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

Optimal. Leaf size=62 \[ -\frac{b^2 \left (a+b x^3\right )^9}{1485 a^3 x^{27}}+\frac{b \left (a+b x^3\right )^9}{165 a^2 x^{30}}-\frac{\left (a+b x^3\right )^9}{33 a x^{33}} \]

[Out]

-(a + b*x^3)^9/(33*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)

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Rubi [A]  time = 0.0267899, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {266, 45, 37} \[ -\frac{b^2 \left (a+b x^3\right )^9}{1485 a^3 x^{27}}+\frac{b \left (a+b x^3\right )^9}{165 a^2 x^{30}}-\frac{\left (a+b x^3\right )^9}{33 a x^{33}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^8/x^34,x]

[Out]

-(a + b*x^3)^9/(33*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 266

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]]

Rule 45

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 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]

Rubi steps

\begin{align*} \int \frac{\left (a+b x^3\right )^8}{x^{34}} \, dx &=\frac{1}{3} \operatorname{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) \operatorname{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 \operatorname{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]  time = 0.0097051, size = 108, normalized size = 1.74 \[ -\frac{28 a^6 b^2}{27 x^{27}}-\frac{7 a^5 b^3}{3 x^{24}}-\frac{10 a^4 b^4}{3 x^{21}}-\frac{28 a^3 b^5}{9 x^{18}}-\frac{28 a^2 b^6}{15 x^{15}}-\frac{4 a^7 b}{15 x^{30}}-\frac{a^8}{33 x^{33}}-\frac{2 a b^7}{3 x^{12}}-\frac{b^8}{9 x^9} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^8/x^34,x]

[Out]

-a^8/(33*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)

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Maple [A]  time = 0.007, size = 91, normalized size = 1.5 \begin{align*} -{\frac{7\,{a}^{5}{b}^{3}}{3\,{x}^{24}}}-{\frac{10\,{a}^{4}{b}^{4}}{3\,{x}^{21}}}-{\frac{28\,{a}^{2}{b}^{6}}{15\,{x}^{15}}}-{\frac{{a}^{8}}{33\,{x}^{33}}}-{\frac{28\,{a}^{3}{b}^{5}}{9\,{x}^{18}}}-{\frac{4\,{a}^{7}b}{15\,{x}^{30}}}-{\frac{28\,{a}^{6}{b}^{2}}{27\,{x}^{27}}}-{\frac{2\,a{b}^{7}}{3\,{x}^{12}}}-{\frac{{b}^{8}}{9\,{x}^{9}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.963001, size = 124, normalized size = 2. \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [A]  time = 1.64572, size = 228, normalized size = 3.68 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Sympy [A]  time = 2.10594, size = 99, normalized size = 1.6 \begin{align*} - \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [A]  time = 1.11371, size = 124, normalized size = 2. \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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