3.5.37 \(\int \frac {(a+b x^2)^{9/2}}{x^{18}} \, dx\) [437]

Optimal. Leaf size=92 \[ -\frac {\left (a+b x^2\right )^{11/2}}{17 a x^{17}}+\frac {2 b \left (a+b x^2\right )^{11/2}}{85 a^2 x^{15}}-\frac {8 b^2 \left (a+b x^2\right )^{11/2}}{1105 a^3 x^{13}}+\frac {16 b^3 \left (a+b x^2\right )^{11/2}}{12155 a^4 x^{11}} \]

[Out]

-1/17*(b*x^2+a)^(11/2)/a/x^17+2/85*b*(b*x^2+a)^(11/2)/a^2/x^15-8/1105*b^2*(b*x^2+a)^(11/2)/a^3/x^13+16/12155*b
^3*(b*x^2+a)^(11/2)/a^4/x^11

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Rubi [A]
time = 0.02, antiderivative size = 92, normalized size of antiderivative = 1.00, 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 = {277, 270} \begin {gather*} \frac {16 b^3 \left (a+b x^2\right )^{11/2}}{12155 a^4 x^{11}}-\frac {8 b^2 \left (a+b x^2\right )^{11/2}}{1105 a^3 x^{13}}+\frac {2 b \left (a+b x^2\right )^{11/2}}{85 a^2 x^{15}}-\frac {\left (a+b x^2\right )^{11/2}}{17 a x^{17}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^(9/2)/x^18,x]

[Out]

-1/17*(a + b*x^2)^(11/2)/(a*x^17) + (2*b*(a + b*x^2)^(11/2))/(85*a^2*x^15) - (8*b^2*(a + b*x^2)^(11/2))/(1105*
a^3*x^13) + (16*b^3*(a + b*x^2)^(11/2))/(12155*a^4*x^11)

Rule 270

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 277

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^{18}} \, dx &=-\frac {\left (a+b x^2\right )^{11/2}}{17 a x^{17}}-\frac {(6 b) \int \frac {\left (a+b x^2\right )^{9/2}}{x^{16}} \, dx}{17 a}\\ &=-\frac {\left (a+b x^2\right )^{11/2}}{17 a x^{17}}+\frac {2 b \left (a+b x^2\right )^{11/2}}{85 a^2 x^{15}}+\frac {\left (8 b^2\right ) \int \frac {\left (a+b x^2\right )^{9/2}}{x^{14}} \, dx}{85 a^2}\\ &=-\frac {\left (a+b x^2\right )^{11/2}}{17 a x^{17}}+\frac {2 b \left (a+b x^2\right )^{11/2}}{85 a^2 x^{15}}-\frac {8 b^2 \left (a+b x^2\right )^{11/2}}{1105 a^3 x^{13}}-\frac {\left (16 b^3\right ) \int \frac {\left (a+b x^2\right )^{9/2}}{x^{12}} \, dx}{1105 a^3}\\ &=-\frac {\left (a+b x^2\right )^{11/2}}{17 a x^{17}}+\frac {2 b \left (a+b x^2\right )^{11/2}}{85 a^2 x^{15}}-\frac {8 b^2 \left (a+b x^2\right )^{11/2}}{1105 a^3 x^{13}}+\frac {16 b^3 \left (a+b x^2\right )^{11/2}}{12155 a^4 x^{11}}\\ \end {align*}

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Mathematica [A]
time = 0.15, size = 53, normalized size = 0.58 \begin {gather*} \frac {\left (a+b x^2\right )^{11/2} \left (-715 a^3+286 a^2 b x^2-88 a b^2 x^4+16 b^3 x^6\right )}{12155 a^4 x^{17}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^(9/2)/x^18,x]

[Out]

((a + b*x^2)^(11/2)*(-715*a^3 + 286*a^2*b*x^2 - 88*a*b^2*x^4 + 16*b^3*x^6))/(12155*a^4*x^17)

