3.276 \(\int \frac{x^{14} (c+d x^3+e x^6+f x^9)}{(a+b x^3)^3} \, dx\)

Optimal. Leaf size=266 \[ \frac{x^6 \left (6 a^2 b e-10 a^3 f-3 a b^2 d+b^3 c\right )}{6 b^6}-\frac{a x^3 \left (10 a^2 b e-15 a^3 f-6 a b^2 d+3 b^3 c\right )}{3 b^7}+\frac{a^3 \left (6 a^2 b e-7 a^3 f-5 a b^2 d+4 b^3 c\right )}{3 b^8 \left (a+b x^3\right )}-\frac{a^4 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 b^8 \left (a+b x^3\right )^2}+\frac{a^2 \log \left (a+b x^3\right ) \left (15 a^2 b e-21 a^3 f-10 a b^2 d+6 b^3 c\right )}{3 b^8}+\frac{x^9 \left (6 a^2 f-3 a b e+b^2 d\right )}{9 b^5}+\frac{x^{12} (b e-3 a f)}{12 b^4}+\frac{f x^{15}}{15 b^3} \]

[Out]

-(a*(3*b^3*c - 6*a*b^2*d + 10*a^2*b*e - 15*a^3*f)*x^3)/(3*b^7) + ((b^3*c - 3*a*b^2*d + 6*a^2*b*e - 10*a^3*f)*x
^6)/(6*b^6) + ((b^2*d - 3*a*b*e + 6*a^2*f)*x^9)/(9*b^5) + ((b*e - 3*a*f)*x^12)/(12*b^4) + (f*x^15)/(15*b^3) -
(a^4*(b^3*c - a*b^2*d + a^2*b*e - a^3*f))/(6*b^8*(a + b*x^3)^2) + (a^3*(4*b^3*c - 5*a*b^2*d + 6*a^2*b*e - 7*a^
3*f))/(3*b^8*(a + b*x^3)) + (a^2*(6*b^3*c - 10*a*b^2*d + 15*a^2*b*e - 21*a^3*f)*Log[a + b*x^3])/(3*b^8)

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Rubi [A]  time = 0.436041, antiderivative size = 266, 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} \[ \frac{x^6 \left (6 a^2 b e-10 a^3 f-3 a b^2 d+b^3 c\right )}{6 b^6}-\frac{a x^3 \left (10 a^2 b e-15 a^3 f-6 a b^2 d+3 b^3 c\right )}{3 b^7}+\frac{a^3 \left (6 a^2 b e-7 a^3 f-5 a b^2 d+4 b^3 c\right )}{3 b^8 \left (a+b x^3\right )}-\frac{a^4 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 b^8 \left (a+b x^3\right )^2}+\frac{a^2 \log \left (a+b x^3\right ) \left (15 a^2 b e-21 a^3 f-10 a b^2 d+6 b^3 c\right )}{3 b^8}+\frac{x^9 \left (6 a^2 f-3 a b e+b^2 d\right )}{9 b^5}+\frac{x^{12} (b e-3 a f)}{12 b^4}+\frac{f x^{15}}{15 b^3} \]

Antiderivative was successfully verified.

[In]

Int[(x^14*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^3,x]

[Out]

-(a*(3*b^3*c - 6*a*b^2*d + 10*a^2*b*e - 15*a^3*f)*x^3)/(3*b^7) + ((b^3*c - 3*a*b^2*d + 6*a^2*b*e - 10*a^3*f)*x
^6)/(6*b^6) + ((b^2*d - 3*a*b*e + 6*a^2*f)*x^9)/(9*b^5) + ((b*e - 3*a*f)*x^12)/(12*b^4) + (f*x^15)/(15*b^3) -
(a^4*(b^3*c - a*b^2*d + a^2*b*e - a^3*f))/(6*b^8*(a + b*x^3)^2) + (a^3*(4*b^3*c - 5*a*b^2*d + 6*a^2*b*e - 7*a^
3*f))/(3*b^8*(a + b*x^3)) + (a^2*(6*b^3*c - 10*a*b^2*d + 15*a^2*b*e - 21*a^3*f)*Log[a + b*x^3])/(3*b^8)

