3.407 \(\int \frac {(a+b x^2)^{5/2}}{x^{16}} \, dx\)

Optimal. Leaf size=116 \[ -\frac {128 b^4 \left (a+b x^2\right )^{7/2}}{45045 a^5 x^7}+\frac {64 b^3 \left (a+b x^2\right )^{7/2}}{6435 a^4 x^9}-\frac {16 b^2 \left (a+b x^2\right )^{7/2}}{715 a^3 x^{11}}+\frac {8 b \left (a+b x^2\right )^{7/2}}{195 a^2 x^{13}}-\frac {\left (a+b x^2\right )^{7/2}}{15 a x^{15}} \]

[Out]

-1/15*(b*x^2+a)^(7/2)/a/x^15+8/195*b*(b*x^2+a)^(7/2)/a^2/x^13-16/715*b^2*(b*x^2+a)^(7/2)/a^3/x^11+64/6435*b^3*
(b*x^2+a)^(7/2)/a^4/x^9-128/45045*b^4*(b*x^2+a)^(7/2)/a^5/x^7

________________________________________________________________________________________

Rubi [A]  time = 0.04, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {271, 264} \[ -\frac {128 b^4 \left (a+b x^2\right )^{7/2}}{45045 a^5 x^7}+\frac {64 b^3 \left (a+b x^2\right )^{7/2}}{6435 a^4 x^9}-\frac {16 b^2 \left (a+b x^2\right )^{7/2}}{715 a^3 x^{11}}+\frac {8 b \left (a+b x^2\right )^{7/2}}{195 a^2 x^{13}}-\frac {\left (a+b x^2\right )^{7/2}}{15 a x^{15}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^(5/2)/x^16,x]

[Out]

-(a + b*x^2)^(7/2)/(15*a*x^15) + (8*b*(a + b*x^2)^(7/2))/(195*a^2*x^13) - (16*b^2*(a + b*x^2)^(7/2))/(715*a^3*
x^11) + (64*b^3*(a + b*x^2)^(7/2))/(6435*a^4*x^9) - (128*b^4*(a + b*x^2)^(7/2))/(45045*a^5*x^7)

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 )^{5/2}}{x^{16}} \, dx &=-\frac {\left (a+b x^2\right )^{7/2}}{15 a x^{15}}-\frac {(8 b) \int \frac {\left (a+b x^2\right )^{5/2}}{x^{14}} \, dx}{15 a}\\ &=-\frac {\left (a+b x^2\right )^{7/2}}{15 a x^{15}}+\frac {8 b \left (a+b x^2\right )^{7/2}}{195 a^2 x^{13}}+\frac {\left (16 b^2\right ) \int \frac {\left (a+b x^2\right )^{5/2}}{x^{12}} \, dx}{65 a^2}\\ &=-\frac {\left (a+b x^2\right )^{7/2}}{15 a x^{15}}+\frac {8 b \left (a+b x^2\right )^{7/2}}{195 a^2 x^{13}}-\frac {16 b^2 \left (a+b x^2\right )^{7/2}}{715 a^3 x^{11}}-\frac {\left (64 b^3\right ) \int \frac {\left (a+b x^2\right )^{5/2}}{x^{10}} \, dx}{715 a^3}\\ &=-\frac {\left (a+b x^2\right )^{7/2}}{15 a x^{15}}+\frac {8 b \left (a+b x^2\right )^{7/2}}{195 a^2 x^{13}}-\frac {16 b^2 \left (a+b x^2\right )^{7/2}}{715 a^3 x^{11}}+\frac {64 b^3 \left (a+b x^2\right )^{7/2}}{6435 a^4 x^9}+\frac {\left (128 b^4\right ) \int \frac {\left (a+b x^2\right )^{5/2}}{x^8} \, dx}{6435 a^4}\\ &=-\frac {\left (a+b x^2\right )^{7/2}}{15 a x^{15}}+\frac {8 b \left (a+b x^2\right )^{7/2}}{195 a^2 x^{13}}-\frac {16 b^2 \left (a+b x^2\right )^{7/2}}{715 a^3 x^{11}}+\frac {64 b^3 \left (a+b x^2\right )^{7/2}}{6435 a^4 x^9}-\frac {128 b^4 \left (a+b x^2\right )^{7/2}}{45045 a^5 x^7}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 64, normalized size = 0.55 \[ -\frac {\left (a+b x^2\right )^{7/2} \left (3003 a^4-1848 a^3 b x^2+1008 a^2 b^2 x^4-448 a b^3 x^6+128 b^4 x^8\right )}{45045 a^5 x^{15}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^(5/2)/x^16,x]

