\(\int \frac {(a+b x^3)^8}{x^{34}} \, dx\) [303]

Optimal result

Integrand size = 13, antiderivative size = 62 \[ \int \frac {\left (a+b x^3\right )^8}{x^{34}} \, dx=-\frac {\left (a+b x^3\right )^9}{33 a x^{33}}+\frac {b \left (a+b x^3\right )^9}{165 a^2 x^{30}}-\frac {b^2 \left (a+b x^3\right )^9}{1485 a^3 x^{27}} \]

[Out]

-1/33*(b*x^3+a)^9/a/x^33+1/165*b*(b*x^3+a)^9/a^2/x^30-1/1485*b^2*(b*x^3+a)^9/a^3/x^27

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {272, 47, 37} \[ \int \frac {\left (a+b x^3\right )^8}{x^{34}} \, dx=-\frac {b^2 \left (a+b x^3\right )^9}{1485 a^3 x^{27}}+\frac {b \left (a+b x^3\right )^9}{165 a^2 x^{30}}-\frac {\left (a+b x^3\right )^9}{33 a x^{33}} \]

[In]

Int[(a + b*x^3)^8/x^34,x]

[Out]

-1/33*(a + b*x^3)^9/(a*x^33) + (b*(a + b*x^3)^9)/(165*a^2*x^30) - (b^2*(a + b*x^3)^9)/(1485*a^3*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*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {(a+b x)^8}{x^{12}} \, dx,x,x^3\right ) \\ & = -\frac {\left (a+b x^3\right )^9}{33 a x^{33}}-\frac {(2 b) \text {Subst}\left (\int \frac {(a+b x)^8}{x^{11}} \, dx,x,x^3\right )}{33 a} \\ & = -\frac {\left (a+b x^3\right )^9}{33 a x^{33}}+\frac {b \left (a+b x^3\right )^9}{165 a^2 x^{30}}+\frac {b^2 \text {Subst}\left (\int \frac {(a+b x)^8}{x^{10}} \, dx,x,x^3\right )}{165 a^2} \\ & = -\frac {\left (a+b x^3\right )^9}{33 a x^{33}}+\frac {b \left (a+b x^3\right )^9}{165 a^2 x^{30}}-\frac {b^2 \left (a+b x^3\right )^9}{1485 a^3 x^{27}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.74 \[ \int \frac {\left (a+b x^3\right )^8}{x^{34}} \, dx=-\frac {a^8}{33 x^{33}}-\frac {4 a^7 b}{15 x^{30}}-\frac {28 a^6 b^2}{27 x^{27}}-\frac {7 a^5 b^3}{3 x^{24}}-\frac {10 a^4 b^4}{3 x^{21}}-\frac {28 a^3 b^5}{9 x^{18}}-\frac {28 a^2 b^6}{15 x^{15}}-\frac {2 a b^7}{3 x^{12}}-\frac {b^8}{9 x^9} \]

[In]

Integrate[(a + b*x^3)^8/x^34,x]

[Out]

-1/33*a^8/x^33 - (4*a^7*b)/(15*x^30) - (28*a^6*b^2)/(27*x^27) - (7*a^5*b^3)/(3*x^24) - (10*a^4*b^4)/(3*x^21) -
(28*a^3*b^5)/(9*x^18) - (28*a^2*b^6)/(15*x^15) - (2*a*b^7)/(3*x^12) - b^8/(9*x^9)

Maple [A] (verified)