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^{14} \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x^4 \left (c+d x+e x^2+f x^3\right )}{(a+b x)^3} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{a \left (-3 b^3 c+6 a b^2 d-10 a^2 b e+15 a^3 f\right )}{b^7}+\frac{\left (b^3 c-3 a b^2 d+6 a^2 b e-10 a^3 f\right ) x}{b^6}+\frac{\left (b^2 d-3 a b e+6 a^2 f\right ) x^2}{b^5}+\frac{(b e-3 a f) x^3}{b^4}+\frac{f x^4}{b^3}-\frac{a^4 \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right )}{b^7 (a+b x)^3}+\frac{a^3 \left (-4 b^3 c+5 a b^2 d-6 a^2 b e+7 a^3 f\right )}{b^7 (a+b x)^2}-\frac{a^2 \left (-6 b^3 c+10 a b^2 d-15 a^2 b e+21 a^3 f\right )}{b^7 (a+b x)}\right ) \, dx,x,x^3\right )\\ &=-\frac{a \left (3 b^3 c-6 a b^2 d+10 a^2 b e-15 a^3 f\right ) x^3}{3 b^7}+\frac{\left (b^3 c-3 a b^2 d+6 a^2 b e-10 a^3 f\right ) x^6}{6 b^6}+\frac{\left (b^2 d-3 a b e+6 a^2 f\right ) x^9}{9 b^5}+\frac{(b e-3 a f) x^{12}}{12 b^4}+\frac{f x^{15}}{15 b^3}-\frac{a^4 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )}{6 b^8 \left (a+b x^3\right )^2}+\frac{a^3 \left (4 b^3 c-5 a b^2 d+6 a^2 b e-7 a^3 f\right )}{3 b^8 \left (a+b x^3\right )}+\frac{a^2 \left (6 b^3 c-10 a b^2 d+15 a^2 b e-21 a^3 f\right ) \log \left (a+b x^3\right )}{3 b^8}\\ \end{align*}

Mathematica [A]  time = 0.142904, size = 246, normalized size = 0.92 \[ \frac{30 b^2 x^6 \left (6 a^2 b e-10 a^3 f-3 a b^2 d+b^3 c\right )+60 a b x^3 \left (-10 a^2 b e+15 a^3 f+6 a b^2 d-3 b^3 c\right )-\frac{60 a^3 \left (-6 a^2 b e+7 a^3 f+5 a b^2 d-4 b^3 c\right )}{a+b x^3}+\frac{30 a^4 \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )}{\left (a+b x^3\right )^2}+60 a^2 \log \left (a+b x^3\right ) \left (15 a^2 b e-21 a^3 f-10 a b^2 d+6 b^3 c\right )+20 b^3 x^9 \left (6 a^2 f-3 a b e+b^2 d\right )+15 b^4 x^{12} (b e-3 a f)+12 b^5 f x^{15}}{180 b^8} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^14*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^3,x]

[Out]

(60*a*b*(-3*b^3*c + 6*a*b^2*d - 10*a^2*b*e + 15*a^3*f)*x^3 + 30*b^2*(b^3*c - 3*a*b^2*d + 6*a^2*b*e - 10*a^3*f)
*x^6 + 20*b^3*(b^2*d - 3*a*b*e + 6*a^2*f)*x^9 + 15*b^4*(b*e - 3*a*f)*x^12 + 12*b^5*f*x^15 + (30*a^4*(-(b^3*c)
+ a*b^2*d - a^2*b*e + a^3*f))/(a + b*x^3)^2 - (60*a^3*(-4*b^3*c + 5*a*b^2*d - 6*a^2*b*e + 7*a^3*f))/(a + b*x^3
) + 60*a^2*(6*b^3*c - 10*a*b^2*d + 15*a^2*b*e - 21*a^3*f)*Log[a + b*x^3])/(180*b^8)

