3.278 \(\int \frac {x^8 (c+d x^3+e x^6+f x^9)}{(a+b x^3)^3} \, dx\)

Optimal. Leaf size=186 \[ \frac {x^3 \left (6 a^2 f-3 a b e+b^2 d\right )}{3 b^5}+\frac {a \left (-5 a^3 f+4 a^2 b e-3 a b^2 d+2 b^3 c\right )}{3 b^6 \left (a+b x^3\right )}-\frac {a^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^6 \left (a+b x^3\right )^2}+\frac {\log \left (a+b x^3\right ) \left (-10 a^3 f+6 a^2 b e-3 a b^2 d+b^3 c\right )}{3 b^6}+\frac {x^6 (b e-3 a f)}{6 b^4}+\frac {f x^9}{9 b^3} \]

[Out]

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

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Rubi [A]  time = 0.27, antiderivative size = 186, normalized size of antiderivative = 1.00, 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} \[ \frac {a \left (4 a^2 b e-5 a^3 f-3 a b^2 d+2 b^3 c\right )}{3 b^6 \left (a+b x^3\right )}-\frac {a^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 b^6 \left (a+b x^3\right )^2}+\frac {\log \left (a+b x^3\right ) \left (6 a^2 b e-10 a^3 f-3 a b^2 d+b^3 c\right )}{3 b^6}+\frac {x^3 \left (6 a^2 f-3 a b e+b^2 d\right )}{3 b^5}+\frac {x^6 (b e-3 a f)}{6 b^4}+\frac {f x^9}{9 b^3} \]

Antiderivative was successfully verified.

[In]

Int[(x^8*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^3,x]

[Out]

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

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]

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]]

Rubi steps

\begin {align*} \int \frac {x^8 \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^2 \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 {b^2 d-3 a b e+6 a^2 f}{b^5}+\frac {(b e-3 a f) x}{b^4}+\frac {f x^2}{b^3}-\frac {a^2 \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right )}{b^5 (a+b x)^3}+\frac {a \left (-2 b^3 c+3 a b^2 d-4 a^2 b e+5 a^3 f\right )}{b^5 (a+b x)^2}+\frac {b^3 c-3 a b^2 d+6 a^2 b e-10 a^3 f}{b^5 (a+b x)}\right ) \, dx,x,x^3\right )\\ &=\frac {\left (b^2 d-3 a b e+6 a^2 f\right ) x^3}{3 b^5}+\frac {(b e-3 a f) x^6}{6 b^4}+\frac {f x^9}{9 b^3}-\frac {a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )}{6 b^6 \left (a+b x^3\right )^2}+\frac {a \left (2 b^3 c-3 a b^2 d+4 a^2 b e-5 a^3 f\right )}{3 b^6 \left (a+b x^3\right )}+\frac {\left (b^3 c-3 a b^2 d+6 a^2 b e-10 a^3 f\right ) \log \left (a+b x^3\right )}{3 b^6}\\ \end {align*}

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Mathematica [A]  time = 0.17, size = 170, normalized size = 0.91 \[ \frac {6 b x^3 \left (6 a^2 f-3 a b e+b^2 d\right )-\frac {6 a \left (5 a^3 f-4 a^2 b e+3 a b^2 d-2 b^3 c\right )}{a+b x^3}+\frac {3 a^2 \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{\left (a+b x^3\right )^2}+6 \log \left (a+b x^3\right ) \left (-10 a^3 f+6 a^2 b e-3 a b^2 d+b^3 c\right )+3 b^2 x^6 (b e-3 a f)+2 b^3 f x^9}{18 b^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^8*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^3,x]

[Out]

(6*b*(b^2*d - 3*a*b*e + 6*a^2*f)*x^3 + 3*b^2*(b*e - 3*a*f)*x^6 + 2*b^3*f*x^9 + (3*a^2*(-(b^3*c) + a*b^2*d - a^
2*b*e + a^3*f))/(a + b*x^3)^2 - (6*a*(-2*b^3*c + 3*a*b^2*d - 4*a^2*b*e + 5*a^3*f))/(a + b*x^3) + 6*(b^3*c - 3*
a*b^2*d + 6*a^2*b*e - 10*a^3*f)*Log[a + b*x^3])/(18*b^6)

