3.192 \(\int \frac{x^{25}}{(a+b x^2)^{10}} \, dx\)

Optimal. Leaf size=216 \[ -\frac{a^{12}}{18 b^{13} \left (a+b x^2\right )^9}+\frac{3 a^{11}}{4 b^{13} \left (a+b x^2\right )^8}-\frac{33 a^{10}}{7 b^{13} \left (a+b x^2\right )^7}+\frac{55 a^9}{3 b^{13} \left (a+b x^2\right )^6}-\frac{99 a^8}{2 b^{13} \left (a+b x^2\right )^5}+\frac{99 a^7}{b^{13} \left (a+b x^2\right )^4}-\frac{154 a^6}{b^{13} \left (a+b x^2\right )^3}+\frac{198 a^5}{b^{13} \left (a+b x^2\right )^2}-\frac{495 a^4}{2 b^{13} \left (a+b x^2\right )}+\frac{55 a^2 x^2}{2 b^{12}}-\frac{110 a^3 \log \left (a+b x^2\right )}{b^{13}}-\frac{5 a x^4}{2 b^{11}}+\frac{x^6}{6 b^{10}} \]

[Out]

(55*a^2*x^2)/(2*b^12) - (5*a*x^4)/(2*b^11) + x^6/(6*b^10) - a^12/(18*b^13*(a + b*x^2)^9) + (3*a^11)/(4*b^13*(a
+ b*x^2)^8) - (33*a^10)/(7*b^13*(a + b*x^2)^7) + (55*a^9)/(3*b^13*(a + b*x^2)^6) - (99*a^8)/(2*b^13*(a + b*x^
2)^5) + (99*a^7)/(b^13*(a + b*x^2)^4) - (154*a^6)/(b^13*(a + b*x^2)^3) + (198*a^5)/(b^13*(a + b*x^2)^2) - (495
*a^4)/(2*b^13*(a + b*x^2)) - (110*a^3*Log[a + b*x^2])/b^13

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Rubi [A]  time = 0.255003, antiderivative size = 216, 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} \[ -\frac{a^{12}}{18 b^{13} \left (a+b x^2\right )^9}+\frac{3 a^{11}}{4 b^{13} \left (a+b x^2\right )^8}-\frac{33 a^{10}}{7 b^{13} \left (a+b x^2\right )^7}+\frac{55 a^9}{3 b^{13} \left (a+b x^2\right )^6}-\frac{99 a^8}{2 b^{13} \left (a+b x^2\right )^5}+\frac{99 a^7}{b^{13} \left (a+b x^2\right )^4}-\frac{154 a^6}{b^{13} \left (a+b x^2\right )^3}+\frac{198 a^5}{b^{13} \left (a+b x^2\right )^2}-\frac{495 a^4}{2 b^{13} \left (a+b x^2\right )}+\frac{55 a^2 x^2}{2 b^{12}}-\frac{110 a^3 \log \left (a+b x^2\right )}{b^{13}}-\frac{5 a x^4}{2 b^{11}}+\frac{x^6}{6 b^{10}} \]

Antiderivative was successfully verified.

[In]

Int[x^25/(a + b*x^2)^10,x]

[Out]

(55*a^2*x^2)/(2*b^12) - (5*a*x^4)/(2*b^11) + x^6/(6*b^10) - a^12/(18*b^13*(a + b*x^2)^9) + (3*a^11)/(4*b^13*(a
+ b*x^2)^8) - (33*a^10)/(7*b^13*(a + b*x^2)^7) + (55*a^9)/(3*b^13*(a + b*x^2)^6) - (99*a^8)/(2*b^13*(a + b*x^
2)^5) + (99*a^7)/(b^13*(a + b*x^2)^4) - (154*a^6)/(b^13*(a + b*x^2)^3) + (198*a^5)/(b^13*(a + b*x^2)^2) - (495
*a^4)/(2*b^13*(a + b*x^2)) - (110*a^3*Log[a + b*x^2])/b^13

