3.2.17 \(\int \frac {(a+b x^2)^8}{x^8} \, dx\) [117]

Optimal. Leaf size=102 \[ -\frac {a^8}{7 x^7}-\frac {8 a^7 b}{5 x^5}-\frac {28 a^6 b^2}{3 x^3}-\frac {56 a^5 b^3}{x}+70 a^4 b^4 x+\frac {56}{3} a^3 b^5 x^3+\frac {28}{5} a^2 b^6 x^5+\frac {8}{7} a b^7 x^7+\frac {b^8 x^9}{9} \]

[Out]

-1/7*a^8/x^7-8/5*a^7*b/x^5-28/3*a^6*b^2/x^3-56*a^5*b^3/x+70*a^4*b^4*x+56/3*a^3*b^5*x^3+28/5*a^2*b^6*x^5+8/7*a*
b^7*x^7+1/9*b^8*x^9

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Rubi [A]
time = 0.02, antiderivative size = 102, 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 = {276} \begin {gather*} -\frac {a^8}{7 x^7}-\frac {8 a^7 b}{5 x^5}-\frac {28 a^6 b^2}{3 x^3}-\frac {56 a^5 b^3}{x}+70 a^4 b^4 x+\frac {56}{3} a^3 b^5 x^3+\frac {28}{5} a^2 b^6 x^5+\frac {8}{7} a b^7 x^7+\frac {b^8 x^9}{9} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^8/x^8,x]

[Out]

-1/7*a^8/x^7 - (8*a^7*b)/(5*x^5) - (28*a^6*b^2)/(3*x^3) - (56*a^5*b^3)/x + 70*a^4*b^4*x + (56*a^3*b^5*x^3)/3 +
(28*a^2*b^6*x^5)/5 + (8*a*b^7*x^7)/7 + (b^8*x^9)/9

Rule 276

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 )^8}{x^8} \, dx &=\int \left (70 a^4 b^4+\frac {a^8}{x^8}+\frac {8 a^7 b}{x^6}+\frac {28 a^6 b^2}{x^4}+\frac {56 a^5 b^3}{x^2}+56 a^3 b^5 x^2+28 a^2 b^6 x^4+8 a b^7 x^6+b^8 x^8\right ) \, dx\\ &=-\frac {a^8}{7 x^7}-\frac {8 a^7 b}{5 x^5}-\frac {28 a^6 b^2}{3 x^3}-\frac {56 a^5 b^3}{x}+70 a^4 b^4 x+\frac {56}{3} a^3 b^5 x^3+\frac {28}{5} a^2 b^6 x^5+\frac {8}{7} a b^7 x^7+\frac {b^8 x^9}{9}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 102, normalized size = 1.00 \begin {gather*} -\frac {a^8}{7 x^7}-\frac {8 a^7 b}{5 x^5}-\frac {28 a^6 b^2}{3 x^3}-\frac {56 a^5 b^3}{x}+70 a^4 b^4 x+\frac {56}{3} a^3 b^5 x^3+\frac {28}{5} a^2 b^6 x^5+\frac {8}{7} a b^7 x^7+\frac {b^8 x^9}{9} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^8/x^8,x]

[Out]

-1/7*a^8/x^7 - (8*a^7*b)/(5*x^5) - (28*a^6*b^2)/(3*x^3) - (56*a^5*b^3)/x + 70*a^4*b^4*x + (56*a^3*b^5*x^3)/3 +
(28*a^2*b^6*x^5)/5 + (8*a*b^7*x^7)/7 + (b^8*x^9)/9

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Maple [A]
time = 0.03, size = 89, normalized size = 0.87

method	result	size
default	\(-\frac {a^{8}}{7 x^{7}}-\frac {8 a^{7} b}{5 x^{5}}-\frac {28 a^{6} b^{2}}{3 x^{3}}-\frac {56 a^{5} b^{3}}{x}+70 a^{4} b^{4} x +\frac {56 a^{3} b^{5} x^{3}}{3}+\frac {28 a^{2} b^{6} x^{5}}{5}+\frac {8 a \,b^{7} x^{7}}{7}+\frac {b^{8} x^{9}}{9}\)	\(89\)
risch	\(\frac {b^{8} x^{9}}{9}+\frac {8 a \,b^{7} x^{7}}{7}+\frac {28 a^{2} b^{6} x^{5}}{5}+\frac {56 a^{3} b^{5} x^{3}}{3}+70 a^{4} b^{4} x +\frac {-56 a^{5} b^{3} x^{6}-\frac {28}{3} a^{6} b^{2} x^{4}-\frac {8}{5} a^{7} b \,x^{2}-\frac {1}{7} a^{8}}{x^{7}}\)	\(91\)
norman	\(\frac {-\frac {1}{7} a^{8}-\frac {8}{5} a^{7} b \,x^{2}-\frac {28}{3} a^{6} b^{2} x^{4}-56 a^{5} b^{3} x^{6}+70 a^{4} b^{4} x^{8}+\frac {56}{3} a^{3} b^{5} x^{10}+\frac {28}{5} a^{2} b^{6} x^{12}+\frac {8}{7} a \,b^{7} x^{14}+\frac {1}{9} b^{8} x^{16}}{x^{7}}\)	\(92\)
gosper	\(-\frac {-35 b^{8} x^{16}-360 a \,b^{7} x^{14}-1764 a^{2} b^{6} x^{12}-5880 a^{3} b^{5} x^{10}-22050 a^{4} b^{4} x^{8}+17640 a^{5} b^{3} x^{6}+2940 a^{6} b^{2} x^{4}+504 a^{7} b \,x^{2}+45 a^{8}}{315 x^{7}}\)	\(93\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^8/x^8,x,method=_RETURNVERBOSE)

