3.43 \(\int \frac{\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^{11}} \, dx\)

Optimal. Leaf size=113 \[ -\frac{a^5 A}{10 x^{10}}-\frac{a^4 (a B+5 A b)}{8 x^8}-\frac{5 a^3 b (a B+2 A b)}{6 x^6}-\frac{5 a^2 b^2 (a B+A b)}{2 x^4}+b^4 \log (x) (5 a B+A b)-\frac{5 a b^3 (2 a B+A b)}{2 x^2}+\frac{1}{2} b^5 B x^2 \]

[Out]

-(a^5*A)/(10*x^10) - (a^4*(5*A*b + a*B))/(8*x^8) - (5*a^3*b*(2*A*b + a*B))/(6*x^
6) - (5*a^2*b^2*(A*b + a*B))/(2*x^4) - (5*a*b^3*(A*b + 2*a*B))/(2*x^2) + (b^5*B*
x^2)/2 + b^4*(A*b + 5*a*B)*Log[x]

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Rubi [A]  time = 0.245703, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ -\frac{a^5 A}{10 x^{10}}-\frac{a^4 (a B+5 A b)}{8 x^8}-\frac{5 a^3 b (a B+2 A b)}{6 x^6}-\frac{5 a^2 b^2 (a B+A b)}{2 x^4}+b^4 \log (x) (5 a B+A b)-\frac{5 a b^3 (2 a B+A b)}{2 x^2}+\frac{1}{2} b^5 B x^2 \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^2)^5*(A + B*x^2))/x^11,x]

[Out]

-(a^5*A)/(10*x^10) - (a^4*(5*A*b + a*B))/(8*x^8) - (5*a^3*b*(2*A*b + a*B))/(6*x^
6) - (5*a^2*b^2*(A*b + a*B))/(2*x^4) - (5*a*b^3*(A*b + 2*a*B))/(2*x^2) + (b^5*B*
x^2)/2 + b^4*(A*b + 5*a*B)*Log[x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{A a^{5}}{10 x^{10}} - \frac{a^{4} \left (5 A b + B a\right )}{8 x^{8}} - \frac{5 a^{3} b \left (2 A b + B a\right )}{6 x^{6}} - \frac{5 a^{2} b^{2} \left (A b + B a\right )}{2 x^{4}} - \frac{5 a b^{3} \left (A b + 2 B a\right )}{2 x^{2}} + \frac{b^{5} \int ^{x^{2}} B\, dx}{2} + \frac{b^{4} \left (A b + 5 B a\right ) \log{\left (x^{2} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**5*(B*x**2+A)/x**11,x)

[Out]

-A*a**5/(10*x**10) - a**4*(5*A*b + B*a)/(8*x**8) - 5*a**3*b*(2*A*b + B*a)/(6*x**
6) - 5*a**2*b**2*(A*b + B*a)/(2*x**4) - 5*a*b**3*(A*b + 2*B*a)/(2*x**2) + b**5*I
ntegral(B, (x, x**2))/2 + b**4*(A*b + 5*B*a)*log(x**2)/2

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Mathematica [A]  time = 0.103834, size = 116, normalized size = 1.03 \[ b^4 \log (x) (5 a B+A b)-\frac{3 a^5 \left (4 A+5 B x^2\right )+25 a^4 b x^2 \left (3 A+4 B x^2\right )+100 a^3 b^2 x^4 \left (2 A+3 B x^2\right )+300 a^2 b^3 x^6 \left (A+2 B x^2\right )+300 a A b^4 x^8-60 b^5 B x^{12}}{120 x^{10}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x^2)^5*(A + B*x^2))/x^11,x]

[Out]

-(300*a*A*b^4*x^8 - 60*b^5*B*x^12 + 300*a^2*b^3*x^6*(A + 2*B*x^2) + 100*a^3*b^2*
x^4*(2*A + 3*B*x^2) + 25*a^4*b*x^2*(3*A + 4*B*x^2) + 3*a^5*(4*A + 5*B*x^2))/(120
*x^10) + b^4*(A*b + 5*a*B)*Log[x]

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Maple [A]  time = 0.012, size = 123, normalized size = 1.1 \[{\frac{{b}^{5}B{x}^{2}}{2}}+A\ln \left ( x \right ){b}^{5}+5\,B\ln \left ( x \right ) a{b}^{4}-{\frac{5\,{a}^{3}{b}^{2}A}{3\,{x}^{6}}}-{\frac{5\,{a}^{4}bB}{6\,{x}^{6}}}-{\frac{5\,{a}^{2}{b}^{3}A}{2\,{x}^{4}}}-{\frac{5\,{a}^{3}{b}^{2}B}{2\,{x}^{4}}}-{\frac{A{a}^{5}}{10\,{x}^{10}}}-{\frac{5\,{a}^{4}bA}{8\,{x}^{8}}}-{\frac{{a}^{5}B}{8\,{x}^{8}}}-{\frac{5\,a{b}^{4}A}{2\,{x}^{2}}}-5\,{\frac{{a}^{2}{b}^{3}B}{{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^5*(B*x^2+A)/x^11,x)

