3.253 \(\int \frac{x^5 (c+d x^3+e x^6+f x^9)}{(a+b x^3)^2} \, dx\)

Optimal. Leaf size=140 \[ \frac{a \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 b^5 \left (a+b x^3\right )}+\frac{\log \left (a+b x^3\right ) \left (3 a^2 b e-4 a^3 f-2 a b^2 d+b^3 c\right )}{3 b^5}+\frac{x^3 \left (3 a^2 f-2 a b e+b^2 d\right )}{3 b^4}+\frac{x^6 (b e-2 a f)}{6 b^3}+\frac{f x^9}{9 b^2} \]

[Out]

((b^2*d - 2*a*b*e + 3*a^2*f)*x^3)/(3*b^4) + ((b*e - 2*a*f)*x^6)/(6*b^3) + (f*x^9)/(9*b^2) + (a*(b^3*c - a*b^2*
d + a^2*b*e - a^3*f))/(3*b^5*(a + b*x^3)) + ((b^3*c - 2*a*b^2*d + 3*a^2*b*e - 4*a^3*f)*Log[a + b*x^3])/(3*b^5)

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Rubi [A]  time = 0.199417, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {1821, 1620} \[ \frac{a \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 b^5 \left (a+b x^3\right )}+\frac{\log \left (a+b x^3\right ) \left (3 a^2 b e-4 a^3 f-2 a b^2 d+b^3 c\right )}{3 b^5}+\frac{x^3 \left (3 a^2 f-2 a b e+b^2 d\right )}{3 b^4}+\frac{x^6 (b e-2 a f)}{6 b^3}+\frac{f x^9}{9 b^2} \]

Antiderivative was successfully verified.

[In]

Int[(x^5*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^2,x]

[Out]

((b^2*d - 2*a*b*e + 3*a^2*f)*x^3)/(3*b^4) + ((b*e - 2*a*f)*x^6)/(6*b^3) + (f*x^9)/(9*b^2) + (a*(b^3*c - a*b^2*
d + a^2*b*e - a^3*f))/(3*b^5*(a + b*x^3)) + ((b^3*c - 2*a*b^2*d + 3*a^2*b*e - 4*a^3*f)*Log[a + b*x^3])/(3*b^5)

Rule 1821

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] -
1)*SubstFor[x^n, Pq, x]*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && PolyQ[Pq, x^n] && Intege
rQ[Simplify[(m + 1)/n]]

Rule 1620

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)
^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) &&
GtQ[Expon[Px, x], 2]

Rubi steps

\begin{align*} \int \frac{x^5 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^2} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x \left (c+d x+e x^2+f x^3\right )}{(a+b x)^2} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{b^2 d-2 a b e+3 a^2 f}{b^4}+\frac{(b e-2 a f) x}{b^3}+\frac{f x^2}{b^2}+\frac{a \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right )}{b^4 (a+b x)^2}+\frac{b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f}{b^4 (a+b x)}\right ) \, dx,x,x^3\right )\\ &=\frac{\left (b^2 d-2 a b e+3 a^2 f\right ) x^3}{3 b^4}+\frac{(b e-2 a f) x^6}{6 b^3}+\frac{f x^9}{9 b^2}+\frac{a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )}{3 b^5 \left (a+b x^3\right )}+\frac{\left (b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f\right ) \log \left (a+b x^3\right )}{3 b^5}\\ \end{align*}

Mathematica [A]  time = 0.0984255, size = 129, normalized size = 0.92 \[ \frac{\frac{6 a \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{a+b x^3}+6 \log \left (a+b x^3\right ) \left (3 a^2 b e-4 a^3 f-2 a b^2 d+b^3 c\right )+6 b x^3 \left (3 a^2 f-2 a b e+b^2 d\right )+3 b^2 x^6 (b e-2 a f)+2 b^3 f x^9}{18 b^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^5*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^2,x]

[Out]

(6*b*(b^2*d - 2*a*b*e + 3*a^2*f)*x^3 + 3*b^2*(b*e - 2*a*f)*x^6 + 2*b^3*f*x^9 + (6*a*(b^3*c - a*b^2*d + a^2*b*e
- a^3*f))/(a + b*x^3) + 6*(b^3*c - 2*a*b^2*d + 3*a^2*b*e - 4*a^3*f)*Log[a + b*x^3])/(18*b^5)

