3.114 \(\int \frac{(a+b x^2)^8}{x^2} \, dx\)

Optimal. Leaf size=100 \[ \frac{28}{11} a^2 b^6 x^{11}+\frac{56}{9} a^3 b^5 x^9+10 a^4 b^4 x^7+\frac{56}{5} a^5 b^3 x^5+\frac{28}{3} a^6 b^2 x^3+8 a^7 b x-\frac{a^8}{x}+\frac{8}{13} a b^7 x^{13}+\frac{b^8 x^{15}}{15} \]

[Out]

-(a^8/x) + 8*a^7*b*x + (28*a^6*b^2*x^3)/3 + (56*a^5*b^3*x^5)/5 + 10*a^4*b^4*x^7 + (56*a^3*b^5*x^9)/9 + (28*a^2
*b^6*x^11)/11 + (8*a*b^7*x^13)/13 + (b^8*x^15)/15

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Rubi [A]  time = 0.0386043, antiderivative size = 100, 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} \[ \frac{28}{11} a^2 b^6 x^{11}+\frac{56}{9} a^3 b^5 x^9+10 a^4 b^4 x^7+\frac{56}{5} a^5 b^3 x^5+\frac{28}{3} a^6 b^2 x^3+8 a^7 b x-\frac{a^8}{x}+\frac{8}{13} a b^7 x^{13}+\frac{b^8 x^{15}}{15} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^8/x^2,x]

[Out]

-(a^8/x) + 8*a^7*b*x + (28*a^6*b^2*x^3)/3 + (56*a^5*b^3*x^5)/5 + 10*a^4*b^4*x^7 + (56*a^3*b^5*x^9)/9 + (28*a^2
*b^6*x^11)/11 + (8*a*b^7*x^13)/13 + (b^8*x^15)/15

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 )^8}{x^2} \, dx &=\int \left (8 a^7 b+\frac{a^8}{x^2}+28 a^6 b^2 x^2+56 a^5 b^3 x^4+70 a^4 b^4 x^6+56 a^3 b^5 x^8+28 a^2 b^6 x^{10}+8 a b^7 x^{12}+b^8 x^{14}\right ) \, dx\\ &=-\frac{a^8}{x}+8 a^7 b x+\frac{28}{3} a^6 b^2 x^3+\frac{56}{5} a^5 b^3 x^5+10 a^4 b^4 x^7+\frac{56}{9} a^3 b^5 x^9+\frac{28}{11} a^2 b^6 x^{11}+\frac{8}{13} a b^7 x^{13}+\frac{b^8 x^{15}}{15}\\ \end{align*}

Mathematica [A]  time = 0.014578, size = 100, normalized size = 1. \[ \frac{28}{11} a^2 b^6 x^{11}+\frac{56}{9} a^3 b^5 x^9+10 a^4 b^4 x^7+\frac{56}{5} a^5 b^3 x^5+\frac{28}{3} a^6 b^2 x^3+8 a^7 b x-\frac{a^8}{x}+\frac{8}{13} a b^7 x^{13}+\frac{b^8 x^{15}}{15} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^8/x^2,x]

[Out]

-(a^8/x) + 8*a^7*b*x + (28*a^6*b^2*x^3)/3 + (56*a^5*b^3*x^5)/5 + 10*a^4*b^4*x^7 + (56*a^3*b^5*x^9)/9 + (28*a^2
*b^6*x^11)/11 + (8*a*b^7*x^13)/13 + (b^8*x^15)/15

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Maple [A]  time = 0.003, size = 89, normalized size = 0.9 \begin{align*} -{\frac{{a}^{8}}{x}}+8\,{a}^{7}bx+{\frac{28\,{a}^{6}{b}^{2}{x}^{3}}{3}}+{\frac{56\,{a}^{5}{b}^{3}{x}^{5}}{5}}+10\,{a}^{4}{b}^{4}{x}^{7}+{\frac{56\,{a}^{3}{b}^{5}{x}^{9}}{9}}+{\frac{28\,{a}^{2}{b}^{6}{x}^{11}}{11}}+{\frac{8\,a{b}^{7}{x}^{13}}{13}}+{\frac{{b}^{8}{x}^{15}}{15}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a^8/x+8*a^7*b*x+28/3*a^6*b^2*x^3+56/5*a^5*b^3*x^5+10*a^4*b^4*x^7+56/9*a^3*b^5*x^9+28/11*a^2*b^6*x^11+8/13*a*b
^7*x^13+1/15*b^8*x^15

