3.406 \(\int \frac{(a+b x^2)^{5/2}}{x^{14}} \, dx\)

Optimal. Leaf size=92 \[ \frac{16 b^3 \left (a+b x^2\right )^{7/2}}{3003 a^4 x^7}-\frac{8 b^2 \left (a+b x^2\right )^{7/2}}{429 a^3 x^9}+\frac{6 b \left (a+b x^2\right )^{7/2}}{143 a^2 x^{11}}-\frac{\left (a+b x^2\right )^{7/2}}{13 a x^{13}} \]

[Out]

-(a + b*x^2)^(7/2)/(13*a*x^13) + (6*b*(a + b*x^2)^(7/2))/(143*a^2*x^11) - (8*b^2*(a + b*x^2)^(7/2))/(429*a^3*x
^9) + (16*b^3*(a + b*x^2)^(7/2))/(3003*a^4*x^7)

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Rubi [A]  time = 0.028226, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {271, 264} \[ \frac{16 b^3 \left (a+b x^2\right )^{7/2}}{3003 a^4 x^7}-\frac{8 b^2 \left (a+b x^2\right )^{7/2}}{429 a^3 x^9}+\frac{6 b \left (a+b x^2\right )^{7/2}}{143 a^2 x^{11}}-\frac{\left (a+b x^2\right )^{7/2}}{13 a x^{13}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^(5/2)/x^14,x]

[Out]

-(a + b*x^2)^(7/2)/(13*a*x^13) + (6*b*(a + b*x^2)^(7/2))/(143*a^2*x^11) - (8*b^2*(a + b*x^2)^(7/2))/(429*a^3*x
^9) + (16*b^3*(a + b*x^2)^(7/2))/(3003*a^4*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^{14}} \, dx &=-\frac{\left (a+b x^2\right )^{7/2}}{13 a x^{13}}-\frac{(6 b) \int \frac{\left (a+b x^2\right )^{5/2}}{x^{12}} \, dx}{13 a}\\ &=-\frac{\left (a+b x^2\right )^{7/2}}{13 a x^{13}}+\frac{6 b \left (a+b x^2\right )^{7/2}}{143 a^2 x^{11}}+\frac{\left (24 b^2\right ) \int \frac{\left (a+b x^2\right )^{5/2}}{x^{10}} \, dx}{143 a^2}\\ &=-\frac{\left (a+b x^2\right )^{7/2}}{13 a x^{13}}+\frac{6 b \left (a+b x^2\right )^{7/2}}{143 a^2 x^{11}}-\frac{8 b^2 \left (a+b x^2\right )^{7/2}}{429 a^3 x^9}-\frac{\left (16 b^3\right ) \int \frac{\left (a+b x^2\right )^{5/2}}{x^8} \, dx}{429 a^3}\\ &=-\frac{\left (a+b x^2\right )^{7/2}}{13 a x^{13}}+\frac{6 b \left (a+b x^2\right )^{7/2}}{143 a^2 x^{11}}-\frac{8 b^2 \left (a+b x^2\right )^{7/2}}{429 a^3 x^9}+\frac{16 b^3 \left (a+b x^2\right )^{7/2}}{3003 a^4 x^7}\\ \end{align*}

Mathematica [A]  time = 0.0140055, size = 53, normalized size = 0.58 \[ \frac{\left (a+b x^2\right )^{7/2} \left (126 a^2 b x^2-231 a^3-56 a b^2 x^4+16 b^3 x^6\right )}{3003 a^4 x^{13}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^(5/2)/x^14,x]

[Out]

((a + b*x^2)^(7/2)*(-231*a^3 + 126*a^2*b*x^2 - 56*a*b^2*x^4 + 16*b^3*x^6))/(3003*a^4*x^13)

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Maple [A]  time = 0.004, size = 50, normalized size = 0.5 \begin{align*} -{\frac{-16\,{b}^{3}{x}^{6}+56\,a{b}^{2}{x}^{4}-126\,{a}^{2}b{x}^{2}+231\,{a}^{3}}{3003\,{x}^{13}{a}^{4}} \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^14,x)

[Out]

-1/3003*(b*x^2+a)^(7/2)*(-16*b^3*x^6+56*a*b^2*x^4-126*a^2*b*x^2+231*a^3)/x^13/a^4

