3.198 \(\int \frac{x^{13}}{\left (a+b x^2\right )^{10}} \, dx\)

Optimal. Leaf size=58 \[ \frac{x^{14}}{504 a^3 \left (a+b x^2\right )^7}+\frac{x^{14}}{72 a^2 \left (a+b x^2\right )^8}+\frac{x^{14}}{18 a \left (a+b x^2\right )^9} \]

[Out]

x^14/(18*a*(a + b*x^2)^9) + x^14/(72*a^2*(a + b*x^2)^8) + x^14/(504*a^3*(a + b*x
^2)^7)

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Rubi [A]  time = 0.0814398, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{x^{14}}{504 a^3 \left (a+b x^2\right )^7}+\frac{x^{14}}{72 a^2 \left (a+b x^2\right )^8}+\frac{x^{14}}{18 a \left (a+b x^2\right )^9} \]

Antiderivative was successfully verified.

[In]  Int[x^13/(a + b*x^2)^10,x]

[Out]

x^14/(18*a*(a + b*x^2)^9) + x^14/(72*a^2*(a + b*x^2)^8) + x^14/(504*a^3*(a + b*x
^2)^7)

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Rubi in Sympy [A]  time = 9.86219, size = 48, normalized size = 0.83 \[ \frac{x^{14}}{18 a \left (a + b x^{2}\right )^{9}} + \frac{x^{14}}{72 a^{2} \left (a + b x^{2}\right )^{8}} + \frac{x^{14}}{504 a^{3} \left (a + b x^{2}\right )^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**13/(b*x**2+a)**10,x)

[Out]

x**14/(18*a*(a + b*x**2)**9) + x**14/(72*a**2*(a + b*x**2)**8) + x**14/(504*a**3
*(a + b*x**2)**7)

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Mathematica [A]  time = 0.031869, size = 79, normalized size = 1.36 \[ -\frac{a^6+9 a^5 b x^2+36 a^4 b^2 x^4+84 a^3 b^3 x^6+126 a^2 b^4 x^8+126 a b^5 x^{10}+84 b^6 x^{12}}{504 b^7 \left (a+b x^2\right )^9} \]

Antiderivative was successfully verified.

[In]  Integrate[x^13/(a + b*x^2)^10,x]

[Out]

-(a^6 + 9*a^5*b*x^2 + 36*a^4*b^2*x^4 + 84*a^3*b^3*x^6 + 126*a^2*b^4*x^8 + 126*a*
b^5*x^10 + 84*b^6*x^12)/(504*b^7*(a + b*x^2)^9)

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Maple [B]  time = 0.014, size = 116, normalized size = 2. \[ -{\frac{3\,{a}^{2}}{2\,{b}^{7} \left ( b{x}^{2}+a \right ) ^{5}}}-{\frac{{a}^{6}}{18\,{b}^{7} \left ( b{x}^{2}+a \right ) ^{9}}}-{\frac{1}{6\, \left ( b{x}^{2}+a \right ) ^{3}{b}^{7}}}-{\frac{15\,{a}^{4}}{14\,{b}^{7} \left ( b{x}^{2}+a \right ) ^{7}}}+{\frac{5\,{a}^{3}}{3\,{b}^{7} \left ( b{x}^{2}+a \right ) ^{6}}}+{\frac{3\,{a}^{5}}{8\,{b}^{7} \left ( b{x}^{2}+a \right ) ^{8}}}+{\frac{3\,a}{4\,{b}^{7} \left ( b{x}^{2}+a \right ) ^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^13/(b*x^2+a)^10,x)

[Out]

-3/2*a^2/b^7/(b*x^2+a)^5-1/18*a^6/b^7/(b*x^2+a)^9-1/6/(b*x^2+a)^3/b^7-15/14*a^4/
b^7/(b*x^2+a)^7+5/3*a^3/b^7/(b*x^2+a)^6+3/8*a^5/b^7/(b*x^2+a)^8+3/4*a/b^7/(b*x^2
+a)^4

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Maxima [A]  time = 1.34643, size = 227, normalized size = 3.91 \[ -\frac{84 \, b^{6} x^{12} + 126 \, a b^{5} x^{10} + 126 \, a^{2} b^{4} x^{8} + 84 \, a^{3} b^{3} x^{6} + 36 \, a^{4} b^{2} x^{4} + 9 \, a^{5} b x^{2} + a^{6}}{504 \,{\left (b^{16} x^{18} + 9 \, a b^{15} x^{16} + 36 \, a^{2} b^{14} x^{14} + 84 \, a^{3} b^{13} x^{12} + 126 \, a^{4} b^{12} x^{10} + 126 \, a^{5} b^{11} x^{8} + 84 \, a^{6} b^{10} x^{6} + 36 \, a^{7} b^{9} x^{4} + 9 \, a^{8} b^{8} x^{2} + a^{9} b^{7}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^13/(b*x^2 + a)^10,x, algorithm="maxima")

