3.405 \(\int \frac{(a+b x^2)^{5/2}}{x^{12}} \, dx\)

Optimal. Leaf size=68 \[ -\frac{8 b^2 \left (a+b x^2\right )^{7/2}}{693 a^3 x^7}+\frac{4 b \left (a+b x^2\right )^{7/2}}{99 a^2 x^9}-\frac{\left (a+b x^2\right )^{7/2}}{11 a x^{11}} \]

[Out]

-(a + b*x^2)^(7/2)/(11*a*x^11) + (4*b*(a + b*x^2)^(7/2))/(99*a^2*x^9) - (8*b^2*(a + b*x^2)^(7/2))/(693*a^3*x^7
)

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Rubi [A]  time = 0.0189424, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {271, 264} \[ -\frac{8 b^2 \left (a+b x^2\right )^{7/2}}{693 a^3 x^7}+\frac{4 b \left (a+b x^2\right )^{7/2}}{99 a^2 x^9}-\frac{\left (a+b x^2\right )^{7/2}}{11 a x^{11}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^(5/2)/x^12,x]

[Out]

-(a + b*x^2)^(7/2)/(11*a*x^11) + (4*b*(a + b*x^2)^(7/2))/(99*a^2*x^9) - (8*b^2*(a + b*x^2)^(7/2))/(693*a^3*x^7
)

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 264

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

Rubi steps

\begin{align*} \int \frac{\left (a+b x^2\right )^{5/2}}{x^{12}} \, dx &=-\frac{\left (a+b x^2\right )^{7/2}}{11 a x^{11}}-\frac{(4 b) \int \frac{\left (a+b x^2\right )^{5/2}}{x^{10}} \, dx}{11 a}\\ &=-\frac{\left (a+b x^2\right )^{7/2}}{11 a x^{11}}+\frac{4 b \left (a+b x^2\right )^{7/2}}{99 a^2 x^9}+\frac{\left (8 b^2\right ) \int \frac{\left (a+b x^2\right )^{5/2}}{x^8} \, dx}{99 a^2}\\ &=-\frac{\left (a+b x^2\right )^{7/2}}{11 a x^{11}}+\frac{4 b \left (a+b x^2\right )^{7/2}}{99 a^2 x^9}-\frac{8 b^2 \left (a+b x^2\right )^{7/2}}{693 a^3 x^7}\\ \end{align*}

Mathematica [A]  time = 0.011223, size = 42, normalized size = 0.62 \[ -\frac{\left (a+b x^2\right )^{7/2} \left (63 a^2-28 a b x^2+8 b^2 x^4\right )}{693 a^3 x^{11}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^(5/2)/x^12,x]

[Out]

-((a + b*x^2)^(7/2)*(63*a^2 - 28*a*b*x^2 + 8*b^2*x^4))/(693*a^3*x^11)

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Maple [A]  time = 0.003, size = 39, normalized size = 0.6 \begin{align*} -{\frac{8\,{b}^{2}{x}^{4}-28\,ab{x}^{2}+63\,{a}^{2}}{693\,{x}^{11}{a}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^(5/2)/x^12,x)

[Out]

-1/693*(b*x^2+a)^(7/2)*(8*b^2*x^4-28*a*b*x^2+63*a^2)/x^11/a^3

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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((b*x^2+a)^(5/2)/x^12,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.71763, size = 163, normalized size = 2.4 \begin{align*} -\frac{{\left (8 \, b^{5} x^{10} - 4 \, a b^{4} x^{8} + 3 \, a^{2} b^{3} x^{6} + 113 \, a^{3} b^{2} x^{4} + 161 \, a^{4} b x^{2} + 63 \, a^{5}\right )} \sqrt{b x^{2} + a}}{693 \, a^{3} x^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(5/2)/x^12,x, algorithm="fricas")

[Out]

-1/693*(8*b^5*x^10 - 4*a*b^4*x^8 + 3*a^2*b^3*x^6 + 113*a^3*b^2*x^4 + 161*a^4*b*x^2 + 63*a^5)*sqrt(b*x^2 + a)/(
a^3*x^11)

