3.3.71 \(\int \frac {(a+b x^3)^5}{x^{22}} \, dx\) [271]

Optimal. Leaf size=40 \[ -\frac {\left (a+b x^3\right )^6}{21 a x^{21}}+\frac {b \left (a+b x^3\right )^6}{126 a^2 x^{18}} \]

[Out]

-1/21*(b*x^3+a)^6/a/x^21+1/126*b*(b*x^3+a)^6/a^2/x^18

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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 )^6}{126 a^2 x^{18}}-\frac {\left (a+b x^3\right )^6}{21 a x^{21}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^5/x^22,x]

[Out]

-1/21*(a + b*x^3)^6/(a*x^21) + (b*(a + b*x^3)^6)/(126*a^2*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*} \int \frac {\left (a+b x^3\right )^5}{x^{22}} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {(a+b x)^5}{x^8} \, dx,x,x^3\right )\\ &=-\frac {\left (a+b x^3\right )^6}{21 a x^{21}}-\frac {b \text {Subst}\left (\int \frac {(a+b x)^5}{x^7} \, dx,x,x^3\right )}{21 a}\\ &=-\frac {\left (a+b x^3\right )^6}{21 a x^{21}}+\frac {b \left (a+b x^3\right )^6}{126 a^2 x^{18}}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 69, normalized size = 1.72 \begin {gather*} -\frac {a^5}{21 x^{21}}-\frac {5 a^4 b}{18 x^{18}}-\frac {2 a^3 b^2}{3 x^{15}}-\frac {5 a^2 b^3}{6 x^{12}}-\frac {5 a b^4}{9 x^9}-\frac {b^5}{6 x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^5/x^22,x]

[Out]

-1/21*a^5/x^21 - (5*a^4*b)/(18*x^18) - (2*a^3*b^2)/(3*x^15) - (5*a^2*b^3)/(6*x^12) - (5*a*b^4)/(9*x^9) - b^5/(
6*x^6)

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Maple [A]
time = 0.13, size = 58, normalized size = 1.45

method	result	size
default	\(-\frac {2 a^{3} b^{2}}{3 x^{15}}-\frac {5 a \,b^{4}}{9 x^{9}}-\frac {5 a^{2} b^{3}}{6 x^{12}}-\frac {b^{5}}{6 x^{6}}-\frac {a^{5}}{21 x^{21}}-\frac {5 a^{4} b}{18 x^{18}}\)	\(58\)
norman	\(\frac {-\frac {1}{21} a^{5}-\frac {5}{18} a^{4} b \,x^{3}-\frac {2}{3} a^{3} b^{2} x^{6}-\frac {5}{6} a^{2} b^{3} x^{9}-\frac {5}{9} a \,b^{4} x^{12}-\frac {1}{6} b^{5} x^{15}}{x^{21}}\)	\(59\)
risch	\(\frac {-\frac {1}{21} a^{5}-\frac {5}{18} a^{4} b \,x^{3}-\frac {2}{3} a^{3} b^{2} x^{6}-\frac {5}{6} a^{2} b^{3} x^{9}-\frac {5}{9} a \,b^{4} x^{12}-\frac {1}{6} b^{5} x^{15}}{x^{21}}\)	\(59\)
gosper	\(-\frac {21 b^{5} x^{15}+70 a \,b^{4} x^{12}+105 a^{2} b^{3} x^{9}+84 a^{3} b^{2} x^{6}+35 a^{4} b \,x^{3}+6 a^{5}}{126 x^{21}}\)	\(60\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^5/x^22,x,method=_RETURNVERBOSE)

[Out]

-2/3*a^3*b^2/x^15-5/9*a*b^4/x^9-5/6*a^2*b^3/x^12-1/6*b^5/x^6-1/21*a^5/x^21-5/18*a^4*b/x^18

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Maxima [A]
time = 0.30, size = 59, normalized size = 1.48 \begin {gather*} -\frac {21 \, b^{5} x^{15} + 70 \, a b^{4} x^{12} + 105 \, a^{2} b^{3} x^{9} + 84 \, a^{3} b^{2} x^{6} + 35 \, a^{4} b x^{3} + 6 \, a^{5}}{126 \, x^{21}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5/x^22,x, algorithm="maxima")

[Out]

-1/126*(21*b^5*x^15 + 70*a*b^4*x^12 + 105*a^2*b^3*x^9 + 84*a^3*b^2*x^6 + 35*a^4*b*x^3 + 6*a^5)/x^21

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Fricas [A]
time = 0.33, size = 59, normalized size = 1.48 \begin {gather*} -\frac {21 \, b^{5} x^{15} + 70 \, a b^{4} x^{12} + 105 \, a^{2} b^{3} x^{9} + 84 \, a^{3} b^{2} x^{6} + 35 \, a^{4} b x^{3} + 6 \, a^{5}}{126 \, x^{21}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5/x^22,x, algorithm="fricas")

[Out]

-1/126*(21*b^5*x^15 + 70*a*b^4*x^12 + 105*a^2*b^3*x^9 + 84*a^3*b^2*x^6 + 35*a^4*b*x^3 + 6*a^5)/x^21

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Sympy [A]
time = 0.27, size = 63, normalized size = 1.58 \begin {gather*} \frac {- 6 a^{5} - 35 a^{4} b x^{3} - 84 a^{3} b^{2} x^{6} - 105 a^{2} b^{3} x^{9} - 70 a b^{4} x^{12} - 21 b^{5} x^{15}}{126 x^{21}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**5/x**22,x)

[Out]

(-6*a**5 - 35*a**4*b*x**3 - 84*a**3*b**2*x**6 - 105*a**2*b**3*x**9 - 70*a*b**4*x**12 - 21*b**5*x**15)/(126*x**
21)

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Giac [A]
time = 1.42, size = 59, normalized size = 1.48 \begin {gather*} -\frac {21 \, b^{5} x^{15} + 70 \, a b^{4} x^{12} + 105 \, a^{2} b^{3} x^{9} + 84 \, a^{3} b^{2} x^{6} + 35 \, a^{4} b x^{3} + 6 \, a^{5}}{126 \, x^{21}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5/x^22,x, algorithm="giac")

[Out]

-1/126*(21*b^5*x^15 + 70*a*b^4*x^12 + 105*a^2*b^3*x^9 + 84*a^3*b^2*x^6 + 35*a^4*b*x^3 + 6*a^5)/x^21

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Mupad [B]
time = 0.95, size = 59, normalized size = 1.48 \begin {gather*} -\frac {\frac {a^5}{21}+\frac {5\,a^4\,b\,x^3}{18}+\frac {2\,a^3\,b^2\,x^6}{3}+\frac {5\,a^2\,b^3\,x^9}{6}+\frac {5\,a\,b^4\,x^{12}}{9}+\frac {b^5\,x^{15}}{6}}{x^{21}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^3)^5/x^22,x)

[Out]

-(a^5/21 + (b^5*x^15)/6 + (5*a^4*b*x^3)/18 + (5*a*b^4*x^12)/9 + (2*a^3*b^2*x^6)/3 + (5*a^2*b^3*x^9)/6)/x^21

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