[Out]

-1/504*(84*b^6*x^12 + 126*a*b^5*x^10 + 126*a^2*b^4*x^8 + 84*a^3*b^3*x^6 + 36*a^4
*b^2*x^4 + 9*a^5*b*x^2 + a^6)/(b^16*x^18 + 9*a*b^15*x^16 + 36*a^2*b^14*x^14 + 84
*a^3*b^13*x^12 + 126*a^4*b^12*x^10 + 126*a^5*b^11*x^8 + 84*a^6*b^10*x^6 + 36*a^7
*b^9*x^4 + 9*a^8*b^8*x^2 + a^9*b^7)

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Fricas [A]  time = 0.204256, size = 227, normalized size = 3.91 \[ -\frac{84 \, b^{6} x^{12} + 126 \, a b^{5} x^{10} + 126 \, a^{2} b^{4} x^{8} + 84 \, a^{3} b^{3} x^{6} + 36 \, a^{4} b^{2} x^{4} + 9 \, a^{5} b x^{2} + a^{6}}{504 \,{\left (b^{16} x^{18} + 9 \, a b^{15} x^{16} + 36 \, a^{2} b^{14} x^{14} + 84 \, a^{3} b^{13} x^{12} + 126 \, a^{4} b^{12} x^{10} + 126 \, a^{5} b^{11} x^{8} + 84 \, a^{6} b^{10} x^{6} + 36 \, a^{7} b^{9} x^{4} + 9 \, a^{8} b^{8} x^{2} + a^{9} b^{7}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^13/(b*x^2 + a)^10,x, algorithm="fricas")

[Out]

-1/504*(84*b^6*x^12 + 126*a*b^5*x^10 + 126*a^2*b^4*x^8 + 84*a^3*b^3*x^6 + 36*a^4
*b^2*x^4 + 9*a^5*b*x^2 + a^6)/(b^16*x^18 + 9*a*b^15*x^16 + 36*a^2*b^14*x^14 + 84
*a^3*b^13*x^12 + 126*a^4*b^12*x^10 + 126*a^5*b^11*x^8 + 84*a^6*b^10*x^6 + 36*a^7
*b^9*x^4 + 9*a^8*b^8*x^2 + a^9*b^7)

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Sympy [A]  time = 34.8363, size = 178, normalized size = 3.07 \[ - \frac{a^{6} + 9 a^{5} b x^{2} + 36 a^{4} b^{2} x^{4} + 84 a^{3} b^{3} x^{6} + 126 a^{2} b^{4} x^{8} + 126 a b^{5} x^{10} + 84 b^{6} x^{12}}{504 a^{9} b^{7} + 4536 a^{8} b^{8} x^{2} + 18144 a^{7} b^{9} x^{4} + 42336 a^{6} b^{10} x^{6} + 63504 a^{5} b^{11} x^{8} + 63504 a^{4} b^{12} x^{10} + 42336 a^{3} b^{13} x^{12} + 18144 a^{2} b^{14} x^{14} + 4536 a b^{15} x^{16} + 504 b^{16} x^{18}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**13/(b*x**2+a)**10,x)

[Out]

-(a**6 + 9*a**5*b*x**2 + 36*a**4*b**2*x**4 + 84*a**3*b**3*x**6 + 126*a**2*b**4*x
**8 + 126*a*b**5*x**10 + 84*b**6*x**12)/(504*a**9*b**7 + 4536*a**8*b**8*x**2 + 1
8144*a**7*b**9*x**4 + 42336*a**6*b**10*x**6 + 63504*a**5*b**11*x**8 + 63504*a**4
*b**12*x**10 + 42336*a**3*b**13*x**12 + 18144*a**2*b**14*x**14 + 4536*a*b**15*x*
*16 + 504*b**16*x**18)

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GIAC/XCAS [A]  time = 0.209252, size = 104, normalized size = 1.79 \[ -\frac{84 \, b^{6} x^{12} + 126 \, a b^{5} x^{10} + 126 \, a^{2} b^{4} x^{8} + 84 \, a^{3} b^{3} x^{6} + 36 \, a^{4} b^{2} x^{4} + 9 \, a^{5} b x^{2} + a^{6}}{504 \,{\left (b x^{2} + a\right )}^{9} b^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^13/(b*x^2 + a)^10,x, algorithm="giac")

[Out]

-1/504*(84*b^6*x^12 + 126*a*b^5*x^10 + 126*a^2*b^4*x^8 + 84*a^3*b^3*x^6 + 36*a^4
*b^2*x^4 + 9*a^5*b*x^2 + a^6)/((b*x^2 + a)^9*b^7)