3.4.6 \(\int \frac {(a+b x^3)^8}{x^{43}} \, dx\) [306]

Optimal. Leaf size=108 \[ -\frac {a^8}{42 x^{42}}-\frac {8 a^7 b}{39 x^{39}}-\frac {7 a^6 b^2}{9 x^{36}}-\frac {56 a^5 b^3}{33 x^{33}}-\frac {7 a^4 b^4}{3 x^{30}}-\frac {56 a^3 b^5}{27 x^{27}}-\frac {7 a^2 b^6}{6 x^{24}}-\frac {8 a b^7}{21 x^{21}}-\frac {b^8}{18 x^{18}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.04, antiderivative size = 108, 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^8}{42 x^{42}}-\frac {8 a^7 b}{39 x^{39}}-\frac {7 a^6 b^2}{9 x^{36}}-\frac {56 a^5 b^3}{33 x^{33}}-\frac {7 a^4 b^4}{3 x^{30}}-\frac {56 a^3 b^5}{27 x^{27}}-\frac {7 a^2 b^6}{6 x^{24}}-\frac {8 a b^7}{21 x^{21}}-\frac {b^8}{18 x^{18}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^8/x^43,x]

[Out]

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

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 \frac {\left (a+b x^3\right )^8}{x^{43}} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {(a+b x)^8}{x^{15}} \, dx,x,x^3\right )\\ &=\frac {1}{3} \text {Subst}\left (\int \left (\frac {a^8}{x^{15}}+\frac {8 a^7 b}{x^{14}}+\frac {28 a^6 b^2}{x^{13}}+\frac {56 a^5 b^3}{x^{12}}+\frac {70 a^4 b^4}{x^{11}}+\frac {56 a^3 b^5}{x^{10}}+\frac {28 a^2 b^6}{x^9}+\frac {8 a b^7}{x^8}+\frac {b^8}{x^7}\right ) \, dx,x,x^3\right )\\ &=-\frac {a^8}{42 x^{42}}-\frac {8 a^7 b}{39 x^{39}}-\frac {7 a^6 b^2}{9 x^{36}}-\frac {56 a^5 b^3}{33 x^{33}}-\frac {7 a^4 b^4}{3 x^{30}}-\frac {56 a^3 b^5}{27 x^{27}}-\frac {7 a^2 b^6}{6 x^{24}}-\frac {8 a b^7}{21 x^{21}}-\frac {b^8}{18 x^{18}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.00, size = 108, normalized size = 1.00 \begin {gather*} -\frac {a^8}{42 x^{42}}-\frac {8 a^7 b}{39 x^{39}}-\frac {7 a^6 b^2}{9 x^{36}}-\frac {56 a^5 b^3}{33 x^{33}}-\frac {7 a^4 b^4}{3 x^{30}}-\frac {56 a^3 b^5}{27 x^{27}}-\frac {7 a^2 b^6}{6 x^{24}}-\frac {8 a b^7}{21 x^{21}}-\frac {b^8}{18 x^{18}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^8/x^43,x]

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.13, size = 91, normalized size = 0.84

method	result	size
default	\(-\frac {a^{8}}{42 x^{42}}-\frac {8 a^{7} b}{39 x^{39}}-\frac {7 a^{6} b^{2}}{9 x^{36}}-\frac {56 a^{5} b^{3}}{33 x^{33}}-\frac {7 a^{4} b^{4}}{3 x^{30}}-\frac {56 a^{3} b^{5}}{27 x^{27}}-\frac {7 a^{2} b^{6}}{6 x^{24}}-\frac {8 a \,b^{7}}{21 x^{21}}-\frac {b^{8}}{18 x^{18}}\)	\(91\)
risch	\(\frac {-\frac {56}{27} a^{3} b^{5} x^{15}-\frac {7}{6} a^{2} b^{6} x^{18}-\frac {8}{21} a \,b^{7} x^{21}-\frac {1}{18} b^{8} x^{24}-\frac {8}{39} a^{7} b \,x^{3}-\frac {7}{9} a^{6} b^{2} x^{6}-\frac {56}{33} a^{5} b^{3} x^{9}-\frac {7}{3} a^{4} b^{4} x^{12}-\frac {1}{42} a^{8}}{x^{42}}\)	\(92\)
gosper	\(-\frac {3003 b^{8} x^{24}+20592 a \,b^{7} x^{21}+63063 a^{2} b^{6} x^{18}+112112 a^{3} b^{5} x^{15}+126126 a^{4} b^{4} x^{12}+91728 a^{5} b^{3} x^{9}+42042 a^{6} b^{2} x^{6}+11088 a^{7} b \,x^{3}+1287 a^{8}}{54054 x^{42}}\)	\(93\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^8/x^43,x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.29, size = 92, normalized size = 0.85 \begin {gather*} -\frac {3003 \, b^{8} x^{24} + 20592 \, a b^{7} x^{21} + 63063 \, a^{2} b^{6} x^{18} + 112112 \, a^{3} b^{5} x^{15} + 126126 \, a^{4} b^{4} x^{12} + 91728 \, a^{5} b^{3} x^{9} + 42042 \, a^{6} b^{2} x^{6} + 11088 \, a^{7} b x^{3} + 1287 \, a^{8}}{54054 \, x^{42}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^8/x^43,x, algorithm="maxima")