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Sympy [B]  time = 3.18987, size = 481, normalized size = 7.07 \begin{align*} - \frac{63 a^{7} b^{\frac{9}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{x^{2} \left (693 a^{5} b^{4} x^{8} + 1386 a^{4} b^{5} x^{10} + 693 a^{3} b^{6} x^{12}\right )} - \frac{287 a^{6} b^{\frac{11}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{693 a^{5} b^{4} x^{8} + 1386 a^{4} b^{5} x^{10} + 693 a^{3} b^{6} x^{12}} - \frac{498 a^{5} b^{\frac{13}{2}} x^{2} \sqrt{\frac{a}{b x^{2}} + 1}}{693 a^{5} b^{4} x^{8} + 1386 a^{4} b^{5} x^{10} + 693 a^{3} b^{6} x^{12}} - \frac{390 a^{4} b^{\frac{15}{2}} x^{4} \sqrt{\frac{a}{b x^{2}} + 1}}{693 a^{5} b^{4} x^{8} + 1386 a^{4} b^{5} x^{10} + 693 a^{3} b^{6} x^{12}} - \frac{115 a^{3} b^{\frac{17}{2}} x^{6} \sqrt{\frac{a}{b x^{2}} + 1}}{693 a^{5} b^{4} x^{8} + 1386 a^{4} b^{5} x^{10} + 693 a^{3} b^{6} x^{12}} - \frac{3 a^{2} b^{\frac{19}{2}} x^{8} \sqrt{\frac{a}{b x^{2}} + 1}}{693 a^{5} b^{4} x^{8} + 1386 a^{4} b^{5} x^{10} + 693 a^{3} b^{6} x^{12}} - \frac{12 a b^{\frac{21}{2}} x^{10} \sqrt{\frac{a}{b x^{2}} + 1}}{693 a^{5} b^{4} x^{8} + 1386 a^{4} b^{5} x^{10} + 693 a^{3} b^{6} x^{12}} - \frac{8 b^{\frac{23}{2}} x^{12} \sqrt{\frac{a}{b x^{2}} + 1}}{693 a^{5} b^{4} x^{8} + 1386 a^{4} b^{5} x^{10} + 693 a^{3} b^{6} x^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**(5/2)/x**12,x)

[Out]

-63*a**7*b**(9/2)*sqrt(a/(b*x**2) + 1)/(x**2*(693*a**5*b**4*x**8 + 1386*a**4*b**5*x**10 + 693*a**3*b**6*x**12)
) - 287*a**6*b**(11/2)*sqrt(a/(b*x**2) + 1)/(693*a**5*b**4*x**8 + 1386*a**4*b**5*x**10 + 693*a**3*b**6*x**12)
- 498*a**5*b**(13/2)*x**2*sqrt(a/(b*x**2) + 1)/(693*a**5*b**4*x**8 + 1386*a**4*b**5*x**10 + 693*a**3*b**6*x**1
2) - 390*a**4*b**(15/2)*x**4*sqrt(a/(b*x**2) + 1)/(693*a**5*b**4*x**8 + 1386*a**4*b**5*x**10 + 693*a**3*b**6*x
**12) - 115*a**3*b**(17/2)*x**6*sqrt(a/(b*x**2) + 1)/(693*a**5*b**4*x**8 + 1386*a**4*b**5*x**10 + 693*a**3*b**
6*x**12) - 3*a**2*b**(19/2)*x**8*sqrt(a/(b*x**2) + 1)/(693*a**5*b**4*x**8 + 1386*a**4*b**5*x**10 + 693*a**3*b*
*6*x**12) - 12*a*b**(21/2)*x**10*sqrt(a/(b*x**2) + 1)/(693*a**5*b**4*x**8 + 1386*a**4*b**5*x**10 + 693*a**3*b*
*6*x**12) - 8*b**(23/2)*x**12*sqrt(a/(b*x**2) + 1)/(693*a**5*b**4*x**8 + 1386*a**4*b**5*x**10 + 693*a**3*b**6*
x**12)

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Giac [B]  time = 1.86646, size = 332, normalized size = 4.88 \begin{align*} \frac{16 \,{\left (462 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{16} b^{\frac{11}{2}} + 1155 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{14} a b^{\frac{11}{2}} + 2541 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{12} a^{2} b^{\frac{11}{2}} + 2079 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{10} a^{3} b^{\frac{11}{2}} + 1485 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{8} a^{4} b^{\frac{11}{2}} + 297 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} a^{5} b^{\frac{11}{2}} + 55 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} a^{6} b^{\frac{11}{2}} - 11 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} a^{7} b^{\frac{11}{2}} + a^{8} b^{\frac{11}{2}}\right )}}{693 \,{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )}^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(5/2)/x^12,x, algorithm="giac")

[Out]

16/693*(462*(sqrt(b)*x - sqrt(b*x^2 + a))^16*b^(11/2) + 1155*(sqrt(b)*x - sqrt(b*x^2 + a))^14*a*b^(11/2) + 254
1*(sqrt(b)*x - sqrt(b*x^2 + a))^12*a^2*b^(11/2) + 2079*(sqrt(b)*x - sqrt(b*x^2 + a))^10*a^3*b^(11/2) + 1485*(s
qrt(b)*x - sqrt(b*x^2 + a))^8*a^4*b^(11/2) + 297*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^5*b^(11/2) + 55*(sqrt(b)*x
- sqrt(b*x^2 + a))^4*a^6*b^(11/2) - 11*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^7*b^(11/2) + a^8*b^(11/2))/((sqrt(b)*
x - sqrt(b*x^2 + a))^2 - a)^11