3.531 \(\int \frac {1}{x^4 (a+b x^2)^{9/2}} \, dx\)

Optimal. Leaf size=132 \[ \frac {256 b^2 x}{21 a^6 \sqrt {a+b x^2}}+\frac {128 b^2 x}{21 a^5 \left (a+b x^2\right )^{3/2}}+\frac {32 b^2 x}{7 a^4 \left (a+b x^2\right )^{5/2}}+\frac {80 b^2 x}{21 a^3 \left (a+b x^2\right )^{7/2}}+\frac {10 b}{3 a^2 x \left (a+b x^2\right )^{7/2}}-\frac {1}{3 a x^3 \left (a+b x^2\right )^{7/2}} \]

[Out]

-1/3/a/x^3/(b*x^2+a)^(7/2)+10/3*b/a^2/x/(b*x^2+a)^(7/2)+80/21*b^2*x/a^3/(b*x^2+a)^(7/2)+32/7*b^2*x/a^4/(b*x^2+
a)^(5/2)+128/21*b^2*x/a^5/(b*x^2+a)^(3/2)+256/21*b^2*x/a^6/(b*x^2+a)^(1/2)

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Rubi [A]  time = 0.04, antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {271, 192, 191} \[ \frac {256 b^2 x}{21 a^6 \sqrt {a+b x^2}}+\frac {128 b^2 x}{21 a^5 \left (a+b x^2\right )^{3/2}}+\frac {32 b^2 x}{7 a^4 \left (a+b x^2\right )^{5/2}}+\frac {80 b^2 x}{21 a^3 \left (a+b x^2\right )^{7/2}}+\frac {10 b}{3 a^2 x \left (a+b x^2\right )^{7/2}}-\frac {1}{3 a x^3 \left (a+b x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^4*(a + b*x^2)^(9/2)),x]

[Out]

-1/(3*a*x^3*(a + b*x^2)^(7/2)) + (10*b)/(3*a^2*x*(a + b*x^2)^(7/2)) + (80*b^2*x)/(21*a^3*(a + b*x^2)^(7/2)) +
(32*b^2*x)/(7*a^4*(a + b*x^2)^(5/2)) + (128*b^2*x)/(21*a^5*(a + b*x^2)^(3/2)) + (256*b^2*x)/(21*a^6*Sqrt[a + b
*x^2])

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 192

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, n, p}, x] && ILtQ[Simplify[1/n + p + 1
], 0] && NeQ[p, -1]

Rule 271

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x^(m + 1)*(a + b*x^n)^(p + 1))/(a*(m + 1)), x]
- Dist[(b*(m + n*(p + 1) + 1))/(a*(m + 1)), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