[Out]

-1/45045*((a + b*x^2)^(7/2)*(3003*a^4 - 1848*a^3*b*x^2 + 1008*a^2*b^2*x^4 - 448*a*b^3*x^6 + 128*b^4*x^8))/(a^5
*x^15)

________________________________________________________________________________________

fricas [A]  time = 1.11, size = 93, normalized size = 0.80 \[ -\frac {{\left (128 \, b^{7} x^{14} - 64 \, a b^{6} x^{12} + 48 \, a^{2} b^{5} x^{10} - 40 \, a^{3} b^{4} x^{8} + 35 \, a^{4} b^{3} x^{6} + 4473 \, a^{5} b^{2} x^{4} + 7161 \, a^{6} b x^{2} + 3003 \, a^{7}\right )} \sqrt {b x^{2} + a}}{45045 \, a^{5} x^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(5/2)/x^16,x, algorithm="fricas")

[Out]

-1/45045*(128*b^7*x^14 - 64*a*b^6*x^12 + 48*a^2*b^5*x^10 - 40*a^3*b^4*x^8 + 35*a^4*b^3*x^6 + 4473*a^5*b^2*x^4
+ 7161*a^6*b*x^2 + 3003*a^7)*sqrt(b*x^2 + a)/(a^5*x^15)

________________________________________________________________________________________

giac [B]  time = 1.23, size = 300, normalized size = 2.59 \[ \frac {256 \, {\left (18018 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{20} b^{\frac {15}{2}} + 60060 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{18} a b^{\frac {15}{2}} + 115830 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{16} a^{2} b^{\frac {15}{2}} + 109395 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{14} a^{3} b^{\frac {15}{2}} + 65065 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{12} a^{4} b^{\frac {15}{2}} + 15015 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} a^{5} b^{\frac {15}{2}} + 1365 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} a^{6} b^{\frac {15}{2}} - 455 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{7} b^{\frac {15}{2}} + 105 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{8} b^{\frac {15}{2}} - 15 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{9} b^{\frac {15}{2}} + a^{10} b^{\frac {15}{2}}\right )}}{45045 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(5/2)/x^16,x, algorithm="giac")

[Out]

256/45045*(18018*(sqrt(b)*x - sqrt(b*x^2 + a))^20*b^(15/2) + 60060*(sqrt(b)*x - sqrt(b*x^2 + a))^18*a*b^(15/2)
+ 115830*(sqrt(b)*x - sqrt(b*x^2 + a))^16*a^2*b^(15/2) + 109395*(sqrt(b)*x - sqrt(b*x^2 + a))^14*a^3*b^(15/2)
+ 65065*(sqrt(b)*x - sqrt(b*x^2 + a))^12*a^4*b^(15/2) + 15015*(sqrt(b)*x - sqrt(b*x^2 + a))^10*a^5*b^(15/2) +
1365*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a^6*b^(15/2) - 455*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^7*b^(15/2) + 105*(s
qrt(b)*x - sqrt(b*x^2 + a))^4*a^8*b^(15/2) - 15*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^9*b^(15/2) + a^10*b^(15/2))/
((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^15