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^{25}}{\left (a+b x^2\right )^{10}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^{12}}{(a+b x)^{10}} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{55 a^2}{b^{12}}-\frac{10 a x}{b^{11}}+\frac{x^2}{b^{10}}+\frac{a^{12}}{b^{12} (a+b x)^{10}}-\frac{12 a^{11}}{b^{12} (a+b x)^9}+\frac{66 a^{10}}{b^{12} (a+b x)^8}-\frac{220 a^9}{b^{12} (a+b x)^7}+\frac{495 a^8}{b^{12} (a+b x)^6}-\frac{792 a^7}{b^{12} (a+b x)^5}+\frac{924 a^6}{b^{12} (a+b x)^4}-\frac{792 a^5}{b^{12} (a+b x)^3}+\frac{495 a^4}{b^{12} (a+b x)^2}-\frac{220 a^3}{b^{12} (a+b x)}\right ) \, dx,x,x^2\right )\\ &=\frac{55 a^2 x^2}{2 b^{12}}-\frac{5 a x^4}{2 b^{11}}+\frac{x^6}{6 b^{10}}-\frac{a^{12}}{18 b^{13} \left (a+b x^2\right )^9}+\frac{3 a^{11}}{4 b^{13} \left (a+b x^2\right )^8}-\frac{33 a^{10}}{7 b^{13} \left (a+b x^2\right )^7}+\frac{55 a^9}{3 b^{13} \left (a+b x^2\right )^6}-\frac{99 a^8}{2 b^{13} \left (a+b x^2\right )^5}+\frac{99 a^7}{b^{13} \left (a+b x^2\right )^4}-\frac{154 a^6}{b^{13} \left (a+b x^2\right )^3}+\frac{198 a^5}{b^{13} \left (a+b x^2\right )^2}-\frac{495 a^4}{2 b^{13} \left (a+b x^2\right )}-\frac{110 a^3 \log \left (a+b x^2\right )}{b^{13}}\\ \end{align*}

Mathematica [A]  time = 0.0448646, size = 169, normalized size = 0.78 \[ -\frac{-2772 a^2 b^{10} x^{20}-43218 a^3 b^9 x^{18}-139482 a^4 b^8 x^{16}-58968 a^5 b^7 x^{14}+638568 a^6 b^6 x^{12}+1831032 a^7 b^5 x^{10}+2529576 a^8 b^4 x^8+2074464 a^9 b^3 x^6+1031616 a^{10} b^2 x^4+289089 a^{11} b x^2+27720 a^3 \left (a+b x^2\right )^9 \log \left (a+b x^2\right )+35201 a^{12}+252 a b^{11} x^{22}-42 b^{12} x^{24}}{252 b^{13} \left (a+b x^2\right )^9} \]

Antiderivative was successfully verified.

[In]

Integrate[x^25/(a + b*x^2)^10,x]

[Out]

-(35201*a^12 + 289089*a^11*b*x^2 + 1031616*a^10*b^2*x^4 + 2074464*a^9*b^3*x^6 + 2529576*a^8*b^4*x^8 + 1831032*
a^7*b^5*x^10 + 638568*a^6*b^6*x^12 - 58968*a^5*b^7*x^14 - 139482*a^4*b^8*x^16 - 43218*a^3*b^9*x^18 - 2772*a^2*
b^10*x^20 + 252*a*b^11*x^22 - 42*b^12*x^24 + 27720*a^3*(a + b*x^2)^9*Log[a + b*x^2])/(252*b^13*(a + b*x^2)^9)

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Maple [A]  time = 0.019, size = 199, normalized size = 0.9 \begin{align*}{\frac{55\,{a}^{2}{x}^{2}}{2\,{b}^{12}}}-{\frac{5\,a{x}^{4}}{2\,{b}^{11}}}+{\frac{{x}^{6}}{6\,{b}^{10}}}-{\frac{{a}^{12}}{18\,{b}^{13} \left ( b{x}^{2}+a \right ) ^{9}}}+{\frac{3\,{a}^{11}}{4\,{b}^{13} \left ( b{x}^{2}+a \right ) ^{8}}}-{\frac{33\,{a}^{10}}{7\,{b}^{13} \left ( b{x}^{2}+a \right ) ^{7}}}+{\frac{55\,{a}^{9}}{3\,{b}^{13} \left ( b{x}^{2}+a \right ) ^{6}}}-{\frac{99\,{a}^{8}}{2\,{b}^{13} \left ( b{x}^{2}+a \right ) ^{5}}}+99\,{\frac{{a}^{7}}{{b}^{13} \left ( b{x}^{2}+a \right ) ^{4}}}-154\,{\frac{{a}^{6}}{{b}^{13} \left ( b{x}^{2}+a \right ) ^{3}}}+198\,{\frac{{a}^{5}}{{b}^{13} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{495\,{a}^{4}}{2\,{b}^{13} \left ( b{x}^{2}+a \right ) }}-110\,{\frac{{a}^{3}\ln \left ( b{x}^{2}+a \right ) }{{b}^{13}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^25/(b*x^2+a)^10,x)

[Out]

55/2*a^2*x^2/b^12-5/2*a*x^4/b^11+1/6*x^6/b^10-1/18*a^12/b^13/(b*x^2+a)^9+3/4*a^11/b^13/(b*x^2+a)^8-33/7*a^10/b
^13/(b*x^2+a)^7+55/3*a^9/b^13/(b*x^2+a)^6-99/2*a^8/b^13/(b*x^2+a)^5+99*a^7/b^13/(b*x^2+a)^4-154*a^6/b^13/(b*x^
2+a)^3+198*a^5/b^13/(b*x^2+a)^2-495/2*a^4/b^13/(b*x^2+a)-110*a^3*ln(b*x^2+a)/b^13