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Maple [A]  time = 0.015, size = 361, normalized size = 1.4 \begin{align*}{\frac{f{x}^{15}}{15\,{b}^{3}}}+5\,{\frac{{a}^{4}f{x}^{3}}{{b}^{7}}}-{\frac{10\,{a}^{3}e{x}^{3}}{3\,{b}^{6}}}+2\,{\frac{{a}^{2}d{x}^{3}}{{b}^{5}}}-{\frac{ac{x}^{3}}{{b}^{4}}}+{\frac{{a}^{7}f}{6\,{b}^{8} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{{a}^{6}e}{6\,{b}^{7} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{{a}^{5}d}{6\,{b}^{6} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{{a}^{4}c}{6\,{b}^{5} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{7\,{a}^{6}f}{3\,{b}^{8} \left ( b{x}^{3}+a \right ) }}+2\,{\frac{{a}^{5}e}{{b}^{7} \left ( b{x}^{3}+a \right ) }}-{\frac{5\,{a}^{4}d}{3\,{b}^{6} \left ( b{x}^{3}+a \right ) }}+{\frac{4\,{a}^{3}c}{3\,{b}^{5} \left ( b{x}^{3}+a \right ) }}-7\,{\frac{{a}^{5}\ln \left ( b{x}^{3}+a \right ) f}{{b}^{8}}}+5\,{\frac{{a}^{4}\ln \left ( b{x}^{3}+a \right ) e}{{b}^{7}}}-{\frac{10\,{a}^{3}\ln \left ( b{x}^{3}+a \right ) d}{3\,{b}^{6}}}+2\,{\frac{{a}^{2}\ln \left ( b{x}^{3}+a \right ) c}{{b}^{5}}}+{\frac{{x}^{12}e}{12\,{b}^{3}}}+{\frac{{x}^{9}d}{9\,{b}^{3}}}+{\frac{{x}^{6}c}{6\,{b}^{3}}}-{\frac{{x}^{12}af}{4\,{b}^{4}}}+{\frac{2\,{x}^{9}{a}^{2}f}{3\,{b}^{5}}}-{\frac{{x}^{9}ae}{3\,{b}^{4}}}-{\frac{5\,{a}^{3}f{x}^{6}}{3\,{b}^{6}}}+{\frac{{a}^{2}e{x}^{6}}{{b}^{5}}}-{\frac{ad{x}^{6}}{2\,{b}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^14*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^3,x)

[Out]

1/15*f*x^15/b^3+5/b^7*a^4*f*x^3-10/3/b^6*a^3*e*x^3+2/b^5*a^2*d*x^3-1/b^4*a*c*x^3+1/6*a^7/b^8/(b*x^3+a)^2*f-1/6
*a^6/b^7/(b*x^3+a)^2*e+1/6*a^5/b^6/(b*x^3+a)^2*d-1/6*a^4/b^5/(b*x^3+a)^2*c-7/3*a^6/b^8/(b*x^3+a)*f+2*a^5/b^7/(
b*x^3+a)*e-5/3*a^4/b^6/(b*x^3+a)*d+4/3*a^3/b^5/(b*x^3+a)*c-7*a^5/b^8*ln(b*x^3+a)*f+5*a^4/b^7*ln(b*x^3+a)*e-10/
3*a^3/b^6*ln(b*x^3+a)*d+2*a^2/b^5*ln(b*x^3+a)*c+1/12/b^3*x^12*e+1/9/b^3*x^9*d+1/6/b^3*x^6*c-1/4/b^4*x^12*a*f+2
/3/b^5*x^9*a^2*f-1/3/b^4*x^9*a*e-5/3/b^6*x^6*a^3*f+1/b^5*x^6*a^2*e-1/2/b^4*x^6*a*d

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Maxima [A]  time = 0.954004, size = 371, normalized size = 1.39 \begin{align*} \frac{7 \, a^{4} b^{3} c - 9 \, a^{5} b^{2} d + 11 \, a^{6} b e - 13 \, a^{7} f + 2 \,{\left (4 \, a^{3} b^{4} c - 5 \, a^{4} b^{3} d + 6 \, a^{5} b^{2} e - 7 \, a^{6} b f\right )} x^{3}}{6 \,{\left (b^{10} x^{6} + 2 \, a b^{9} x^{3} + a^{2} b^{8}\right )}} + \frac{12 \, b^{4} f x^{15} + 15 \,{\left (b^{4} e - 3 \, a b^{3} f\right )} x^{12} + 20 \,{\left (b^{4} d - 3 \, a b^{3} e + 6 \, a^{2} b^{2} f\right )} x^{9} + 30 \,{\left (b^{4} c - 3 \, a b^{3} d + 6 \, a^{2} b^{2} e - 10 \, a^{3} b f\right )} x^{6} - 60 \,{\left (3 \, a b^{3} c - 6 \, a^{2} b^{2} d + 10 \, a^{3} b e - 15 \, a^{4} f\right )} x^{3}}{180 \, b^{7}} + \frac{{\left (6 \, a^{2} b^{3} c - 10 \, a^{3} b^{2} d + 15 \, a^{4} b e - 21 \, a^{5} f\right )} \log \left (b x^{3} + a\right )}{3 \, b^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^14*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^3,x, algorithm="maxima")

