\(\int \frac {(a+b x^3)^5}{x^{25}} \, dx\) [272]

Optimal result

Integrand size = 13, antiderivative size = 62 \[ \int \frac {\left (a+b x^3\right )^5}{x^{25}} \, dx=-\frac {\left (a+b x^3\right )^6}{24 a x^{24}}+\frac {b \left (a+b x^3\right )^6}{84 a^2 x^{21}}-\frac {b^2 \left (a+b x^3\right )^6}{504 a^3 x^{18}} \]

[Out]

-1/24*(b*x^3+a)^6/a/x^24+1/84*b*(b*x^3+a)^6/a^2/x^21-1/504*b^2*(b*x^3+a)^6/a^3/x^18

Rubi [A] (verified)

Time = 0.02 (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} \[ \int \frac {\left (a+b x^3\right )^5}{x^{25}} \, dx=-\frac {b^2 \left (a+b x^3\right )^6}{504 a^3 x^{18}}+\frac {b \left (a+b x^3\right )^6}{84 a^2 x^{21}}-\frac {\left (a+b x^3\right )^6}{24 a x^{24}} \]

[In]

Int[(a + b*x^3)^5/x^25,x]

[Out]

-1/24*(a + b*x^3)^6/(a*x^24) + (b*(a + b*x^3)^6)/(84*a^2*x^21) - (b^2*(a + b*x^3)^6)/(504*a^3*x^18)

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)^5}{x^9} \, dx,x,x^3\right ) \\ & = -\frac {\left (a+b x^3\right )^6}{24 a x^{24}}-\frac {b \text {Subst}\left (\int \frac {(a+b x)^5}{x^8} \, dx,x,x^3\right )}{12 a} \\ & = -\frac {\left (a+b x^3\right )^6}{24 a x^{24}}+\frac {b \left (a+b x^3\right )^6}{84 a^2 x^{21}}+\frac {b^2 \text {Subst}\left (\int \frac {(a+b x)^5}{x^7} \, dx,x,x^3\right )}{84 a^2} \\ & = -\frac {\left (a+b x^3\right )^6}{24 a x^{24}}+\frac {b \left (a+b x^3\right )^6}{84 a^2 x^{21}}-\frac {b^2 \left (a+b x^3\right )^6}{504 a^3 x^{18}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.11 \[ \int \frac {\left (a+b x^3\right )^5}{x^{25}} \, dx=-\frac {a^5}{24 x^{24}}-\frac {5 a^4 b}{21 x^{21}}-\frac {5 a^3 b^2}{9 x^{18}}-\frac {2 a^2 b^3}{3 x^{15}}-\frac {5 a b^4}{12 x^{12}}-\frac {b^5}{9 x^9} \]

[In]

Integrate[(a + b*x^3)^5/x^25,x]

[Out]

-1/24*a^5/x^24 - (5*a^4*b)/(21*x^21) - (5*a^3*b^2)/(9*x^18) - (2*a^2*b^3)/(3*x^15) - (5*a*b^4)/(12*x^12) - b^5
/(9*x^9)

Maple [A] (verified)

Time = 3.53 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.94

method	result	size
default	\(-\frac {5 a \,b^{4}}{12 x^{12}}-\frac {5 a^{4} b}{21 x^{21}}-\frac {2 a^{2} b^{3}}{3 x^{15}}-\frac {5 a^{3} b^{2}}{9 x^{18}}-\frac {b^{5}}{9 x^{9}}-\frac {a^{5}}{24 x^{24}}\)	\(58\)
norman	\(\frac {-\frac {1}{24} a^{5}-\frac {2}{3} a^{2} b^{3} x^{9}-\frac {5}{12} a \,b^{4} x^{12}-\frac {1}{9} b^{5} x^{15}-\frac {5}{21} a^{4} b \,x^{3}-\frac {5}{9} a^{3} b^{2} x^{6}}{x^{24}}\)	\(59\)
risch	\(\frac {-\frac {1}{24} a^{5}-\frac {2}{3} a^{2} b^{3} x^{9}-\frac {5}{12} a \,b^{4} x^{12}-\frac {1}{9} b^{5} x^{15}-\frac {5}{21} a^{4} b \,x^{3}-\frac {5}{9} a^{3} b^{2} x^{6}}{x^{24}}\)	\(59\)
gosper	\(-\frac {56 b^{5} x^{15}+210 a \,b^{4} x^{12}+336 a^{2} b^{3} x^{9}+280 a^{3} b^{2} x^{6}+120 a^{4} b \,x^{3}+21 a^{5}}{504 x^{24}}\)	\(60\)
parallelrisch	\(\frac {-56 b^{5} x^{15}-210 a \,b^{4} x^{12}-336 a^{2} b^{3} x^{9}-280 a^{3} b^{2} x^{6}-120 a^{4} b \,x^{3}-21 a^{5}}{504 x^{24}}\)	\(60\)

