3.2.95 \(\int \frac {x^{19}}{(a+b x^2)^{10}} \, dx\) [195]

Optimal. Leaf size=179 \[ \frac {a^9}{18 b^{10} \left (a+b x^2\right )^9}-\frac {9 a^8}{16 b^{10} \left (a+b x^2\right )^8}+\frac {18 a^7}{7 b^{10} \left (a+b x^2\right )^7}-\frac {7 a^6}{b^{10} \left (a+b x^2\right )^6}+\frac {63 a^5}{5 b^{10} \left (a+b x^2\right )^5}-\frac {63 a^4}{4 b^{10} \left (a+b x^2\right )^4}+\frac {14 a^3}{b^{10} \left (a+b x^2\right )^3}-\frac {9 a^2}{b^{10} \left (a+b x^2\right )^2}+\frac {9 a}{2 b^{10} \left (a+b x^2\right )}+\frac {\log \left (a+b x^2\right )}{2 b^{10}} \]

[Out]

1/18*a^9/b^10/(b*x^2+a)^9-9/16*a^8/b^10/(b*x^2+a)^8+18/7*a^7/b^10/(b*x^2+a)^7-7*a^6/b^10/(b*x^2+a)^6+63/5*a^5/
b^10/(b*x^2+a)^5-63/4*a^4/b^10/(b*x^2+a)^4+14*a^3/b^10/(b*x^2+a)^3-9*a^2/b^10/(b*x^2+a)^2+9/2*a/b^10/(b*x^2+a)
+1/2*ln(b*x^2+a)/b^10

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Rubi [A]
time = 0.11, antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {272, 45} \begin {gather*} \frac {a^9}{18 b^{10} \left (a+b x^2\right )^9}-\frac {9 a^8}{16 b^{10} \left (a+b x^2\right )^8}+\frac {18 a^7}{7 b^{10} \left (a+b x^2\right )^7}-\frac {7 a^6}{b^{10} \left (a+b x^2\right )^6}+\frac {63 a^5}{5 b^{10} \left (a+b x^2\right )^5}-\frac {63 a^4}{4 b^{10} \left (a+b x^2\right )^4}+\frac {14 a^3}{b^{10} \left (a+b x^2\right )^3}-\frac {9 a^2}{b^{10} \left (a+b x^2\right )^2}+\frac {9 a}{2 b^{10} \left (a+b x^2\right )}+\frac {\log \left (a+b x^2\right )}{2 b^{10}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^19/(a + b*x^2)^10,x]

[Out]

a^9/(18*b^10*(a + b*x^2)^9) - (9*a^8)/(16*b^10*(a + b*x^2)^8) + (18*a^7)/(7*b^10*(a + b*x^2)^7) - (7*a^6)/(b^1
0*(a + b*x^2)^6) + (63*a^5)/(5*b^10*(a + b*x^2)^5) - (63*a^4)/(4*b^10*(a + b*x^2)^4) + (14*a^3)/(b^10*(a + b*x
^2)^3) - (9*a^2)/(b^10*(a + b*x^2)^2) + (9*a)/(2*b^10*(a + b*x^2)) + Log[a + b*x^2]/(2*b^10)

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int \frac {x^{19}}{\left (a+b x^2\right )^{10}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x^9}{(a+b x)^{10}} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (-\frac {a^9}{b^9 (a+b x)^{10}}+\frac {9 a^8}{b^9 (a+b x)^9}-\frac {36 a^7}{b^9 (a+b x)^8}+\frac {84 a^6}{b^9 (a+b x)^7}-\frac {126 a^5}{b^9 (a+b x)^6}+\frac {126 a^4}{b^9 (a+b x)^5}-\frac {84 a^3}{b^9 (a+b x)^4}+\frac {36 a^2}{b^9 (a+b x)^3}-\frac {9 a}{b^9 (a+b x)^2}+\frac {1}{b^9 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=\frac {a^9}{18 b^{10} \left (a+b x^2\right )^9}-\frac {9 a^8}{16 b^{10} \left (a+b x^2\right )^8}+\frac {18 a^7}{7 b^{10} \left (a+b x^2\right )^7}-\frac {7 a^6}{b^{10} \left (a+b x^2\right )^6}+\frac {63 a^5}{5 b^{10} \left (a+b x^2\right )^5}-\frac {63 a^4}{4 b^{10} \left (a+b x^2\right )^4}+\frac {14 a^3}{b^{10} \left (a+b x^2\right )^3}-\frac {9 a^2}{b^{10} \left (a+b x^2\right )^2}+\frac {9 a}{2 b^{10} \left (a+b x^2\right )}+\frac {\log \left (a+b x^2\right )}{2 b^{10}}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 116, normalized size = 0.65 \begin {gather*} \frac {\frac {a \left (7129 a^8+61641 a^7 b x^2+235224 a^6 b^2 x^4+518616 a^5 b^3 x^6+725004 a^4 b^4 x^8+661500 a^3 b^5 x^{10}+388080 a^2 b^6 x^{12}+136080 a b^7 x^{14}+22680 b^8 x^{16}\right )}{\left (a+b x^2\right )^9}+2520 \log \left (a+b x^2\right )}{5040 b^{10}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^19/(a + b*x^2)^10,x]

