3.385 \(\int \frac{(a+b x^2)^{3/2}}{x^{10}} \, dx\)

Optimal. Leaf size=68 \[ -\frac{8 b^2 \left (a+b x^2\right )^{5/2}}{315 a^3 x^5}+\frac{4 b \left (a+b x^2\right )^{5/2}}{63 a^2 x^7}-\frac{\left (a+b x^2\right )^{5/2}}{9 a x^9} \]

[Out]

-(a + b*x^2)^(5/2)/(9*a*x^9) + (4*b*(a + b*x^2)^(5/2))/(63*a^2*x^7) - (8*b^2*(a + b*x^2)^(5/2))/(315*a^3*x^5)

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Rubi [A]  time = 0.0207039, 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} \[ -\frac{8 b^2 \left (a+b x^2\right )^{5/2}}{315 a^3 x^5}+\frac{4 b \left (a+b x^2\right )^{5/2}}{63 a^2 x^7}-\frac{\left (a+b x^2\right )^{5/2}}{9 a x^9} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^(3/2)/x^10,x]

[Out]

-(a + b*x^2)^(5/2)/(9*a*x^9) + (4*b*(a + b*x^2)^(5/2))/(63*a^2*x^7) - (8*b^2*(a + b*x^2)^(5/2))/(315*a^3*x^5)

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 )^{3/2}}{x^{10}} \, dx &=-\frac{\left (a+b x^2\right )^{5/2}}{9 a x^9}-\frac{(4 b) \int \frac{\left (a+b x^2\right )^{3/2}}{x^8} \, dx}{9 a}\\ &=-\frac{\left (a+b x^2\right )^{5/2}}{9 a x^9}+\frac{4 b \left (a+b x^2\right )^{5/2}}{63 a^2 x^7}+\frac{\left (8 b^2\right ) \int \frac{\left (a+b x^2\right )^{3/2}}{x^6} \, dx}{63 a^2}\\ &=-\frac{\left (a+b x^2\right )^{5/2}}{9 a x^9}+\frac{4 b \left (a+b x^2\right )^{5/2}}{63 a^2 x^7}-\frac{8 b^2 \left (a+b x^2\right )^{5/2}}{315 a^3 x^5}\\ \end{align*}

Mathematica [A]  time = 0.0105376, size = 42, normalized size = 0.62 \[ -\frac{\left (a+b x^2\right )^{5/2} \left (35 a^2-20 a b x^2+8 b^2 x^4\right )}{315 a^3 x^9} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^(3/2)/x^10,x]

[Out]

-((a + b*x^2)^(5/2)*(35*a^2 - 20*a*b*x^2 + 8*b^2*x^4))/(315*a^3*x^9)

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Maple [A]  time = 0.004, size = 39, normalized size = 0.6 \begin{align*} -{\frac{8\,{b}^{2}{x}^{4}-20\,ab{x}^{2}+35\,{a}^{2}}{315\,{x}^{9}{a}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^(3/2)/x^10,x)

[Out]

-1/315*(b*x^2+a)^(5/2)*(8*b^2*x^4-20*a*b*x^2+35*a^2)/x^9/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)^(3/2)/x^10,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.59179, size = 135, normalized size = 1.99 \begin{align*} -\frac{{\left (8 \, b^{4} x^{8} - 4 \, a b^{3} x^{6} + 3 \, a^{2} b^{2} x^{4} + 50 \, a^{3} b x^{2} + 35 \, a^{4}\right )} \sqrt{b x^{2} + a}}{315 \, a^{3} x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(3/2)/x^10,x, algorithm="fricas")

[Out]

-1/315*(8*b^4*x^8 - 4*a*b^3*x^6 + 3*a^2*b^2*x^4 + 50*a^3*b*x^2 + 35*a^4)*sqrt(b*x^2 + a)/(a^3*x^9)