[Out]

1/2*b^5*B*x^2+A*ln(x)*b^5+5*B*ln(x)*a*b^4-5/3*a^3*b^2/x^6*A-5/6*a^4*b/x^6*B-5/2*
a^2*b^3/x^4*A-5/2*a^3*b^2/x^4*B-1/10*a^5*A/x^10-5/8*a^4/x^8*A*b-1/8*a^5/x^8*B-5/
2*a*b^4/x^2*A-5*a^2*b^3/x^2*B

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Maxima [A]  time = 1.33405, size = 166, normalized size = 1.47 \[ \frac{1}{2} \, B b^{5} x^{2} + \frac{1}{2} \,{\left (5 \, B a b^{4} + A b^{5}\right )} \log \left (x^{2}\right ) - \frac{300 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 300 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + 12 \, A a^{5} + 100 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 15 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{120 \, x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^5/x^11,x, algorithm="maxima")

[Out]

1/2*B*b^5*x^2 + 1/2*(5*B*a*b^4 + A*b^5)*log(x^2) - 1/120*(300*(2*B*a^2*b^3 + A*a
*b^4)*x^8 + 300*(B*a^3*b^2 + A*a^2*b^3)*x^6 + 12*A*a^5 + 100*(B*a^4*b + 2*A*a^3*
b^2)*x^4 + 15*(B*a^5 + 5*A*a^4*b)*x^2)/x^10

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Fricas [A]  time = 0.237817, size = 166, normalized size = 1.47 \[ \frac{60 \, B b^{5} x^{12} + 120 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} \log \left (x\right ) - 300 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} - 300 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} - 12 \, A a^{5} - 100 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} - 15 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{120 \, x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^5/x^11,x, algorithm="fricas")

[Out]

1/120*(60*B*b^5*x^12 + 120*(5*B*a*b^4 + A*b^5)*x^10*log(x) - 300*(2*B*a^2*b^3 +
A*a*b^4)*x^8 - 300*(B*a^3*b^2 + A*a^2*b^3)*x^6 - 12*A*a^5 - 100*(B*a^4*b + 2*A*a
^3*b^2)*x^4 - 15*(B*a^5 + 5*A*a^4*b)*x^2)/x^10

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Sympy [A]  time = 21.584, size = 122, normalized size = 1.08 \[ \frac{B b^{5} x^{2}}{2} + b^{4} \left (A b + 5 B a\right ) \log{\left (x \right )} - \frac{12 A a^{5} + x^{8} \left (300 A a b^{4} + 600 B a^{2} b^{3}\right ) + x^{6} \left (300 A a^{2} b^{3} + 300 B a^{3} b^{2}\right ) + x^{4} \left (200 A a^{3} b^{2} + 100 B a^{4} b\right ) + x^{2} \left (75 A a^{4} b + 15 B a^{5}\right )}{120 x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**5*(B*x**2+A)/x**11,x)

[Out]

B*b**5*x**2/2 + b**4*(A*b + 5*B*a)*log(x) - (12*A*a**5 + x**8*(300*A*a*b**4 + 60
0*B*a**2*b**3) + x**6*(300*A*a**2*b**3 + 300*B*a**3*b**2) + x**4*(200*A*a**3*b**
2 + 100*B*a**4*b) + x**2*(75*A*a**4*b + 15*B*a**5))/(120*x**10)

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GIAC/XCAS [A]  time = 0.227729, size = 198, normalized size = 1.75 \[ \frac{1}{2} \, B b^{5} x^{2} + \frac{1}{2} \,{\left (5 \, B a b^{4} + A b^{5}\right )}{\rm ln}\left (x^{2}\right ) - \frac{685 \, B a b^{4} x^{10} + 137 \, A b^{5} x^{10} + 600 \, B a^{2} b^{3} x^{8} + 300 \, A a b^{4} x^{8} + 300 \, B a^{3} b^{2} x^{6} + 300 \, A a^{2} b^{3} x^{6} + 100 \, B a^{4} b x^{4} + 200 \, A a^{3} b^{2} x^{4} + 15 \, B a^{5} x^{2} + 75 \, A a^{4} b x^{2} + 12 \, A a^{5}}{120 \, x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^5/x^11,x, algorithm="giac")

[Out]

1/2*B*b^5*x^2 + 1/2*(5*B*a*b^4 + A*b^5)*ln(x^2) - 1/120*(685*B*a*b^4*x^10 + 137*
A*b^5*x^10 + 600*B*a^2*b^3*x^8 + 300*A*a*b^4*x^8 + 300*B*a^3*b^2*x^6 + 300*A*a^2
*b^3*x^6 + 100*B*a^4*b*x^4 + 200*A*a^3*b^2*x^4 + 15*B*a^5*x^2 + 75*A*a^4*b*x^2 +
12*A*a^5)/x^10