[Out]

1/6*(7*a^4*b^3*c - 9*a^5*b^2*d + 11*a^6*b*e - 13*a^7*f + 2*(4*a^3*b^4*c - 5*a^4*b^3*d + 6*a^5*b^2*e - 7*a^6*b*
f)*x^3)/(b^10*x^6 + 2*a*b^9*x^3 + a^2*b^8) + 1/180*(12*b^4*f*x^15 + 15*(b^4*e - 3*a*b^3*f)*x^12 + 20*(b^4*d -
3*a*b^3*e + 6*a^2*b^2*f)*x^9 + 30*(b^4*c - 3*a*b^3*d + 6*a^2*b^2*e - 10*a^3*b*f)*x^6 - 60*(3*a*b^3*c - 6*a^2*b
^2*d + 10*a^3*b*e - 15*a^4*f)*x^3)/b^7 + 1/3*(6*a^2*b^3*c - 10*a^3*b^2*d + 15*a^4*b*e - 21*a^5*f)*log(b*x^3 +
a)/b^8

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Fricas [A]  time = 1.23426, size = 883, normalized size = 3.32 \begin{align*} \frac{12 \, b^{7} f x^{21} + 3 \,{\left (5 \, b^{7} e - 7 \, a b^{6} f\right )} x^{18} + 2 \,{\left (10 \, b^{7} d - 15 \, a b^{6} e + 21 \, a^{2} b^{5} f\right )} x^{15} + 5 \,{\left (6 \, b^{7} c - 10 \, a b^{6} d + 15 \, a^{2} b^{5} e - 21 \, a^{3} b^{4} f\right )} x^{12} - 20 \,{\left (6 \, a b^{6} c - 10 \, a^{2} b^{5} d + 15 \, a^{3} b^{4} e - 21 \, a^{4} b^{3} f\right )} x^{9} + 210 \, a^{4} b^{3} c - 270 \, a^{5} b^{2} d + 330 \, a^{6} b e - 390 \, a^{7} f - 30 \,{\left (11 \, a^{2} b^{5} c - 21 \, a^{3} b^{4} d + 34 \, a^{4} b^{3} e - 50 \, a^{5} b^{2} f\right )} x^{6} + 60 \,{\left (a^{3} b^{4} c + a^{4} b^{3} d - 4 \, a^{5} b^{2} e + 8 \, a^{6} b f\right )} x^{3} + 60 \,{\left (6 \, a^{4} b^{3} c - 10 \, a^{5} b^{2} d + 15 \, a^{6} b e - 21 \, a^{7} f +{\left (6 \, a^{2} b^{5} c - 10 \, a^{3} b^{4} d + 15 \, a^{4} b^{3} e - 21 \, a^{5} b^{2} f\right )} x^{6} + 2 \,{\left (6 \, a^{3} b^{4} c - 10 \, a^{4} b^{3} d + 15 \, a^{5} b^{2} e - 21 \, a^{6} b f\right )} x^{3}\right )} \log \left (b x^{3} + a\right )}{180 \,{\left (b^{10} x^{6} + 2 \, a b^{9} x^{3} + a^{2} b^{8}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^14*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^3,x, algorithm="fricas")

[Out]