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Maple [A]
time = 1.54, size = 85, normalized size = 0.92

method	result	size
gosper	\(-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}} \left (-16 b^{3} x^{6}+88 a \,b^{2} x^{4}-286 a^{2} b \,x^{2}+715 a^{3}\right )}{12155 x^{17} a^{4}}\)	\(50\)
default	\(-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}}}{17 a \,x^{17}}-\frac {6 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}}}{15 a \,x^{15}}-\frac {4 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}}}{13 a \,x^{13}}+\frac {2 b \left (b \,x^{2}+a \right )^{\frac {11}{2}}}{143 a^{2} x^{11}}\right )}{15 a}\right )}{17 a}\)	\(85\)
trager	\(-\frac {\left (-16 b^{8} x^{16}+8 a \,b^{7} x^{14}-6 a^{2} b^{6} x^{12}+5 a^{3} b^{5} x^{10}+1515 a^{4} b^{4} x^{8}+4714 a^{5} b^{3} x^{6}+5808 a^{6} b^{2} x^{4}+3289 a^{7} b \,x^{2}+715 a^{8}\right ) \sqrt {b \,x^{2}+a}}{12155 a^{4} x^{17}}\)	\(105\)
risch	\(-\frac {\left (-16 b^{8} x^{16}+8 a \,b^{7} x^{14}-6 a^{2} b^{6} x^{12}+5 a^{3} b^{5} x^{10}+1515 a^{4} b^{4} x^{8}+4714 a^{5} b^{3} x^{6}+5808 a^{6} b^{2} x^{4}+3289 a^{7} b \,x^{2}+715 a^{8}\right ) \sqrt {b \,x^{2}+a}}{12155 a^{4} x^{17}}\)	\(105\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^(9/2)/x^18,x,method=_RETURNVERBOSE)

[Out]

-1/17*(b*x^2+a)^(11/2)/a/x^17-6/17*b/a*(-1/15*(b*x^2+a)^(11/2)/a/x^15-4/15*b/a*(-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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Maxima [A]
time = 0.30, size = 76, normalized size = 0.83 \begin {gather*} \frac {16 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} b^{3}}{12155 \, a^{4} x^{11}} - \frac {8 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} b^{2}}{1105 \, a^{3} x^{13}} + \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} b}{85 \, a^{2} x^{15}} - \frac {{\left (b x^{2} + a\right )}^{\frac {11}{2}}}{17 \, a x^{17}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(9/2)/x^18,x, algorithm="maxima")

[Out]

16/12155*(b*x^2 + a)^(11/2)*b^3/(a^4*x^11) - 8/1105*(b*x^2 + a)^(11/2)*b^2/(a^3*x^13) + 2/85*(b*x^2 + a)^(11/2
)*b/(a^2*x^15) - 1/17*(b*x^2 + a)^(11/2)/(a*x^17)

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Fricas [A]
time = 1.45, size = 104, normalized size = 1.13 \begin {gather*} \frac {{\left (16 \, b^{8} x^{16} - 8 \, a b^{7} x^{14} + 6 \, a^{2} b^{6} x^{12} - 5 \, a^{3} b^{5} x^{10} - 1515 \, a^{4} b^{4} x^{8} - 4714 \, a^{5} b^{3} x^{6} - 5808 \, a^{6} b^{2} x^{4} - 3289 \, a^{7} b x^{2} - 715 \, a^{8}\right )} \sqrt {b x^{2} + a}}{12155 \, a^{4} x^{17}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(9/2)/x^18,x, algorithm="fricas")

[Out]