Rubi steps

\begin {align*} \int \frac {1}{x^4 \left (a+b x^2\right )^{9/2}} \, dx &=-\frac {1}{3 a x^3 \left (a+b x^2\right )^{7/2}}-\frac {(10 b) \int \frac {1}{x^2 \left (a+b x^2\right )^{9/2}} \, dx}{3 a}\\ &=-\frac {1}{3 a x^3 \left (a+b x^2\right )^{7/2}}+\frac {10 b}{3 a^2 x \left (a+b x^2\right )^{7/2}}+\frac {\left (80 b^2\right ) \int \frac {1}{\left (a+b x^2\right )^{9/2}} \, dx}{3 a^2}\\ &=-\frac {1}{3 a x^3 \left (a+b x^2\right )^{7/2}}+\frac {10 b}{3 a^2 x \left (a+b x^2\right )^{7/2}}+\frac {80 b^2 x}{21 a^3 \left (a+b x^2\right )^{7/2}}+\frac {\left (160 b^2\right ) \int \frac {1}{\left (a+b x^2\right )^{7/2}} \, dx}{7 a^3}\\ &=-\frac {1}{3 a x^3 \left (a+b x^2\right )^{7/2}}+\frac {10 b}{3 a^2 x \left (a+b x^2\right )^{7/2}}+\frac {80 b^2 x}{21 a^3 \left (a+b x^2\right )^{7/2}}+\frac {32 b^2 x}{7 a^4 \left (a+b x^2\right )^{5/2}}+\frac {\left (128 b^2\right ) \int \frac {1}{\left (a+b x^2\right )^{5/2}} \, dx}{7 a^4}\\ &=-\frac {1}{3 a x^3 \left (a+b x^2\right )^{7/2}}+\frac {10 b}{3 a^2 x \left (a+b x^2\right )^{7/2}}+\frac {80 b^2 x}{21 a^3 \left (a+b x^2\right )^{7/2}}+\frac {32 b^2 x}{7 a^4 \left (a+b x^2\right )^{5/2}}+\frac {128 b^2 x}{21 a^5 \left (a+b x^2\right )^{3/2}}+\frac {\left (256 b^2\right ) \int \frac {1}{\left (a+b x^2\right )^{3/2}} \, dx}{21 a^5}\\ &=-\frac {1}{3 a x^3 \left (a+b x^2\right )^{7/2}}+\frac {10 b}{3 a^2 x \left (a+b x^2\right )^{7/2}}+\frac {80 b^2 x}{21 a^3 \left (a+b x^2\right )^{7/2}}+\frac {32 b^2 x}{7 a^4 \left (a+b x^2\right )^{5/2}}+\frac {128 b^2 x}{21 a^5 \left (a+b x^2\right )^{3/2}}+\frac {256 b^2 x}{21 a^6 \sqrt {a+b x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 75, normalized size = 0.57 \[ \frac {-7 a^5+70 a^4 b x^2+560 a^3 b^2 x^4+1120 a^2 b^3 x^6+896 a b^4 x^8+256 b^5 x^{10}}{21 a^6 x^3 \left (a+b x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*(a + b*x^2)^(9/2)),x]

[Out]

(-7*a^5 + 70*a^4*b*x^2 + 560*a^3*b^2*x^4 + 1120*a^2*b^3*x^6 + 896*a*b^4*x^8 + 256*b^5*x^10)/(21*a^6*x^3*(a + b
*x^2)^(7/2))

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fricas [A]  time = 1.06, size = 116, normalized size = 0.88 \[ \frac {{\left (256 \, b^{5} x^{10} + 896 \, a b^{4} x^{8} + 1120 \, a^{2} b^{3} x^{6} + 560 \, a^{3} b^{2} x^{4} + 70 \, a^{4} b x^{2} - 7 \, a^{5}\right )} \sqrt {b x^{2} + a}}{21 \, {\left (a^{6} b^{4} x^{11} + 4 \, a^{7} b^{3} x^{9} + 6 \, a^{8} b^{2} x^{7} + 4 \, a^{9} b x^{5} + a^{10} x^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(b*x^2+a)^(9/2),x, algorithm="fricas")

[Out]

1/21*(256*b^5*x^10 + 896*a*b^4*x^8 + 1120*a^2*b^3*x^6 + 560*a^3*b^2*x^4 + 70*a^4*b*x^2 - 7*a^5)*sqrt(b*x^2 + a
)/(a^6*b^4*x^11 + 4*a^7*b^3*x^9 + 6*a^8*b^2*x^7 + 4*a^9*b*x^5 + a^10*x^3)

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giac [A]  time = 1.16, size = 147, normalized size = 1.11 \[ \frac {{\left ({\left (x^{2} {\left (\frac {158 \, b^{5} x^{2}}{a^{6}} + \frac {511 \, b^{4}}{a^{5}}\right )} + \frac {560 \, b^{3}}{a^{4}}\right )} x^{2} + \frac {210 \, b^{2}}{a^{3}}\right )} x}{21 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}}} - \frac {4 \, {\left (6 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} b^{\frac {3}{2}} - 15 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a b^{\frac {3}{2}} + 7 \, a^{2} b^{\frac {3}{2}}\right )}}{3 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{3} a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(b*x^2+a)^(9/2),x, algorithm="giac")

[Out]

1/21*((x^2*(158*b^5*x^2/a^6 + 511*b^4/a^5) + 560*b^3/a^4)*x^2 + 210*b^2/a^3)*x/(b*x^2 + a)^(7/2) - 4/3*(6*(sqr
t(b)*x - sqrt(b*x^2 + a))^4*b^(3/2) - 15*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a*b^(3/2) + 7*a^2*b^(3/2))/(((sqrt(b)
*x - sqrt(b*x^2 + a))^2 - a)^3*a^5)