[In]

int((b*x^3+a)^5/x^25,x,method=_RETURNVERBOSE)

[Out]

-5/12*a*b^4/x^12-5/21*a^4*b/x^21-2/3*a^2*b^3/x^15-5/9*a^3*b^2/x^18-1/9*b^5/x^9-1/24*a^5/x^24

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.95 \[ \int \frac {\left (a+b x^3\right )^5}{x^{25}} \, dx=-\frac {56 \, b^{5} x^{15} + 210 \, a b^{4} x^{12} + 336 \, a^{2} b^{3} x^{9} + 280 \, a^{3} b^{2} x^{6} + 120 \, a^{4} b x^{3} + 21 \, a^{5}}{504 \, x^{24}} \]

[In]

integrate((b*x^3+a)^5/x^25,x, algorithm="fricas")

[Out]

-1/504*(56*b^5*x^15 + 210*a*b^4*x^12 + 336*a^2*b^3*x^9 + 280*a^3*b^2*x^6 + 120*a^4*b*x^3 + 21*a^5)/x^24

Sympy [A] (verification not implemented)

Time = 0.33 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.02 \[ \int \frac {\left (a+b x^3\right )^5}{x^{25}} \, dx=\frac {- 21 a^{5} - 120 a^{4} b x^{3} - 280 a^{3} b^{2} x^{6} - 336 a^{2} b^{3} x^{9} - 210 a b^{4} x^{12} - 56 b^{5} x^{15}}{504 x^{24}} \]

[In]

integrate((b*x**3+a)**5/x**25,x)

[Out]

(-21*a**5 - 120*a**4*b*x**3 - 280*a**3*b**2*x**6 - 336*a**2*b**3*x**9 - 210*a*b**4*x**12 - 56*b**5*x**15)/(504
*x**24)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.95 \[ \int \frac {\left (a+b x^3\right )^5}{x^{25}} \, dx=-\frac {56 \, b^{5} x^{15} + 210 \, a b^{4} x^{12} + 336 \, a^{2} b^{3} x^{9} + 280 \, a^{3} b^{2} x^{6} + 120 \, a^{4} b x^{3} + 21 \, a^{5}}{504 \, x^{24}} \]

[In]

integrate((b*x^3+a)^5/x^25,x, algorithm="maxima")

[Out]

-1/504*(56*b^5*x^15 + 210*a*b^4*x^12 + 336*a^2*b^3*x^9 + 280*a^3*b^2*x^6 + 120*a^4*b*x^3 + 21*a^5)/x^24

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.95 \[ \int \frac {\left (a+b x^3\right )^5}{x^{25}} \, dx=-\frac {56 \, b^{5} x^{15} + 210 \, a b^{4} x^{12} + 336 \, a^{2} b^{3} x^{9} + 280 \, a^{3} b^{2} x^{6} + 120 \, a^{4} b x^{3} + 21 \, a^{5}}{504 \, x^{24}} \]

[In]

integrate((b*x^3+a)^5/x^25,x, algorithm="giac")

[Out]

-1/504*(56*b^5*x^15 + 210*a*b^4*x^12 + 336*a^2*b^3*x^9 + 280*a^3*b^2*x^6 + 120*a^4*b*x^3 + 21*a^5)/x^24

Mupad [B] (verification not implemented)

Time = 5.39 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.95 \[ \int \frac {\left (a+b x^3\right )^5}{x^{25}} \, dx=-\frac {\frac {a^5}{24}+\frac {5\,a^4\,b\,x^3}{21}+\frac {5\,a^3\,b^2\,x^6}{9}+\frac {2\,a^2\,b^3\,x^9}{3}+\frac {5\,a\,b^4\,x^{12}}{12}+\frac {b^5\,x^{15}}{9}}{x^{24}} \]

[In]

int((a + b*x^3)^5/x^25,x)

[Out]

-(a^5/24 + (b^5*x^15)/9 + (5*a^4*b*x^3)/21 + (5*a*b^4*x^12)/12 + (5*a^3*b^2*x^6)/9 + (2*a^2*b^3*x^9)/3)/x^24