{b}}-5 a g+20 b d\right )}{54 a^{11/3} \sqrt [3]{b}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (5 \sqrt [3]{b} (4 b d-a g)-2 \sqrt [3]{a} (7 b e-a h)\right )}{27 a^{11/3} b^{2/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-2 a^{4/3} h+14 \sqrt [3]{a} b e-5 a \sqrt [3]{b} g+20 b^{4/3} d\right )}{9 \sqrt{3} a^{11/3} b^{2/3}}+\frac{(3 b c-a f) \log \left (a+b x^3\right )}{3 a^4}-\frac{\log (x) (3 b c-a f)}{a^4}-\frac{x \left (-3 b x^2 \left (\frac{5 b c}{a}-3 f\right )+2 x (5 b e-2 a h)-5 a g+11 b d\right )}{18 a^3 \left (a+b x^3\right )}-\frac{c}{3 a^3 x^3}-\frac{d}{2 a^3 x^2}-\frac{e}{a^3 x}-\frac{x \left (-b x^2 \left (\frac{b c}{a}-f\right )+x (b e-a h)-a g+b d\right )}{6 a^2 \left (a+b x^3\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 49.5221, size = 88, normalized size = 0.22 \[ \frac{x \left (\frac{6 f}{x^{4}} + \frac{6 g}{x^{3}} + \frac{6 h}{x^{2}}\right )}{18 a b \left (a + b x^{3}\right )} - \frac{x \left (\frac{a f}{x^{4}} + \frac{a g}{x^{3}} + \frac{a h}{x^{2}} - \frac{b c}{x^{4}} - \frac{b d}{x^{3}} - \frac{b e}{x^{2}}\right )}{6 a b \left (a + b x^{3}\right )^{2}} \]

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**4/(b*x**3+a)**3,x)

[Out]

x*(6*f/x**4 + 6*g/x**3 + 6*h/x**2)/(18*a*b*(a + b*x**3)) - x*(a*f/x**4 + a*g/x**
3 + a*h/x**2 - b*c/x**4 - b*d/x**3 - b*e/x**2)/(6*a*b*(a + b*x**3)**2)

_______________________________________________________________________________________

Mathematica [A]  time = 1.87989, size = 352, normalized size = 0.89 \[ \frac{\frac{\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (2 a^{4/3} h-14 \sqrt [3]{a} b e-5 a \sqrt [3]{b} g+20 b^{4/3} d\right )}{b^{2/3}}-\frac{2 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (2 a^{4/3} h-14 \sqrt [3]{a} b e-5 a \sqrt [3]{b} g+20 b^{4/3} d\right )}{b^{2/3}}+\frac{2 \sqrt{3} \sqrt [3]{a} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (-2 a^{4/3} h+14 \sqrt [3]{a} b e-5 a \sqrt [3]{b} g+20 b^{4/3} d\right )}{b^{2/3}}+\frac{a^2 (9 a (f+x (g+h x))-9 b (c+x (d+e x)))}{\left (a+b x^3\right )^2}+\frac{3 a (6 a f+a x (5 g+4 h x)-12 b c-b x (11 d+10 e x))}{a+b x^3}+18 (3 b c-a f) \log \left (a+b x^3\right )+54 \log (x) (a f-3 b c)-\frac{18 a c}{x^3}-\frac{27 a d}{x^2}-\frac{54 a e}{x}}{54 a^4} \]

Antiderivative was successfully verified.

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

[Out]

((-18*a*c)/x^3 - (27*a*d)/x^2 - (54*a*e)/x + (3*a*(-12*b*c + 6*a*f - b*x*(11*d +
10*e*x) + a*x*(5*g + 4*h*x)))/(a + b*x^3) + (a^2*(-9*b*(c + x*(d + e*x)) + 9*a*
(f + x*(g + h*x))))/(a + b*x^3)^2 + (2*Sqrt[3]*a^(1/3)*(20*b^(4/3)*d + 14*a^(1/3
)*b*e - 5*a*b^(1/3)*g - 2*a^(4/3)*h)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]]
)/b^(2/3) + 54*(-3*b*c + a*f)*Log[x] - (2*a^(1/3)*(20*b^(4/3)*d - 14*a^(1/3)*b*e
- 5*a*b^(1/3)*g + 2*a^(4/3)*h)*Log[a^(1/3) + b^(1/3)*x])/b^(2/3) + (a^(1/3)*(20
*b^(4/3)*d - 14*a^(1/3)*b*e - 5*a*b^(1/3)*g + 2*a^(4/3)*h)*Log[a^(2/3) - a^(1/3)
*b^(1/3)*x + b^(2/3)*x^2])/b^(2/3) + 18*(3*b*c - a*f)*Log[a + b*x^3])/(54*a^4)