Time = 3.73 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.47

method	result	size
default	\(-\frac {2 a \,b^{7}}{3 x^{12}}-\frac {4 a^{7} b}{15 x^{30}}-\frac {10 a^{4} b^{4}}{3 x^{21}}-\frac {28 a^{2} b^{6}}{15 x^{15}}-\frac {28 a^{3} b^{5}}{9 x^{18}}-\frac {b^{8}}{9 x^{9}}-\frac {7 a^{5} b^{3}}{3 x^{24}}-\frac {a^{8}}{33 x^{33}}-\frac {28 a^{6} b^{2}}{27 x^{27}}\)	\(91\)
norman	\(\frac {-\frac {1}{33} a^{8}-\frac {1}{9} b^{8} x^{24}-\frac {28}{9} a^{3} b^{5} x^{15}-\frac {28}{15} a^{2} b^{6} x^{18}-\frac {2}{3} a \,b^{7} x^{21}-\frac {4}{15} x^{3} b \,a^{7}-\frac {28}{27} a^{6} b^{2} x^{6}-\frac {7}{3} x^{9} b^{3} a^{5}-\frac {10}{3} a^{4} b^{4} x^{12}}{x^{33}}\)	\(92\)
risch	\(\frac {-\frac {1}{33} a^{8}-\frac {1}{9} b^{8} x^{24}-\frac {28}{9} a^{3} b^{5} x^{15}-\frac {28}{15} a^{2} b^{6} x^{18}-\frac {2}{3} a \,b^{7} x^{21}-\frac {4}{15} x^{3} b \,a^{7}-\frac {28}{27} a^{6} b^{2} x^{6}-\frac {7}{3} x^{9} b^{3} a^{5}-\frac {10}{3} a^{4} b^{4} x^{12}}{x^{33}}\)	\(92\)
gosper	\(-\frac {165 b^{8} x^{24}+990 a \,b^{7} x^{21}+2772 a^{2} b^{6} x^{18}+4620 a^{3} b^{5} x^{15}+4950 a^{4} b^{4} x^{12}+3465 x^{9} b^{3} a^{5}+1540 a^{6} b^{2} x^{6}+396 x^{3} b \,a^{7}+45 a^{8}}{1485 x^{33}}\)	\(93\)
parallelrisch	\(\frac {-165 b^{8} x^{24}-990 a \,b^{7} x^{21}-2772 a^{2} b^{6} x^{18}-4620 a^{3} b^{5} x^{15}-4950 a^{4} b^{4} x^{12}-3465 x^{9} b^{3} a^{5}-1540 a^{6} b^{2} x^{6}-396 x^{3} b \,a^{7}-45 a^{8}}{1485 x^{33}}\)	\(93\)

[In]

int((b*x^3+a)^8/x^34,x,method=_RETURNVERBOSE)

[Out]

-2/3*a*b^7/x^12-4/15*a^7*b/x^30-10/3*a^4*b^4/x^21-28/15*a^2*b^6/x^15-28/9*a^3*b^5/x^18-1/9*b^8/x^9-7/3*a^5*b^3
/x^24-1/33*a^8/x^33-28/27*a^6*b^2/x^27

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.48 \[ \int \frac {\left (a+b x^3\right )^8}{x^{34}} \, dx=-\frac {165 \, b^{8} x^{24} + 990 \, a b^{7} x^{21} + 2772 \, a^{2} b^{6} x^{18} + 4620 \, a^{3} b^{5} x^{15} + 4950 \, a^{4} b^{4} x^{12} + 3465 \, a^{5} b^{3} x^{9} + 1540 \, a^{6} b^{2} x^{6} + 396 \, a^{7} b x^{3} + 45 \, a^{8}}{1485 \, x^{33}} \]

[In]

integrate((b*x^3+a)^8/x^34,x, algorithm="fricas")

[Out]

-1/1485*(165*b^8*x^24 + 990*a*b^7*x^21 + 2772*a^2*b^6*x^18 + 4620*a^3*b^5*x^15 + 4950*a^4*b^4*x^12 + 3465*a^5*
b^3*x^9 + 1540*a^6*b^2*x^6 + 396*a^7*b*x^3 + 45*a^8)/x^33

Sympy [A] (verification not implemented)