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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^14,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.86854, size = 189, normalized size = 2.05 \begin{align*} \frac{{\left (16 \, b^{6} x^{12} - 8 \, a b^{5} x^{10} + 6 \, a^{2} b^{4} x^{8} - 5 \, a^{3} b^{3} x^{6} - 371 \, a^{4} b^{2} x^{4} - 567 \, a^{5} b x^{2} - 231 \, a^{6}\right )} \sqrt{b x^{2} + a}}{3003 \, a^{4} x^{13}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(5/2)/x^14,x, algorithm="fricas")

[Out]

1/3003*(16*b^6*x^12 - 8*a*b^5*x^10 + 6*a^2*b^4*x^8 - 5*a^3*b^3*x^6 - 371*a^4*b^2*x^4 - 567*a^5*b*x^2 - 231*a^6
)*sqrt(b*x^2 + a)/(a^4*x^13)

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Sympy [B]  time = 4.4519, size = 721, normalized size = 7.84 \begin{align*} - \frac{231 a^{9} b^{\frac{19}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{3003 a^{7} b^{9} x^{12} + 9009 a^{6} b^{10} x^{14} + 9009 a^{5} b^{11} x^{16} + 3003 a^{4} b^{12} x^{18}} - \frac{1260 a^{8} b^{\frac{21}{2}} x^{2} \sqrt{\frac{a}{b x^{2}} + 1}}{3003 a^{7} b^{9} x^{12} + 9009 a^{6} b^{10} x^{14} + 9009 a^{5} b^{11} x^{16} + 3003 a^{4} b^{12} x^{18}} - \frac{2765 a^{7} b^{\frac{23}{2}} x^{4} \sqrt{\frac{a}{b x^{2}} + 1}}{3003 a^{7} b^{9} x^{12} + 9009 a^{6} b^{10} x^{14} + 9009 a^{5} b^{11} x^{16} + 3003 a^{4} b^{12} x^{18}} - \frac{3050 a^{6} b^{\frac{25}{2}} x^{6} \sqrt{\frac{a}{b x^{2}} + 1}}{3003 a^{7} b^{9} x^{12} + 9009 a^{6} b^{10} x^{14} + 9009 a^{5} b^{11} x^{16} + 3003 a^{4} b^{12} x^{18}} - \frac{1689 a^{5} b^{\frac{27}{2}} x^{8} \sqrt{\frac{a}{b x^{2}} + 1}}{3003 a^{7} b^{9} x^{12} + 9009 a^{6} b^{10} x^{14} + 9009 a^{5} b^{11} x^{16} + 3003 a^{4} b^{12} x^{18}} - \frac{376 a^{4} b^{\frac{29}{2}} x^{10} \sqrt{\frac{a}{b x^{2}} + 1}}{3003 a^{7} b^{9} x^{12} + 9009 a^{6} b^{10} x^{14} + 9009 a^{5} b^{11} x^{16} + 3003 a^{4} b^{12} x^{18}} + \frac{5 a^{3} b^{\frac{31}{2}} x^{12} \sqrt{\frac{a}{b x^{2}} + 1}}{3003 a^{7} b^{9} x^{12} + 9009 a^{6} b^{10} x^{14} + 9009 a^{5} b^{11} x^{16} + 3003 a^{4} b^{12} x^{18}} + \frac{30 a^{2} b^{\frac{33}{2}} x^{14} \sqrt{\frac{a}{b x^{2}} + 1}}{3003 a^{7} b^{9} x^{12} + 9009 a^{6} b^{10} x^{14} + 9009 a^{5} b^{11} x^{16} + 3003 a^{4} b^{12} x^{18}} + \frac{40 a b^{\frac{35}{2}} x^{16} \sqrt{\frac{a}{b x^{2}} + 1}}{3003 a^{7} b^{9} x^{12} + 9009 a^{6} b^{10} x^{14} + 9009 a^{5} b^{11} x^{16} + 3003 a^{4} b^{12} x^{18}} + \frac{16 b^{\frac{37}{2}} x^{18} \sqrt{\frac{a}{b x^{2}} + 1}}{3003 a^{7} b^{9} x^{12} + 9009 a^{6} b^{10} x^{14} + 9009 a^{5} b^{11} x^{16} + 3003 a^{4} b^{12} x^{18}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**(5/2)/x**14,x)

[Out]