________________________________________________________________________________________

maple [A]  time = 0.01, size = 61, normalized size = 0.53 \[ -\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}} \left (128 b^{4} x^{8}-448 a \,b^{3} x^{6}+1008 a^{2} b^{2} x^{4}-1848 a^{3} b \,x^{2}+3003 a^{4}\right )}{45045 a^{5} x^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^(5/2)/x^16,x)

[Out]

-1/45045*(b*x^2+a)^(7/2)*(128*b^4*x^8-448*a*b^3*x^6+1008*a^2*b^2*x^4-1848*a^3*b*x^2+3003*a^4)/x^15/a^5

________________________________________________________________________________________

maxima [A]  time = 1.45, size = 96, normalized size = 0.83 \[ -\frac {128 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{4}}{45045 \, a^{5} x^{7}} + \frac {64 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{3}}{6435 \, a^{4} x^{9}} - \frac {16 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2}}{715 \, a^{3} x^{11}} + \frac {8 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b}{195 \, a^{2} x^{13}} - \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}}}{15 \, a x^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(5/2)/x^16,x, algorithm="maxima")

[Out]

-128/45045*(b*x^2 + a)^(7/2)*b^4/(a^5*x^7) + 64/6435*(b*x^2 + a)^(7/2)*b^3/(a^4*x^9) - 16/715*(b*x^2 + a)^(7/2
)*b^2/(a^3*x^11) + 8/195*(b*x^2 + a)^(7/2)*b/(a^2*x^13) - 1/15*(b*x^2 + a)^(7/2)/(a*x^15)

________________________________________________________________________________________

mupad [B]  time = 6.37, size = 151, normalized size = 1.30 \[ \frac {8\,b^4\,\sqrt {b\,x^2+a}}{9009\,a^2\,x^7}-\frac {71\,b^2\,\sqrt {b\,x^2+a}}{715\,x^{11}}-\frac {b^3\,\sqrt {b\,x^2+a}}{1287\,a\,x^9}-\frac {a^2\,\sqrt {b\,x^2+a}}{15\,x^{15}}-\frac {16\,b^5\,\sqrt {b\,x^2+a}}{15015\,a^3\,x^5}+\frac {64\,b^6\,\sqrt {b\,x^2+a}}{45045\,a^4\,x^3}-\frac {128\,b^7\,\sqrt {b\,x^2+a}}{45045\,a^5\,x}-\frac {31\,a\,b\,\sqrt {b\,x^2+a}}{195\,x^{13}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^(5/2)/x^16,x)

[Out]

(8*b^4*(a + b*x^2)^(1/2))/(9009*a^2*x^7) - (71*b^2*(a + b*x^2)^(1/2))/(715*x^11) - (b^3*(a + b*x^2)^(1/2))/(12
87*a*x^9) - (a^2*(a + b*x^2)^(1/2))/(15*x^15) - (16*b^5*(a + b*x^2)^(1/2))/(15015*a^3*x^5) + (64*b^6*(a + b*x^
2)^(1/2))/(45045*a^4*x^3) - (128*b^7*(a + b*x^2)^(1/2))/(45045*a^5*x) - (31*a*b*(a + b*x^2)^(1/2))/(195*x^13)