1/180*(12*b^7*f*x^21 + 3*(5*b^7*e - 7*a*b^6*f)*x^18 + 2*(10*b^7*d - 15*a*b^6*e + 21*a^2*b^5*f)*x^15 + 5*(6*b^7
*c - 10*a*b^6*d + 15*a^2*b^5*e - 21*a^3*b^4*f)*x^12 - 20*(6*a*b^6*c - 10*a^2*b^5*d + 15*a^3*b^4*e - 21*a^4*b^3
*f)*x^9 + 210*a^4*b^3*c - 270*a^5*b^2*d + 330*a^6*b*e - 390*a^7*f - 30*(11*a^2*b^5*c - 21*a^3*b^4*d + 34*a^4*b
^3*e - 50*a^5*b^2*f)*x^6 + 60*(a^3*b^4*c + a^4*b^3*d - 4*a^5*b^2*e + 8*a^6*b*f)*x^3 + 60*(6*a^4*b^3*c - 10*a^5
*b^2*d + 15*a^6*b*e - 21*a^7*f + (6*a^2*b^5*c - 10*a^3*b^4*d + 15*a^4*b^3*e - 21*a^5*b^2*f)*x^6 + 2*(6*a^3*b^4
*c - 10*a^4*b^3*d + 15*a^5*b^2*e - 21*a^6*b*f)*x^3)*log(b*x^3 + a))/(b^10*x^6 + 2*a*b^9*x^3 + a^2*b^8)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**14*(f*x**9+e*x**6+d*x**3+c)/(b*x**3+a)**3,x)

[Out]

Timed out

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Giac [A]  time = 1.07827, size = 471, normalized size = 1.77 \begin{align*} \frac{{\left (6 \, a^{2} b^{3} c - 10 \, a^{3} b^{2} d - 21 \, a^{5} f + 15 \, a^{4} b e\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{8}} - \frac{18 \, a^{2} b^{5} c x^{6} - 30 \, a^{3} b^{4} d x^{6} - 63 \, a^{5} b^{2} f x^{6} + 45 \, a^{4} b^{3} x^{6} e + 28 \, a^{3} b^{4} c x^{3} - 50 \, a^{4} b^{3} d x^{3} - 112 \, a^{6} b f x^{3} + 78 \, a^{5} b^{2} x^{3} e + 11 \, a^{4} b^{3} c - 21 \, a^{5} b^{2} d - 50 \, a^{7} f + 34 \, a^{6} b e}{6 \,{\left (b x^{3} + a\right )}^{2} b^{8}} + \frac{12 \, b^{12} f x^{15} - 45 \, a b^{11} f x^{12} + 15 \, b^{12} x^{12} e + 20 \, b^{12} d x^{9} + 120 \, a^{2} b^{10} f x^{9} - 60 \, a b^{11} x^{9} e + 30 \, b^{12} c x^{6} - 90 \, a b^{11} d x^{6} - 300 \, a^{3} b^{9} f x^{6} + 180 \, a^{2} b^{10} x^{6} e - 180 \, a b^{11} c x^{3} + 360 \, a^{2} b^{10} d x^{3} + 900 \, a^{4} b^{8} f x^{3} - 600 \, a^{3} b^{9} x^{3} e}{180 \, b^{15}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^14*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^3,x, algorithm="giac")

[Out]

1/3*(6*a^2*b^3*c - 10*a^3*b^2*d - 21*a^5*f + 15*a^4*b*e)*log(abs(b*x^3 + a))/b^8 - 1/6*(18*a^2*b^5*c*x^6 - 30*
a^3*b^4*d*x^6 - 63*a^5*b^2*f*x^6 + 45*a^4*b^3*x^6*e + 28*a^3*b^4*c*x^3 - 50*a^4*b^3*d*x^3 - 112*a^6*b*f*x^3 +
78*a^5*b^2*x^3*e + 11*a^4*b^3*c - 21*a^5*b^2*d - 50*a^7*f + 34*a^6*b*e)/((b*x^3 + a)^2*b^8) + 1/180*(12*b^12*f
*x^15 - 45*a*b^11*f*x^12 + 15*b^12*x^12*e + 20*b^12*d*x^9 + 120*a^2*b^10*f*x^9 - 60*a*b^11*x^9*e + 30*b^12*c*x
^6 - 90*a*b^11*d*x^6 - 300*a^3*b^9*f*x^6 + 180*a^2*b^10*x^6*e - 180*a*b^11*c*x^3 + 360*a^2*b^10*d*x^3 + 900*a^
4*b^8*f*x^3 - 600*a^3*b^9*x^3*e)/b^15