3.408 \(\int \frac {(a+b x^2)^{5/2}}{x^{18}} \, dx\)

Optimal. Leaf size=140 \[ \frac {256 b^5 \left (a+b x^2\right )^{7/2}}{153153 a^6 x^7}-\frac {128 b^4 \left (a+b x^2\right )^{7/2}}{21879 a^5 x^9}+\frac {32 b^3 \left (a+b x^2\right )^{7/2}}{2431 a^4 x^{11}}-\frac {16 b^2 \left (a+b x^2\right )^{7/2}}{663 a^3 x^{13}}+\frac {2 b \left (a+b x^2\right )^{7/2}}{51 a^2 x^{15}}-\frac {\left (a+b x^2\right )^{7/2}}{17 a x^{17}} \]

[Out]

-1/17*(b*x^2+a)^(7/2)/a/x^17+2/51*b*(b*x^2+a)^(7/2)/a^2/x^15-16/663*b^2*(b*x^2+a)^(7/2)/a^3/x^13+32/2431*b^3*(
b*x^2+a)^(7/2)/a^4/x^11-128/21879*b^4*(b*x^2+a)^(7/2)/a^5/x^9+256/153153*b^5*(b*x^2+a)^(7/2)/a^6/x^7

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Rubi [A]  time = 0.06, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {271, 264} \[ \frac {256 b^5 \left (a+b x^2\right )^{7/2}}{153153 a^6 x^7}-\frac {128 b^4 \left (a+b x^2\right )^{7/2}}{21879 a^5 x^9}+\frac {32 b^3 \left (a+b x^2\right )^{7/2}}{2431 a^4 x^{11}}-\frac {16 b^2 \left (a+b x^2\right )^{7/2}}{663 a^3 x^{13}}+\frac {2 b \left (a+b x^2\right )^{7/2}}{51 a^2 x^{15}}-\frac {\left (a+b x^2\right )^{7/2}}{17 a x^{17}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^(5/2)/x^18,x]

[Out]

-(a + b*x^2)^(7/2)/(17*a*x^17) + (2*b*(a + b*x^2)^(7/2))/(51*a^2*x^15) - (16*b^2*(a + b*x^2)^(7/2))/(663*a^3*x
^13) + (32*b^3*(a + b*x^2)^(7/2))/(2431*a^4*x^11) - (128*b^4*(a + b*x^2)^(7/2))/(21879*a^5*x^9) + (256*b^5*(a
+ b*x^2)^(7/2))/(153153*a^6*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^{18}} \, dx &=-\frac {\left (a+b x^2\right )^{7/2}}{17 a x^{17}}-\frac {(10 b) \int \frac {\left (a+b x^2\right )^{5/2}}{x^{16}} \, dx}{17 a}\\ &=-\frac {\left (a+b x^2\right )^{7/2}}{17 a x^{17}}+\frac {2 b \left (a+b x^2\right )^{7/2}}{51 a^2 x^{15}}+\frac {\left (16 b^2\right ) \int \frac {\left (a+b x^2\right )^{5/2}}{x^{14}} \, dx}{51 a^2}\\ &=-\frac {\left (a+b x^2\right )^{7/2}}{17 a x^{17}}+\frac {2 b \left (a+b x^2\right )^{7/2}}{51 a^2 x^{15}}-\frac {16 b^2 \left (a+b x^2\right )^{7/2}}{663 a^3 x^{13}}-\frac {\left (32 b^3\right ) \int \frac {\left (a+b x^2\right )^{5/2}}{x^{12}} \, dx}{221 a^3}\\ &=-\frac {\left (a+b x^2\right )^{7/2}}{17 a x^{17}}+\frac {2 b \left (a+b x^2\right )^{7/2}}{51 a^2 x^{15}}-\frac {16 b^2 \left (a+b x^2\right )^{7/2}}{663 a^3 x^{13}}+\frac {32 b^3 \left (a+b x^2\right )^{7/2}}{2431 a^4 x^{11}}+\frac {\left (128 b^4\right ) \int \frac {\left (a+b x^2\right )^{5/2}}{x^{10}} \, dx}{2431 a^4}\\ &=-\frac {\left (a+b x^2\right )^{7/2}}{17 a x^{17}}+\frac {2 b \left (a+b x^2\right )^{7/2}}{51 a^2 x^{15}}-\frac {16 b^2 \left (a+b x^2\right )^{7/2}}{663 a^3 x^{13}}+\frac {32 b^3 \left (a+b x^2\right )^{7/2}}{2431 a^4 x^{11}}-\frac {128 b^4 \left (a+b x^2\right )^{7/2}}{21879 a^5 x^9}-\frac {\left (256 b^5\right ) \int \frac {\left (a+b x^2\right )^{5/2}}{x^8} \, dx}{21879 a^5}\\ &=-\frac {\left (a+b x^2\right )^{7/2}}{17 a x^{17}}+\frac {2 b \left (a+b x^2\right )^{7/2}}{51 a^2 x^{15}}-\frac {16 b^2 \left (a+b x^2\right )^{7/2}}{663 a^3 x^{13}}+\frac {32 b^3 \left (a+b x^2\right )^{7/2}}{2431 a^4 x^{11}}-\frac {128 b^4 \left (a+b x^2\right )^{7/2}}{21879 a^5 x^9}+\frac {256 b^5 \left (a+b x^2\right )^{7/2}}{153153 a^6 x^7}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 75, normalized size = 0.54 \[ \frac {\left (a+b x^2\right )^{7/2} \left (-9009 a^5+6006 a^4 b x^2-3696 a^3 b^2 x^4+2016 a^2 b^3 x^6-896 a b^4 x^8+256 b^5 x^{10}\right )}{153153 a^6 x^{17}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^(5/2)/x^18,x]

