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

Optimal. Leaf size=325 \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (\sqrt [3]{b} (2 a f+b c)-\sqrt [3]{a} (5 a g+b d)\right )}{54 a^{5/3} b^{8/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (2 a f+b c)-\sqrt [3]{a} (5 a g+b d)\right )}{27 a^{5/3} b^{8/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (5 a^{4/3} g+\sqrt [3]{a} b d+2 a \sqrt [3]{b} f+b^{4/3} c\right )}{9 \sqrt{3} a^{5/3} b^{8/3}}+\frac{h \log \left (a+b x^3\right )}{3 b^3}+\frac{x \left (2 x (b d-4 a g)+3 x^2 (b e-3 a h)-7 a f+b c\right )}{18 a b^2 \left (a+b x^3\right )}-\frac{x \left (x (b d-a g)+x^2 (b e-a h)-a f+b c\right )}{6 b^2 \left (a+b x^3\right )^2} \]

[Out]

-(x*(b*c - a*f + (b*d - a*g)*x + (b*e - a*h)*x^2))/(6*b^2*(a + b*x^3)^2) + (x*(b
*c - 7*a*f + 2*(b*d - 4*a*g)*x + 3*(b*e - 3*a*h)*x^2))/(18*a*b^2*(a + b*x^3)) -
((b^(4/3)*c + a^(1/3)*b*d + 2*a*b^(1/3)*f + 5*a^(4/3)*g)*ArcTan[(a^(1/3) - 2*b^(
1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(5/3)*b^(8/3)) + ((b^(1/3)*(b*c + 2*a*f
) - a^(1/3)*(b*d + 5*a*g))*Log[a^(1/3) + b^(1/3)*x])/(27*a^(5/3)*b^(8/3)) - ((b^
(1/3)*(b*c + 2*a*f) - a^(1/3)*(b*d + 5*a*g))*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b
^(2/3)*x^2])/(54*a^(5/3)*b^(8/3)) + (h*Log[a + b*x^3])/(3*b^3)

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Rubi [A]  time = 1.32227, antiderivative size = 325, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.237 \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (\sqrt [3]{b} (2 a f+b c)-\sqrt [3]{a} (5 a g+b d)\right )}{54 a^{5/3} b^{8/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (2 a f+b c)-\sqrt [3]{a} (5 a g+b d)\right )}{27 a^{5/3} b^{8/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (5 a^{4/3} g+\sqrt [3]{a} b d+2 a \sqrt [3]{b} f+b^{4/3} c\right )}{9 \sqrt{3} a^{5/3} b^{8/3}}+\frac{h \log \left (a+b x^3\right )}{3 b^3}+\frac{x \left (2 x (b d-4 a g)+3 x^2 (b e-3 a h)-7 a f+b c\right )}{18 a b^2 \left (a+b x^3\right )}-\frac{x \left (x (b d-a g)+x^2 (b e-a h)-a f+b c\right )}{6 b^2 \left (a+b x^3\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

-(x*(b*c - a*f + (b*d - a*g)*x + (b*e - a*h)*x^2))/(6*b^2*(a + b*x^3)^2) + (x*(b
*c - 7*a*f + 2*(b*d - 4*a*g)*x + 3*(b*e - 3*a*h)*x^2))/(18*a*b^2*(a + b*x^3)) -
((b^(4/3)*c + a^(1/3)*b*d + 2*a*b^(1/3)*f + 5*a^(4/3)*g)*ArcTan[(a^(1/3) - 2*b^(
1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(5/3)*b^(8/3)) + ((b^(1/3)*(b*c + 2*a*f
) - a^(1/3)*(b*d + 5*a*g))*Log[a^(1/3) + b^(1/3)*x])/(27*a^(5/3)*b^(8/3)) - ((b^
(1/3)*(b*c + 2*a*f) - a^(1/3)*(b*d + 5*a*g))*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b
^(2/3)*x^2])/(54*a^(5/3)*b^(8/3)) + (h*Log[a + b*x^3])/(3*b^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*(h*x**5+g*x**4+f*x**3+e*x**2+d*x+c)/(b*x**3+a)**3,x)