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fricas [A]  time = 0.84, size = 295, normalized size = 1.59 \[ \frac {2 \, b^{5} f x^{15} + {\left (3 \, b^{5} e - 5 \, a b^{4} f\right )} x^{12} + 2 \, {\left (3 \, b^{5} d - 6 \, a b^{4} e + 10 \, a^{2} b^{3} f\right )} x^{9} + 3 \, {\left (4 \, a b^{4} d - 11 \, a^{2} b^{3} e + 21 \, a^{3} b^{2} f\right )} x^{6} + 9 \, a^{2} b^{3} c - 15 \, a^{3} b^{2} d + 21 \, a^{4} b e - 27 \, a^{5} f + 6 \, {\left (2 \, a b^{4} c - 2 \, a^{2} b^{3} d + a^{3} b^{2} e + a^{4} b f\right )} x^{3} + 6 \, {\left ({\left (b^{5} c - 3 \, a b^{4} d + 6 \, a^{2} b^{3} e - 10 \, a^{3} b^{2} f\right )} x^{6} + a^{2} b^{3} c - 3 \, a^{3} b^{2} d + 6 \, a^{4} b e - 10 \, a^{5} f + 2 \, {\left (a b^{4} c - 3 \, a^{2} b^{3} d + 6 \, a^{3} b^{2} e - 10 \, a^{4} b f\right )} x^{3}\right )} \log \left (b x^{3} + a\right )}{18 \, {\left (b^{8} x^{6} + 2 \, a b^{7} x^{3} + a^{2} b^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^3,x, algorithm="fricas")

[Out]

1/18*(2*b^5*f*x^15 + (3*b^5*e - 5*a*b^4*f)*x^12 + 2*(3*b^5*d - 6*a*b^4*e + 10*a^2*b^3*f)*x^9 + 3*(4*a*b^4*d -
11*a^2*b^3*e + 21*a^3*b^2*f)*x^6 + 9*a^2*b^3*c - 15*a^3*b^2*d + 21*a^4*b*e - 27*a^5*f + 6*(2*a*b^4*c - 2*a^2*b
^3*d + a^3*b^2*e + a^4*b*f)*x^3 + 6*((b^5*c - 3*a*b^4*d + 6*a^2*b^3*e - 10*a^3*b^2*f)*x^6 + a^2*b^3*c - 3*a^3*
b^2*d + 6*a^4*b*e - 10*a^5*f + 2*(a*b^4*c - 3*a^2*b^3*d + 6*a^3*b^2*e - 10*a^4*b*f)*x^3)*log(b*x^3 + a))/(b^8*
x^6 + 2*a*b^7*x^3 + a^2*b^6)

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giac [A]  time = 0.22, size = 236, normalized size = 1.27 \[ \frac {{\left (b^{3} c - 3 \, a b^{2} d - 10 \, a^{3} f + 6 \, a^{2} b e\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{6}} - \frac {3 \, b^{5} c x^{6} - 9 \, a b^{4} d x^{6} - 30 \, a^{3} b^{2} f x^{6} + 18 \, a^{2} b^{3} x^{6} e + 2 \, a b^{4} c x^{3} - 12 \, a^{2} b^{3} d x^{3} - 50 \, a^{4} b f x^{3} + 28 \, a^{3} b^{2} x^{3} e - 4 \, a^{3} b^{2} d - 21 \, a^{5} f + 11 \, a^{4} b e}{6 \, {\left (b x^{3} + a\right )}^{2} b^{6}} + \frac {2 \, b^{6} f x^{9} - 9 \, a b^{5} f x^{6} + 3 \, b^{6} x^{6} e + 6 \, b^{6} d x^{3} + 36 \, a^{2} b^{4} f x^{3} - 18 \, a b^{5} x^{3} e}{18 \, b^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^3,x, algorithm="giac")

[Out]

1/3*(b^3*c - 3*a*b^2*d - 10*a^3*f + 6*a^2*b*e)*log(abs(b*x^3 + a))/b^6 - 1/6*(3*b^5*c*x^6 - 9*a*b^4*d*x^6 - 30
*a^3*b^2*f*x^6 + 18*a^2*b^3*x^6*e + 2*a*b^4*c*x^3 - 12*a^2*b^3*d*x^3 - 50*a^4*b*f*x^3 + 28*a^3*b^2*x^3*e - 4*a
^3*b^2*d - 21*a^5*f + 11*a^4*b*e)/((b*x^3 + a)^2*b^6) + 1/18*(2*b^6*f*x^9 - 9*a*b^5*f*x^6 + 3*b^6*x^6*e + 6*b^
6*d*x^3 + 36*a^2*b^4*f*x^3 - 18*a*b^5*x^3*e)/b^9

