3.4.2 \(\int \frac {(a+b x^3)^8}{x^{31}} \, dx\) [302]

Optimal. Leaf size=40 \[ -\frac {\left (a+b x^3\right )^9}{30 a x^{30}}+\frac {b \left (a+b x^3\right )^9}{270 a^2 x^{27}} \]

[Out]

-1/30*(b*x^3+a)^9/a/x^30+1/270*b*(b*x^3+a)^9/a^2/x^27

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Rubi [A]
time = 0.01, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {272, 47, 37} \begin {gather*} \frac {b \left (a+b x^3\right )^9}{270 a^2 x^{27}}-\frac {\left (a+b x^3\right )^9}{30 a x^{30}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^8/x^31,x]

[Out]

-1/30*(a + b*x^3)^9/(a*x^30) + (b*(a + b*x^3)^9)/(270*a^2*x^27)

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*(Simplify[m + n + 2]/((b*c - a*d)*(m + 1))), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

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 \frac {\left (a+b x^3\right )^8}{x^{31}} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {(a+b x)^8}{x^{11}} \, dx,x,x^3\right )\\ &=-\frac {\left (a+b x^3\right )^9}{30 a x^{30}}-\frac {b \text {Subst}\left (\int \frac {(a+b x)^8}{x^{10}} \, dx,x,x^3\right )}{30 a}\\ &=-\frac {\left (a+b x^3\right )^9}{30 a x^{30}}+\frac {b \left (a+b x^3\right )^9}{270 a^2 x^{27}}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(108\) vs. \(2(40)=80\).
time = 0.00, size = 108, normalized size = 2.70 \begin {gather*} -\frac {a^8}{30 x^{30}}-\frac {8 a^7 b}{27 x^{27}}-\frac {7 a^6 b^2}{6 x^{24}}-\frac {8 a^5 b^3}{3 x^{21}}-\frac {35 a^4 b^4}{9 x^{18}}-\frac {56 a^3 b^5}{15 x^{15}}-\frac {7 a^2 b^6}{3 x^{12}}-\frac {8 a b^7}{9 x^9}-\frac {b^8}{6 x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^8/x^31,x]

[Out]

-1/30*a^8/x^30 - (8*a^7*b)/(27*x^27) - (7*a^6*b^2)/(6*x^24) - (8*a^5*b^3)/(3*x^21) - (35*a^4*b^4)/(9*x^18) - (
56*a^3*b^5)/(15*x^15) - (7*a^2*b^6)/(3*x^12) - (8*a*b^7)/(9*x^9) - b^8/(6*x^6)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(90\) vs. \(2(36)=72\).
time = 0.12, size = 91, normalized size = 2.28

method	result	size
default	\(-\frac {56 a^{3} b^{5}}{15 x^{15}}-\frac {8 a \,b^{7}}{9 x^{9}}-\frac {7 a^{2} b^{6}}{3 x^{12}}-\frac {b^{8}}{6 x^{6}}-\frac {8 a^{5} b^{3}}{3 x^{21}}-\frac {7 a^{6} b^{2}}{6 x^{24}}-\frac {35 a^{4} b^{4}}{9 x^{18}}-\frac {8 b \,a^{7}}{27 x^{27}}-\frac {a^{8}}{30 x^{30}}\)	\(91\)
norman	\(\frac {-\frac {8}{9} a \,b^{7} x^{21}-\frac {1}{6} b^{8} x^{24}-\frac {56}{15} a^{3} b^{5} x^{15}-\frac {7}{3} a^{2} b^{6} x^{18}-\frac {35}{9} a^{4} b^{4} x^{12}-\frac {8}{27} a^{7} b \,x^{3}-\frac {7}{6} a^{6} b^{2} x^{6}-\frac {8}{3} a^{5} b^{3} x^{9}-\frac {1}{30} a^{8}}{x^{30}}\)	\(92\)
risch	\(\frac {-\frac {8}{9} a \,b^{7} x^{21}-\frac {1}{6} b^{8} x^{24}-\frac {56}{15} a^{3} b^{5} x^{15}-\frac {7}{3} a^{2} b^{6} x^{18}-\frac {35}{9} a^{4} b^{4} x^{12}-\frac {8}{27} a^{7} b \,x^{3}-\frac {7}{6} a^{6} b^{2} x^{6}-\frac {8}{3} a^{5} b^{3} x^{9}-\frac {1}{30} a^{8}}{x^{30}}\)	\(92\)
gosper	\(-\frac {45 b^{8} x^{24}+240 a \,b^{7} x^{21}+630 a^{2} b^{6} x^{18}+1008 a^{3} b^{5} x^{15}+1050 a^{4} b^{4} x^{12}+720 a^{5} b^{3} x^{9}+315 a^{6} b^{2} x^{6}+80 a^{7} b \,x^{3}+9 a^{8}}{270 x^{30}}\)	\(93\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^8/x^31,x,method=_RETURNVERBOSE)

[Out]

-56/15*a^3*b^5/x^15-8/9*a*b^7/x^9-7/3*a^2*b^6/x^12-1/6*b^8/x^6-8/3*a^5*b^3/x^21-7/6*a^6*b^2/x^24-35/9*a^4*b^4/
x^18-8/27*b*a^7/x^27-1/30*a^8/x^30

