3.435 \(\int \frac{(a+b x^2)^{9/2}}{x^{14}} \, dx\)

Optimal. Leaf size=44 \[ \frac{2 b \left (a+b x^2\right )^{11/2}}{143 a^2 x^{11}}-\frac{\left (a+b x^2\right )^{11/2}}{13 a x^{13}} \]

[Out]

-(a + b*x^2)^(11/2)/(13*a*x^13) + (2*b*(a + b*x^2)^(11/2))/(143*a^2*x^11)

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Rubi [A]  time = 0.0112088, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {271, 264} \[ \frac{2 b \left (a+b x^2\right )^{11/2}}{143 a^2 x^{11}}-\frac{\left (a+b x^2\right )^{11/2}}{13 a x^{13}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^(9/2)/x^14,x]

[Out]

-(a + b*x^2)^(11/2)/(13*a*x^13) + (2*b*(a + b*x^2)^(11/2))/(143*a^2*x^11)

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 )^{9/2}}{x^{14}} \, dx &=-\frac{\left (a+b x^2\right )^{11/2}}{13 a x^{13}}-\frac{(2 b) \int \frac{\left (a+b x^2\right )^{9/2}}{x^{12}} \, dx}{13 a}\\ &=-\frac{\left (a+b x^2\right )^{11/2}}{13 a x^{13}}+\frac{2 b \left (a+b x^2\right )^{11/2}}{143 a^2 x^{11}}\\ \end{align*}

Mathematica [A]  time = 0.0133952, size = 31, normalized size = 0.7 \[ \frac{\left (a+b x^2\right )^{11/2} \left (2 b x^2-11 a\right )}{143 a^2 x^{13}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^(9/2)/x^14,x]

[Out]

((a + b*x^2)^(11/2)*(-11*a + 2*b*x^2))/(143*a^2*x^13)

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Maple [A]  time = 0.004, size = 28, normalized size = 0.6 \begin{align*} -{\frac{-2\,b{x}^{2}+11\,a}{143\,{x}^{13}{a}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^(9/2)/x^14,x)

[Out]

-1/143*(b*x^2+a)^(11/2)*(-2*b*x^2+11*a)/x^13/a^2

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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)^(9/2)/x^14,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.12072, size = 184, normalized size = 4.18 \begin{align*} \frac{{\left (2 \, b^{6} x^{12} - a b^{5} x^{10} - 35 \, a^{2} b^{4} x^{8} - 90 \, a^{3} b^{3} x^{6} - 100 \, a^{4} b^{2} x^{4} - 53 \, a^{5} b x^{2} - 11 \, a^{6}\right )} \sqrt{b x^{2} + a}}{143 \, a^{2} x^{13}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(9/2)/x^14,x, algorithm="fricas")

[Out]

1/143*(2*b^6*x^12 - a*b^5*x^10 - 35*a^2*b^4*x^8 - 90*a^3*b^3*x^6 - 100*a^4*b^2*x^4 - 53*a^5*b*x^2 - 11*a^6)*sq
rt(b*x^2 + a)/(a^2*x^13)

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Sympy [B]  time = 6.51651, size = 175, normalized size = 3.98 \begin{align*} - \frac{a^{4} \sqrt{b} \sqrt{\frac{a}{b x^{2}} + 1}}{13 x^{12}} - \frac{53 a^{3} b^{\frac{3}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{143 x^{10}} - \frac{100 a^{2} b^{\frac{5}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{143 x^{8}} - \frac{90 a b^{\frac{7}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{143 x^{6}} - \frac{35 b^{\frac{9}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{143 x^{4}} - \frac{b^{\frac{11}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{143 a x^{2}} + \frac{2 b^{\frac{13}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{143 a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**(9/2)/x**14,x)

[Out]

-a**4*sqrt(b)*sqrt(a/(b*x**2) + 1)/(13*x**12) - 53*a**3*b**(3/2)*sqrt(a/(b*x**2) + 1)/(143*x**10) - 100*a**2*b
**(5/2)*sqrt(a/(b*x**2) + 1)/(143*x**8) - 90*a*b**(7/2)*sqrt(a/(b*x**2) + 1)/(143*x**6) - 35*b**(9/2)*sqrt(a/(
b*x**2) + 1)/(143*x**4) - b**(11/2)*sqrt(a/(b*x**2) + 1)/(143*a*x**2) + 2*b**(13/2)*sqrt(a/(b*x**2) + 1)/(143*
a**2)

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Giac [B]  time = 1.68215, size = 443, normalized size = 10.07 \begin{align*} \frac{4 \,{\left (143 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{22} b^{\frac{13}{2}} + 429 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{20} a b^{\frac{13}{2}} + 2145 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{18} a^{2} b^{\frac{13}{2}} + 3003 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{16} a^{3} b^{\frac{13}{2}} + 6006 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{14} a^{4} b^{\frac{13}{2}} + 4290 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{12} a^{5} b^{\frac{13}{2}} + 4290 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{10} a^{6} b^{\frac{13}{2}} + 1430 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{8} a^{7} b^{\frac{13}{2}} + 715 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} a^{8} b^{\frac{13}{2}} + 65 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} a^{9} b^{\frac{13}{2}} + 13 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} a^{10} b^{\frac{13}{2}} - a^{11} b^{\frac{13}{2}}\right )}}{143 \,{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )}^{13}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(9/2)/x^14,x, algorithm="giac")

[Out]

4/143*(143*(sqrt(b)*x - sqrt(b*x^2 + a))^22*b^(13/2) + 429*(sqrt(b)*x - sqrt(b*x^2 + a))^20*a*b^(13/2) + 2145*
(sqrt(b)*x - sqrt(b*x^2 + a))^18*a^2*b^(13/2) + 3003*(sqrt(b)*x - sqrt(b*x^2 + a))^16*a^3*b^(13/2) + 6006*(sqr
t(b)*x - sqrt(b*x^2 + a))^14*a^4*b^(13/2) + 4290*(sqrt(b)*x - sqrt(b*x^2 + a))^12*a^5*b^(13/2) + 4290*(sqrt(b)
*x - sqrt(b*x^2 + a))^10*a^6*b^(13/2) + 1430*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a^7*b^(13/2) + 715*(sqrt(b)*x - s
qrt(b*x^2 + a))^6*a^8*b^(13/2) + 65*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^9*b^(13/2) + 13*(sqrt(b)*x - sqrt(b*x^2
+ a))^2*a^10*b^(13/2) - a^11*b^(13/2))/((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^13