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

Optimal. Leaf size=132 \[ \frac {x^6 \left (a^2 f-a b e+b^2 d\right )}{6 b^3}-\frac {a \log \left (a+b x^3\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 b^5}+\frac {x^3 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 b^4}+\frac {x^9 (b e-a f)}{9 b^2}+\frac {f x^{12}}{12 b} \]

[Out]

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

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Rubi [A]  time = 0.18, antiderivative size = 132, 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} \[ \frac {x^3 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 b^4}-\frac {a \log \left (a+b x^3\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 b^5}+\frac {x^6 \left (a^2 f-a b e+b^2 d\right )}{6 b^3}+\frac {x^9 (b e-a f)}{9 b^2}+\frac {f x^{12}}{12 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]  time = 0.07, size = 119, normalized size = 0.90 \[ \frac {12 a \log \left (a+b x^3\right ) \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )+b x^3 \left (-12 a^3 f+6 a^2 b \left (2 e+f x^3\right )-2 a b^2 \left (6 d+3 e x^3+2 f x^6\right )+b^3 \left (12 c+6 d x^3+4 e x^6+3 f x^9\right )\right )}{36 b^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*x^3*(-12*a^3*f + 6*a^2*b*(2*e + f*x^3) - 2*a*b^2*(6*d + 3*e*x^3 + 2*f*x^6) + b^3*(12*c + 6*d*x^3 + 4*e*x^6
+ 3*f*x^9)) + 12*a*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*Log[a + b*x^3])/(36*b^5)

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fricas [A]  time = 0.59, size = 130, normalized size = 0.98 \[ \frac {3 \, b^{4} f x^{12} + 4 \, {\left (b^{4} e - a b^{3} f\right )} x^{9} + 6 \, {\left (b^{4} d - a b^{3} e + a^{2} b^{2} f\right )} x^{6} + 12 \, {\left (b^{4} c - a b^{3} d + a^{2} b^{2} e - a^{3} b f\right )} x^{3} - 12 \, {\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} \log \left (b x^{3} + a\right )}{36 \, b^{5}} \]

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),x, algorithm="fricas")

[Out]

1/36*(3*b^4*f*x^12 + 4*(b^4*e - a*b^3*f)*x^9 + 6*(b^4*d - a*b^3*e + a^2*b^2*f)*x^6 + 12*(b^4*c - a*b^3*d + a^2
*b^2*e - a^3*b*f)*x^3 - 12*(a*b^3*c - a^2*b^2*d + a^3*b*e - a^4*f)*log(b*x^3 + a))/b^5

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giac [A]  time = 0.17, size = 148, normalized size = 1.12 \[ \frac {3 \, b^{3} f x^{12} - 4 \, a b^{2} f x^{9} + 4 \, b^{3} x^{9} e + 6 \, b^{3} d x^{6} + 6 \, a^{2} b f x^{6} - 6 \, a b^{2} x^{6} e + 12 \, b^{3} c x^{3} - 12 \, a b^{2} d x^{3} - 12 \, a^{3} f x^{3} + 12 \, a^{2} b x^{3} e}{36 \, b^{4}} - \frac {{\left (a b^{3} c - a^{2} b^{2} d - a^{4} f + a^{3} b e\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{5}} \]

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),x, algorithm="giac")

[Out]

1/36*(3*b^3*f*x^12 - 4*a*b^2*f*x^9 + 4*b^3*x^9*e + 6*b^3*d*x^6 + 6*a^2*b*f*x^6 - 6*a*b^2*x^6*e + 12*b^3*c*x^3
- 12*a*b^2*d*x^3 - 12*a^3*f*x^3 + 12*a^2*b*x^3*e)/b^4 - 1/3*(a*b^3*c - a^2*b^2*d - a^4*f + a^3*b*e)*log(abs(b*
x^3 + a))/b^5

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maple [A]  time = 0.05, size = 170, normalized size = 1.29 \[ \frac {f \,x^{12}}{12 b}-\frac {a f \,x^{9}}{9 b^{2}}+\frac {e \,x^{9}}{9 b}+\frac {a^{2} f \,x^{6}}{6 b^{3}}-\frac {a e \,x^{6}}{6 b^{2}}+\frac {d \,x^{6}}{6 b}-\frac {a^{3} f \,x^{3}}{3 b^{4}}+\frac {a^{2} e \,x^{3}}{3 b^{3}}-\frac {a d \,x^{3}}{3 b^{2}}+\frac {c \,x^{3}}{3 b}+\frac {a^{4} f \ln \left (b \,x^{3}+a \right )}{3 b^{5}}-\frac {a^{3} e \ln \left (b \,x^{3}+a \right )}{3 b^{4}}+\frac {a^{2} d \ln \left (b \,x^{3}+a \right )}{3 b^{3}}-\frac {a c \ln \left (b \,x^{3}+a \right )}{3 b^{2}} \]

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),x)

[Out]

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

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maxima [A]  time = 1.37, size = 129, normalized size = 0.98 \[ \frac {3 \, b^{3} f x^{12} + 4 \, {\left (b^{3} e - a b^{2} f\right )} x^{9} + 6 \, {\left (b^{3} d - a b^{2} e + a^{2} b f\right )} x^{6} + 12 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{3}}{36 \, b^{4}} - \frac {{\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} \log \left (b x^{3} + a\right )}{3 \, b^{5}} \]

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),x, algorithm="maxima")

[Out]

1/36*(3*b^3*f*x^12 + 4*(b^3*e - a*b^2*f)*x^9 + 6*(b^3*d - a*b^2*e + a^2*b*f)*x^6 + 12*(b^3*c - a*b^2*d + a^2*b
*e - a^3*f)*x^3)/b^4 - 1/3*(a*b^3*c - a^2*b^2*d + a^3*b*e - a^4*f)*log(b*x^3 + a)/b^5

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mupad [B]  time = 4.93, size = 141, normalized size = 1.07 \[ x^9\,\left (\frac {e}{9\,b}-\frac {a\,f}{9\,b^2}\right )+x^6\,\left (\frac {d}{6\,b}-\frac {a\,\left (\frac {e}{b}-\frac {a\,f}{b^2}\right )}{6\,b}\right )+x^3\,\left (\frac {c}{3\,b}-\frac {a\,\left (\frac {d}{b}-\frac {a\,\left (\frac {e}{b}-\frac {a\,f}{b^2}\right )}{b}\right )}{3\,b}\right )+\frac {f\,x^{12}}{12\,b}+\frac {\ln \left (b\,x^3+a\right )\,\left (f\,a^4-e\,a^3\,b+d\,a^2\,b^2-c\,a\,b^3\right )}{3\,b^5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 1.05, size = 128, normalized size = 0.97 \[ \frac {a \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log {\left (a + b x^{3} \right )}}{3 b^{5}} + x^{9} \left (- \frac {a f}{9 b^{2}} + \frac {e}{9 b}\right ) + x^{6} \left (\frac {a^{2} f}{6 b^{3}} - \frac {a e}{6 b^{2}} + \frac {d}{6 b}\right ) + x^{3} \left (- \frac {a^{3} f}{3 b^{4}} + \frac {a^{2} e}{3 b^{3}} - \frac {a d}{3 b^{2}} + \frac {c}{3 b}\right ) + \frac {f x^{12}}{12 b} \]

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),x)

[Out]

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

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