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Maple [A]  time = 0.01, size = 192, normalized size = 1.4 \begin{align*}{\frac{f{x}^{9}}{9\,{b}^{2}}}-{\frac{{x}^{6}af}{3\,{b}^{3}}}+{\frac{e{x}^{6}}{6\,{b}^{2}}}+{\frac{{a}^{2}f{x}^{3}}{{b}^{4}}}-{\frac{2\,ae{x}^{3}}{3\,{b}^{3}}}+{\frac{d{x}^{3}}{3\,{b}^{2}}}-{\frac{4\,\ln \left ( b{x}^{3}+a \right ){a}^{3}f}{3\,{b}^{5}}}+{\frac{\ln \left ( b{x}^{3}+a \right ){a}^{2}e}{{b}^{4}}}-{\frac{2\,\ln \left ( b{x}^{3}+a \right ) ad}{3\,{b}^{3}}}+{\frac{\ln \left ( b{x}^{3}+a \right ) c}{3\,{b}^{2}}}-{\frac{{a}^{4}f}{3\,{b}^{5} \left ( b{x}^{3}+a \right ) }}+{\frac{{a}^{3}e}{3\,{b}^{4} \left ( b{x}^{3}+a \right ) }}-{\frac{{a}^{2}d}{3\,{b}^{3} \left ( b{x}^{3}+a \right ) }}+{\frac{ac}{3\,{b}^{2} \left ( b{x}^{3}+a \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^2,x)

[Out]

1/9*f*x^9/b^2-1/3/b^3*x^6*a*f+1/6/b^2*x^6*e+1/b^4*a^2*f*x^3-2/3/b^3*a*e*x^3+1/3/b^2*d*x^3-4/3/b^5*ln(b*x^3+a)*
a^3*f+1/b^4*ln(b*x^3+a)*a^2*e-2/3/b^3*ln(b*x^3+a)*a*d+1/3/b^2*ln(b*x^3+a)*c-1/3/b^5*a^4/(b*x^3+a)*f+1/3/b^4*a^
3/(b*x^3+a)*e-1/3/b^3*a^2/(b*x^3+a)*d+1/3/b^2*a/(b*x^3+a)*c

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Maxima [A]  time = 0.958848, size = 186, normalized size = 1.33 \begin{align*} \frac{a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f}{3 \,{\left (b^{6} x^{3} + a b^{5}\right )}} + \frac{2 \, b^{2} f x^{9} + 3 \,{\left (b^{2} e - 2 \, a b f\right )} x^{6} + 6 \,{\left (b^{2} d - 2 \, a b e + 3 \, a^{2} f\right )} x^{3}}{18 \, b^{4}} + \frac{{\left (b^{3} c - 2 \, a b^{2} d + 3 \, a^{2} b e - 4 \, a^{3} f\right )} \log \left (b x^{3} + a\right )}{3 \, b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^2,x, algorithm="maxima")

[Out]

1/3*(a*b^3*c - a^2*b^2*d + a^3*b*e - a^4*f)/(b^6*x^3 + a*b^5) + 1/18*(2*b^2*f*x^9 + 3*(b^2*e - 2*a*b*f)*x^6 +
6*(b^2*d - 2*a*b*e + 3*a^2*f)*x^3)/b^4 + 1/3*(b^3*c - 2*a*b^2*d + 3*a^2*b*e - 4*a^3*f)*log(b*x^3 + a)/b^5

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Fricas [A]  time = 1.30494, size = 432, normalized size = 3.09 \begin{align*} \frac{2 \, b^{4} f x^{12} +{\left (3 \, b^{4} e - 4 \, a b^{3} f\right )} x^{9} + 3 \,{\left (2 \, b^{4} d - 3 \, a b^{3} e + 4 \, a^{2} b^{2} f\right )} x^{6} + 6 \, a b^{3} c - 6 \, a^{2} b^{2} d + 6 \, a^{3} b e - 6 \, a^{4} f + 6 \,{\left (a b^{3} d - 2 \, a^{2} b^{2} e + 3 \, a^{3} b f\right )} x^{3} + 6 \,{\left (a b^{3} c - 2 \, a^{2} b^{2} d + 3 \, a^{3} b e - 4 \, a^{4} f +{\left (b^{4} c - 2 \, a b^{3} d + 3 \, a^{2} b^{2} e - 4 \, a^{3} b f\right )} x^{3}\right )} \log \left (b x^{3} + a\right )}{18 \,{\left (b^{6} x^{3} + a b^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^2,x, algorithm="fricas")