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maple [A]  time = 0.01, size = 72, normalized size = 0.55 \[ -\frac {-256 b^{5} x^{10}-896 a \,b^{4} x^{8}-1120 a^{2} b^{3} x^{6}-560 a^{3} b^{2} x^{4}-70 a^{4} b \,x^{2}+7 a^{5}}{21 \left (b \,x^{2}+a \right )^{\frac {7}{2}} a^{6} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^4/(b*x^2+a)^(9/2),x)

[Out]

-1/21*(-256*b^5*x^10-896*a*b^4*x^8-1120*a^2*b^3*x^6-560*a^3*b^2*x^4-70*a^4*b*x^2+7*a^5)/x^3/(b*x^2+a)^(7/2)/a^
6

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maxima [A]  time = 1.42, size = 108, normalized size = 0.82 \[ \frac {256 \, b^{2} x}{21 \, \sqrt {b x^{2} + a} a^{6}} + \frac {128 \, b^{2} x}{21 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{5}} + \frac {32 \, b^{2} x}{7 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{4}} + \frac {80 \, b^{2} x}{21 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{3}} + \frac {10 \, b}{3 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{2} x} - \frac {1}{3 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(b*x^2+a)^(9/2),x, algorithm="maxima")

[Out]

256/21*b^2*x/(sqrt(b*x^2 + a)*a^6) + 128/21*b^2*x/((b*x^2 + a)^(3/2)*a^5) + 32/7*b^2*x/((b*x^2 + a)^(5/2)*a^4)
+ 80/21*b^2*x/((b*x^2 + a)^(7/2)*a^3) + 10/3*b/((b*x^2 + a)^(7/2)*a^2*x) - 1/3/((b*x^2 + a)^(7/2)*a*x^3)

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mupad [B]  time = 4.80, size = 97, normalized size = 0.73 \[ \frac {\frac {128\,b}{21\,a^5}+\frac {256\,b^2\,x^2}{21\,a^6}}{x\,\sqrt {b\,x^2+a}}-\frac {\frac {1}{3\,a^2}+\frac {19\,b\,x^2}{21\,a^3}}{x^3\,{\left (b\,x^2+a\right )}^{5/2}}-\frac {32\,b}{21\,a^4\,x\,{\left (b\,x^2+a\right )}^{3/2}}+\frac {b^2\,x}{7\,a^3\,{\left (b\,x^2+a\right )}^{7/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^4*(a + b*x^2)^(9/2)),x)

[Out]

((128*b)/(21*a^5) + (256*b^2*x^2)/(21*a^6))/(x*(a + b*x^2)^(1/2)) - (1/(3*a^2) + (19*b*x^2)/(21*a^3))/(x^3*(a
+ b*x^2)^(5/2)) - (32*b)/(21*a^4*x*(a + b*x^2)^(3/2)) + (b^2*x)/(7*a^3*(a + b*x^2)^(7/2))

