3.76 \(\int \frac{(a+b x^2)^5}{x^4} \, dx\)

Optimal. Leaf size=60 \[ \frac{10}{3} a^2 b^3 x^3+10 a^3 b^2 x-\frac{5 a^4 b}{x}-\frac{a^5}{3 x^3}+a b^4 x^5+\frac{b^5 x^7}{7} \]

[Out]

-a^5/(3*x^3) - (5*a^4*b)/x + 10*a^3*b^2*x + (10*a^2*b^3*x^3)/3 + a*b^4*x^5 + (b^5*x^7)/7

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Rubi [A]  time = 0.0217368, antiderivative size = 60, 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} \[ \frac{10}{3} a^2 b^3 x^3+10 a^3 b^2 x-\frac{5 a^4 b}{x}-\frac{a^5}{3 x^3}+a b^4 x^5+\frac{b^5 x^7}{7} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^5/x^4,x]

[Out]

-a^5/(3*x^3) - (5*a^4*b)/x + 10*a^3*b^2*x + (10*a^2*b^3*x^3)/3 + a*b^4*x^5 + (b^5*x^7)/7

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

Mathematica [A]  time = 0.0038595, size = 60, normalized size = 1. \[ \frac{10}{3} a^2 b^3 x^3+10 a^3 b^2 x-\frac{5 a^4 b}{x}-\frac{a^5}{3 x^3}+a b^4 x^5+\frac{b^5 x^7}{7} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^5/x^4,x]

[Out]

-a^5/(3*x^3) - (5*a^4*b)/x + 10*a^3*b^2*x + (10*a^2*b^3*x^3)/3 + a*b^4*x^5 + (b^5*x^7)/7

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Maple [A]  time = 0.006, size = 55, normalized size = 0.9 \begin{align*} -{\frac{{a}^{5}}{3\,{x}^{3}}}-5\,{\frac{{a}^{4}b}{x}}+10\,{a}^{3}{b}^{2}x+{\frac{10\,{a}^{2}{b}^{3}{x}^{3}}{3}}+a{b}^{4}{x}^{5}+{\frac{{b}^{5}{x}^{7}}{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^5/x^4,x)

[Out]

-1/3*a^5/x^3-5*a^4*b/x+10*a^3*b^2*x+10/3*a^2*b^3*x^3+a*b^4*x^5+1/7*b^5*x^7

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Maxima [A]  time = 2.77038, size = 74, normalized size = 1.23 \begin{align*} \frac{1}{7} \, b^{5} x^{7} + a b^{4} x^{5} + \frac{10}{3} \, a^{2} b^{3} x^{3} + 10 \, a^{3} b^{2} x - \frac{15 \, a^{4} b x^{2} + a^{5}}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5/x^4,x, algorithm="maxima")

[Out]

1/7*b^5*x^7 + a*b^4*x^5 + 10/3*a^2*b^3*x^3 + 10*a^3*b^2*x - 1/3*(15*a^4*b*x^2 + a^5)/x^3

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Fricas [A]  time = 1.09077, size = 131, normalized size = 2.18 \begin{align*} \frac{3 \, b^{5} x^{10} + 21 \, a b^{4} x^{8} + 70 \, a^{2} b^{3} x^{6} + 210 \, a^{3} b^{2} x^{4} - 105 \, a^{4} b x^{2} - 7 \, a^{5}}{21 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5/x^4,x, algorithm="fricas")

[Out]

1/21*(3*b^5*x^10 + 21*a*b^4*x^8 + 70*a^2*b^3*x^6 + 210*a^3*b^2*x^4 - 105*a^4*b*x^2 - 7*a^5)/x^3

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Sympy [A]  time = 0.318195, size = 58, normalized size = 0.97 \begin{align*} 10 a^{3} b^{2} x + \frac{10 a^{2} b^{3} x^{3}}{3} + a b^{4} x^{5} + \frac{b^{5} x^{7}}{7} - \frac{a^{5} + 15 a^{4} b x^{2}}{3 x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**5/x**4,x)

[Out]

10*a**3*b**2*x + 10*a**2*b**3*x**3/3 + a*b**4*x**5 + b**5*x**7/7 - (a**5 + 15*a**4*b*x**2)/(3*x**3)

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Giac [A]  time = 2.04291, size = 74, normalized size = 1.23 \begin{align*} \frac{1}{7} \, b^{5} x^{7} + a b^{4} x^{5} + \frac{10}{3} \, a^{2} b^{3} x^{3} + 10 \, a^{3} b^{2} x - \frac{15 \, a^{4} b x^{2} + a^{5}}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5/x^4,x, algorithm="giac")

[Out]

1/7*b^5*x^7 + a*b^4*x^5 + 10/3*a^2*b^3*x^3 + 10*a^3*b^2*x - 1/3*(15*a^4*b*x^2 + a^5)/x^3