[Out]

Timed out

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Mathematica [A]  time = 0.539033, size = 315, normalized size = 0.97 \[ \frac{\frac{\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (5 a^{4/3} g+\sqrt [3]{a} b d-2 a \sqrt [3]{b} f-b^{4/3} c\right )}{a^{5/3}}+\frac{2 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-5 a^{4/3} g-\sqrt [3]{a} b d+2 a \sqrt [3]{b} f+b^{4/3} c\right )}{a^{5/3}}-\frac{2 \sqrt{3} \sqrt [3]{b} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (5 a^{4/3} g+\sqrt [3]{a} b d+2 a \sqrt [3]{b} f+b^{4/3} c\right )}{a^{5/3}}-\frac{9 \left (a^2 h-a b (e+x (f+g x))+b^2 x (c+d x)\right )}{\left (a+b x^3\right )^2}+\frac{36 a^2 h-3 a b (6 e+x (7 f+8 g x))+3 b^2 x (c+2 d x)}{a \left (a+b x^3\right )}+18 h \log \left (a+b x^3\right )}{54 b^3} \]

Antiderivative was successfully verified.

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

[Out]

((-9*(a^2*h + b^2*x*(c + d*x) - a*b*(e + x*(f + g*x))))/(a + b*x^3)^2 + (36*a^2*
h + 3*b^2*x*(c + 2*d*x) - 3*a*b*(6*e + x*(7*f + 8*g*x)))/(a*(a + b*x^3)) - (2*Sq
rt[3]*b^(1/3)*(b^(4/3)*c + a^(1/3)*b*d + 2*a*b^(1/3)*f + 5*a^(4/3)*g)*ArcTan[(1
- (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(5/3) + (2*b^(1/3)*(b^(4/3)*c - a^(1/3)*b*d
+ 2*a*b^(1/3)*f - 5*a^(4/3)*g)*Log[a^(1/3) + b^(1/3)*x])/a^(5/3) + (b^(1/3)*(-(
b^(4/3)*c) + a^(1/3)*b*d - 2*a*b^(1/3)*f + 5*a^(4/3)*g)*Log[a^(2/3) - a^(1/3)*b^
(1/3)*x + b^(2/3)*x^2])/a^(5/3) + 18*h*Log[a + b*x^3])/(54*b^3)

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Maple [A]  time = 0.017, size = 520, normalized size = 1.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.226425, size = 512, normalized size = 1.58 \[ \frac{h{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{3}} + \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{2} c + 2 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b f - \left (-a b^{2}\right )^{\frac{2}{3}} b d - 5 \, \left (-a b^{2}\right )^{\frac{2}{3}} a g\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{27 \, a^{2} b^{4}} + \frac{2 \,{\left (b^{2} d - 4 \, a b g\right )} x^{5} +{\left (b^{2} c - 7 \, a b f\right )} x^{4} + 6 \,{\left (2 \, a^{2} h - a b e\right )} x^{3} -{\left (a b d + 5 \, a^{2} g\right )} x^{2} - 2 \,{\left (a b c + 2 \, a^{2} f\right )} x + \frac{3 \,{\left (3 \, a^{3} h - a^{2} b e\right )}}{b}}{18 \,{\left (b x^{3} + a\right )}^{2} a b^{2}} + \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{2} c + 2 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b f + \left (-a b^{2}\right )^{\frac{2}{3}} b d + 5 \, \left (-a b^{2}\right )^{\frac{2}{3}} a g\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{54 \, a^{2} b^{4}} - \frac{{\left (a b^{4} d \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 5 \, a^{2} b^{3} g \left (-\frac{a}{b}\right )^{\frac{1}{3}} + a b^{4} c + 2 \, a^{2} b^{3} f\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{27 \, a^{3} b^{5}} \]

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, algorithm="giac")

[Out]

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