Time = 0.63 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.60 \[ \int \frac {\left (a+b x^3\right )^8}{x^{34}} \, dx=\frac {- 45 a^{8} - 396 a^{7} b x^{3} - 1540 a^{6} b^{2} x^{6} - 3465 a^{5} b^{3} x^{9} - 4950 a^{4} b^{4} x^{12} - 4620 a^{3} b^{5} x^{15} - 2772 a^{2} b^{6} x^{18} - 990 a b^{7} x^{21} - 165 b^{8} x^{24}}{1485 x^{33}} \]

[In]

integrate((b*x**3+a)**8/x**34,x)

[Out]

(-45*a**8 - 396*a**7*b*x**3 - 1540*a**6*b**2*x**6 - 3465*a**5*b**3*x**9 - 4950*a**4*b**4*x**12 - 4620*a**3*b**
5*x**15 - 2772*a**2*b**6*x**18 - 990*a*b**7*x**21 - 165*b**8*x**24)/(1485*x**33)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.48 \[ \int \frac {\left (a+b x^3\right )^8}{x^{34}} \, dx=-\frac {165 \, b^{8} x^{24} + 990 \, a b^{7} x^{21} + 2772 \, a^{2} b^{6} x^{18} + 4620 \, a^{3} b^{5} x^{15} + 4950 \, a^{4} b^{4} x^{12} + 3465 \, a^{5} b^{3} x^{9} + 1540 \, a^{6} b^{2} x^{6} + 396 \, a^{7} b x^{3} + 45 \, a^{8}}{1485 \, x^{33}} \]

[In]

integrate((b*x^3+a)^8/x^34,x, algorithm="maxima")

[Out]

-1/1485*(165*b^8*x^24 + 990*a*b^7*x^21 + 2772*a^2*b^6*x^18 + 4620*a^3*b^5*x^15 + 4950*a^4*b^4*x^12 + 3465*a^5*
b^3*x^9 + 1540*a^6*b^2*x^6 + 396*a^7*b*x^3 + 45*a^8)/x^33

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.48 \[ \int \frac {\left (a+b x^3\right )^8}{x^{34}} \, dx=-\frac {165 \, b^{8} x^{24} + 990 \, a b^{7} x^{21} + 2772 \, a^{2} b^{6} x^{18} + 4620 \, a^{3} b^{5} x^{15} + 4950 \, a^{4} b^{4} x^{12} + 3465 \, a^{5} b^{3} x^{9} + 1540 \, a^{6} b^{2} x^{6} + 396 \, a^{7} b x^{3} + 45 \, a^{8}}{1485 \, x^{33}} \]

[In]

integrate((b*x^3+a)^8/x^34,x, algorithm="giac")

[Out]

-1/1485*(165*b^8*x^24 + 990*a*b^7*x^21 + 2772*a^2*b^6*x^18 + 4620*a^3*b^5*x^15 + 4950*a^4*b^4*x^12 + 3465*a^5*
b^3*x^9 + 1540*a^6*b^2*x^6 + 396*a^7*b*x^3 + 45*a^8)/x^33

Mupad [B] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.48 \[ \int \frac {\left (a+b x^3\right )^8}{x^{34}} \, dx=-\frac {\frac {a^8}{33}+\frac {4\,a^7\,b\,x^3}{15}+\frac {28\,a^6\,b^2\,x^6}{27}+\frac {7\,a^5\,b^3\,x^9}{3}+\frac {10\,a^4\,b^4\,x^{12}}{3}+\frac {28\,a^3\,b^5\,x^{15}}{9}+\frac {28\,a^2\,b^6\,x^{18}}{15}+\frac {2\,a\,b^7\,x^{21}}{3}+\frac {b^8\,x^{24}}{9}}{x^{33}} \]

[In]

int((a + b*x^3)^8/x^34,x)

[Out]

-(a^8/33 + (b^8*x^24)/9 + (4*a^7*b*x^3)/15 + (2*a*b^7*x^21)/3 + (28*a^6*b^2*x^6)/27 + (7*a^5*b^3*x^9)/3 + (10*
a^4*b^4*x^12)/3 + (28*a^3*b^5*x^15)/9 + (28*a^2*b^6*x^18)/15)/x^33