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 92 vs. \(2 (36) = 72\).
time = 0.29, size = 92, normalized size = 2.30 \begin {gather*} -\frac {45 \, b^{8} x^{24} + 240 \, a b^{7} x^{21} + 630 \, a^{2} b^{6} x^{18} + 1008 \, a^{3} b^{5} x^{15} + 1050 \, a^{4} b^{4} x^{12} + 720 \, a^{5} b^{3} x^{9} + 315 \, a^{6} b^{2} x^{6} + 80 \, a^{7} b x^{3} + 9 \, a^{8}}{270 \, x^{30}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^8/x^31,x, algorithm="maxima")

[Out]

-1/270*(45*b^8*x^24 + 240*a*b^7*x^21 + 630*a^2*b^6*x^18 + 1008*a^3*b^5*x^15 + 1050*a^4*b^4*x^12 + 720*a^5*b^3*
x^9 + 315*a^6*b^2*x^6 + 80*a^7*b*x^3 + 9*a^8)/x^30

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 92 vs. \(2 (36) = 72\).
time = 0.37, size = 92, normalized size = 2.30 \begin {gather*} -\frac {45 \, b^{8} x^{24} + 240 \, a b^{7} x^{21} + 630 \, a^{2} b^{6} x^{18} + 1008 \, a^{3} b^{5} x^{15} + 1050 \, a^{4} b^{4} x^{12} + 720 \, a^{5} b^{3} x^{9} + 315 \, a^{6} b^{2} x^{6} + 80 \, a^{7} b x^{3} + 9 \, a^{8}}{270 \, x^{30}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^8/x^31,x, algorithm="fricas")

[Out]

-1/270*(45*b^8*x^24 + 240*a*b^7*x^21 + 630*a^2*b^6*x^18 + 1008*a^3*b^5*x^15 + 1050*a^4*b^4*x^12 + 720*a^5*b^3*
x^9 + 315*a^6*b^2*x^6 + 80*a^7*b*x^3 + 9*a^8)/x^30

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 99 vs. \(2 (32) = 64\).
time = 0.52, size = 99, normalized size = 2.48 \begin {gather*} \frac {- 9 a^{8} - 80 a^{7} b x^{3} - 315 a^{6} b^{2} x^{6} - 720 a^{5} b^{3} x^{9} - 1050 a^{4} b^{4} x^{12} - 1008 a^{3} b^{5} x^{15} - 630 a^{2} b^{6} x^{18} - 240 a b^{7} x^{21} - 45 b^{8} x^{24}}{270 x^{30}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**8/x**31,x)

[Out]

(-9*a**8 - 80*a**7*b*x**3 - 315*a**6*b**2*x**6 - 720*a**5*b**3*x**9 - 1050*a**4*b**4*x**12 - 1008*a**3*b**5*x*
*15 - 630*a**2*b**6*x**18 - 240*a*b**7*x**21 - 45*b**8*x**24)/(270*x**30)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 92 vs. \(2 (36) = 72\).
time = 1.63, size = 92, normalized size = 2.30 \begin {gather*} -\frac {45 \, b^{8} x^{24} + 240 \, a b^{7} x^{21} + 630 \, a^{2} b^{6} x^{18} + 1008 \, a^{3} b^{5} x^{15} + 1050 \, a^{4} b^{4} x^{12} + 720 \, a^{5} b^{3} x^{9} + 315 \, a^{6} b^{2} x^{6} + 80 \, a^{7} b x^{3} + 9 \, a^{8}}{270 \, x^{30}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^8/x^31,x, algorithm="giac")

[Out]

-1/270*(45*b^8*x^24 + 240*a*b^7*x^21 + 630*a^2*b^6*x^18 + 1008*a^3*b^5*x^15 + 1050*a^4*b^4*x^12 + 720*a^5*b^3*
x^9 + 315*a^6*b^2*x^6 + 80*a^7*b*x^3 + 9*a^8)/x^30

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Mupad [B]
time = 0.98, size = 92, normalized size = 2.30 \begin {gather*} -\frac {\frac {a^8}{30}+\frac {8\,a^7\,b\,x^3}{27}+\frac {7\,a^6\,b^2\,x^6}{6}+\frac {8\,a^5\,b^3\,x^9}{3}+\frac {35\,a^4\,b^4\,x^{12}}{9}+\frac {56\,a^3\,b^5\,x^{15}}{15}+\frac {7\,a^2\,b^6\,x^{18}}{3}+\frac {8\,a\,b^7\,x^{21}}{9}+\frac {b^8\,x^{24}}{6}}{x^{30}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^3)^8/x^31,x)

[Out]

-(a^8/30 + (b^8*x^24)/6 + (8*a^7*b*x^3)/27 + (8*a*b^7*x^21)/9 + (7*a^6*b^2*x^6)/6 + (8*a^5*b^3*x^9)/3 + (35*a^
4*b^4*x^12)/9 + (56*a^3*b^5*x^15)/15 + (7*a^2*b^6*x^18)/3)/x^30

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