3.3.58 \(\int x^{17} (a+b x^3)^5 \, dx\) [258]

Optimal. Leaf size=69 \[ \frac {a^5 x^{18}}{18}+\frac {5}{21} a^4 b x^{21}+\frac {5}{12} a^3 b^2 x^{24}+\frac {10}{27} a^2 b^3 x^{27}+\frac {1}{6} a b^4 x^{30}+\frac {b^5 x^{33}}{33} \]

[Out]

1/18*a^5*x^18+5/21*a^4*b*x^21+5/12*a^3*b^2*x^24+10/27*a^2*b^3*x^27+1/6*a*b^4*x^30+1/33*b^5*x^33

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Rubi [A]
time = 0.03, antiderivative size = 69, 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 x^{18}}{18}+\frac {5}{21} a^4 b x^{21}+\frac {5}{12} a^3 b^2 x^{24}+\frac {10}{27} a^2 b^3 x^{27}+\frac {1}{6} a b^4 x^{30}+\frac {b^5 x^{33}}{33} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^17*(a + b*x^3)^5,x]

[Out]

(a^5*x^18)/18 + (5*a^4*b*x^21)/21 + (5*a^3*b^2*x^24)/12 + (10*a^2*b^3*x^27)/27 + (a*b^4*x^30)/6 + (b^5*x^33)/3
3

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 )^5 \, dx &=\frac {1}{3} \text {Subst}\left (\int x^5 (a+b x)^5 \, dx,x,x^3\right )\\ &=\frac {1}{3} \text {Subst}\left (\int \left (a^5 x^5+5 a^4 b x^6+10 a^3 b^2 x^7+10 a^2 b^3 x^8+5 a b^4 x^9+b^5 x^{10}\right ) \, dx,x,x^3\right )\\ &=\frac {a^5 x^{18}}{18}+\frac {5}{21} a^4 b x^{21}+\frac {5}{12} a^3 b^2 x^{24}+\frac {10}{27} a^2 b^3 x^{27}+\frac {1}{6} a b^4 x^{30}+\frac {b^5 x^{33}}{33}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 69, normalized size = 1.00 \begin {gather*} \frac {a^5 x^{18}}{18}+\frac {5}{21} a^4 b x^{21}+\frac {5}{12} a^3 b^2 x^{24}+\frac {10}{27} a^2 b^3 x^{27}+\frac {1}{6} a b^4 x^{30}+\frac {b^5 x^{33}}{33} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^17*(a + b*x^3)^5,x]

[Out]

(a^5*x^18)/18 + (5*a^4*b*x^21)/21 + (5*a^3*b^2*x^24)/12 + (10*a^2*b^3*x^27)/27 + (a*b^4*x^30)/6 + (b^5*x^33)/3
3

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Maple [A]
time = 0.13, size = 58, normalized size = 0.84

method	result	size
gosper	\(\frac {1}{18} a^{5} x^{18}+\frac {5}{21} a^{4} b \,x^{21}+\frac {5}{12} a^{3} b^{2} x^{24}+\frac {10}{27} a^{2} b^{3} x^{27}+\frac {1}{6} a \,b^{4} x^{30}+\frac {1}{33} b^{5} x^{33}\)	\(58\)
default	\(\frac {1}{18} a^{5} x^{18}+\frac {5}{21} a^{4} b \,x^{21}+\frac {5}{12} a^{3} b^{2} x^{24}+\frac {10}{27} a^{2} b^{3} x^{27}+\frac {1}{6} a \,b^{4} x^{30}+\frac {1}{33} b^{5} x^{33}\)	\(58\)
norman	\(\frac {1}{18} a^{5} x^{18}+\frac {5}{21} a^{4} b \,x^{21}+\frac {5}{12} a^{3} b^{2} x^{24}+\frac {10}{27} a^{2} b^{3} x^{27}+\frac {1}{6} a \,b^{4} x^{30}+\frac {1}{33} b^{5} x^{33}\)	\(58\)
risch	\(\frac {1}{18} a^{5} x^{18}+\frac {5}{21} a^{4} b \,x^{21}+\frac {5}{12} a^{3} b^{2} x^{24}+\frac {10}{27} a^{2} b^{3} x^{27}+\frac {1}{6} a \,b^{4} x^{30}+\frac {1}{33} b^{5} x^{33}\)	\(58\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^17*(b*x^3+a)^5,x,method=_RETURNVERBOSE)

