3.304 \(\int \frac{(a+b x^3)^8}{x^{37}} \, dx\)

Optimal. Leaf size=84 \[ \frac{b^3 \left (a+b x^3\right )^9}{5940 a^4 x^{27}}-\frac{b^2 \left (a+b x^3\right )^9}{660 a^3 x^{30}}+\frac{b \left (a+b x^3\right )^9}{132 a^2 x^{33}}-\frac{\left (a+b x^3\right )^9}{36 a x^{36}} \]

[Out]

-(a + b*x^3)^9/(36*a*x^36) + (b*(a + b*x^3)^9)/(132*a^2*x^33) - (b^2*(a + b*x^3)^9)/(660*a^3*x^30) + (b^3*(a +
b*x^3)^9)/(5940*a^4*x^27)

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Rubi [A]  time = 0.0393221, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {266, 45, 37} \[ \frac{b^3 \left (a+b x^3\right )^9}{5940 a^4 x^{27}}-\frac{b^2 \left (a+b x^3\right )^9}{660 a^3 x^{30}}+\frac{b \left (a+b x^3\right )^9}{132 a^2 x^{33}}-\frac{\left (a+b x^3\right )^9}{36 a x^{36}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^8/x^37,x]

[Out]

-(a + b*x^3)^9/(36*a*x^36) + (b*(a + b*x^3)^9)/(132*a^2*x^33) - (b^2*(a + b*x^3)^9)/(660*a^3*x^30) + (b^3*(a +
b*x^3)^9)/(5940*a^4*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^{37}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^{13}} \, dx,x,x^3\right )\\ &=-\frac{\left (a+b x^3\right )^9}{36 a x^{36}}-\frac{b \operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^{12}} \, dx,x,x^3\right )}{12 a}\\ &=-\frac{\left (a+b x^3\right )^9}{36 a x^{36}}+\frac{b \left (a+b x^3\right )^9}{132 a^2 x^{33}}+\frac{b^2 \operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^{11}} \, dx,x,x^3\right )}{66 a^2}\\ &=-\frac{\left (a+b x^3\right )^9}{36 a x^{36}}+\frac{b \left (a+b x^3\right )^9}{132 a^2 x^{33}}-\frac{b^2 \left (a+b x^3\right )^9}{660 a^3 x^{30}}-\frac{b^3 \operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^{10}} \, dx,x,x^3\right )}{660 a^3}\\ &=-\frac{\left (a+b x^3\right )^9}{36 a x^{36}}+\frac{b \left (a+b x^3\right )^9}{132 a^2 x^{33}}-\frac{b^2 \left (a+b x^3\right )^9}{660 a^3 x^{30}}+\frac{b^3 \left (a+b x^3\right )^9}{5940 a^4 x^{27}}\\ \end{align*}

Mathematica [A]  time = 0.0046114, size = 108, normalized size = 1.29 \[ -\frac{14 a^6 b^2}{15 x^{30}}-\frac{56 a^5 b^3}{27 x^{27}}-\frac{35 a^4 b^4}{12 x^{24}}-\frac{8 a^3 b^5}{3 x^{21}}-\frac{14 a^2 b^6}{9 x^{18}}-\frac{8 a^7 b}{33 x^{33}}-\frac{a^8}{36 x^{36}}-\frac{8 a b^7}{15 x^{15}}-\frac{b^8}{12 x^{12}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^8/x^37,x]

[Out]

-a^8/(36*x^36) - (8*a^7*b)/(33*x^33) - (14*a^6*b^2)/(15*x^30) - (56*a^5*b^3)/(27*x^27) - (35*a^4*b^4)/(12*x^24
) - (8*a^3*b^5)/(3*x^21) - (14*a^2*b^6)/(9*x^18) - (8*a*b^7)/(15*x^15) - b^8/(12*x^12)

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Maple [A]  time = 0.008, size = 91, normalized size = 1.1 \begin{align*} -{\frac{8\,{a}^{3}{b}^{5}}{3\,{x}^{21}}}-{\frac{8\,a{b}^{7}}{15\,{x}^{15}}}-{\frac{35\,{a}^{4}{b}^{4}}{12\,{x}^{24}}}-{\frac{8\,{a}^{7}b}{33\,{x}^{33}}}-{\frac{14\,{a}^{6}{b}^{2}}{15\,{x}^{30}}}-{\frac{14\,{a}^{2}{b}^{6}}{9\,{x}^{18}}}-{\frac{{b}^{8}}{12\,{x}^{12}}}-{\frac{{a}^{8}}{36\,{x}^{36}}}-{\frac{56\,{a}^{5}{b}^{3}}{27\,{x}^{27}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^8/x^37,x)