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sympy [B]  time = 3.84, size = 668, normalized size = 5.06 \[ - \frac {7 a^{6} b^{\frac {51}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{21 a^{11} b^{25} x^{2} + 105 a^{10} b^{26} x^{4} + 210 a^{9} b^{27} x^{6} + 210 a^{8} b^{28} x^{8} + 105 a^{7} b^{29} x^{10} + 21 a^{6} b^{30} x^{12}} + \frac {63 a^{5} b^{\frac {53}{2}} x^{2} \sqrt {\frac {a}{b x^{2}} + 1}}{21 a^{11} b^{25} x^{2} + 105 a^{10} b^{26} x^{4} + 210 a^{9} b^{27} x^{6} + 210 a^{8} b^{28} x^{8} + 105 a^{7} b^{29} x^{10} + 21 a^{6} b^{30} x^{12}} + \frac {630 a^{4} b^{\frac {55}{2}} x^{4} \sqrt {\frac {a}{b x^{2}} + 1}}{21 a^{11} b^{25} x^{2} + 105 a^{10} b^{26} x^{4} + 210 a^{9} b^{27} x^{6} + 210 a^{8} b^{28} x^{8} + 105 a^{7} b^{29} x^{10} + 21 a^{6} b^{30} x^{12}} + \frac {1680 a^{3} b^{\frac {57}{2}} x^{6} \sqrt {\frac {a}{b x^{2}} + 1}}{21 a^{11} b^{25} x^{2} + 105 a^{10} b^{26} x^{4} + 210 a^{9} b^{27} x^{6} + 210 a^{8} b^{28} x^{8} + 105 a^{7} b^{29} x^{10} + 21 a^{6} b^{30} x^{12}} + \frac {2016 a^{2} b^{\frac {59}{2}} x^{8} \sqrt {\frac {a}{b x^{2}} + 1}}{21 a^{11} b^{25} x^{2} + 105 a^{10} b^{26} x^{4} + 210 a^{9} b^{27} x^{6} + 210 a^{8} b^{28} x^{8} + 105 a^{7} b^{29} x^{10} + 21 a^{6} b^{30} x^{12}} + \frac {1152 a b^{\frac {61}{2}} x^{10} \sqrt {\frac {a}{b x^{2}} + 1}}{21 a^{11} b^{25} x^{2} + 105 a^{10} b^{26} x^{4} + 210 a^{9} b^{27} x^{6} + 210 a^{8} b^{28} x^{8} + 105 a^{7} b^{29} x^{10} + 21 a^{6} b^{30} x^{12}} + \frac {256 b^{\frac {63}{2}} x^{12} \sqrt {\frac {a}{b x^{2}} + 1}}{21 a^{11} b^{25} x^{2} + 105 a^{10} b^{26} x^{4} + 210 a^{9} b^{27} x^{6} + 210 a^{8} b^{28} x^{8} + 105 a^{7} b^{29} x^{10} + 21 a^{6} b^{30} x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**4/(b*x**2+a)**(9/2),x)

[Out]

-7*a**6*b**(51/2)*sqrt(a/(b*x**2) + 1)/(21*a**11*b**25*x**2 + 105*a**10*b**26*x**4 + 210*a**9*b**27*x**6 + 210
*a**8*b**28*x**8 + 105*a**7*b**29*x**10 + 21*a**6*b**30*x**12) + 63*a**5*b**(53/2)*x**2*sqrt(a/(b*x**2) + 1)/(
21*a**11*b**25*x**2 + 105*a**10*b**26*x**4 + 210*a**9*b**27*x**6 + 210*a**8*b**28*x**8 + 105*a**7*b**29*x**10
+ 21*a**6*b**30*x**12) + 630*a**4*b**(55/2)*x**4*sqrt(a/(b*x**2) + 1)/(21*a**11*b**25*x**2 + 105*a**10*b**26*x
**4 + 210*a**9*b**27*x**6 + 210*a**8*b**28*x**8 + 105*a**7*b**29*x**10 + 21*a**6*b**30*x**12) + 1680*a**3*b**(
57/2)*x**6*sqrt(a/(b*x**2) + 1)/(21*a**11*b**25*x**2 + 105*a**10*b**26*x**4 + 210*a**9*b**27*x**6 + 210*a**8*b
**28*x**8 + 105*a**7*b**29*x**10 + 21*a**6*b**30*x**12) + 2016*a**2*b**(59/2)*x**8*sqrt(a/(b*x**2) + 1)/(21*a*
*11*b**25*x**2 + 105*a**10*b**26*x**4 + 210*a**9*b**27*x**6 + 210*a**8*b**28*x**8 + 105*a**7*b**29*x**10 + 21*
a**6*b**30*x**12) + 1152*a*b**(61/2)*x**10*sqrt(a/(b*x**2) + 1)/(21*a**11*b**25*x**2 + 105*a**10*b**26*x**4 +
210*a**9*b**27*x**6 + 210*a**8*b**28*x**8 + 105*a**7*b**29*x**10 + 21*a**6*b**30*x**12) + 256*b**(63/2)*x**12*
sqrt(a/(b*x**2) + 1)/(21*a**11*b**25*x**2 + 105*a**10*b**26*x**4 + 210*a**9*b**27*x**6 + 210*a**8*b**28*x**8 +
105*a**7*b**29*x**10 + 21*a**6*b**30*x**12)

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