[Out]

((a*(7129*a^8 + 61641*a^7*b*x^2 + 235224*a^6*b^2*x^4 + 518616*a^5*b^3*x^6 + 725004*a^4*b^4*x^8 + 661500*a^3*b^
5*x^10 + 388080*a^2*b^6*x^12 + 136080*a*b^7*x^14 + 22680*b^8*x^16))/(a + b*x^2)^9 + 2520*Log[a + b*x^2])/(5040
*b^10)

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Maple [A]
time = 0.10, size = 166, normalized size = 0.93

method	result	size
norman	\(\frac {\frac {7129 a^{9}}{5040 b^{10}}+\frac {9 a \,x^{16}}{2 b^{2}}+\frac {27 a^{2} x^{14}}{b^{3}}+\frac {77 a^{3} x^{12}}{b^{4}}+\frac {525 a^{4} x^{10}}{4 b^{5}}+\frac {2877 a^{5} x^{8}}{20 b^{6}}+\frac {1029 a^{6} x^{6}}{10 b^{7}}+\frac {3267 a^{7} x^{4}}{70 b^{8}}+\frac {6849 a^{8} x^{2}}{560 b^{9}}}{\left (b \,x^{2}+a \right )^{9}}+\frac {\ln \left (b \,x^{2}+a \right )}{2 b^{10}}\)	\(120\)
risch	\(\frac {\frac {7129 a^{9}}{5040 b^{10}}+\frac {9 a \,x^{16}}{2 b^{2}}+\frac {27 a^{2} x^{14}}{b^{3}}+\frac {77 a^{3} x^{12}}{b^{4}}+\frac {525 a^{4} x^{10}}{4 b^{5}}+\frac {2877 a^{5} x^{8}}{20 b^{6}}+\frac {1029 a^{6} x^{6}}{10 b^{7}}+\frac {3267 a^{7} x^{4}}{70 b^{8}}+\frac {6849 a^{8} x^{2}}{560 b^{9}}}{\left (b \,x^{2}+a \right )^{9}}+\frac {\ln \left (b \,x^{2}+a \right )}{2 b^{10}}\)	\(120\)
default	\(\frac {a^{9}}{18 b^{10} \left (b \,x^{2}+a \right )^{9}}-\frac {9 a^{8}}{16 b^{10} \left (b \,x^{2}+a \right )^{8}}+\frac {18 a^{7}}{7 b^{10} \left (b \,x^{2}+a \right )^{7}}-\frac {7 a^{6}}{b^{10} \left (b \,x^{2}+a \right )^{6}}+\frac {63 a^{5}}{5 b^{10} \left (b \,x^{2}+a \right )^{5}}-\frac {63 a^{4}}{4 b^{10} \left (b \,x^{2}+a \right )^{4}}+\frac {14 a^{3}}{b^{10} \left (b \,x^{2}+a \right )^{3}}-\frac {9 a^{2}}{b^{10} \left (b \,x^{2}+a \right )^{2}}+\frac {9 a}{2 b^{10} \left (b \,x^{2}+a \right )}+\frac {\ln \left (b \,x^{2}+a \right )}{2 b^{10}}\)	\(166\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^19/(b*x^2+a)^10,x,method=_RETURNVERBOSE)

[Out]

1/18*a^9/b^10/(b*x^2+a)^9-9/16*a^8/b^10/(b*x^2+a)^8+18/7*a^7/b^10/(b*x^2+a)^7-7*a^6/b^10/(b*x^2+a)^6+63/5*a^5/
b^10/(b*x^2+a)^5-63/4*a^4/b^10/(b*x^2+a)^4+14*a^3/b^10/(b*x^2+a)^3-9*a^2/b^10/(b*x^2+a)^2+9/2*a/b^10/(b*x^2+a)
+1/2*ln(b*x^2+a)/b^10