[Out]

((a + b*x^2)^(7/2)*(-9009*a^5 + 6006*a^4*b*x^2 - 3696*a^3*b^2*x^4 + 2016*a^2*b^3*x^6 - 896*a*b^4*x^8 + 256*b^5
*x^10))/(153153*a^6*x^17)

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fricas [A]  time = 1.41, size = 104, normalized size = 0.74 \[ \frac {{\left (256 \, b^{8} x^{16} - 128 \, a b^{7} x^{14} + 96 \, a^{2} b^{6} x^{12} - 80 \, a^{3} b^{5} x^{10} + 70 \, a^{4} b^{4} x^{8} - 63 \, a^{5} b^{3} x^{6} - 12705 \, a^{6} b^{2} x^{4} - 21021 \, a^{7} b x^{2} - 9009 \, a^{8}\right )} \sqrt {b x^{2} + a}}{153153 \, a^{6} x^{17}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(5/2)/x^18,x, algorithm="fricas")

[Out]

1/153153*(256*b^8*x^16 - 128*a*b^7*x^14 + 96*a^2*b^6*x^12 - 80*a^3*b^5*x^10 + 70*a^4*b^4*x^8 - 63*a^5*b^3*x^6
- 12705*a^6*b^2*x^4 - 21021*a^7*b*x^2 - 9009*a^8)*sqrt(b*x^2 + a)/(a^6*x^17)

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giac [B]  time = 1.10, size = 328, normalized size = 2.34 \[ \frac {512 \, {\left (102102 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{22} b^{\frac {17}{2}} + 364650 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{20} a b^{\frac {17}{2}} + 692835 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{18} a^{2} b^{\frac {17}{2}} + 668525 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{16} a^{3} b^{\frac {17}{2}} + 384098 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{14} a^{4} b^{\frac {17}{2}} + 89726 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{12} a^{5} b^{\frac {17}{2}} + 6188 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} a^{6} b^{\frac {17}{2}} - 2380 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} a^{7} b^{\frac {17}{2}} + 680 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{8} b^{\frac {17}{2}} - 136 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{9} b^{\frac {17}{2}} + 17 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{10} b^{\frac {17}{2}} - a^{11} b^{\frac {17}{2}}\right )}}{153153 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{17}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(5/2)/x^18,x, algorithm="giac")

[Out]

512/153153*(102102*(sqrt(b)*x - sqrt(b*x^2 + a))^22*b^(17/2) + 364650*(sqrt(b)*x - sqrt(b*x^2 + a))^20*a*b^(17
/2) + 692835*(sqrt(b)*x - sqrt(b*x^2 + a))^18*a^2*b^(17/2) + 668525*(sqrt(b)*x - sqrt(b*x^2 + a))^16*a^3*b^(17
/2) + 384098*(sqrt(b)*x - sqrt(b*x^2 + a))^14*a^4*b^(17/2) + 89726*(sqrt(b)*x - sqrt(b*x^2 + a))^12*a^5*b^(17/
2) + 6188*(sqrt(b)*x - sqrt(b*x^2 + a))^10*a^6*b^(17/2) - 2380*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a^7*b^(17/2) +
680*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^8*b^(17/2) - 136*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^9*b^(17/2) + 17*(sqrt
(b)*x - sqrt(b*x^2 + a))^2*a^10*b^(17/2) - a^11*b^(17/2))/((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^17

