3.2.18 \(\int \frac {(a+b x^2)^8}{x^{10}} \, dx\) [118]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.03, 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}{9 x^9}-\frac {8 a^7 b}{7 x^7}-\frac {28 a^6 b^2}{5 x^5}-\frac {56 a^5 b^3}{3 x^3}-\frac {70 a^4 b^4}{x}+56 a^3 b^5 x+\frac {28}{3} a^2 b^6 x^3+\frac {8}{5} a b^7 x^5+\frac {b^8 x^7}{7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^8/x^10,x]

[Out]

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

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^{10}} \, dx &=\int \left (56 a^3 b^5+\frac {a^8}{x^{10}}+\frac {8 a^7 b}{x^8}+\frac {28 a^6 b^2}{x^6}+\frac {56 a^5 b^3}{x^4}+\frac {70 a^4 b^4}{x^2}+28 a^2 b^6 x^2+8 a b^7 x^4+b^8 x^6\right ) \, dx\\ &=-\frac {a^8}{9 x^9}-\frac {8 a^7 b}{7 x^7}-\frac {28 a^6 b^2}{5 x^5}-\frac {56 a^5 b^3}{3 x^3}-\frac {70 a^4 b^4}{x}+56 a^3 b^5 x+\frac {28}{3} a^2 b^6 x^3+\frac {8}{5} a b^7 x^5+\frac {b^8 x^7}{7}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.01, size = 102, normalized size = 1.00 \begin {gather*} -\frac {a^8}{9 x^9}-\frac {8 a^7 b}{7 x^7}-\frac {28 a^6 b^2}{5 x^5}-\frac {56 a^5 b^3}{3 x^3}-\frac {70 a^4 b^4}{x}+56 a^3 b^5 x+\frac {28}{3} a^2 b^6 x^3+\frac {8}{5} a b^7 x^5+\frac {b^8 x^7}{7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^8/x^10,x]

[Out]

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

________________________________________________________________________________________

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^8/x^10,x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.27, size = 91, normalized size = 0.89 \begin {gather*} \frac {1}{7} \, b^{8} x^{7} + \frac {8}{5} \, a b^{7} x^{5} + \frac {28}{3} \, a^{2} b^{6} x^{3} + 56 \, a^{3} b^{5} x - \frac {22050 \, a^{4} b^{4} x^{8} + 5880 \, a^{5} b^{3} x^{6} + 1764 \, a^{6} b^{2} x^{4} + 360 \, a^{7} b x^{2} + 35 \, a^{8}}{315 \, x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^10,x, algorithm="maxima")

[Out]

1/7*b^8*x^7 + 8/5*a*b^7*x^5 + 28/3*a^2*b^6*x^3 + 56*a^3*b^5*x - 1/315*(22050*a^4*b^4*x^8 + 5880*a^5*b^3*x^6 +
1764*a^6*b^2*x^4 + 360*a^7*b*x^2 + 35*a^8)/x^9

________________________________________________________________________________________

Fricas [A]
time = 1.16, size = 92, normalized size = 0.90 \begin {gather*} \frac {45 \, b^{8} x^{16} + 504 \, a b^{7} x^{14} + 2940 \, a^{2} b^{6} x^{12} + 17640 \, a^{3} b^{5} x^{10} - 22050 \, a^{4} b^{4} x^{8} - 5880 \, a^{5} b^{3} x^{6} - 1764 \, a^{6} b^{2} x^{4} - 360 \, a^{7} b x^{2} - 35 \, a^{8}}{315 \, x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^10,x, algorithm="fricas")

[Out]

1/315*(45*b^8*x^16 + 504*a*b^7*x^14 + 2940*a^2*b^6*x^12 + 17640*a^3*b^5*x^10 - 22050*a^4*b^4*x^8 - 5880*a^5*b^
3*x^6 - 1764*a^6*b^2*x^4 - 360*a^7*b*x^2 - 35*a^8)/x^9

________________________________________________________________________________________

Sympy [A]
time = 0.16, size = 100, normalized size = 0.98 \begin {gather*} 56 a^{3} b^{5} x + \frac {28 a^{2} b^{6} x^{3}}{3} + \frac {8 a b^{7} x^{5}}{5} + \frac {b^{8} x^{7}}{7} + \frac {- 35 a^{8} - 360 a^{7} b x^{2} - 1764 a^{6} b^{2} x^{4} - 5880 a^{5} b^{3} x^{6} - 22050 a^{4} b^{4} x^{8}}{315 x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**8/x**10,x)

[Out]

56*a**3*b**5*x + 28*a**2*b**6*x**3/3 + 8*a*b**7*x**5/5 + b**8*x**7/7 + (-35*a**8 - 360*a**7*b*x**2 - 1764*a**6
*b**2*x**4 - 5880*a**5*b**3*x**6 - 22050*a**4*b**4*x**8)/(315*x**9)

________________________________________________________________________________________

Giac [A]
time = 1.14, size = 91, normalized size = 0.89 \begin {gather*} \frac {1}{7} \, b^{8} x^{7} + \frac {8}{5} \, a b^{7} x^{5} + \frac {28}{3} \, a^{2} b^{6} x^{3} + 56 \, a^{3} b^{5} x - \frac {22050 \, a^{4} b^{4} x^{8} + 5880 \, a^{5} b^{3} x^{6} + 1764 \, a^{6} b^{2} x^{4} + 360 \, a^{7} b x^{2} + 35 \, a^{8}}{315 \, x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^10,x, algorithm="giac")

[Out]

1/7*b^8*x^7 + 8/5*a*b^7*x^5 + 28/3*a^2*b^6*x^3 + 56*a^3*b^5*x - 1/315*(22050*a^4*b^4*x^8 + 5880*a^5*b^3*x^6 +
1764*a^6*b^2*x^4 + 360*a^7*b*x^2 + 35*a^8)/x^9

________________________________________________________________________________________

Mupad [B]
time = 0.05, size = 91, normalized size = 0.89 \begin {gather*} \frac {b^8\,x^7}{7}-\frac {\frac {a^8}{9}+\frac {8\,a^7\,b\,x^2}{7}+\frac {28\,a^6\,b^2\,x^4}{5}+\frac {56\,a^5\,b^3\,x^6}{3}+70\,a^4\,b^4\,x^8}{x^9}+56\,a^3\,b^5\,x+\frac {8\,a\,b^7\,x^5}{5}+\frac {28\,a^2\,b^6\,x^3}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^8/x^10,x)

[Out]

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

________________________________________________________________________________________