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

Optimal. Leaf size=96 \[ \frac {x^3 \left (a^2 f-a b e+b^2 d\right )}{3 b^3}+\frac {\log \left (a+b x^3\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 b^4}+\frac {x^6 (b e-a f)}{6 b^2}+\frac {f x^9}{9 b} \]

[Out]

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

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Rubi [A]  time = 0.14, antiderivative size = 96, 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 = {1819, 1850} \[ \frac {\log \left (a+b x^3\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 b^4}+\frac {x^3 \left (a^2 f-a b e+b^2 d\right )}{3 b^3}+\frac {x^6 (b e-a f)}{6 b^2}+\frac {f x^9}{9 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 1819

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1)
, Pq, x]*(a + b*x^Simplify[n/(m + 1)])^p, x], x, x^(m + 1)], x] /; FreeQ[{a, b, m, n, p}, x] && NeQ[m, -1] &&
IGtQ[Simplify[n/(m + 1)], 0] && PolyQ[Pq, x^(m + 1)]

Rule 1850

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^n)^p, x], x] /; FreeQ[
{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

Rubi steps

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

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Mathematica [A]  time = 0.05, size = 88, normalized size = 0.92 \[ \frac {b x^3 \left (6 a^2 f-3 a b \left (2 e+f x^3\right )+b^2 \left (6 d+3 e x^3+2 f x^6\right )\right )+6 \log \left (a+b x^3\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{18 b^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.74, size = 92, normalized size = 0.96 \[ \frac {2 \, b^{3} f x^{9} + 3 \, {\left (b^{3} e - a b^{2} f\right )} x^{6} + 6 \, {\left (b^{3} d - a b^{2} e + a^{2} b f\right )} x^{3} + 6 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \log \left (b x^{3} + a\right )}{18 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.20, size = 101, normalized size = 1.05 \[ \frac {2 \, b^{2} f x^{9} - 3 \, a b f x^{6} + 3 \, b^{2} x^{6} e + 6 \, b^{2} d x^{3} + 6 \, a^{2} f x^{3} - 6 \, a b x^{3} e}{18 \, b^{3}} + \frac {{\left (b^{3} c - a b^{2} d - a^{3} f + a^{2} b e\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.04, size = 124, normalized size = 1.29 \[ \frac {f \,x^{9}}{9 b}-\frac {a f \,x^{6}}{6 b^{2}}+\frac {e \,x^{6}}{6 b}+\frac {a^{2} f \,x^{3}}{3 b^{3}}-\frac {a e \,x^{3}}{3 b^{2}}+\frac {d \,x^{3}}{3 b}-\frac {a^{3} f \ln \left (b \,x^{3}+a \right )}{3 b^{4}}+\frac {a^{2} e \ln \left (b \,x^{3}+a \right )}{3 b^{3}}-\frac {a d \ln \left (b \,x^{3}+a \right )}{3 b^{2}}+\frac {c \ln \left (b \,x^{3}+a \right )}{3 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.39, size = 91, normalized size = 0.95 \[ \frac {2 \, b^{2} f x^{9} + 3 \, {\left (b^{2} e - a b f\right )} x^{6} + 6 \, {\left (b^{2} d - a b e + a^{2} f\right )} x^{3}}{18 \, b^{3}} + \frac {{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \log \left (b x^{3} + a\right )}{3 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 4.83, size = 96, normalized size = 1.00 \[ x^6\,\left (\frac {e}{6\,b}-\frac {a\,f}{6\,b^2}\right )+x^3\,\left (\frac {d}{3\,b}-\frac {a\,\left (\frac {e}{b}-\frac {a\,f}{b^2}\right )}{3\,b}\right )+\frac {\ln \left (b\,x^3+a\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{3\,b^4}+\frac {f\,x^9}{9\,b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 1.13, size = 88, normalized size = 0.92 \[ x^{6} \left (- \frac {a f}{6 b^{2}} + \frac {e}{6 b}\right ) + x^{3} \left (\frac {a^{2} f}{3 b^{3}} - \frac {a e}{3 b^{2}} + \frac {d}{3 b}\right ) + \frac {f x^{9}}{9 b} - \frac {\left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log {\left (a + b x^{3} \right )}}{3 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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