3.286 \(\int x^{17} (a+b x^3)^8 \, dx\)

Optimal. Leaf size=110 \[ \frac{5 a^2 \left (a+b x^3\right )^{12}}{18 b^6}-\frac{10 a^3 \left (a+b x^3\right )^{11}}{33 b^6}+\frac{a^4 \left (a+b x^3\right )^{10}}{6 b^6}-\frac{a^5 \left (a+b x^3\right )^9}{27 b^6}+\frac{\left (a+b x^3\right )^{14}}{42 b^6}-\frac{5 a \left (a+b x^3\right )^{13}}{39 b^6} \]

[Out]

-(a^5*(a + b*x^3)^9)/(27*b^6) + (a^4*(a + b*x^3)^10)/(6*b^6) - (10*a^3*(a + b*x^3)^11)/(33*b^6) + (5*a^2*(a +
b*x^3)^12)/(18*b^6) - (5*a*(a + b*x^3)^13)/(39*b^6) + (a + b*x^3)^14/(42*b^6)

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Rubi [A]  time = 0.172613, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {266, 43} \[ \frac{5 a^2 \left (a+b x^3\right )^{12}}{18 b^6}-\frac{10 a^3 \left (a+b x^3\right )^{11}}{33 b^6}+\frac{a^4 \left (a+b x^3\right )^{10}}{6 b^6}-\frac{a^5 \left (a+b x^3\right )^9}{27 b^6}+\frac{\left (a+b x^3\right )^{14}}{42 b^6}-\frac{5 a \left (a+b x^3\right )^{13}}{39 b^6} \]

Antiderivative was successfully verified.

[In]

Int[x^17*(a + b*x^3)^8,x]

[Out]

-(a^5*(a + b*x^3)^9)/(27*b^6) + (a^4*(a + b*x^3)^10)/(6*b^6) - (10*a^3*(a + b*x^3)^11)/(33*b^6) + (5*a^2*(a +
b*x^3)^12)/(18*b^6) - (5*a*(a + b*x^3)^13)/(39*b^6) + (a + b*x^3)^14/(42*b^6)

Rule 266

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int x^{17} \left (a+b x^3\right )^8 \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int x^5 (a+b x)^8 \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (-\frac{a^5 (a+b x)^8}{b^5}+\frac{5 a^4 (a+b x)^9}{b^5}-\frac{10 a^3 (a+b x)^{10}}{b^5}+\frac{10 a^2 (a+b x)^{11}}{b^5}-\frac{5 a (a+b x)^{12}}{b^5}+\frac{(a+b x)^{13}}{b^5}\right ) \, dx,x,x^3\right )\\ &=-\frac{a^5 \left (a+b x^3\right )^9}{27 b^6}+\frac{a^4 \left (a+b x^3\right )^{10}}{6 b^6}-\frac{10 a^3 \left (a+b x^3\right )^{11}}{33 b^6}+\frac{5 a^2 \left (a+b x^3\right )^{12}}{18 b^6}-\frac{5 a \left (a+b x^3\right )^{13}}{39 b^6}+\frac{\left (a+b x^3\right )^{14}}{42 b^6}\\ \end{align*}

Mathematica [A]  time = 0.0028087, size = 108, normalized size = 0.98 \[ \frac{7}{9} a^2 b^6 x^{36}+\frac{56}{33} a^3 b^5 x^{33}+\frac{7}{3} a^4 b^4 x^{30}+\frac{56}{27} a^5 b^3 x^{27}+\frac{7}{6} a^6 b^2 x^{24}+\frac{8}{21} a^7 b x^{21}+\frac{a^8 x^{18}}{18}+\frac{8}{39} a b^7 x^{39}+\frac{b^8 x^{42}}{42} \]

Antiderivative was successfully verified.

[In]

Integrate[x^17*(a + b*x^3)^8,x]

[Out]

(a^8*x^18)/18 + (8*a^7*b*x^21)/21 + (7*a^6*b^2*x^24)/6 + (56*a^5*b^3*x^27)/27 + (7*a^4*b^4*x^30)/3 + (56*a^3*b
^5*x^33)/33 + (7*a^2*b^6*x^36)/9 + (8*a*b^7*x^39)/39 + (b^8*x^42)/42

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Maple [A]  time = 0.002, size = 91, normalized size = 0.8 \begin{align*}{\frac{{b}^{8}{x}^{42}}{42}}+{\frac{8\,a{b}^{7}{x}^{39}}{39}}+{\frac{7\,{b}^{6}{a}^{2}{x}^{36}}{9}}+{\frac{56\,{a}^{3}{b}^{5}{x}^{33}}{33}}+{\frac{7\,{a}^{4}{b}^{4}{x}^{30}}{3}}+{\frac{56\,{a}^{5}{b}^{3}{x}^{27}}{27}}+{\frac{7\,{a}^{6}{b}^{2}{x}^{24}}{6}}+{\frac{8\,{a}^{7}b{x}^{21}}{21}}+{\frac{{a}^{8}{x}^{18}}{18}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^17*(b*x^3+a)^8,x)