[Out]

-1/54054*(3003*b^8*x^24 + 20592*a*b^7*x^21 + 63063*a^2*b^6*x^18 + 112112*a^3*b^5*x^15 + 126126*a^4*b^4*x^12 +
91728*a^5*b^3*x^9 + 42042*a^6*b^2*x^6 + 11088*a^7*b*x^3 + 1287*a^8)/x^42

________________________________________________________________________________________

Fricas [A]
time = 0.38, size = 92, normalized size = 0.85 \begin {gather*} -\frac {3003 \, b^{8} x^{24} + 20592 \, a b^{7} x^{21} + 63063 \, a^{2} b^{6} x^{18} + 112112 \, a^{3} b^{5} x^{15} + 126126 \, a^{4} b^{4} x^{12} + 91728 \, a^{5} b^{3} x^{9} + 42042 \, a^{6} b^{2} x^{6} + 11088 \, a^{7} b x^{3} + 1287 \, a^{8}}{54054 \, x^{42}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^8/x^43,x, algorithm="fricas")

[Out]

-1/54054*(3003*b^8*x^24 + 20592*a*b^7*x^21 + 63063*a^2*b^6*x^18 + 112112*a^3*b^5*x^15 + 126126*a^4*b^4*x^12 +
91728*a^5*b^3*x^9 + 42042*a^6*b^2*x^6 + 11088*a^7*b*x^3 + 1287*a^8)/x^42

________________________________________________________________________________________

Sympy [A]
time = 0.67, size = 99, normalized size = 0.92 \begin {gather*} \frac {- 1287 a^{8} - 11088 a^{7} b x^{3} - 42042 a^{6} b^{2} x^{6} - 91728 a^{5} b^{3} x^{9} - 126126 a^{4} b^{4} x^{12} - 112112 a^{3} b^{5} x^{15} - 63063 a^{2} b^{6} x^{18} - 20592 a b^{7} x^{21} - 3003 b^{8} x^{24}}{54054 x^{42}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**8/x**43,x)

[Out]

(-1287*a**8 - 11088*a**7*b*x**3 - 42042*a**6*b**2*x**6 - 91728*a**5*b**3*x**9 - 126126*a**4*b**4*x**12 - 11211
2*a**3*b**5*x**15 - 63063*a**2*b**6*x**18 - 20592*a*b**7*x**21 - 3003*b**8*x**24)/(54054*x**42)

________________________________________________________________________________________

Giac [A]
time = 2.00, size = 92, normalized size = 0.85 \begin {gather*} -\frac {3003 \, b^{8} x^{24} + 20592 \, a b^{7} x^{21} + 63063 \, a^{2} b^{6} x^{18} + 112112 \, a^{3} b^{5} x^{15} + 126126 \, a^{4} b^{4} x^{12} + 91728 \, a^{5} b^{3} x^{9} + 42042 \, a^{6} b^{2} x^{6} + 11088 \, a^{7} b x^{3} + 1287 \, a^{8}}{54054 \, x^{42}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^8/x^43,x, algorithm="giac")

[Out]

-1/54054*(3003*b^8*x^24 + 20592*a*b^7*x^21 + 63063*a^2*b^6*x^18 + 112112*a^3*b^5*x^15 + 126126*a^4*b^4*x^12 +
91728*a^5*b^3*x^9 + 42042*a^6*b^2*x^6 + 11088*a^7*b*x^3 + 1287*a^8)/x^42

________________________________________________________________________________________

Mupad [B]
time = 0.08, size = 92, normalized size = 0.85 \begin {gather*} -\frac {\frac {a^8}{42}+\frac {8\,a^7\,b\,x^3}{39}+\frac {7\,a^6\,b^2\,x^6}{9}+\frac {56\,a^5\,b^3\,x^9}{33}+\frac {7\,a^4\,b^4\,x^{12}}{3}+\frac {56\,a^3\,b^5\,x^{15}}{27}+\frac {7\,a^2\,b^6\,x^{18}}{6}+\frac {8\,a\,b^7\,x^{21}}{21}+\frac {b^8\,x^{24}}{18}}{x^{42}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^3)^8/x^43,x)

[Out]

-(a^8/42 + (b^8*x^24)/18 + (8*a^7*b*x^3)/39 + (8*a*b^7*x^21)/21 + (7*a^6*b^2*x^6)/9 + (56*a^5*b^3*x^9)/33 + (7
*a^4*b^4*x^12)/3 + (56*a^3*b^5*x^15)/27 + (7*a^2*b^6*x^18)/6)/x^42

________________________________________________________________________________________