[Out]

1/18*a^5*x^18+5/21*a^4*b*x^21+5/12*a^3*b^2*x^24+10/27*a^2*b^3*x^27+1/6*a*b^4*x^30+1/33*b^5*x^33

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Maxima [A]
time = 0.30, size = 57, normalized size = 0.83 \begin {gather*} \frac {1}{33} \, b^{5} x^{33} + \frac {1}{6} \, a b^{4} x^{30} + \frac {10}{27} \, a^{2} b^{3} x^{27} + \frac {5}{12} \, a^{3} b^{2} x^{24} + \frac {5}{21} \, a^{4} b x^{21} + \frac {1}{18} \, a^{5} x^{18} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^17*(b*x^3+a)^5,x, algorithm="maxima")

[Out]

1/33*b^5*x^33 + 1/6*a*b^4*x^30 + 10/27*a^2*b^3*x^27 + 5/12*a^3*b^2*x^24 + 5/21*a^4*b*x^21 + 1/18*a^5*x^18

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Fricas [A]
time = 0.35, size = 57, normalized size = 0.83 \begin {gather*} \frac {1}{33} \, b^{5} x^{33} + \frac {1}{6} \, a b^{4} x^{30} + \frac {10}{27} \, a^{2} b^{3} x^{27} + \frac {5}{12} \, a^{3} b^{2} x^{24} + \frac {5}{21} \, a^{4} b x^{21} + \frac {1}{18} \, a^{5} x^{18} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^17*(b*x^3+a)^5,x, algorithm="fricas")

[Out]

1/33*b^5*x^33 + 1/6*a*b^4*x^30 + 10/27*a^2*b^3*x^27 + 5/12*a^3*b^2*x^24 + 5/21*a^4*b*x^21 + 1/18*a^5*x^18

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Sympy [A]
time = 0.01, size = 65, normalized size = 0.94 \begin {gather*} \frac {a^{5} x^{18}}{18} + \frac {5 a^{4} b x^{21}}{21} + \frac {5 a^{3} b^{2} x^{24}}{12} + \frac {10 a^{2} b^{3} x^{27}}{27} + \frac {a b^{4} x^{30}}{6} + \frac {b^{5} x^{33}}{33} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**17*(b*x**3+a)**5,x)

[Out]

a**5*x**18/18 + 5*a**4*b*x**21/21 + 5*a**3*b**2*x**24/12 + 10*a**2*b**3*x**27/27 + a*b**4*x**30/6 + b**5*x**33
/33

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Giac [A]
time = 0.93, size = 57, normalized size = 0.83 \begin {gather*} \frac {1}{33} \, b^{5} x^{33} + \frac {1}{6} \, a b^{4} x^{30} + \frac {10}{27} \, a^{2} b^{3} x^{27} + \frac {5}{12} \, a^{3} b^{2} x^{24} + \frac {5}{21} \, a^{4} b x^{21} + \frac {1}{18} \, a^{5} x^{18} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^17*(b*x^3+a)^5,x, algorithm="giac")

[Out]

1/33*b^5*x^33 + 1/6*a*b^4*x^30 + 10/27*a^2*b^3*x^27 + 5/12*a^3*b^2*x^24 + 5/21*a^4*b*x^21 + 1/18*a^5*x^18

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Mupad [B]
time = 0.02, size = 57, normalized size = 0.83 \begin {gather*} \frac {a^5\,x^{18}}{18}+\frac {5\,a^4\,b\,x^{21}}{21}+\frac {5\,a^3\,b^2\,x^{24}}{12}+\frac {10\,a^2\,b^3\,x^{27}}{27}+\frac {a\,b^4\,x^{30}}{6}+\frac {b^5\,x^{33}}{33} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^17*(a + b*x^3)^5,x)

[Out]

(a^5*x^18)/18 + (b^5*x^33)/33 + (5*a^4*b*x^21)/21 + (a*b^4*x^30)/6 + (5*a^3*b^2*x^24)/12 + (10*a^2*b^3*x^27)/2
7

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