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]

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

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Rubi [A]  time = 0.01, antiderivative size = 44, normalized size of antiderivative = 1.00, 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 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]

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 {\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*}

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Mathematica [A]  time = 0.01, size = 31, normalized size = 0.70 \[ \frac {\left (a+b x^2\right )^{11/2} \left (2 b x^2-11 a\right )}{143 a^2 x^{13}} \]

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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fricas [B]  time = 0.94, size = 82, normalized size = 1.86 \[ \frac {{\left (2 \, b^{6} x^{12} - a b^{5} x^{10} - 35 \, a^{2} b^{4} x^{8} - 90 \, a^{3} b^{3} x^{6} - 100 \, a^{4} b^{2} x^{4} - 53 \, a^{5} b x^{2} - 11 \, a^{6}\right )} \sqrt {b x^{2} + a}}{143 \, a^{2} x^{13}} \]

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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giac [B]  time = 1.09, size = 328, normalized size = 7.45 \[ \frac {4 \, {\left (143 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{22} b^{\frac {13}{2}} + 429 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{20} a b^{\frac {13}{2}} + 2145 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{18} a^{2} b^{\frac {13}{2}} + 3003 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{16} a^{3} b^{\frac {13}{2}} + 6006 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{14} a^{4} b^{\frac {13}{2}} + 4290 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{12} a^{5} b^{\frac {13}{2}} + 4290 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} a^{6} b^{\frac {13}{2}} + 1430 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} a^{7} b^{\frac {13}{2}} + 715 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{8} b^{\frac {13}{2}} + 65 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{9} b^{\frac {13}{2}} + 13 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{10} b^{\frac {13}{2}} - a^{11} b^{\frac {13}{2}}\right )}}{143 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{13}} \]

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

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maple [A]  time = 0.01, size = 28, normalized size = 0.64 \[ -\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}} \left (-2 b \,x^{2}+11 a \right )}{143 a^{2} x^{13}} \]

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 [A]  time = 1.57, size = 36, normalized size = 0.82 \[ \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} b}{143 \, a^{2} x^{11}} - \frac {{\left (b x^{2} + a\right )}^{\frac {11}{2}}}{13 \, a x^{13}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 6.85, size = 131, normalized size = 2.98 \[ \frac {2\,b^6\,\sqrt {b\,x^2+a}}{143\,a^2\,x}-\frac {35\,b^4\,\sqrt {b\,x^2+a}}{143\,x^5}-\frac {90\,a\,b^3\,\sqrt {b\,x^2+a}}{143\,x^7}-\frac {53\,a^3\,b\,\sqrt {b\,x^2+a}}{143\,x^{11}}-\frac {b^5\,\sqrt {b\,x^2+a}}{143\,a\,x^3}-\frac {a^4\,\sqrt {b\,x^2+a}}{13\,x^{13}}-\frac {100\,a^2\,b^2\,\sqrt {b\,x^2+a}}{143\,x^9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^(9/2)/x^14,x)

[Out]

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

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sympy [B]  time = 2.98, size = 175, normalized size = 3.98 \[ - \frac {a^{4} \sqrt {b} \sqrt {\frac {a}{b x^{2}} + 1}}{13 x^{12}} - \frac {53 a^{3} b^{\frac {3}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{143 x^{10}} - \frac {100 a^{2} b^{\frac {5}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{143 x^{8}} - \frac {90 a b^{\frac {7}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{143 x^{6}} - \frac {35 b^{\frac {9}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{143 x^{4}} - \frac {b^{\frac {11}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{143 a x^{2}} + \frac {2 b^{\frac {13}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{143 a^{2}} \]

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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