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maple [A]  time = 0.06, size = 266, normalized size = 1.43 \[ \frac {f \,x^{9}}{9 b^{3}}-\frac {a f \,x^{6}}{2 b^{4}}+\frac {e \,x^{6}}{6 b^{3}}+\frac {2 a^{2} f \,x^{3}}{b^{5}}-\frac {a e \,x^{3}}{b^{4}}+\frac {d \,x^{3}}{3 b^{3}}+\frac {a^{5} f}{6 \left (b \,x^{3}+a \right )^{2} b^{6}}-\frac {a^{4} e}{6 \left (b \,x^{3}+a \right )^{2} b^{5}}+\frac {a^{3} d}{6 \left (b \,x^{3}+a \right )^{2} b^{4}}-\frac {a^{2} c}{6 \left (b \,x^{3}+a \right )^{2} b^{3}}-\frac {5 a^{4} f}{3 \left (b \,x^{3}+a \right ) b^{6}}+\frac {4 a^{3} e}{3 \left (b \,x^{3}+a \right ) b^{5}}-\frac {10 a^{3} f \ln \left (b \,x^{3}+a \right )}{3 b^{6}}-\frac {a^{2} d}{\left (b \,x^{3}+a \right ) b^{4}}+\frac {2 a^{2} e \ln \left (b \,x^{3}+a \right )}{b^{5}}+\frac {2 a c}{3 \left (b \,x^{3}+a \right ) b^{3}}-\frac {a d \ln \left (b \,x^{3}+a \right )}{b^{4}}+\frac {c \ln \left (b \,x^{3}+a \right )}{3 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^3,x)

[Out]

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

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maxima [A]  time = 1.35, size = 191, normalized size = 1.03 \[ \frac {3 \, a^{2} b^{3} c - 5 \, a^{3} b^{2} d + 7 \, a^{4} b e - 9 \, a^{5} f + 2 \, {\left (2 \, a b^{4} c - 3 \, a^{2} b^{3} d + 4 \, a^{3} b^{2} e - 5 \, a^{4} b f\right )} x^{3}}{6 \, {\left (b^{8} x^{6} + 2 \, a b^{7} x^{3} + a^{2} b^{6}\right )}} + \frac {2 \, b^{2} f x^{9} + 3 \, {\left (b^{2} e - 3 \, a b f\right )} x^{6} + 6 \, {\left (b^{2} d - 3 \, a b e + 6 \, a^{2} f\right )} x^{3}}{18 \, b^{5}} + \frac {{\left (b^{3} c - 3 \, a b^{2} d + 6 \, a^{2} b e - 10 \, a^{3} f\right )} \log \left (b x^{3} + a\right )}{3 \, b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^3,x, algorithm="maxima")

[Out]

1/6*(3*a^2*b^3*c - 5*a^3*b^2*d + 7*a^4*b*e - 9*a^5*f + 2*(2*a*b^4*c - 3*a^2*b^3*d + 4*a^3*b^2*e - 5*a^4*b*f)*x
^3)/(b^8*x^6 + 2*a*b^7*x^3 + a^2*b^6) + 1/18*(2*b^2*f*x^9 + 3*(b^2*e - 3*a*b*f)*x^6 + 6*(b^2*d - 3*a*b*e + 6*a
^2*f)*x^3)/b^5 + 1/3*(b^3*c - 3*a*b^2*d + 6*a^2*b*e - 10*a^3*f)*log(b*x^3 + a)/b^6

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mupad [B]  time = 4.92, size = 204, normalized size = 1.10 \[ x^6\,\left (\frac {e}{6\,b^3}-\frac {a\,f}{2\,b^4}\right )-\frac {x^3\,\left (\frac {5\,f\,a^4}{3}-\frac {4\,e\,a^3\,b}{3}+d\,a^2\,b^2-\frac {2\,c\,a\,b^3}{3}\right )+\frac {9\,f\,a^5-7\,e\,a^4\,b+5\,d\,a^3\,b^2-3\,c\,a^2\,b^3}{6\,b}}{a^2\,b^5+2\,a\,b^6\,x^3+b^7\,x^6}-x^3\,\left (\frac {a^2\,f}{b^5}-\frac {d}{3\,b^3}+\frac {a\,\left (\frac {e}{b^3}-\frac {3\,a\,f}{b^4}\right )}{b}\right )+\frac {\ln \left (b\,x^3+a\right )\,\left (-10\,f\,a^3+6\,e\,a^2\,b-3\,d\,a\,b^2+c\,b^3\right )}{3\,b^6}+\frac {f\,x^9}{9\,b^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^8*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^3,x)

[Out]

x^6*(e/(6*b^3) - (a*f)/(2*b^4)) - (x^3*((5*a^4*f)/3 + a^2*b^2*d - (2*a*b^3*c)/3 - (4*a^3*b*e)/3) + (9*a^5*f -
3*a^2*b^3*c + 5*a^3*b^2*d - 7*a^4*b*e)/(6*b))/(a^2*b^5 + b^7*x^6 + 2*a*b^6*x^3) - x^3*((a^2*f)/b^5 - d/(3*b^3)
+ (a*(e/b^3 - (3*a*f)/b^4))/b) + (log(a + b*x^3)*(b^3*c - 10*a^3*f - 3*a*b^2*d + 6*a^2*b*e))/(3*b^6) + (f*x^9
)/(9*b^3)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**8*(f*x**9+e*x**6+d*x**3+c)/(b*x**3+a)**3,x)

[Out]

Timed out

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