[Out]

1/18*(2*b^4*f*x^12 + (3*b^4*e - 4*a*b^3*f)*x^9 + 3*(2*b^4*d - 3*a*b^3*e + 4*a^2*b^2*f)*x^6 + 6*a*b^3*c - 6*a^2
*b^2*d + 6*a^3*b*e - 6*a^4*f + 6*(a*b^3*d - 2*a^2*b^2*e + 3*a^3*b*f)*x^3 + 6*(a*b^3*c - 2*a^2*b^2*d + 3*a^3*b*
e - 4*a^4*f + (b^4*c - 2*a*b^3*d + 3*a^2*b^2*e - 4*a^3*b*f)*x^3)*log(b*x^3 + a))/(b^6*x^3 + a*b^5)

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Sympy [A]  time = 10.6441, size = 138, normalized size = 0.99 \begin{align*} - \frac{a^{4} f - a^{3} b e + a^{2} b^{2} d - a b^{3} c}{3 a b^{5} + 3 b^{6} x^{3}} + \frac{f x^{9}}{9 b^{2}} - \frac{x^{6} \left (2 a f - b e\right )}{6 b^{3}} + \frac{x^{3} \left (3 a^{2} f - 2 a b e + b^{2} d\right )}{3 b^{4}} - \frac{\left (4 a^{3} f - 3 a^{2} b e + 2 a b^{2} d - b^{3} c\right ) \log{\left (a + b x^{3} \right )}}{3 b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(f*x**9+e*x**6+d*x**3+c)/(b*x**3+a)**2,x)

[Out]

-(a**4*f - a**3*b*e + a**2*b**2*d - a*b**3*c)/(3*a*b**5 + 3*b**6*x**3) + f*x**9/(9*b**2) - x**6*(2*a*f - b*e)/
(6*b**3) + x**3*(3*a**2*f - 2*a*b*e + b**2*d)/(3*b**4) - (4*a**3*f - 3*a**2*b*e + 2*a*b**2*d - b**3*c)*log(a +
b*x**3)/(3*b**5)

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Giac [A]  time = 1.06383, size = 293, normalized size = 2.09 \begin{align*} \frac{\frac{{\left (b x^{3} + a\right )}^{3}{\left (2 \, f - \frac{3 \,{\left (4 \, a b f - b^{2} e\right )}}{{\left (b x^{3} + a\right )} b} + \frac{6 \,{\left (b^{4} d + 6 \, a^{2} b^{2} f - 3 \, a b^{3} e\right )}}{{\left (b x^{3} + a\right )}^{2} b^{2}}\right )}}{b^{4}} - \frac{6 \,{\left (b^{3} c - 2 \, a b^{2} d - 4 \, a^{3} f + 3 \, a^{2} b e\right )} \log \left (\frac{{\left | b x^{3} + a \right |}}{{\left (b x^{3} + a\right )}^{2}{\left | b \right |}}\right )}{b^{4}} + \frac{6 \,{\left (\frac{a b^{6} c}{b x^{3} + a} - \frac{a^{2} b^{5} d}{b x^{3} + a} - \frac{a^{4} b^{3} f}{b x^{3} + a} + \frac{a^{3} b^{4} e}{b x^{3} + a}\right )}}{b^{7}}}{18 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^2,x, algorithm="giac")

[Out]

1/18*((b*x^3 + a)^3*(2*f - 3*(4*a*b*f - b^2*e)/((b*x^3 + a)*b) + 6*(b^4*d + 6*a^2*b^2*f - 3*a*b^3*e)/((b*x^3 +
a)^2*b^2))/b^4 - 6*(b^3*c - 2*a*b^2*d - 4*a^3*f + 3*a^2*b*e)*log(abs(b*x^3 + a)/((b*x^3 + a)^2*abs(b)))/b^4 +
6*(a*b^6*c/(b*x^3 + a) - a^2*b^5*d/(b*x^3 + a) - a^4*b^3*f/(b*x^3 + a) + a^3*b^4*e/(b*x^3 + a))/b^7)/b