3.266 \(\int \frac{(a+b x^3)^5}{x^7} \, dx\)

Optimal. Leaf size=66 \[ \frac{10}{3} a^2 b^3 x^3+10 a^3 b^2 \log (x)-\frac{5 a^4 b}{3 x^3}-\frac{a^5}{6 x^6}+\frac{5}{6} a b^4 x^6+\frac{b^5 x^9}{9} \]

[Out]

-a^5/(6*x^6) - (5*a^4*b)/(3*x^3) + (10*a^2*b^3*x^3)/3 + (5*a*b^4*x^6)/6 + (b^5*x^9)/9 + 10*a^3*b^2*Log[x]

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Rubi [A]  time = 0.0344692, antiderivative size = 66, 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} \[ \frac{10}{3} a^2 b^3 x^3+10 a^3 b^2 \log (x)-\frac{5 a^4 b}{3 x^3}-\frac{a^5}{6 x^6}+\frac{5}{6} a b^4 x^6+\frac{b^5 x^9}{9} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^5/x^7,x]

[Out]

-a^5/(6*x^6) - (5*a^4*b)/(3*x^3) + (10*a^2*b^3*x^3)/3 + (5*a*b^4*x^6)/6 + (b^5*x^9)/9 + 10*a^3*b^2*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 )^5}{x^7} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{(a+b x)^5}{x^3} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (10 a^2 b^3+\frac{a^5}{x^3}+\frac{5 a^4 b}{x^2}+\frac{10 a^3 b^2}{x}+5 a b^4 x+b^5 x^2\right ) \, dx,x,x^3\right )\\ &=-\frac{a^5}{6 x^6}-\frac{5 a^4 b}{3 x^3}+\frac{10}{3} a^2 b^3 x^3+\frac{5}{6} a b^4 x^6+\frac{b^5 x^9}{9}+10 a^3 b^2 \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0046279, size = 66, normalized size = 1. \[ \frac{10}{3} a^2 b^3 x^3+10 a^3 b^2 \log (x)-\frac{5 a^4 b}{3 x^3}-\frac{a^5}{6 x^6}+\frac{5}{6} a b^4 x^6+\frac{b^5 x^9}{9} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^5/x^7,x]

[Out]

-a^5/(6*x^6) - (5*a^4*b)/(3*x^3) + (10*a^2*b^3*x^3)/3 + (5*a*b^4*x^6)/6 + (b^5*x^9)/9 + 10*a^3*b^2*Log[x]

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Maple [A]  time = 0.007, size = 57, normalized size = 0.9 \begin{align*} -{\frac{{a}^{5}}{6\,{x}^{6}}}-{\frac{5\,{a}^{4}b}{3\,{x}^{3}}}+{\frac{10\,{a}^{2}{b}^{3}{x}^{3}}{3}}+{\frac{5\,a{b}^{4}{x}^{6}}{6}}+{\frac{{b}^{5}{x}^{9}}{9}}+10\,{a}^{3}{b}^{2}\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^5/x^7,x)

[Out]

-1/6*a^5/x^6-5/3*a^4*b/x^3+10/3*a^2*b^3*x^3+5/6*a*b^4*x^6+1/9*b^5*x^9+10*a^3*b^2*ln(x)

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Maxima [A]  time = 0.972796, size = 80, normalized size = 1.21 \begin{align*} \frac{1}{9} \, b^{5} x^{9} + \frac{5}{6} \, a b^{4} x^{6} + \frac{10}{3} \, a^{2} b^{3} x^{3} + \frac{10}{3} \, a^{3} b^{2} \log \left (x^{3}\right ) - \frac{10 \, a^{4} b x^{3} + a^{5}}{6 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5/x^7,x, algorithm="maxima")

[Out]

1/9*b^5*x^9 + 5/6*a*b^4*x^6 + 10/3*a^2*b^3*x^3 + 10/3*a^3*b^2*log(x^3) - 1/6*(10*a^4*b*x^3 + a^5)/x^6

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Fricas [A]  time = 1.67031, size = 140, normalized size = 2.12 \begin{align*} \frac{2 \, b^{5} x^{15} + 15 \, a b^{4} x^{12} + 60 \, a^{2} b^{3} x^{9} + 180 \, a^{3} b^{2} x^{6} \log \left (x\right ) - 30 \, a^{4} b x^{3} - 3 \, a^{5}}{18 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5/x^7,x, algorithm="fricas")

[Out]

1/18*(2*b^5*x^15 + 15*a*b^4*x^12 + 60*a^2*b^3*x^9 + 180*a^3*b^2*x^6*log(x) - 30*a^4*b*x^3 - 3*a^5)/x^6

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Sympy [A]  time = 0.506625, size = 63, normalized size = 0.95 \begin{align*} 10 a^{3} b^{2} \log{\left (x \right )} + \frac{10 a^{2} b^{3} x^{3}}{3} + \frac{5 a b^{4} x^{6}}{6} + \frac{b^{5} x^{9}}{9} - \frac{a^{5} + 10 a^{4} b x^{3}}{6 x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**5/x**7,x)

[Out]

10*a**3*b**2*log(x) + 10*a**2*b**3*x**3/3 + 5*a*b**4*x**6/6 + b**5*x**9/9 - (a**5 + 10*a**4*b*x**3)/(6*x**6)

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Giac [A]  time = 1.08184, size = 93, normalized size = 1.41 \begin{align*} \frac{1}{9} \, b^{5} x^{9} + \frac{5}{6} \, a b^{4} x^{6} + \frac{10}{3} \, a^{2} b^{3} x^{3} + 10 \, a^{3} b^{2} \log \left ({\left | x \right |}\right ) - \frac{30 \, a^{3} b^{2} x^{6} + 10 \, a^{4} b x^{3} + a^{5}}{6 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5/x^7,x, algorithm="giac")

[Out]

1/9*b^5*x^9 + 5/6*a*b^4*x^6 + 10/3*a^2*b^3*x^3 + 10*a^3*b^2*log(abs(x)) - 1/6*(30*a^3*b^2*x^6 + 10*a^4*b*x^3 +
a^5)/x^6