1/12155*(16*b^8*x^16 - 8*a*b^7*x^14 + 6*a^2*b^6*x^12 - 5*a^3*b^5*x^10 - 1515*a^4*b^4*x^8 - 4714*a^5*b^3*x^6 -
5808*a^6*b^2*x^4 - 3289*a^7*b*x^2 - 715*a^8)*sqrt(b*x^2 + a)/(a^4*x^17)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 867 vs. \(2 (85) = 170\).
time = 2.61, size = 867, normalized size = 9.42 \begin {gather*} - \frac {715 a^{11} b^{\frac {19}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{12155 a^{7} b^{9} x^{16} + 36465 a^{6} b^{10} x^{18} + 36465 a^{5} b^{11} x^{20} + 12155 a^{4} b^{12} x^{22}} - \frac {5434 a^{10} b^{\frac {21}{2}} x^{2} \sqrt {\frac {a}{b x^{2}} + 1}}{12155 a^{7} b^{9} x^{16} + 36465 a^{6} b^{10} x^{18} + 36465 a^{5} b^{11} x^{20} + 12155 a^{4} b^{12} x^{22}} - \frac {17820 a^{9} b^{\frac {23}{2}} x^{4} \sqrt {\frac {a}{b x^{2}} + 1}}{12155 a^{7} b^{9} x^{16} + 36465 a^{6} b^{10} x^{18} + 36465 a^{5} b^{11} x^{20} + 12155 a^{4} b^{12} x^{22}} - \frac {32720 a^{8} b^{\frac {25}{2}} x^{6} \sqrt {\frac {a}{b x^{2}} + 1}}{12155 a^{7} b^{9} x^{16} + 36465 a^{6} b^{10} x^{18} + 36465 a^{5} b^{11} x^{20} + 12155 a^{4} b^{12} x^{22}} - \frac {36370 a^{7} b^{\frac {27}{2}} x^{8} \sqrt {\frac {a}{b x^{2}} + 1}}{12155 a^{7} b^{9} x^{16} + 36465 a^{6} b^{10} x^{18} + 36465 a^{5} b^{11} x^{20} + 12155 a^{4} b^{12} x^{22}} - \frac {24500 a^{6} b^{\frac {29}{2}} x^{10} \sqrt {\frac {a}{b x^{2}} + 1}}{12155 a^{7} b^{9} x^{16} + 36465 a^{6} b^{10} x^{18} + 36465 a^{5} b^{11} x^{20} + 12155 a^{4} b^{12} x^{22}} - \frac {9268 a^{5} b^{\frac {31}{2}} x^{12} \sqrt {\frac {a}{b x^{2}} + 1}}{12155 a^{7} b^{9} x^{16} + 36465 a^{6} b^{10} x^{18} + 36465 a^{5} b^{11} x^{20} + 12155 a^{4} b^{12} x^{22}} - \frac {1520 a^{4} b^{\frac {33}{2}} x^{14} \sqrt {\frac {a}{b x^{2}} + 1}}{12155 a^{7} b^{9} x^{16} + 36465 a^{6} b^{10} x^{18} + 36465 a^{5} b^{11} x^{20} + 12155 a^{4} b^{12} x^{22}} + \frac {5 a^{3} b^{\frac {35}{2}} x^{16} \sqrt {\frac {a}{b x^{2}} + 1}}{12155 a^{7} b^{9} x^{16} + 36465 a^{6} b^{10} x^{18} + 36465 a^{5} b^{11} x^{20} + 12155 a^{4} b^{12} x^{22}} + \frac {30 a^{2} b^{\frac {37}{2}} x^{18} \sqrt {\frac {a}{b x^{2}} + 1}}{12155 a^{7} b^{9} x^{16} + 36465 a^{6} b^{10} x^{18} + 36465 a^{5} b^{11} x^{20} + 12155 a^{4} b^{12} x^{22}} + \frac {40 a b^{\frac {39}{2}} x^{20} \sqrt {\frac {a}{b x^{2}} + 1}}{12155 a^{7} b^{9} x^{16} + 36465 a^{6} b^{10} x^{18} + 36465 a^{5} b^{11} x^{20} + 12155 a^{4} b^{12} x^{22}} + \frac {16 b^{\frac {41}{2}} x^{22} \sqrt {\frac {a}{b x^{2}} + 1}}{12155 a^{7} b^{9} x^{16} + 36465 a^{6} b^{10} x^{18} + 36465 a^{5} b^{11} x^{20} + 12155 a^{4} b^{12} x^{22}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**(9/2)/x**18,x)

