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

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

[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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Rubi [A]  time = 0.02, antiderivative size = 60, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.00, size = 60, normalized size = 1.00 \[ -\frac {a^5}{3 x^3}-\frac {5 a^4 b}{x}+10 a^3 b^2 x+\frac {10}{3} a^2 b^3 x^3+a b^4 x^5+\frac {b^5 x^7}{7} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^5/x^4,x]

[Out]

-1/3*a^5/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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fricas [A]  time = 0.80, size = 59, normalized size = 0.98 \[ \frac {3 \, b^{5} x^{10} + 21 \, a b^{4} x^{8} + 70 \, a^{2} b^{3} x^{6} + 210 \, a^{3} b^{2} x^{4} - 105 \, a^{4} b x^{2} - 7 \, a^{5}}{21 \, x^{3}} \]

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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giac [A]  time = 1.13, size = 55, normalized size = 0.92 \[ \frac {1}{7} \, b^{5} x^{7} + a b^{4} x^{5} + \frac {10}{3} \, a^{2} b^{3} x^{3} + 10 \, a^{3} b^{2} x - \frac {15 \, a^{4} b x^{2} + a^{5}}{3 \, x^{3}} \]

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

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maple [A]  time = 0.00, size = 55, normalized size = 0.92 \[ \frac {b^{5} x^{7}}{7}+a \,b^{4} x^{5}+\frac {10 a^{2} b^{3} x^{3}}{3}+10 a^{3} b^{2} x -\frac {5 a^{4} b}{x}-\frac {a^{5}}{3 x^{3}} \]

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 = 1.36, size = 55, normalized size = 0.92 \[ \frac {1}{7} \, b^{5} x^{7} + a b^{4} x^{5} + \frac {10}{3} \, a^{2} b^{3} x^{3} + 10 \, a^{3} b^{2} x - \frac {15 \, a^{4} b x^{2} + a^{5}}{3 \, x^{3}} \]

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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mupad [B]  time = 0.03, size = 57, normalized size = 0.95 \[ \frac {b^5\,x^7}{7}-\frac {\frac {a^5}{3}+5\,b\,a^4\,x^2}{x^3}+10\,a^3\,b^2\,x+a\,b^4\,x^5+\frac {10\,a^2\,b^3\,x^3}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^5/x^4,x)

[Out]

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

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sympy [A]  time = 0.17, size = 60, normalized size = 1.00 \[ 10 a^{3} b^{2} x + \frac {10 a^{2} b^{3} x^{3}}{3} + a b^{4} x^{5} + \frac {b^{5} x^{7}}{7} + \frac {- a^{5} - 15 a^{4} b x^{2}}{3 x^{3}} \]

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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