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Maxima [A]  time = 2.87715, size = 327, normalized size = 1.51 \begin{align*} -\frac{62370 \, a^{4} b^{8} x^{16} + 449064 \, a^{5} b^{7} x^{14} + 1435896 \, a^{6} b^{6} x^{12} + 2652804 \, a^{7} b^{5} x^{10} + 3089394 \, a^{8} b^{4} x^{8} + 2318316 \, a^{9} b^{3} x^{6} + 1093356 \, a^{10} b^{2} x^{4} + 296019 \, a^{11} b x^{2} + 35201 \, a^{12}}{252 \,{\left (b^{22} x^{18} + 9 \, a b^{21} x^{16} + 36 \, a^{2} b^{20} x^{14} + 84 \, a^{3} b^{19} x^{12} + 126 \, a^{4} b^{18} x^{10} + 126 \, a^{5} b^{17} x^{8} + 84 \, a^{6} b^{16} x^{6} + 36 \, a^{7} b^{15} x^{4} + 9 \, a^{8} b^{14} x^{2} + a^{9} b^{13}\right )}} - \frac{110 \, a^{3} \log \left (b x^{2} + a\right )}{b^{13}} + \frac{b^{2} x^{6} - 15 \, a b x^{4} + 165 \, a^{2} x^{2}}{6 \, b^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^25/(b*x^2+a)^10,x, algorithm="maxima")

[Out]

-1/252*(62370*a^4*b^8*x^16 + 449064*a^5*b^7*x^14 + 1435896*a^6*b^6*x^12 + 2652804*a^7*b^5*x^10 + 3089394*a^8*b
^4*x^8 + 2318316*a^9*b^3*x^6 + 1093356*a^10*b^2*x^4 + 296019*a^11*b*x^2 + 35201*a^12)/(b^22*x^18 + 9*a*b^21*x^
16 + 36*a^2*b^20*x^14 + 84*a^3*b^19*x^12 + 126*a^4*b^18*x^10 + 126*a^5*b^17*x^8 + 84*a^6*b^16*x^6 + 36*a^7*b^1
5*x^4 + 9*a^8*b^14*x^2 + a^9*b^13) - 110*a^3*log(b*x^2 + a)/b^13 + 1/6*(b^2*x^6 - 15*a*b*x^4 + 165*a^2*x^2)/b^
12

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Fricas [A]  time = 1.27799, size = 842, normalized size = 3.9 \begin{align*} \frac{42 \, b^{12} x^{24} - 252 \, a b^{11} x^{22} + 2772 \, a^{2} b^{10} x^{20} + 43218 \, a^{3} b^{9} x^{18} + 139482 \, a^{4} b^{8} x^{16} + 58968 \, a^{5} b^{7} x^{14} - 638568 \, a^{6} b^{6} x^{12} - 1831032 \, a^{7} b^{5} x^{10} - 2529576 \, a^{8} b^{4} x^{8} - 2074464 \, a^{9} b^{3} x^{6} - 1031616 \, a^{10} b^{2} x^{4} - 289089 \, a^{11} b x^{2} - 35201 \, a^{12} - 27720 \,{\left (a^{3} b^{9} x^{18} + 9 \, a^{4} b^{8} x^{16} + 36 \, a^{5} b^{7} x^{14} + 84 \, a^{6} b^{6} x^{12} + 126 \, a^{7} b^{5} x^{10} + 126 \, a^{8} b^{4} x^{8} + 84 \, a^{9} b^{3} x^{6} + 36 \, a^{10} b^{2} x^{4} + 9 \, a^{11} b x^{2} + a^{12}\right )} \log \left (b x^{2} + a\right )}{252 \,{\left (b^{22} x^{18} + 9 \, a b^{21} x^{16} + 36 \, a^{2} b^{20} x^{14} + 84 \, a^{3} b^{19} x^{12} + 126 \, a^{4} b^{18} x^{10} + 126 \, a^{5} b^{17} x^{8} + 84 \, a^{6} b^{16} x^{6} + 36 \, a^{7} b^{15} x^{4} + 9 \, a^{8} b^{14} x^{2} + a^{9} b^{13}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^25/(b*x^2+a)^10,x, algorithm="fricas")

[Out]

