3.279 \(\int \frac{(a+b x^3)^5}{x^2} \, dx\)

Optimal. Leaf size=65 \[ \frac{5}{4} a^2 b^3 x^8+2 a^3 b^2 x^5+\frac{5}{2} a^4 b x^2-\frac{a^5}{x}+\frac{5}{11} a b^4 x^{11}+\frac{b^5 x^{14}}{14} \]

[Out]

-(a^5/x) + (5*a^4*b*x^2)/2 + 2*a^3*b^2*x^5 + (5*a^2*b^3*x^8)/4 + (5*a*b^4*x^11)/11 + (b^5*x^14)/14

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Rubi [A]  time = 0.0224125, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {270} \[ \frac{5}{4} a^2 b^3 x^8+2 a^3 b^2 x^5+\frac{5}{2} a^4 b x^2-\frac{a^5}{x}+\frac{5}{11} a b^4 x^{11}+\frac{b^5 x^{14}}{14} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^5/x^2,x]

[Out]

-(a^5/x) + (5*a^4*b*x^2)/2 + 2*a^3*b^2*x^5 + (5*a^2*b^3*x^8)/4 + (5*a*b^4*x^11)/11 + (b^5*x^14)/14

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \frac{\left (a+b x^3\right )^5}{x^2} \, dx &=\int \left (\frac{a^5}{x^2}+5 a^4 b x+10 a^3 b^2 x^4+10 a^2 b^3 x^7+5 a b^4 x^{10}+b^5 x^{13}\right ) \, dx\\ &=-\frac{a^5}{x}+\frac{5}{2} a^4 b x^2+2 a^3 b^2 x^5+\frac{5}{4} a^2 b^3 x^8+\frac{5}{11} a b^4 x^{11}+\frac{b^5 x^{14}}{14}\\ \end{align*}

Mathematica [A]  time = 0.0039549, size = 65, normalized size = 1. \[ \frac{5}{4} a^2 b^3 x^8+2 a^3 b^2 x^5+\frac{5}{2} a^4 b x^2-\frac{a^5}{x}+\frac{5}{11} a b^4 x^{11}+\frac{b^5 x^{14}}{14} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^5/x^2,x]

[Out]

-(a^5/x) + (5*a^4*b*x^2)/2 + 2*a^3*b^2*x^5 + (5*a^2*b^3*x^8)/4 + (5*a*b^4*x^11)/11 + (b^5*x^14)/14

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Maple [A]  time = 0.004, size = 58, normalized size = 0.9 \begin{align*} -{\frac{{a}^{5}}{x}}+{\frac{5\,{a}^{4}b{x}^{2}}{2}}+2\,{a}^{3}{b}^{2}{x}^{5}+{\frac{5\,{a}^{2}{b}^{3}{x}^{8}}{4}}+{\frac{5\,a{b}^{4}{x}^{11}}{11}}+{\frac{{b}^{5}{x}^{14}}{14}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^5/x^2,x)

[Out]

-a^5/x+5/2*a^4*b*x^2+2*a^3*b^2*x^5+5/4*a^2*b^3*x^8+5/11*a*b^4*x^11+1/14*b^5*x^14

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Maxima [A]  time = 0.953222, size = 77, normalized size = 1.18 \begin{align*} \frac{1}{14} \, b^{5} x^{14} + \frac{5}{11} \, a b^{4} x^{11} + \frac{5}{4} \, a^{2} b^{3} x^{8} + 2 \, a^{3} b^{2} x^{5} + \frac{5}{2} \, a^{4} b x^{2} - \frac{a^{5}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5/x^2,x, algorithm="maxima")

[Out]

1/14*b^5*x^14 + 5/11*a*b^4*x^11 + 5/4*a^2*b^3*x^8 + 2*a^3*b^2*x^5 + 5/2*a^4*b*x^2 - a^5/x

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Fricas [A]  time = 1.71899, size = 138, normalized size = 2.12 \begin{align*} \frac{22 \, b^{5} x^{15} + 140 \, a b^{4} x^{12} + 385 \, a^{2} b^{3} x^{9} + 616 \, a^{3} b^{2} x^{6} + 770 \, a^{4} b x^{3} - 308 \, a^{5}}{308 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5/x^2,x, algorithm="fricas")

[Out]

1/308*(22*b^5*x^15 + 140*a*b^4*x^12 + 385*a^2*b^3*x^9 + 616*a^3*b^2*x^6 + 770*a^4*b*x^3 - 308*a^5)/x

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Sympy [A]  time = 0.378884, size = 61, normalized size = 0.94 \begin{align*} - \frac{a^{5}}{x} + \frac{5 a^{4} b x^{2}}{2} + 2 a^{3} b^{2} x^{5} + \frac{5 a^{2} b^{3} x^{8}}{4} + \frac{5 a b^{4} x^{11}}{11} + \frac{b^{5} x^{14}}{14} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**5/x**2,x)

[Out]

-a**5/x + 5*a**4*b*x**2/2 + 2*a**3*b**2*x**5 + 5*a**2*b**3*x**8/4 + 5*a*b**4*x**11/11 + b**5*x**14/14

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Giac [A]  time = 1.12274, size = 77, normalized size = 1.18 \begin{align*} \frac{1}{14} \, b^{5} x^{14} + \frac{5}{11} \, a b^{4} x^{11} + \frac{5}{4} \, a^{2} b^{3} x^{8} + 2 \, a^{3} b^{2} x^{5} + \frac{5}{2} \, a^{4} b x^{2} - \frac{a^{5}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5/x^2,x, algorithm="giac")

[Out]

1/14*b^5*x^14 + 5/11*a*b^4*x^11 + 5/4*a^2*b^3*x^8 + 2*a^3*b^2*x^5 + 5/2*a^4*b*x^2 - a^5/x