________________________________________________________________________________________

sympy [B]  time = 3.49, size = 1012, normalized size = 8.72 \[ - \frac {3003 a^{11} b^{\frac {33}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{45045 a^{9} b^{16} x^{14} + 180180 a^{8} b^{17} x^{16} + 270270 a^{7} b^{18} x^{18} + 180180 a^{6} b^{19} x^{20} + 45045 a^{5} b^{20} x^{22}} - \frac {19173 a^{10} b^{\frac {35}{2}} x^{2} \sqrt {\frac {a}{b x^{2}} + 1}}{45045 a^{9} b^{16} x^{14} + 180180 a^{8} b^{17} x^{16} + 270270 a^{7} b^{18} x^{18} + 180180 a^{6} b^{19} x^{20} + 45045 a^{5} b^{20} x^{22}} - \frac {51135 a^{9} b^{\frac {37}{2}} x^{4} \sqrt {\frac {a}{b x^{2}} + 1}}{45045 a^{9} b^{16} x^{14} + 180180 a^{8} b^{17} x^{16} + 270270 a^{7} b^{18} x^{18} + 180180 a^{6} b^{19} x^{20} + 45045 a^{5} b^{20} x^{22}} - \frac {72905 a^{8} b^{\frac {39}{2}} x^{6} \sqrt {\frac {a}{b x^{2}} + 1}}{45045 a^{9} b^{16} x^{14} + 180180 a^{8} b^{17} x^{16} + 270270 a^{7} b^{18} x^{18} + 180180 a^{6} b^{19} x^{20} + 45045 a^{5} b^{20} x^{22}} - \frac {58585 a^{7} b^{\frac {41}{2}} x^{8} \sqrt {\frac {a}{b x^{2}} + 1}}{45045 a^{9} b^{16} x^{14} + 180180 a^{8} b^{17} x^{16} + 270270 a^{7} b^{18} x^{18} + 180180 a^{6} b^{19} x^{20} + 45045 a^{5} b^{20} x^{22}} - \frac {25151 a^{6} b^{\frac {43}{2}} x^{10} \sqrt {\frac {a}{b x^{2}} + 1}}{45045 a^{9} b^{16} x^{14} + 180180 a^{8} b^{17} x^{16} + 270270 a^{7} b^{18} x^{18} + 180180 a^{6} b^{19} x^{20} + 45045 a^{5} b^{20} x^{22}} - \frac {4501 a^{5} b^{\frac {45}{2}} x^{12} \sqrt {\frac {a}{b x^{2}} + 1}}{45045 a^{9} b^{16} x^{14} + 180180 a^{8} b^{17} x^{16} + 270270 a^{7} b^{18} x^{18} + 180180 a^{6} b^{19} x^{20} + 45045 a^{5} b^{20} x^{22}} - \frac {35 a^{4} b^{\frac {47}{2}} x^{14} \sqrt {\frac {a}{b x^{2}} + 1}}{45045 a^{9} b^{16} x^{14} + 180180 a^{8} b^{17} x^{16} + 270270 a^{7} b^{18} x^{18} + 180180 a^{6} b^{19} x^{20} + 45045 a^{5} b^{20} x^{22}} - \frac {280 a^{3} b^{\frac {49}{2}} x^{16} \sqrt {\frac {a}{b x^{2}} + 1}}{45045 a^{9} b^{16} x^{14} + 180180 a^{8} b^{17} x^{16} + 270270 a^{7} b^{18} x^{18} + 180180 a^{6} b^{19} x^{20} + 45045 a^{5} b^{20} x^{22}} - \frac {560 a^{2} b^{\frac {51}{2}} x^{18} \sqrt {\frac {a}{b x^{2}} + 1}}{45045 a^{9} b^{16} x^{14} + 180180 a^{8} b^{17} x^{16} + 270270 a^{7} b^{18} x^{18} + 180180 a^{6} b^{19} x^{20} + 45045 a^{5} b^{20} x^{22}} - \frac {448 a b^{\frac {53}{2}} x^{20} \sqrt {\frac {a}{b x^{2}} + 1}}{45045 a^{9} b^{16} x^{14} + 180180 a^{8} b^{17} x^{16} + 270270 a^{7} b^{18} x^{18} + 180180 a^{6} b^{19} x^{20} + 45045 a^{5} b^{20} x^{22}} - \frac {128 b^{\frac {55}{2}} x^{22} \sqrt {\frac {a}{b x^{2}} + 1}}{45045 a^{9} b^{16} x^{14} + 180180 a^{8} b^{17} x^{16} + 270270 a^{7} b^{18} x^{18} + 180180 a^{6} b^{19} x^{20} + 45045 a^{5} b^{20} x^{22}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**(5/2)/x**16,x)

