3.265 \(\int \frac{(a+b x^3)^5}{x^4} \, dx\)

Optimal. Leaf size=66 \[ \frac{5}{3} a^2 b^3 x^6+\frac{10}{3} a^3 b^2 x^3+5 a^4 b \log (x)-\frac{a^5}{3 x^3}+\frac{5}{9} a b^4 x^9+\frac{b^5 x^{12}}{12} \]

[Out]

-a^5/(3*x^3) + (10*a^3*b^2*x^3)/3 + (5*a^2*b^3*x^6)/3 + (5*a*b^4*x^9)/9 + (b^5*x^12)/12 + 5*a^4*b*Log[x]

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Rubi [A]  time = 0.0363388, 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} \[ \frac{5}{3} a^2 b^3 x^6+\frac{10}{3} a^3 b^2 x^3+5 a^4 b \log (x)-\frac{a^5}{3 x^3}+\frac{5}{9} a b^4 x^9+\frac{b^5 x^{12}}{12} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^5/x^4,x]

[Out]

-a^5/(3*x^3) + (10*a^3*b^2*x^3)/3 + (5*a^2*b^3*x^6)/3 + (5*a*b^4*x^9)/9 + (b^5*x^12)/12 + 5*a^4*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 )^5}{x^4} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{(a+b x)^5}{x^2} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (10 a^3 b^2+\frac{a^5}{x^2}+\frac{5 a^4 b}{x}+10 a^2 b^3 x+5 a b^4 x^2+b^5 x^3\right ) \, dx,x,x^3\right )\\ &=-\frac{a^5}{3 x^3}+\frac{10}{3} a^3 b^2 x^3+\frac{5}{3} a^2 b^3 x^6+\frac{5}{9} a b^4 x^9+\frac{b^5 x^{12}}{12}+5 a^4 b \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0046972, size = 66, normalized size = 1. \[ \frac{5}{3} a^2 b^3 x^6+\frac{10}{3} a^3 b^2 x^3+5 a^4 b \log (x)-\frac{a^5}{3 x^3}+\frac{5}{9} a b^4 x^9+\frac{b^5 x^{12}}{12} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^5/x^4,x]

[Out]

-a^5/(3*x^3) + (10*a^3*b^2*x^3)/3 + (5*a^2*b^3*x^6)/3 + (5*a*b^4*x^9)/9 + (b^5*x^12)/12 + 5*a^4*b*Log[x]

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Maple [A]  time = 0.005, size = 57, normalized size = 0.9 \begin{align*} -{\frac{{a}^{5}}{3\,{x}^{3}}}+{\frac{10\,{a}^{3}{b}^{2}{x}^{3}}{3}}+{\frac{5\,{a}^{2}{b}^{3}{x}^{6}}{3}}+{\frac{5\,a{b}^{4}{x}^{9}}{9}}+{\frac{{b}^{5}{x}^{12}}{12}}+5\,{a}^{4}b\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^5/x^4,x)

[Out]

-1/3*a^5/x^3+10/3*a^3*b^2*x^3+5/3*a^2*b^3*x^6+5/9*a*b^4*x^9+1/12*b^5*x^12+5*a^4*b*ln(x)

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Maxima [A]  time = 0.981415, size = 78, normalized size = 1.18 \begin{align*} \frac{1}{12} \, b^{5} x^{12} + \frac{5}{9} \, a b^{4} x^{9} + \frac{5}{3} \, a^{2} b^{3} x^{6} + \frac{10}{3} \, a^{3} b^{2} x^{3} + \frac{5}{3} \, a^{4} b \log \left (x^{3}\right ) - \frac{a^{5}}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5/x^4,x, algorithm="maxima")

[Out]

1/12*b^5*x^12 + 5/9*a*b^4*x^9 + 5/3*a^2*b^3*x^6 + 10/3*a^3*b^2*x^3 + 5/3*a^4*b*log(x^3) - 1/3*a^5/x^3

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Fricas [A]  time = 1.72968, size = 143, normalized size = 2.17 \begin{align*} \frac{3 \, b^{5} x^{15} + 20 \, a b^{4} x^{12} + 60 \, a^{2} b^{3} x^{9} + 120 \, a^{3} b^{2} x^{6} + 180 \, a^{4} b x^{3} \log \left (x\right ) - 12 \, a^{5}}{36 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5/x^4,x, algorithm="fricas")

[Out]

1/36*(3*b^5*x^15 + 20*a*b^4*x^12 + 60*a^2*b^3*x^9 + 120*a^3*b^2*x^6 + 180*a^4*b*x^3*log(x) - 12*a^5)/x^3

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Sympy [A]  time = 0.391699, size = 65, normalized size = 0.98 \begin{align*} - \frac{a^{5}}{3 x^{3}} + 5 a^{4} b \log{\left (x \right )} + \frac{10 a^{3} b^{2} x^{3}}{3} + \frac{5 a^{2} b^{3} x^{6}}{3} + \frac{5 a b^{4} x^{9}}{9} + \frac{b^{5} x^{12}}{12} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**5/x**4,x)

[Out]

-a**5/(3*x**3) + 5*a**4*b*log(x) + 10*a**3*b**2*x**3/3 + 5*a**2*b**3*x**6/3 + 5*a*b**4*x**9/9 + b**5*x**12/12

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Giac [A]  time = 1.10896, size = 90, normalized size = 1.36 \begin{align*} \frac{1}{12} \, b^{5} x^{12} + \frac{5}{9} \, a b^{4} x^{9} + \frac{5}{3} \, a^{2} b^{3} x^{6} + \frac{10}{3} \, a^{3} b^{2} x^{3} + 5 \, a^{4} b \log \left ({\left | x \right |}\right ) - \frac{5 \, a^{4} b x^{3} + a^{5}}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5/x^4,x, algorithm="giac")

[Out]

1/12*b^5*x^12 + 5/9*a*b^4*x^9 + 5/3*a^2*b^3*x^6 + 10/3*a^3*b^2*x^3 + 5*a^4*b*log(abs(x)) - 1/3*(5*a^4*b*x^3 +
a^5)/x^3