3.3.86 \(\int x^{17} (a+b x^3)^8 \, dx\) [286]

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

[Out]

-1/27*a^5*(b*x^3+a)^9/b^6+1/6*a^4*(b*x^3+a)^10/b^6-10/33*a^3*(b*x^3+a)^11/b^6+5/18*a^2*(b*x^3+a)^12/b^6-5/39*a
*(b*x^3+a)^13/b^6+1/42*(b*x^3+a)^14/b^6

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Rubi [A]
time = 0.12, antiderivative size = 110, normalized size of antiderivative = 1.00, 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 = {272, 45} \begin {gather*} -\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 {\left (a+b x^3\right )^{14}}{42 b^6}-\frac {5 a \left (a+b x^3\right )^{13}}{39 b^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^17*(a + b*x^3)^8,x]

[Out]

-1/27*(a^5*(a + b*x^3)^9)/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 45

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])

Rule 272

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]]

Rubi steps

\begin {align*} \int x^{17} \left (a+b x^3\right )^8 \, dx &=\frac {1}{3} \text {Subst}\left (\int x^5 (a+b x)^8 \, dx,x,x^3\right )\\ &=\frac {1}{3} \text {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*}

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Mathematica [A]
time = 0.00, size = 108, normalized size = 0.98 \begin {gather*} \frac {a^8 x^{18}}{18}+\frac {8}{21} a^7 b x^{21}+\frac {7}{6} a^6 b^2 x^{24}+\frac {56}{27} a^5 b^3 x^{27}+\frac {7}{3} a^4 b^4 x^{30}+\frac {56}{33} a^3 b^5 x^{33}+\frac {7}{9} a^2 b^6 x^{36}+\frac {8}{39} a b^7 x^{39}+\frac {b^8 x^{42}}{42} \end {gather*}

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 = 0.14, size = 91, normalized size = 0.83

method	result	size
gosper	\(\frac {8}{39} a \,b^{7} x^{39}+\frac {1}{42} b^{8} x^{42}+\frac {8}{21} b \,a^{7} x^{21}+\frac {7}{6} a^{6} b^{2} x^{24}+\frac {56}{27} a^{5} b^{3} x^{27}+\frac {7}{3} a^{4} b^{4} x^{30}+\frac {56}{33} a^{3} b^{5} x^{33}+\frac {7}{9} a^{2} b^{6} x^{36}+\frac {1}{18} a^{8} x^{18}\)	\(91\)
default	\(\frac {8}{39} a \,b^{7} x^{39}+\frac {1}{42} b^{8} x^{42}+\frac {8}{21} b \,a^{7} x^{21}+\frac {7}{6} a^{6} b^{2} x^{24}+\frac {56}{27} a^{5} b^{3} x^{27}+\frac {7}{3} a^{4} b^{4} x^{30}+\frac {56}{33} a^{3} b^{5} x^{33}+\frac {7}{9} a^{2} b^{6} x^{36}+\frac {1}{18} a^{8} x^{18}\)	\(91\)
risch	\(\frac {8}{39} a \,b^{7} x^{39}+\frac {1}{42} b^{8} x^{42}+\frac {8}{21} b \,a^{7} x^{21}+\frac {7}{6} a^{6} b^{2} x^{24}+\frac {56}{27} a^{5} b^{3} x^{27}+\frac {7}{3} a^{4} b^{4} x^{30}+\frac {56}{33} a^{3} b^{5} x^{33}+\frac {7}{9} a^{2} b^{6} x^{36}+\frac {1}{18} a^{8} x^{18}\)	\(91\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.30, size = 90, normalized size = 0.82 \begin {gather*} \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 {gather*}

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 = 0.33, size = 90, normalized size = 0.82 \begin {gather*} \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 {gather*}

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*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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Sympy [A]
time = 0.02, size = 107, normalized size = 0.97 \begin {gather*} \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 {gather*}

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 = 1.71, size = 90, normalized size = 0.82 \begin {gather*} \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 {gather*}

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

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Mupad [B]
time = 0.09, size = 90, normalized size = 0.82 \begin {gather*} \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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^17*(a + b*x^3)^8,x)

[Out]

(a^8*x^18)/18 + (b^8*x^42)/42 + (8*a^7*b*x^21)/21 + (8*a*b^7*x^39)/39 + (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

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