[Out]

-715*a**11*b**(19/2)*sqrt(a/(b*x**2) + 1)/(12155*a**7*b**9*x**16 + 36465*a**6*b**10*x**18 + 36465*a**5*b**11*x
**20 + 12155*a**4*b**12*x**22) - 5434*a**10*b**(21/2)*x**2*sqrt(a/(b*x**2) + 1)/(12155*a**7*b**9*x**16 + 36465
*a**6*b**10*x**18 + 36465*a**5*b**11*x**20 + 12155*a**4*b**12*x**22) - 17820*a**9*b**(23/2)*x**4*sqrt(a/(b*x**
2) + 1)/(12155*a**7*b**9*x**16 + 36465*a**6*b**10*x**18 + 36465*a**5*b**11*x**20 + 12155*a**4*b**12*x**22) - 3
2720*a**8*b**(25/2)*x**6*sqrt(a/(b*x**2) + 1)/(12155*a**7*b**9*x**16 + 36465*a**6*b**10*x**18 + 36465*a**5*b**
11*x**20 + 12155*a**4*b**12*x**22) - 36370*a**7*b**(27/2)*x**8*sqrt(a/(b*x**2) + 1)/(12155*a**7*b**9*x**16 + 3
6465*a**6*b**10*x**18 + 36465*a**5*b**11*x**20 + 12155*a**4*b**12*x**22) - 24500*a**6*b**(29/2)*x**10*sqrt(a/(
b*x**2) + 1)/(12155*a**7*b**9*x**16 + 36465*a**6*b**10*x**18 + 36465*a**5*b**11*x**20 + 12155*a**4*b**12*x**22
) - 9268*a**5*b**(31/2)*x**12*sqrt(a/(b*x**2) + 1)/(12155*a**7*b**9*x**16 + 36465*a**6*b**10*x**18 + 36465*a**
5*b**11*x**20 + 12155*a**4*b**12*x**22) - 1520*a**4*b**(33/2)*x**14*sqrt(a/(b*x**2) + 1)/(12155*a**7*b**9*x**1
6 + 36465*a**6*b**10*x**18 + 36465*a**5*b**11*x**20 + 12155*a**4*b**12*x**22) + 5*a**3*b**(35/2)*x**16*sqrt(a/
(b*x**2) + 1)/(12155*a**7*b**9*x**16 + 36465*a**6*b**10*x**18 + 36465*a**5*b**11*x**20 + 12155*a**4*b**12*x**2
2) + 30*a**2*b**(37/2)*x**18*sqrt(a/(b*x**2) + 1)/(12155*a**7*b**9*x**16 + 36465*a**6*b**10*x**18 + 36465*a**5
*b**11*x**20 + 12155*a**4*b**12*x**22) + 40*a*b**(39/2)*x**20*sqrt(a/(b*x**2) + 1)/(12155*a**7*b**9*x**16 + 36
465*a**6*b**10*x**18 + 36465*a**5*b**11*x**20 + 12155*a**4*b**12*x**22) + 16*b**(41/2)*x**22*sqrt(a/(b*x**2) +
1)/(12155*a**7*b**9*x**16 + 36465*a**6*b**10*x**18 + 36465*a**5*b**11*x**20 + 12155*a**4*b**12*x**22)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 382 vs. \(2 (76) = 152\).
time = 0.81, size = 382, normalized size = 4.15 \begin {gather*} \frac {32 \, {\left (12155 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{26} b^{\frac {17}{2}} + 65637 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{24} a b^{\frac {17}{2}} + 233376 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{22} a^{2} b^{\frac {17}{2}} + 466752 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{20} a^{3} b^{\frac {17}{2}} + 692835 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{18} a^{4} b^{\frac {17}{2}} + 668525 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{16} a^{5} b^{\frac {17}{2}} + 486200 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{14} a^{6} b^{\frac {17}{2}} + 221000 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{12} a^{7} b^{\frac {17}{2}} + 71825 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} a^{8} b^{\frac {17}{2}} + 9775 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} a^{9} b^{\frac {17}{2}} + 680 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{10} b^{\frac {17}{2}} - 136 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{11} b^{\frac {17}{2}} + 17 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{12} b^{\frac {17}{2}} - a^{13} b^{\frac {17}{2}}\right )}}{12155 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{17}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(9/2)/x^18,x, algorithm="giac")