[Out]

-1/7*a^8/x^7-8/5*a^7*b/x^5-28/3*a^6*b^2/x^3-56*a^5*b^3/x+70*a^4*b^4*x+56/3*a^3*b^5*x^3+28/5*a^2*b^6*x^5+8/7*a*
b^7*x^7+1/9*b^8*x^9

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Maxima [A]
time = 0.28, size = 91, normalized size = 0.89 \begin {gather*} \frac {1}{9} \, b^{8} x^{9} + \frac {8}{7} \, a b^{7} x^{7} + \frac {28}{5} \, a^{2} b^{6} x^{5} + \frac {56}{3} \, a^{3} b^{5} x^{3} + 70 \, a^{4} b^{4} x - \frac {5880 \, a^{5} b^{3} x^{6} + 980 \, a^{6} b^{2} x^{4} + 168 \, a^{7} b x^{2} + 15 \, a^{8}}{105 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^8,x, algorithm="maxima")

[Out]

1/9*b^8*x^9 + 8/7*a*b^7*x^7 + 28/5*a^2*b^6*x^5 + 56/3*a^3*b^5*x^3 + 70*a^4*b^4*x - 1/105*(5880*a^5*b^3*x^6 + 9
80*a^6*b^2*x^4 + 168*a^7*b*x^2 + 15*a^8)/x^7

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Fricas [A]
time = 0.82, size = 92, normalized size = 0.90 \begin {gather*} \frac {35 \, b^{8} x^{16} + 360 \, a b^{7} x^{14} + 1764 \, a^{2} b^{6} x^{12} + 5880 \, a^{3} b^{5} x^{10} + 22050 \, a^{4} b^{4} x^{8} - 17640 \, a^{5} b^{3} x^{6} - 2940 \, a^{6} b^{2} x^{4} - 504 \, a^{7} b x^{2} - 45 \, a^{8}}{315 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^8,x, algorithm="fricas")

[Out]

1/315*(35*b^8*x^16 + 360*a*b^7*x^14 + 1764*a^2*b^6*x^12 + 5880*a^3*b^5*x^10 + 22050*a^4*b^4*x^8 - 17640*a^5*b^
3*x^6 - 2940*a^6*b^2*x^4 - 504*a^7*b*x^2 - 45*a^8)/x^7

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Sympy [A]
time = 0.12, size = 102, normalized size = 1.00 \begin {gather*} 70 a^{4} b^{4} x + \frac {56 a^{3} b^{5} x^{3}}{3} + \frac {28 a^{2} b^{6} x^{5}}{5} + \frac {8 a b^{7} x^{7}}{7} + \frac {b^{8} x^{9}}{9} + \frac {- 15 a^{8} - 168 a^{7} b x^{2} - 980 a^{6} b^{2} x^{4} - 5880 a^{5} b^{3} x^{6}}{105 x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**8/x**8,x)

[Out]

70*a**4*b**4*x + 56*a**3*b**5*x**3/3 + 28*a**2*b**6*x**5/5 + 8*a*b**7*x**7/7 + b**8*x**9/9 + (-15*a**8 - 168*a
**7*b*x**2 - 980*a**6*b**2*x**4 - 5880*a**5*b**3*x**6)/(105*x**7)

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Giac [A]
time = 0.95, size = 91, normalized size = 0.89 \begin {gather*} \frac {1}{9} \, b^{8} x^{9} + \frac {8}{7} \, a b^{7} x^{7} + \frac {28}{5} \, a^{2} b^{6} x^{5} + \frac {56}{3} \, a^{3} b^{5} x^{3} + 70 \, a^{4} b^{4} x - \frac {5880 \, a^{5} b^{3} x^{6} + 980 \, a^{6} b^{2} x^{4} + 168 \, a^{7} b x^{2} + 15 \, a^{8}}{105 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^8,x, algorithm="giac")

[Out]

1/9*b^8*x^9 + 8/7*a*b^7*x^7 + 28/5*a^2*b^6*x^5 + 56/3*a^3*b^5*x^3 + 70*a^4*b^4*x - 1/105*(5880*a^5*b^3*x^6 + 9
80*a^6*b^2*x^4 + 168*a^7*b*x^2 + 15*a^8)/x^7

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Mupad [B]
time = 4.80, size = 91, normalized size = 0.89 \begin {gather*} \frac {b^8\,x^9}{9}-\frac {\frac {a^8}{7}+\frac {8\,a^7\,b\,x^2}{5}+\frac {28\,a^6\,b^2\,x^4}{3}+56\,a^5\,b^3\,x^6}{x^7}+70\,a^4\,b^4\,x+\frac {8\,a\,b^7\,x^7}{7}+\frac {56\,a^3\,b^5\,x^3}{3}+\frac {28\,a^2\,b^6\,x^5}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^8/x^8,x)

[Out]

(b^8*x^9)/9 - (a^8/7 + (8*a^7*b*x^2)/5 + (28*a^6*b^2*x^4)/3 + 56*a^5*b^3*x^6)/x^7 + 70*a^4*b^4*x + (8*a*b^7*x^
7)/7 + (56*a^3*b^5*x^3)/3 + (28*a^2*b^6*x^5)/5

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