3.293 \(\int \frac{(a+b x^3)^8}{x^4} \, dx\)

Optimal. Leaf size=105 \[ \frac{28}{15} a^2 b^6 x^{15}+\frac{14}{3} a^3 b^5 x^{12}+\frac{70}{9} a^4 b^4 x^9+\frac{28}{3} a^5 b^3 x^6+\frac{28}{3} a^6 b^2 x^3+8 a^7 b \log (x)-\frac{a^8}{3 x^3}+\frac{4}{9} a b^7 x^{18}+\frac{b^8 x^{21}}{21} \]

[Out]

-a^8/(3*x^3) + (28*a^6*b^2*x^3)/3 + (28*a^5*b^3*x^6)/3 + (70*a^4*b^4*x^9)/9 + (14*a^3*b^5*x^12)/3 + (28*a^2*b^
6*x^15)/15 + (4*a*b^7*x^18)/9 + (b^8*x^21)/21 + 8*a^7*b*Log[x]

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Rubi [A]  time = 0.0582046, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {266, 43} \[ \frac{28}{15} a^2 b^6 x^{15}+\frac{14}{3} a^3 b^5 x^{12}+\frac{70}{9} a^4 b^4 x^9+\frac{28}{3} a^5 b^3 x^6+\frac{28}{3} a^6 b^2 x^3+8 a^7 b \log (x)-\frac{a^8}{3 x^3}+\frac{4}{9} a b^7 x^{18}+\frac{b^8 x^{21}}{21} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^8/x^4,x]

[Out]

-a^8/(3*x^3) + (28*a^6*b^2*x^3)/3 + (28*a^5*b^3*x^6)/3 + (70*a^4*b^4*x^9)/9 + (14*a^3*b^5*x^12)/3 + (28*a^2*b^
6*x^15)/15 + (4*a*b^7*x^18)/9 + (b^8*x^21)/21 + 8*a^7*b*Log[x]

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 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{\left (a+b x^3\right )^8}{x^4} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^2} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (28 a^6 b^2+\frac{a^8}{x^2}+\frac{8 a^7 b}{x}+56 a^5 b^3 x+70 a^4 b^4 x^2+56 a^3 b^5 x^3+28 a^2 b^6 x^4+8 a b^7 x^5+b^8 x^6\right ) \, dx,x,x^3\right )\\ &=-\frac{a^8}{3 x^3}+\frac{28}{3} a^6 b^2 x^3+\frac{28}{3} a^5 b^3 x^6+\frac{70}{9} a^4 b^4 x^9+\frac{14}{3} a^3 b^5 x^{12}+\frac{28}{15} a^2 b^6 x^{15}+\frac{4}{9} a b^7 x^{18}+\frac{b^8 x^{21}}{21}+8 a^7 b \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0088502, size = 105, normalized size = 1. \[ \frac{28}{15} a^2 b^6 x^{15}+\frac{14}{3} a^3 b^5 x^{12}+\frac{70}{9} a^4 b^4 x^9+\frac{28}{3} a^5 b^3 x^6+\frac{28}{3} a^6 b^2 x^3+8 a^7 b \log (x)-\frac{a^8}{3 x^3}+\frac{4}{9} a b^7 x^{18}+\frac{b^8 x^{21}}{21} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^8/x^4,x]

[Out]

-a^8/(3*x^3) + (28*a^6*b^2*x^3)/3 + (28*a^5*b^3*x^6)/3 + (70*a^4*b^4*x^9)/9 + (14*a^3*b^5*x^12)/3 + (28*a^2*b^
6*x^15)/15 + (4*a*b^7*x^18)/9 + (b^8*x^21)/21 + 8*a^7*b*Log[x]

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Maple [A]  time = 0.004, size = 90, normalized size = 0.9 \begin{align*} -{\frac{{a}^{8}}{3\,{x}^{3}}}+{\frac{28\,{a}^{6}{b}^{2}{x}^{3}}{3}}+{\frac{28\,{a}^{5}{b}^{3}{x}^{6}}{3}}+{\frac{70\,{a}^{4}{b}^{4}{x}^{9}}{9}}+{\frac{14\,{a}^{3}{b}^{5}{x}^{12}}{3}}+{\frac{28\,{a}^{2}{b}^{6}{x}^{15}}{15}}+{\frac{4\,a{b}^{7}{x}^{18}}{9}}+{\frac{{b}^{8}{x}^{21}}{21}}+8\,{a}^{7}b\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^8/x^4,x)

[Out]