_______________________________________________________________________________________

Maple [B]  time = 0.028, size = 680, 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^4/(b*x^3+a)^3,x)

[Out]

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

_______________________________________________________________________________________

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^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

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^4),x, algorithm="fricas")

[Out]

Exception raised: NotImplementedError

_______________________________________________________________________________________

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**4/(b*x**3+a)**3,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.229167, size = 612, normalized size = 1.55 \[ \frac{{\left (3 \, b c - a f\right )}{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{4}} - \frac{{\left (3 \, b c - a f\right )}{\rm ln}\left ({\left | x \right |}\right )}{a^{4}} - \frac{\sqrt{3}{\left (20 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{2} d - 5 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b g + 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} a h - 14 \, \left (-a b^{2}\right )^{\frac{2}{3}} b e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{27 \, a^{4} b^{2}} - \frac{{\left (20 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{2} d - 5 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b g - 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} a h + 14 \, \left (-a b^{2}\right )^{\frac{2}{3}} b e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{54 \, a^{4} b^{2}} - \frac{{\left (2 \, a^{6} b h \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 14 \, a^{5} b^{2} \left (-\frac{a}{b}\right )^{\frac{1}{3}} e - 20 \, a^{5} b^{2} d + 5 \, a^{6} b g\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{27 \, a^{9} b} + \frac{4 \,{\left (a^{2} b h - 7 \, a b^{2} e\right )} x^{8} - 5 \,{\left (4 \, a b^{2} d - a^{2} b g\right )} x^{7} - 6 \,{\left (3 \, a b^{2} c - a^{2} b f\right )} x^{6} + 7 \,{\left (a^{3} h - 7 \, a^{2} b e\right )} x^{5} - 18 \, a^{3} x^{2} e - 9 \, a^{3} d x - 8 \,{\left (4 \, a^{2} b d - a^{3} g\right )} x^{4} - 6 \, a^{3} c - 9 \,{\left (3 \, a^{2} b c - a^{3} f\right )} x^{3}}{18 \,{\left (b x^{3} + a\right )}^{2} a^{4} x^{3}} \]

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^4),x, algorithm="giac")

[Out]

1/3*(3*b*c - a*f)*ln(abs(b*x^3 + a))/a^4 - (3*b*c - a*f)*ln(abs(x))/a^4 - 1/27*s
qrt(3)*(20*(-a*b^2)^(1/3)*b^2*d - 5*(-a*b^2)^(1/3)*a*b*g + 2*(-a*b^2)^(2/3)*a*h
- 14*(-a*b^2)^(2/3)*b*e)*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*d - 5*(-a*b^2)^(1/3)*a*b*g - 2*(-a*b^2)^(
2/3)*a*h + 14*(-a*b^2)^(2/3)*b*e)*ln(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^4*b
^2) - 1/27*(2*a^6*b*h*(-a/b)^(1/3) - 14*a^5*b^2*(-a/b)^(1/3)*e - 20*a^5*b^2*d +
5*a^6*b*g)*(-a/b)^(1/3)*ln(abs(x - (-a/b)^(1/3)))/(a^9*b) + 1/18*(4*(a^2*b*h - 7
*a*b^2*e)*x^8 - 5*(4*a*b^2*d - a^2*b*g)*x^7 - 6*(3*a*b^2*c - a^2*b*f)*x^6 + 7*(a
^3*h - 7*a^2*b*e)*x^5 - 18*a^3*x^2*e - 9*a^3*d*x - 8*(4*a^2*b*d - a^3*g)*x^4 - 6
*a^3*c - 9*(3*a^2*b*c - a^3*f)*x^3)/((b*x^3 + a)^2*a^4*x^3)