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maple [A]  time = 0.01, size = 72, normalized size = 0.51 \[ -\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}} \left (-256 b^{5} x^{10}+896 a \,b^{4} x^{8}-2016 a^{2} b^{3} x^{6}+3696 a^{3} b^{2} x^{4}-6006 a^{4} b \,x^{2}+9009 a^{5}\right )}{153153 a^{6} x^{17}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^(5/2)/x^18,x)

[Out]

-1/153153*(b*x^2+a)^(7/2)*(-256*b^5*x^10+896*a*b^4*x^8-2016*a^2*b^3*x^6+3696*a^3*b^2*x^4-6006*a^4*b*x^2+9009*a
^5)/x^17/a^6

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maxima [A]  time = 1.54, size = 116, normalized size = 0.83 \[ \frac {256 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{5}}{153153 \, a^{6} x^{7}} - \frac {128 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{4}}{21879 \, a^{5} x^{9}} + \frac {32 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{3}}{2431 \, a^{4} x^{11}} - \frac {16 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2}}{663 \, a^{3} x^{13}} + \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b}{51 \, a^{2} x^{15}} - \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}}}{17 \, a x^{17}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(5/2)/x^18,x, algorithm="maxima")

[Out]

256/153153*(b*x^2 + a)^(7/2)*b^5/(a^6*x^7) - 128/21879*(b*x^2 + a)^(7/2)*b^4/(a^5*x^9) + 32/2431*(b*x^2 + a)^(
7/2)*b^3/(a^4*x^11) - 16/663*(b*x^2 + a)^(7/2)*b^2/(a^3*x^13) + 2/51*(b*x^2 + a)^(7/2)*b/(a^2*x^15) - 1/17*(b*
x^2 + a)^(7/2)/(a*x^17)

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mupad [B]  time = 6.87, size = 171, normalized size = 1.22 \[ \frac {10\,b^4\,\sqrt {b\,x^2+a}}{21879\,a^2\,x^9}-\frac {55\,b^2\,\sqrt {b\,x^2+a}}{663\,x^{13}}-\frac {b^3\,\sqrt {b\,x^2+a}}{2431\,a\,x^{11}}-\frac {a^2\,\sqrt {b\,x^2+a}}{17\,x^{17}}-\frac {80\,b^5\,\sqrt {b\,x^2+a}}{153153\,a^3\,x^7}+\frac {32\,b^6\,\sqrt {b\,x^2+a}}{51051\,a^4\,x^5}-\frac {128\,b^7\,\sqrt {b\,x^2+a}}{153153\,a^5\,x^3}+\frac {256\,b^8\,\sqrt {b\,x^2+a}}{153153\,a^6\,x}-\frac {7\,a\,b\,\sqrt {b\,x^2+a}}{51\,x^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^(5/2)/x^18,x)

[Out]

(10*b^4*(a + b*x^2)^(1/2))/(21879*a^2*x^9) - (55*b^2*(a + b*x^2)^(1/2))/(663*x^13) - (b^3*(a + b*x^2)^(1/2))/(
2431*a*x^11) - (a^2*(a + b*x^2)^(1/2))/(17*x^17) - (80*b^5*(a + b*x^2)^(1/2))/(153153*a^3*x^7) + (32*b^6*(a +
b*x^2)^(1/2))/(51051*a^4*x^5) - (128*b^7*(a + b*x^2)^(1/2))/(153153*a^5*x^3) + (256*b^8*(a + b*x^2)^(1/2))/(15
3153*a^6*x) - (7*a*b*(a + b*x^2)^(1/2))/(51*x^15)

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sympy [B]  time = 4.42, size = 1346, normalized size = 9.61 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**(5/2)/x**18,x)

[Out]