[Out]

-8/3*a^3*b^5/x^21-8/15*a*b^7/x^15-35/12*a^4*b^4/x^24-8/33*a^7*b/x^33-14/15*a^6*b^2/x^30-14/9*a^2*b^6/x^18-1/12
*b^8/x^12-1/36*a^8/x^36-56/27*a^5*b^3/x^27

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Maxima [A]  time = 0.976479, size = 124, normalized size = 1.48 \begin{align*} -\frac{495 \, b^{8} x^{24} + 3168 \, a b^{7} x^{21} + 9240 \, a^{2} b^{6} x^{18} + 15840 \, a^{3} b^{5} x^{15} + 17325 \, a^{4} b^{4} x^{12} + 12320 \, a^{5} b^{3} x^{9} + 5544 \, a^{6} b^{2} x^{6} + 1440 \, a^{7} b x^{3} + 165 \, a^{8}}{5940 \, x^{36}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^8/x^37,x, algorithm="maxima")

[Out]

-1/5940*(495*b^8*x^24 + 3168*a*b^7*x^21 + 9240*a^2*b^6*x^18 + 15840*a^3*b^5*x^15 + 17325*a^4*b^4*x^12 + 12320*
a^5*b^3*x^9 + 5544*a^6*b^2*x^6 + 1440*a^7*b*x^3 + 165*a^8)/x^36

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Fricas [A]  time = 1.62384, size = 236, normalized size = 2.81 \begin{align*} -\frac{495 \, b^{8} x^{24} + 3168 \, a b^{7} x^{21} + 9240 \, a^{2} b^{6} x^{18} + 15840 \, a^{3} b^{5} x^{15} + 17325 \, a^{4} b^{4} x^{12} + 12320 \, a^{5} b^{3} x^{9} + 5544 \, a^{6} b^{2} x^{6} + 1440 \, a^{7} b x^{3} + 165 \, a^{8}}{5940 \, x^{36}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^8/x^37,x, algorithm="fricas")

[Out]

-1/5940*(495*b^8*x^24 + 3168*a*b^7*x^21 + 9240*a^2*b^6*x^18 + 15840*a^3*b^5*x^15 + 17325*a^4*b^4*x^12 + 12320*
a^5*b^3*x^9 + 5544*a^6*b^2*x^6 + 1440*a^7*b*x^3 + 165*a^8)/x^36

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Sympy [A]  time = 2.13725, size = 99, normalized size = 1.18 \begin{align*} - \frac{165 a^{8} + 1440 a^{7} b x^{3} + 5544 a^{6} b^{2} x^{6} + 12320 a^{5} b^{3} x^{9} + 17325 a^{4} b^{4} x^{12} + 15840 a^{3} b^{5} x^{15} + 9240 a^{2} b^{6} x^{18} + 3168 a b^{7} x^{21} + 495 b^{8} x^{24}}{5940 x^{36}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**8/x**37,x)

[Out]

-(165*a**8 + 1440*a**7*b*x**3 + 5544*a**6*b**2*x**6 + 12320*a**5*b**3*x**9 + 17325*a**4*b**4*x**12 + 15840*a**
3*b**5*x**15 + 9240*a**2*b**6*x**18 + 3168*a*b**7*x**21 + 495*b**8*x**24)/(5940*x**36)

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Giac [A]  time = 1.1207, size = 124, normalized size = 1.48 \begin{align*} -\frac{495 \, b^{8} x^{24} + 3168 \, a b^{7} x^{21} + 9240 \, a^{2} b^{6} x^{18} + 15840 \, a^{3} b^{5} x^{15} + 17325 \, a^{4} b^{4} x^{12} + 12320 \, a^{5} b^{3} x^{9} + 5544 \, a^{6} b^{2} x^{6} + 1440 \, a^{7} b x^{3} + 165 \, a^{8}}{5940 \, x^{36}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^8/x^37,x, algorithm="giac")

[Out]

-1/5940*(495*b^8*x^24 + 3168*a*b^7*x^21 + 9240*a^2*b^6*x^18 + 15840*a^3*b^5*x^15 + 17325*a^4*b^4*x^12 + 12320*
a^5*b^3*x^9 + 5544*a^6*b^2*x^6 + 1440*a^7*b*x^3 + 165*a^8)/x^36