1/252*(42*b^12*x^24 - 252*a*b^11*x^22 + 2772*a^2*b^10*x^20 + 43218*a^3*b^9*x^18 + 139482*a^4*b^8*x^16 + 58968*
a^5*b^7*x^14 - 638568*a^6*b^6*x^12 - 1831032*a^7*b^5*x^10 - 2529576*a^8*b^4*x^8 - 2074464*a^9*b^3*x^6 - 103161
6*a^10*b^2*x^4 - 289089*a^11*b*x^2 - 35201*a^12 - 27720*(a^3*b^9*x^18 + 9*a^4*b^8*x^16 + 36*a^5*b^7*x^14 + 84*
a^6*b^6*x^12 + 126*a^7*b^5*x^10 + 126*a^8*b^4*x^8 + 84*a^9*b^3*x^6 + 36*a^10*b^2*x^4 + 9*a^11*b*x^2 + a^12)*lo
g(b*x^2 + a))/(b^22*x^18 + 9*a*b^21*x^16 + 36*a^2*b^20*x^14 + 84*a^3*b^19*x^12 + 126*a^4*b^18*x^10 + 126*a^5*b
^17*x^8 + 84*a^6*b^16*x^6 + 36*a^7*b^15*x^4 + 9*a^8*b^14*x^2 + a^9*b^13)

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Sympy [A]  time = 8.6922, size = 258, normalized size = 1.19 \begin{align*} - \frac{110 a^{3} \log{\left (a + b x^{2} \right )}}{b^{13}} + \frac{55 a^{2} x^{2}}{2 b^{12}} - \frac{5 a x^{4}}{2 b^{11}} - \frac{35201 a^{12} + 296019 a^{11} b x^{2} + 1093356 a^{10} b^{2} x^{4} + 2318316 a^{9} b^{3} x^{6} + 3089394 a^{8} b^{4} x^{8} + 2652804 a^{7} b^{5} x^{10} + 1435896 a^{6} b^{6} x^{12} + 449064 a^{5} b^{7} x^{14} + 62370 a^{4} b^{8} x^{16}}{252 a^{9} b^{13} + 2268 a^{8} b^{14} x^{2} + 9072 a^{7} b^{15} x^{4} + 21168 a^{6} b^{16} x^{6} + 31752 a^{5} b^{17} x^{8} + 31752 a^{4} b^{18} x^{10} + 21168 a^{3} b^{19} x^{12} + 9072 a^{2} b^{20} x^{14} + 2268 a b^{21} x^{16} + 252 b^{22} x^{18}} + \frac{x^{6}}{6 b^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**25/(b*x**2+a)**10,x)

[Out]

-110*a**3*log(a + b*x**2)/b**13 + 55*a**2*x**2/(2*b**12) - 5*a*x**4/(2*b**11) - (35201*a**12 + 296019*a**11*b*
x**2 + 1093356*a**10*b**2*x**4 + 2318316*a**9*b**3*x**6 + 3089394*a**8*b**4*x**8 + 2652804*a**7*b**5*x**10 + 1
435896*a**6*b**6*x**12 + 449064*a**5*b**7*x**14 + 62370*a**4*b**8*x**16)/(252*a**9*b**13 + 2268*a**8*b**14*x**
2 + 9072*a**7*b**15*x**4 + 21168*a**6*b**16*x**6 + 31752*a**5*b**17*x**8 + 31752*a**4*b**18*x**10 + 21168*a**3
*b**19*x**12 + 9072*a**2*b**20*x**14 + 2268*a*b**21*x**16 + 252*b**22*x**18) + x**6/(6*b**10)

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Giac [A]  time = 2.45032, size = 227, normalized size = 1.05 \begin{align*} -\frac{110 \, a^{3} \log \left ({\left | b x^{2} + a \right |}\right )}{b^{13}} + \frac{78419 \, a^{3} b^{9} x^{18} + 643401 \, a^{4} b^{8} x^{16} + 2374020 \, a^{5} b^{7} x^{14} + 5151300 \, a^{6} b^{6} x^{12} + 7227990 \, a^{7} b^{5} x^{10} + 6791400 \, a^{8} b^{4} x^{8} + 4268880 \, a^{9} b^{3} x^{6} + 1729728 \, a^{10} b^{2} x^{4} + 409752 \, a^{11} b x^{2} + 43218 \, a^{12}}{252 \,{\left (b x^{2} + a\right )}^{9} b^{13}} + \frac{b^{20} x^{6} - 15 \, a b^{19} x^{4} + 165 \, a^{2} b^{18} x^{2}}{6 \, b^{30}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^25/(b*x^2+a)^10,x, algorithm="giac")

[Out]

-110*a^3*log(abs(b*x^2 + a))/b^13 + 1/252*(78419*a^3*b^9*x^18 + 643401*a^4*b^8*x^16 + 2374020*a^5*b^7*x^14 + 5
151300*a^6*b^6*x^12 + 7227990*a^7*b^5*x^10 + 6791400*a^8*b^4*x^8 + 4268880*a^9*b^3*x^6 + 1729728*a^10*b^2*x^4
+ 409752*a^11*b*x^2 + 43218*a^12)/((b*x^2 + a)^9*b^13) + 1/6*(b^20*x^6 - 15*a*b^19*x^4 + 165*a^2*b^18*x^2)/b^3
0