-231*a**9*b**(19/2)*sqrt(a/(b*x**2) + 1)/(3003*a**7*b**9*x**12 + 9009*a**6*b**10*x**14 + 9009*a**5*b**11*x**16
+ 3003*a**4*b**12*x**18) - 1260*a**8*b**(21/2)*x**2*sqrt(a/(b*x**2) + 1)/(3003*a**7*b**9*x**12 + 9009*a**6*b*
*10*x**14 + 9009*a**5*b**11*x**16 + 3003*a**4*b**12*x**18) - 2765*a**7*b**(23/2)*x**4*sqrt(a/(b*x**2) + 1)/(30
03*a**7*b**9*x**12 + 9009*a**6*b**10*x**14 + 9009*a**5*b**11*x**16 + 3003*a**4*b**12*x**18) - 3050*a**6*b**(25
/2)*x**6*sqrt(a/(b*x**2) + 1)/(3003*a**7*b**9*x**12 + 9009*a**6*b**10*x**14 + 9009*a**5*b**11*x**16 + 3003*a**
4*b**12*x**18) - 1689*a**5*b**(27/2)*x**8*sqrt(a/(b*x**2) + 1)/(3003*a**7*b**9*x**12 + 9009*a**6*b**10*x**14 +
9009*a**5*b**11*x**16 + 3003*a**4*b**12*x**18) - 376*a**4*b**(29/2)*x**10*sqrt(a/(b*x**2) + 1)/(3003*a**7*b**
9*x**12 + 9009*a**6*b**10*x**14 + 9009*a**5*b**11*x**16 + 3003*a**4*b**12*x**18) + 5*a**3*b**(31/2)*x**12*sqrt
(a/(b*x**2) + 1)/(3003*a**7*b**9*x**12 + 9009*a**6*b**10*x**14 + 9009*a**5*b**11*x**16 + 3003*a**4*b**12*x**18
) + 30*a**2*b**(33/2)*x**14*sqrt(a/(b*x**2) + 1)/(3003*a**7*b**9*x**12 + 9009*a**6*b**10*x**14 + 9009*a**5*b**
11*x**16 + 3003*a**4*b**12*x**18) + 40*a*b**(35/2)*x**16*sqrt(a/(b*x**2) + 1)/(3003*a**7*b**9*x**12 + 9009*a**
6*b**10*x**14 + 9009*a**5*b**11*x**16 + 3003*a**4*b**12*x**18) + 16*b**(37/2)*x**18*sqrt(a/(b*x**2) + 1)/(3003
*a**7*b**9*x**12 + 9009*a**6*b**10*x**14 + 9009*a**5*b**11*x**16 + 3003*a**4*b**12*x**18)

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Giac [B]  time = 1.88285, size = 370, normalized size = 4.02 \begin{align*} \frac{32 \,{\left (3003 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{18} b^{\frac{13}{2}} + 9009 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{16} a b^{\frac{13}{2}} + 18018 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{14} a^{2} b^{\frac{13}{2}} + 16302 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{12} a^{3} b^{\frac{13}{2}} + 10296 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{10} a^{4} b^{\frac{13}{2}} + 2288 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{8} a^{5} b^{\frac{13}{2}} + 286 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} a^{6} b^{\frac{13}{2}} - 78 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} a^{7} b^{\frac{13}{2}} + 13 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} a^{8} b^{\frac{13}{2}} - a^{9} b^{\frac{13}{2}}\right )}}{3003 \,{\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)^(5/2)/x^14,x, algorithm="giac")

[Out]

32/3003*(3003*(sqrt(b)*x - sqrt(b*x^2 + a))^18*b^(13/2) + 9009*(sqrt(b)*x - sqrt(b*x^2 + a))^16*a*b^(13/2) + 1
8018*(sqrt(b)*x - sqrt(b*x^2 + a))^14*a^2*b^(13/2) + 16302*(sqrt(b)*x - sqrt(b*x^2 + a))^12*a^3*b^(13/2) + 102
96*(sqrt(b)*x - sqrt(b*x^2 + a))^10*a^4*b^(13/2) + 2288*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a^5*b^(13/2) + 286*(sq
rt(b)*x - sqrt(b*x^2 + a))^6*a^6*b^(13/2) - 78*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^7*b^(13/2) + 13*(sqrt(b)*x -
sqrt(b*x^2 + a))^2*a^8*b^(13/2) - a^9*b^(13/2))/((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^13