3.195 \(\int \frac{x^{19}}{(a+b x^2)^{10}} \, dx\)

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]

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)

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Rubi [A]  time = 0.169356, antiderivative size = 179, normalized size of antiderivative = 1., 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 = {266, 43} \[ \frac{a^9}{18 b^{10} \left (a+b x^2\right )^9}-\frac{9 a^8}{16 b^{10} \left (a+b x^2\right )^8}+\frac{18 a^7}{7 b^{10} \left (a+b x^2\right )^7}-\frac{7 a^6}{b^{10} \left (a+b x^2\right )^6}+\frac{63 a^5}{5 b^{10} \left (a+b x^2\right )^5}-\frac{63 a^4}{4 b^{10} \left (a+b x^2\right )^4}+\frac{14 a^3}{b^{10} \left (a+b x^2\right )^3}-\frac{9 a^2}{b^{10} \left (a+b x^2\right )^2}+\frac{9 a}{2 b^{10} \left (a+b x^2\right )}+\frac{\log \left (a+b x^2\right )}{2 b^{10}} \]

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 266

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]]

Rule 43

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])

Rubi steps

\begin{align*} \int \frac{x^{19}}{\left (a+b x^2\right )^{10}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^9}{(a+b x)^{10}} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{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*}

Mathematica [A]  time = 0.0317584, size = 116, normalized size = 0.65 \[ \frac{\frac{a \left (388080 a^2 b^6 x^{12}+661500 a^3 b^5 x^{10}+725004 a^4 b^4 x^8+518616 a^5 b^3 x^6+235224 a^6 b^2 x^4+61641 a^7 b x^2+7129 a^8+136080 a b^7 x^{14}+22680 b^8 x^{16}\right )}{\left (a+b x^2\right )^9}+2520 \log \left (a+b x^2\right )}{5040 b^{10}} \]

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 = 0.011, size = 166, normalized size = 0.9 \begin{align*}{\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}}}-7\,{\frac{{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}}}+14\,{\frac{{a}^{3}}{{b}^{10} \left ( b{x}^{2}+a \right ) ^{3}}}-9\,{\frac{{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}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 2.32338, size = 282, normalized size = 1.58 \begin{align*} \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{align*}

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 = 1.27218, size = 714, normalized size = 3.99 \begin{align*} \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{align*}

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 = 7.94604, size = 219, normalized size = 1.22 \begin{align*} \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{align*}

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 = 1.31202, size = 161, normalized size = 0.9 \begin{align*} \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{align*}

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)