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Maxima [A]
time = 0.30, size = 209, normalized size = 1.17 \begin {gather*} \frac {22680 \, a b^{8} x^{16} + 136080 \, a^{2} b^{7} x^{14} + 388080 \, a^{3} b^{6} x^{12} + 661500 \, a^{4} b^{5} x^{10} + 725004 \, a^{5} b^{4} x^{8} + 518616 \, a^{6} b^{3} x^{6} + 235224 \, a^{7} b^{2} x^{4} + 61641 \, a^{8} b x^{2} + 7129 \, a^{9}}{5040 \, {\left (b^{19} x^{18} + 9 \, a b^{18} x^{16} + 36 \, a^{2} b^{17} x^{14} + 84 \, a^{3} b^{16} x^{12} + 126 \, a^{4} b^{15} x^{10} + 126 \, a^{5} b^{14} x^{8} + 84 \, a^{6} b^{13} x^{6} + 36 \, a^{7} b^{12} x^{4} + 9 \, a^{8} b^{11} x^{2} + a^{9} b^{10}\right )}} + \frac {\log \left (b x^{2} + a\right )}{2 \, b^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^19/(b*x^2+a)^10,x, algorithm="maxima")

[Out]

1/5040*(22680*a*b^8*x^16 + 136080*a^2*b^7*x^14 + 388080*a^3*b^6*x^12 + 661500*a^4*b^5*x^10 + 725004*a^5*b^4*x^
8 + 518616*a^6*b^3*x^6 + 235224*a^7*b^2*x^4 + 61641*a^8*b*x^2 + 7129*a^9)/(b^19*x^18 + 9*a*b^18*x^16 + 36*a^2*
b^17*x^14 + 84*a^3*b^16*x^12 + 126*a^4*b^15*x^10 + 126*a^5*b^14*x^8 + 84*a^6*b^13*x^6 + 36*a^7*b^12*x^4 + 9*a^
8*b^11*x^2 + a^9*b^10) + 1/2*log(b*x^2 + a)/b^10

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Fricas [A]
time = 1.47, size = 300, normalized size = 1.68 \begin {gather*} \frac {22680 \, a b^{8} x^{16} + 136080 \, a^{2} b^{7} x^{14} + 388080 \, a^{3} b^{6} x^{12} + 661500 \, a^{4} b^{5} x^{10} + 725004 \, a^{5} b^{4} x^{8} + 518616 \, a^{6} b^{3} x^{6} + 235224 \, a^{7} b^{2} x^{4} + 61641 \, a^{8} b x^{2} + 7129 \, a^{9} + 2520 \, {\left (b^{9} x^{18} + 9 \, a b^{8} x^{16} + 36 \, a^{2} b^{7} x^{14} + 84 \, a^{3} b^{6} x^{12} + 126 \, a^{4} b^{5} x^{10} + 126 \, a^{5} b^{4} x^{8} + 84 \, a^{6} b^{3} x^{6} + 36 \, a^{7} b^{2} x^{4} + 9 \, a^{8} b x^{2} + a^{9}\right )} \log \left (b x^{2} + a\right )}{5040 \, {\left (b^{19} x^{18} + 9 \, a b^{18} x^{16} + 36 \, a^{2} b^{17} x^{14} + 84 \, a^{3} b^{16} x^{12} + 126 \, a^{4} b^{15} x^{10} + 126 \, a^{5} b^{14} x^{8} + 84 \, a^{6} b^{13} x^{6} + 36 \, a^{7} b^{12} x^{4} + 9 \, a^{8} b^{11} x^{2} + a^{9} b^{10}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^19/(b*x^2+a)^10,x, algorithm="fricas")

[Out]

1/5040*(22680*a*b^8*x^16 + 136080*a^2*b^7*x^14 + 388080*a^3*b^6*x^12 + 661500*a^4*b^5*x^10 + 725004*a^5*b^4*x^
8 + 518616*a^6*b^3*x^6 + 235224*a^7*b^2*x^4 + 61641*a^8*b*x^2 + 7129*a^9 + 2520*(b^9*x^18 + 9*a*b^8*x^16 + 36*
a^2*b^7*x^14 + 84*a^3*b^6*x^12 + 126*a^4*b^5*x^10 + 126*a^5*b^4*x^8 + 84*a^6*b^3*x^6 + 36*a^7*b^2*x^4 + 9*a^8*
b*x^2 + a^9)*log(b*x^2 + a))/(b^19*x^18 + 9*a*b^18*x^16 + 36*a^2*b^17*x^14 + 84*a^3*b^16*x^12 + 126*a^4*b^15*x
^10 + 126*a^5*b^14*x^8 + 84*a^6*b^13*x^6 + 36*a^7*b^12*x^4 + 9*a^8*b^11*x^2 + a^9*b^10)