[Out]

1/42*b^8*x^42+8/39*a*b^7*x^39+7/9*b^6*a^2*x^36+56/33*a^3*b^5*x^33+7/3*a^4*b^4*x^30+56/27*a^5*b^3*x^27+7/6*a^6*
b^2*x^24+8/21*a^7*b*x^21+1/18*a^8*x^18

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Maxima [A]  time = 0.971629, size = 122, normalized size = 1.11 \begin{align*} \frac{1}{42} \, b^{8} x^{42} + \frac{8}{39} \, a b^{7} x^{39} + \frac{7}{9} \, a^{2} b^{6} x^{36} + \frac{56}{33} \, a^{3} b^{5} x^{33} + \frac{7}{3} \, a^{4} b^{4} x^{30} + \frac{56}{27} \, a^{5} b^{3} x^{27} + \frac{7}{6} \, a^{6} b^{2} x^{24} + \frac{8}{21} \, a^{7} b x^{21} + \frac{1}{18} \, a^{8} x^{18} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^17*(b*x^3+a)^8,x, algorithm="maxima")

[Out]

1/42*b^8*x^42 + 8/39*a*b^7*x^39 + 7/9*a^2*b^6*x^36 + 56/33*a^3*b^5*x^33 + 7/3*a^4*b^4*x^30 + 56/27*a^5*b^3*x^2
7 + 7/6*a^6*b^2*x^24 + 8/21*a^7*b*x^21 + 1/18*a^8*x^18

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Fricas [A]  time = 1.53003, size = 224, normalized size = 2.04 \begin{align*} \frac{1}{42} x^{42} b^{8} + \frac{8}{39} x^{39} b^{7} a + \frac{7}{9} x^{36} b^{6} a^{2} + \frac{56}{33} x^{33} b^{5} a^{3} + \frac{7}{3} x^{30} b^{4} a^{4} + \frac{56}{27} x^{27} b^{3} a^{5} + \frac{7}{6} x^{24} b^{2} a^{6} + \frac{8}{21} x^{21} b a^{7} + \frac{1}{18} x^{18} a^{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^17*(b*x^3+a)^8,x, algorithm="fricas")

[Out]

1/42*x^42*b^8 + 8/39*x^39*b^7*a + 7/9*x^36*b^6*a^2 + 56/33*x^33*b^5*a^3 + 7/3*x^30*b^4*a^4 + 56/27*x^27*b^3*a^
5 + 7/6*x^24*b^2*a^6 + 8/21*x^21*b*a^7 + 1/18*x^18*a^8

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Sympy [A]  time = 0.115149, size = 107, normalized size = 0.97 \begin{align*} \frac{a^{8} x^{18}}{18} + \frac{8 a^{7} b x^{21}}{21} + \frac{7 a^{6} b^{2} x^{24}}{6} + \frac{56 a^{5} b^{3} x^{27}}{27} + \frac{7 a^{4} b^{4} x^{30}}{3} + \frac{56 a^{3} b^{5} x^{33}}{33} + \frac{7 a^{2} b^{6} x^{36}}{9} + \frac{8 a b^{7} x^{39}}{39} + \frac{b^{8} x^{42}}{42} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**17*(b*x**3+a)**8,x)

[Out]

a**8*x**18/18 + 8*a**7*b*x**21/21 + 7*a**6*b**2*x**24/6 + 56*a**5*b**3*x**27/27 + 7*a**4*b**4*x**30/3 + 56*a**
3*b**5*x**33/33 + 7*a**2*b**6*x**36/9 + 8*a*b**7*x**39/39 + b**8*x**42/42

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Giac [A]  time = 1.59656, size = 122, normalized size = 1.11 \begin{align*} \frac{1}{42} \, b^{8} x^{42} + \frac{8}{39} \, a b^{7} x^{39} + \frac{7}{9} \, a^{2} b^{6} x^{36} + \frac{56}{33} \, a^{3} b^{5} x^{33} + \frac{7}{3} \, a^{4} b^{4} x^{30} + \frac{56}{27} \, a^{5} b^{3} x^{27} + \frac{7}{6} \, a^{6} b^{2} x^{24} + \frac{8}{21} \, a^{7} b x^{21} + \frac{1}{18} \, a^{8} x^{18} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^17*(b*x^3+a)^8,x, algorithm="giac")

[Out]

1/42*b^8*x^42 + 8/39*a*b^7*x^39 + 7/9*a^2*b^6*x^36 + 56/33*a^3*b^5*x^33 + 7/3*a^4*b^4*x^30 + 56/27*a^5*b^3*x^2
7 + 7/6*a^6*b^2*x^24 + 8/21*a^7*b*x^21 + 1/18*a^8*x^18