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Maxima [A]  time = 1.1504, size = 119, normalized size = 1.19 \begin{align*} \frac{1}{15} \, b^{8} x^{15} + \frac{8}{13} \, a b^{7} x^{13} + \frac{28}{11} \, a^{2} b^{6} x^{11} + \frac{56}{9} \, a^{3} b^{5} x^{9} + 10 \, a^{4} b^{4} x^{7} + \frac{56}{5} \, a^{5} b^{3} x^{5} + \frac{28}{3} \, a^{6} b^{2} x^{3} + 8 \, a^{7} b x - \frac{a^{8}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15*b^8*x^15 + 8/13*a*b^7*x^13 + 28/11*a^2*b^6*x^11 + 56/9*a^3*b^5*x^9 + 10*a^4*b^4*x^7 + 56/5*a^5*b^3*x^5 +
28/3*a^6*b^2*x^3 + 8*a^7*b*x - a^8/x

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Fricas [A]  time = 1.17631, size = 235, normalized size = 2.35 \begin{align*} \frac{429 \, b^{8} x^{16} + 3960 \, a b^{7} x^{14} + 16380 \, a^{2} b^{6} x^{12} + 40040 \, a^{3} b^{5} x^{10} + 64350 \, a^{4} b^{4} x^{8} + 72072 \, a^{5} b^{3} x^{6} + 60060 \, a^{6} b^{2} x^{4} + 51480 \, a^{7} b x^{2} - 6435 \, a^{8}}{6435 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6435*(429*b^8*x^16 + 3960*a*b^7*x^14 + 16380*a^2*b^6*x^12 + 40040*a^3*b^5*x^10 + 64350*a^4*b^4*x^8 + 72072*a
^5*b^3*x^6 + 60060*a^6*b^2*x^4 + 51480*a^7*b*x^2 - 6435*a^8)/x

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Sympy [A]  time = 0.3486, size = 99, normalized size = 0.99 \begin{align*} - \frac{a^{8}}{x} + 8 a^{7} b x + \frac{28 a^{6} b^{2} x^{3}}{3} + \frac{56 a^{5} b^{3} x^{5}}{5} + 10 a^{4} b^{4} x^{7} + \frac{56 a^{3} b^{5} x^{9}}{9} + \frac{28 a^{2} b^{6} x^{11}}{11} + \frac{8 a b^{7} x^{13}}{13} + \frac{b^{8} x^{15}}{15} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a**8/x + 8*a**7*b*x + 28*a**6*b**2*x**3/3 + 56*a**5*b**3*x**5/5 + 10*a**4*b**4*x**7 + 56*a**3*b**5*x**9/9 + 2
8*a**2*b**6*x**11/11 + 8*a*b**7*x**13/13 + b**8*x**15/15

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Giac [A]  time = 2.81734, size = 119, normalized size = 1.19 \begin{align*} \frac{1}{15} \, b^{8} x^{15} + \frac{8}{13} \, a b^{7} x^{13} + \frac{28}{11} \, a^{2} b^{6} x^{11} + \frac{56}{9} \, a^{3} b^{5} x^{9} + 10 \, a^{4} b^{4} x^{7} + \frac{56}{5} \, a^{5} b^{3} x^{5} + \frac{28}{3} \, a^{6} b^{2} x^{3} + 8 \, a^{7} b x - \frac{a^{8}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15*b^8*x^15 + 8/13*a*b^7*x^13 + 28/11*a^2*b^6*x^11 + 56/9*a^3*b^5*x^9 + 10*a^4*b^4*x^7 + 56/5*a^5*b^3*x^5 +
28/3*a^6*b^2*x^3 + 8*a^7*b*x - a^8/x