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Sympy [A]
time = 0.75, size = 219, normalized size = 1.22 \begin {gather*} \frac {7129 a^{9} + 61641 a^{8} b x^{2} + 235224 a^{7} b^{2} x^{4} + 518616 a^{6} b^{3} x^{6} + 725004 a^{5} b^{4} x^{8} + 661500 a^{4} b^{5} x^{10} + 388080 a^{3} b^{6} x^{12} + 136080 a^{2} b^{7} x^{14} + 22680 a b^{8} x^{16}}{5040 a^{9} b^{10} + 45360 a^{8} b^{11} x^{2} + 181440 a^{7} b^{12} x^{4} + 423360 a^{6} b^{13} x^{6} + 635040 a^{5} b^{14} x^{8} + 635040 a^{4} b^{15} x^{10} + 423360 a^{3} b^{16} x^{12} + 181440 a^{2} b^{17} x^{14} + 45360 a b^{18} x^{16} + 5040 b^{19} x^{18}} + \frac {\log {\left (a + b x^{2} \right )}}{2 b^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**19/(b*x**2+a)**10,x)

[Out]

(7129*a**9 + 61641*a**8*b*x**2 + 235224*a**7*b**2*x**4 + 518616*a**6*b**3*x**6 + 725004*a**5*b**4*x**8 + 66150
0*a**4*b**5*x**10 + 388080*a**3*b**6*x**12 + 136080*a**2*b**7*x**14 + 22680*a*b**8*x**16)/(5040*a**9*b**10 + 4
5360*a**8*b**11*x**2 + 181440*a**7*b**12*x**4 + 423360*a**6*b**13*x**6 + 635040*a**5*b**14*x**8 + 635040*a**4*
b**15*x**10 + 423360*a**3*b**16*x**12 + 181440*a**2*b**17*x**14 + 45360*a*b**18*x**16 + 5040*b**19*x**18) + lo
g(a + b*x**2)/(2*b**10)

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Giac [A]
time = 1.73, size = 119, normalized size = 0.66 \begin {gather*} \frac {\log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{10}} - \frac {7129 \, b^{8} x^{18} + 41481 \, a b^{7} x^{16} + 120564 \, a^{2} b^{6} x^{14} + 210756 \, a^{3} b^{5} x^{12} + 236754 \, a^{4} b^{4} x^{10} + 173250 \, a^{5} b^{3} x^{8} + 80220 \, a^{6} b^{2} x^{6} + 21420 \, a^{7} b x^{4} + 2520 \, a^{8} x^{2}}{5040 \, {\left (b x^{2} + a\right )}^{9} b^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^19/(b*x^2+a)^10,x, algorithm="giac")

[Out]

1/2*log(abs(b*x^2 + a))/b^10 - 1/5040*(7129*b^8*x^18 + 41481*a*b^7*x^16 + 120564*a^2*b^6*x^14 + 210756*a^3*b^5
*x^12 + 236754*a^4*b^4*x^10 + 173250*a^5*b^3*x^8 + 80220*a^6*b^2*x^6 + 21420*a^7*b*x^4 + 2520*a^8*x^2)/((b*x^2
+ a)^9*b^9)

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Mupad [B]
time = 5.36, size = 207, normalized size = 1.16 \begin {gather*} \frac {\frac {7129\,a^9}{5040\,b^{10}}+\frac {9\,a\,x^{16}}{2\,b^2}+\frac {27\,a^2\,x^{14}}{b^3}+\frac {77\,a^3\,x^{12}}{b^4}+\frac {525\,a^4\,x^{10}}{4\,b^5}+\frac {2877\,a^5\,x^8}{20\,b^6}+\frac {1029\,a^6\,x^6}{10\,b^7}+\frac {3267\,a^7\,x^4}{70\,b^8}+\frac {6849\,a^8\,x^2}{560\,b^9}}{a^9+9\,a^8\,b\,x^2+36\,a^7\,b^2\,x^4+84\,a^6\,b^3\,x^6+126\,a^5\,b^4\,x^8+126\,a^4\,b^5\,x^{10}+84\,a^3\,b^6\,x^{12}+36\,a^2\,b^7\,x^{14}+9\,a\,b^8\,x^{16}+b^9\,x^{18}}+\frac {\ln \left (b\,x^2+a\right )}{2\,b^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^19/(a + b*x^2)^10,x)

[Out]

((7129*a^9)/(5040*b^10) + (9*a*x^16)/(2*b^2) + (27*a^2*x^14)/b^3 + (77*a^3*x^12)/b^4 + (525*a^4*x^10)/(4*b^5)
+ (2877*a^5*x^8)/(20*b^6) + (1029*a^6*x^6)/(10*b^7) + (3267*a^7*x^4)/(70*b^8) + (6849*a^8*x^2)/(560*b^9))/(a^9
+ b^9*x^18 + 9*a^8*b*x^2 + 9*a*b^8*x^16 + 36*a^7*b^2*x^4 + 84*a^6*b^3*x^6 + 126*a^5*b^4*x^8 + 126*a^4*b^5*x^1
0 + 84*a^3*b^6*x^12 + 36*a^2*b^7*x^14) + log(a + b*x^2)/(2*b^10)

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