[Out]

32/12155*(12155*(sqrt(b)*x - sqrt(b*x^2 + a))^26*b^(17/2) + 65637*(sqrt(b)*x - sqrt(b*x^2 + a))^24*a*b^(17/2)
+ 233376*(sqrt(b)*x - sqrt(b*x^2 + a))^22*a^2*b^(17/2) + 466752*(sqrt(b)*x - sqrt(b*x^2 + a))^20*a^3*b^(17/2)
+ 692835*(sqrt(b)*x - sqrt(b*x^2 + a))^18*a^4*b^(17/2) + 668525*(sqrt(b)*x - sqrt(b*x^2 + a))^16*a^5*b^(17/2)
+ 486200*(sqrt(b)*x - sqrt(b*x^2 + a))^14*a^6*b^(17/2) + 221000*(sqrt(b)*x - sqrt(b*x^2 + a))^12*a^7*b^(17/2)
+ 71825*(sqrt(b)*x - sqrt(b*x^2 + a))^10*a^8*b^(17/2) + 9775*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a^9*b^(17/2) + 68
0*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^10*b^(17/2) - 136*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^11*b^(17/2) + 17*(sqrt
(b)*x - sqrt(b*x^2 + a))^2*a^12*b^(17/2) - a^13*b^(17/2))/((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^17

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Mupad [B]
time = 8.17, size = 171, normalized size = 1.86 \begin {gather*} \frac {6\,b^6\,\sqrt {b\,x^2+a}}{12155\,a^2\,x^5}-\frac {303\,b^4\,\sqrt {b\,x^2+a}}{2431\,x^9}-\frac {4714\,a\,b^3\,\sqrt {b\,x^2+a}}{12155\,x^{11}}-\frac {23\,a^3\,b\,\sqrt {b\,x^2+a}}{85\,x^{15}}-\frac {b^5\,\sqrt {b\,x^2+a}}{2431\,a\,x^7}-\frac {a^4\,\sqrt {b\,x^2+a}}{17\,x^{17}}-\frac {8\,b^7\,\sqrt {b\,x^2+a}}{12155\,a^3\,x^3}+\frac {16\,b^8\,\sqrt {b\,x^2+a}}{12155\,a^4\,x}-\frac {528\,a^2\,b^2\,\sqrt {b\,x^2+a}}{1105\,x^{13}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^(9/2)/x^18,x)

[Out]

(6*b^6*(a + b*x^2)^(1/2))/(12155*a^2*x^5) - (303*b^4*(a + b*x^2)^(1/2))/(2431*x^9) - (4714*a*b^3*(a + b*x^2)^(
1/2))/(12155*x^11) - (23*a^3*b*(a + b*x^2)^(1/2))/(85*x^15) - (b^5*(a + b*x^2)^(1/2))/(2431*a*x^7) - (a^4*(a +
b*x^2)^(1/2))/(17*x^17) - (8*b^7*(a + b*x^2)^(1/2))/(12155*a^3*x^3) + (16*b^8*(a + b*x^2)^(1/2))/(12155*a^4*x
) - (528*a^2*b^2*(a + b*x^2)^(1/2))/(1105*x^13)

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