-1/3*a^8/x^3+28/3*a^6*b^2*x^3+28/3*a^5*b^3*x^6+70/9*a^4*b^4*x^9+14/3*a^3*b^5*x^12+28/15*a^2*b^6*x^15+4/9*a*b^7
*x^18+1/21*b^8*x^21+8*a^7*b*ln(x)

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Maxima [A]  time = 0.955478, size = 123, normalized size = 1.17 \begin{align*} \frac{1}{21} \, b^{8} x^{21} + \frac{4}{9} \, a b^{7} x^{18} + \frac{28}{15} \, a^{2} b^{6} x^{15} + \frac{14}{3} \, a^{3} b^{5} x^{12} + \frac{70}{9} \, a^{4} b^{4} x^{9} + \frac{28}{3} \, a^{5} b^{3} x^{6} + \frac{28}{3} \, a^{6} b^{2} x^{3} + \frac{8}{3} \, a^{7} b \log \left (x^{3}\right ) - \frac{a^{8}}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^8/x^4,x, algorithm="maxima")

[Out]

1/21*b^8*x^21 + 4/9*a*b^7*x^18 + 28/15*a^2*b^6*x^15 + 14/3*a^3*b^5*x^12 + 70/9*a^4*b^4*x^9 + 28/3*a^5*b^3*x^6
+ 28/3*a^6*b^2*x^3 + 8/3*a^7*b*log(x^3) - 1/3*a^8/x^3

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Fricas [A]  time = 1.66007, size = 234, normalized size = 2.23 \begin{align*} \frac{15 \, b^{8} x^{24} + 140 \, a b^{7} x^{21} + 588 \, a^{2} b^{6} x^{18} + 1470 \, a^{3} b^{5} x^{15} + 2450 \, a^{4} b^{4} x^{12} + 2940 \, a^{5} b^{3} x^{9} + 2940 \, a^{6} b^{2} x^{6} + 2520 \, a^{7} b x^{3} \log \left (x\right ) - 105 \, a^{8}}{315 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^8/x^4,x, algorithm="fricas")

[Out]

1/315*(15*b^8*x^24 + 140*a*b^7*x^21 + 588*a^2*b^6*x^18 + 1470*a^3*b^5*x^15 + 2450*a^4*b^4*x^12 + 2940*a^5*b^3*
x^9 + 2940*a^6*b^2*x^6 + 2520*a^7*b*x^3*log(x) - 105*a^8)/x^3

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Sympy [A]  time = 0.493005, size = 105, normalized size = 1. \begin{align*} - \frac{a^{8}}{3 x^{3}} + 8 a^{7} b \log{\left (x \right )} + \frac{28 a^{6} b^{2} x^{3}}{3} + \frac{28 a^{5} b^{3} x^{6}}{3} + \frac{70 a^{4} b^{4} x^{9}}{9} + \frac{14 a^{3} b^{5} x^{12}}{3} + \frac{28 a^{2} b^{6} x^{15}}{15} + \frac{4 a b^{7} x^{18}}{9} + \frac{b^{8} x^{21}}{21} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**8/x**4,x)

[Out]

-a**8/(3*x**3) + 8*a**7*b*log(x) + 28*a**6*b**2*x**3/3 + 28*a**5*b**3*x**6/3 + 70*a**4*b**4*x**9/9 + 14*a**3*b
**5*x**12/3 + 28*a**2*b**6*x**15/15 + 4*a*b**7*x**18/9 + b**8*x**21/21

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Giac [A]  time = 1.13569, size = 135, normalized size = 1.29 \begin{align*} \frac{1}{21} \, b^{8} x^{21} + \frac{4}{9} \, a b^{7} x^{18} + \frac{28}{15} \, a^{2} b^{6} x^{15} + \frac{14}{3} \, a^{3} b^{5} x^{12} + \frac{70}{9} \, a^{4} b^{4} x^{9} + \frac{28}{3} \, a^{5} b^{3} x^{6} + \frac{28}{3} \, a^{6} b^{2} x^{3} + 8 \, a^{7} b \log \left ({\left | x \right |}\right ) - \frac{8 \, a^{7} b x^{3} + a^{8}}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^8/x^4,x, algorithm="giac")

[Out]

1/21*b^8*x^21 + 4/9*a*b^7*x^18 + 28/15*a^2*b^6*x^15 + 14/3*a^3*b^5*x^12 + 70/9*a^4*b^4*x^9 + 28/3*a^5*b^3*x^6
+ 28/3*a^6*b^2*x^3 + 8*a^7*b*log(abs(x)) - 1/3*(8*a^7*b*x^3 + a^8)/x^3