-9009*a**13*b**(51/2)*sqrt(a/(b*x**2) + 1)/(153153*a**11*b**25*x**16 + 765765*a**10*b**26*x**18 + 1531530*a**9
*b**27*x**20 + 1531530*a**8*b**28*x**22 + 765765*a**7*b**29*x**24 + 153153*a**6*b**30*x**26) - 66066*a**12*b**
(53/2)*x**2*sqrt(a/(b*x**2) + 1)/(153153*a**11*b**25*x**16 + 765765*a**10*b**26*x**18 + 1531530*a**9*b**27*x**
20 + 1531530*a**8*b**28*x**22 + 765765*a**7*b**29*x**24 + 153153*a**6*b**30*x**26) - 207900*a**11*b**(55/2)*x*
*4*sqrt(a/(b*x**2) + 1)/(153153*a**11*b**25*x**16 + 765765*a**10*b**26*x**18 + 1531530*a**9*b**27*x**20 + 1531
530*a**8*b**28*x**22 + 765765*a**7*b**29*x**24 + 153153*a**6*b**30*x**26) - 363888*a**10*b**(57/2)*x**6*sqrt(a
/(b*x**2) + 1)/(153153*a**11*b**25*x**16 + 765765*a**10*b**26*x**18 + 1531530*a**9*b**27*x**20 + 1531530*a**8*
b**28*x**22 + 765765*a**7*b**29*x**24 + 153153*a**6*b**30*x**26) - 382550*a**9*b**(59/2)*x**8*sqrt(a/(b*x**2)
+ 1)/(153153*a**11*b**25*x**16 + 765765*a**10*b**26*x**18 + 1531530*a**9*b**27*x**20 + 1531530*a**8*b**28*x**2
2 + 765765*a**7*b**29*x**24 + 153153*a**6*b**30*x**26) - 241524*a**8*b**(61/2)*x**10*sqrt(a/(b*x**2) + 1)/(153
153*a**11*b**25*x**16 + 765765*a**10*b**26*x**18 + 1531530*a**9*b**27*x**20 + 1531530*a**8*b**28*x**22 + 76576
5*a**7*b**29*x**24 + 153153*a**6*b**30*x**26) - 84780*a**7*b**(63/2)*x**12*sqrt(a/(b*x**2) + 1)/(153153*a**11*
b**25*x**16 + 765765*a**10*b**26*x**18 + 1531530*a**9*b**27*x**20 + 1531530*a**8*b**28*x**22 + 765765*a**7*b**
29*x**24 + 153153*a**6*b**30*x**26) - 12768*a**6*b**(65/2)*x**14*sqrt(a/(b*x**2) + 1)/(153153*a**11*b**25*x**1
6 + 765765*a**10*b**26*x**18 + 1531530*a**9*b**27*x**20 + 1531530*a**8*b**28*x**22 + 765765*a**7*b**29*x**24 +
153153*a**6*b**30*x**26) + 63*a**5*b**(67/2)*x**16*sqrt(a/(b*x**2) + 1)/(153153*a**11*b**25*x**16 + 765765*a*
*10*b**26*x**18 + 1531530*a**9*b**27*x**20 + 1531530*a**8*b**28*x**22 + 765765*a**7*b**29*x**24 + 153153*a**6*
b**30*x**26) + 630*a**4*b**(69/2)*x**18*sqrt(a/(b*x**2) + 1)/(153153*a**11*b**25*x**16 + 765765*a**10*b**26*x*
*18 + 1531530*a**9*b**27*x**20 + 1531530*a**8*b**28*x**22 + 765765*a**7*b**29*x**24 + 153153*a**6*b**30*x**26)
+ 1680*a**3*b**(71/2)*x**20*sqrt(a/(b*x**2) + 1)/(153153*a**11*b**25*x**16 + 765765*a**10*b**26*x**18 + 15315
30*a**9*b**27*x**20 + 1531530*a**8*b**28*x**22 + 765765*a**7*b**29*x**24 + 153153*a**6*b**30*x**26) + 2016*a**
2*b**(73/2)*x**22*sqrt(a/(b*x**2) + 1)/(153153*a**11*b**25*x**16 + 765765*a**10*b**26*x**18 + 1531530*a**9*b**
27*x**20 + 1531530*a**8*b**28*x**22 + 765765*a**7*b**29*x**24 + 153153*a**6*b**30*x**26) + 1152*a*b**(75/2)*x*
*24*sqrt(a/(b*x**2) + 1)/(153153*a**11*b**25*x**16 + 765765*a**10*b**26*x**18 + 1531530*a**9*b**27*x**20 + 153
1530*a**8*b**28*x**22 + 765765*a**7*b**29*x**24 + 153153*a**6*b**30*x**26) + 256*b**(77/2)*x**26*sqrt(a/(b*x**
2) + 1)/(153153*a**11*b**25*x**16 + 765765*a**10*b**26*x**18 + 1531530*a**9*b**27*x**20 + 1531530*a**8*b**28*x
**22 + 765765*a**7*b**29*x**24 + 153153*a**6*b**30*x**26)

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