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Sympy [B]  time = 1.97982, size = 420, normalized size = 6.18 \begin{align*} - \frac{35 a^{6} b^{\frac{9}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{315 a^{5} b^{4} x^{8} + 630 a^{4} b^{5} x^{10} + 315 a^{3} b^{6} x^{12}} - \frac{120 a^{5} b^{\frac{11}{2}} x^{2} \sqrt{\frac{a}{b x^{2}} + 1}}{315 a^{5} b^{4} x^{8} + 630 a^{4} b^{5} x^{10} + 315 a^{3} b^{6} x^{12}} - \frac{138 a^{4} b^{\frac{13}{2}} x^{4} \sqrt{\frac{a}{b x^{2}} + 1}}{315 a^{5} b^{4} x^{8} + 630 a^{4} b^{5} x^{10} + 315 a^{3} b^{6} x^{12}} - \frac{52 a^{3} b^{\frac{15}{2}} x^{6} \sqrt{\frac{a}{b x^{2}} + 1}}{315 a^{5} b^{4} x^{8} + 630 a^{4} b^{5} x^{10} + 315 a^{3} b^{6} x^{12}} - \frac{3 a^{2} b^{\frac{17}{2}} x^{8} \sqrt{\frac{a}{b x^{2}} + 1}}{315 a^{5} b^{4} x^{8} + 630 a^{4} b^{5} x^{10} + 315 a^{3} b^{6} x^{12}} - \frac{12 a b^{\frac{19}{2}} x^{10} \sqrt{\frac{a}{b x^{2}} + 1}}{315 a^{5} b^{4} x^{8} + 630 a^{4} b^{5} x^{10} + 315 a^{3} b^{6} x^{12}} - \frac{8 b^{\frac{21}{2}} x^{12} \sqrt{\frac{a}{b x^{2}} + 1}}{315 a^{5} b^{4} x^{8} + 630 a^{4} b^{5} x^{10} + 315 a^{3} b^{6} x^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**(3/2)/x**10,x)

[Out]

-35*a**6*b**(9/2)*sqrt(a/(b*x**2) + 1)/(315*a**5*b**4*x**8 + 630*a**4*b**5*x**10 + 315*a**3*b**6*x**12) - 120*
a**5*b**(11/2)*x**2*sqrt(a/(b*x**2) + 1)/(315*a**5*b**4*x**8 + 630*a**4*b**5*x**10 + 315*a**3*b**6*x**12) - 13
8*a**4*b**(13/2)*x**4*sqrt(a/(b*x**2) + 1)/(315*a**5*b**4*x**8 + 630*a**4*b**5*x**10 + 315*a**3*b**6*x**12) -
52*a**3*b**(15/2)*x**6*sqrt(a/(b*x**2) + 1)/(315*a**5*b**4*x**8 + 630*a**4*b**5*x**10 + 315*a**3*b**6*x**12) -
3*a**2*b**(17/2)*x**8*sqrt(a/(b*x**2) + 1)/(315*a**5*b**4*x**8 + 630*a**4*b**5*x**10 + 315*a**3*b**6*x**12) -
12*a*b**(19/2)*x**10*sqrt(a/(b*x**2) + 1)/(315*a**5*b**4*x**8 + 630*a**4*b**5*x**10 + 315*a**3*b**6*x**12) -
8*b**(21/2)*x**12*sqrt(a/(b*x**2) + 1)/(315*a**5*b**4*x**8 + 630*a**4*b**5*x**10 + 315*a**3*b**6*x**12)

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Giac [B]  time = 2.51133, size = 259, normalized size = 3.81 \begin{align*} \frac{16 \,{\left (210 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{12} b^{\frac{9}{2}} + 315 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{10} a b^{\frac{9}{2}} + 441 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{8} a^{2} b^{\frac{9}{2}} + 126 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} a^{3} b^{\frac{9}{2}} + 36 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} a^{4} b^{\frac{9}{2}} - 9 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} a^{5} b^{\frac{9}{2}} + a^{6} b^{\frac{9}{2}}\right )}}{315 \,{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )}^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(3/2)/x^10,x, algorithm="giac")

[Out]

16/315*(210*(sqrt(b)*x - sqrt(b*x^2 + a))^12*b^(9/2) + 315*(sqrt(b)*x - sqrt(b*x^2 + a))^10*a*b^(9/2) + 441*(s
qrt(b)*x - sqrt(b*x^2 + a))^8*a^2*b^(9/2) + 126*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^3*b^(9/2) + 36*(sqrt(b)*x -
sqrt(b*x^2 + a))^4*a^4*b^(9/2) - 9*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^5*b^(9/2) + a^6*b^(9/2))/((sqrt(b)*x - sq
rt(b*x^2 + a))^2 - a)^9