[Out]

-3003*a**11*b**(33/2)*sqrt(a/(b*x**2) + 1)/(45045*a**9*b**16*x**14 + 180180*a**8*b**17*x**16 + 270270*a**7*b**
18*x**18 + 180180*a**6*b**19*x**20 + 45045*a**5*b**20*x**22) - 19173*a**10*b**(35/2)*x**2*sqrt(a/(b*x**2) + 1)
/(45045*a**9*b**16*x**14 + 180180*a**8*b**17*x**16 + 270270*a**7*b**18*x**18 + 180180*a**6*b**19*x**20 + 45045
*a**5*b**20*x**22) - 51135*a**9*b**(37/2)*x**4*sqrt(a/(b*x**2) + 1)/(45045*a**9*b**16*x**14 + 180180*a**8*b**1
7*x**16 + 270270*a**7*b**18*x**18 + 180180*a**6*b**19*x**20 + 45045*a**5*b**20*x**22) - 72905*a**8*b**(39/2)*x
**6*sqrt(a/(b*x**2) + 1)/(45045*a**9*b**16*x**14 + 180180*a**8*b**17*x**16 + 270270*a**7*b**18*x**18 + 180180*
a**6*b**19*x**20 + 45045*a**5*b**20*x**22) - 58585*a**7*b**(41/2)*x**8*sqrt(a/(b*x**2) + 1)/(45045*a**9*b**16*
x**14 + 180180*a**8*b**17*x**16 + 270270*a**7*b**18*x**18 + 180180*a**6*b**19*x**20 + 45045*a**5*b**20*x**22)
- 25151*a**6*b**(43/2)*x**10*sqrt(a/(b*x**2) + 1)/(45045*a**9*b**16*x**14 + 180180*a**8*b**17*x**16 + 270270*a
**7*b**18*x**18 + 180180*a**6*b**19*x**20 + 45045*a**5*b**20*x**22) - 4501*a**5*b**(45/2)*x**12*sqrt(a/(b*x**2
) + 1)/(45045*a**9*b**16*x**14 + 180180*a**8*b**17*x**16 + 270270*a**7*b**18*x**18 + 180180*a**6*b**19*x**20 +
45045*a**5*b**20*x**22) - 35*a**4*b**(47/2)*x**14*sqrt(a/(b*x**2) + 1)/(45045*a**9*b**16*x**14 + 180180*a**8*
b**17*x**16 + 270270*a**7*b**18*x**18 + 180180*a**6*b**19*x**20 + 45045*a**5*b**20*x**22) - 280*a**3*b**(49/2)
*x**16*sqrt(a/(b*x**2) + 1)/(45045*a**9*b**16*x**14 + 180180*a**8*b**17*x**16 + 270270*a**7*b**18*x**18 + 1801
80*a**6*b**19*x**20 + 45045*a**5*b**20*x**22) - 560*a**2*b**(51/2)*x**18*sqrt(a/(b*x**2) + 1)/(45045*a**9*b**1
6*x**14 + 180180*a**8*b**17*x**16 + 270270*a**7*b**18*x**18 + 180180*a**6*b**19*x**20 + 45045*a**5*b**20*x**22
) - 448*a*b**(53/2)*x**20*sqrt(a/(b*x**2) + 1)/(45045*a**9*b**16*x**14 + 180180*a**8*b**17*x**16 + 270270*a**7
*b**18*x**18 + 180180*a**6*b**19*x**20 + 45045*a**5*b**20*x**22) - 128*b**(55/2)*x**22*sqrt(a/(b*x**2) + 1)/(4
5045*a**9*b**16*x**14 + 180180*a**8*b**17*x**16 + 270270*a**7*b**18*x**18 + 180180*a**6*b**19*x**20 + 45045*a*
*5*b**20*x**22)

________________________________________________________________________________________