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*(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])

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Rubi [A]  time = 0.0406277, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, 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 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]

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 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]

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*}

Mathematica [A]  time = 0.0130469, size = 75, normalized size = 0.57 \[ \frac{1120 a^2 b^3 x^6+560 a^3 b^2 x^4+70 a^4 b x^2-7 a^5+896 a b^4 x^8+256 b^5 x^{10}}{21 a^6 x^3 \left (a+b x^2\right )^{7/2}} \]

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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Maple [A]  time = 0.004, size = 72, normalized size = 0.6 \begin{align*} -{\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\,{x}^{3}{a}^{6}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}} \end{align*}

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 [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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]

Exception raised: ValueError

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Fricas [A]  time = 1.7756, size = 250, normalized size = 1.89 \begin{align*} \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 )}} \end{align*}

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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Sympy [B]  time = 5.32823, size = 668, normalized size = 5.06 \begin{align*} - \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}} \end{align*}

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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Giac [A]  time = 2.37613, size = 198, normalized size = 1.5 \begin{align*} \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}} \end{align*}

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)