3.45 \(\int \frac{\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^{13}} \, dx\)

Optimal. Leaf size=91 \[ -\frac{a^5 B}{10 x^{10}}-\frac{5 a^4 b B}{8 x^8}-\frac{5 a^3 b^2 B}{3 x^6}-\frac{5 a^2 b^3 B}{2 x^4}-\frac{A \left (a+b x^2\right )^6}{12 a x^{12}}-\frac{5 a b^4 B}{2 x^2}+b^5 B \log (x) \]

[Out]

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

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Rubi [A]  time = 0.141153, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ -\frac{a^5 B}{10 x^{10}}-\frac{5 a^4 b B}{8 x^8}-\frac{5 a^3 b^2 B}{3 x^6}-\frac{5 a^2 b^3 B}{2 x^4}-\frac{A \left (a+b x^2\right )^6}{12 a x^{12}}-\frac{5 a b^4 B}{2 x^2}+b^5 B \log (x) \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^2)^5*(A + B*x^2))/x^13,x]

[Out]

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

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Rubi in Sympy [A]  time = 21.5749, size = 95, normalized size = 1.04 \[ - \frac{A \left (a + b x^{2}\right )^{6}}{12 a x^{12}} - \frac{B a^{5}}{10 x^{10}} - \frac{5 B a^{4} b}{8 x^{8}} - \frac{5 B a^{3} b^{2}}{3 x^{6}} - \frac{5 B a^{2} b^{3}}{2 x^{4}} - \frac{5 B a b^{4}}{2 x^{2}} + \frac{B b^{5} \log{\left (x^{2} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**5*(B*x**2+A)/x**13,x)

[Out]

-A*(a + b*x**2)**6/(12*a*x**12) - B*a**5/(10*x**10) - 5*B*a**4*b/(8*x**8) - 5*B*
a**3*b**2/(3*x**6) - 5*B*a**2*b**3/(2*x**4) - 5*B*a*b**4/(2*x**2) + B*b**5*log(x
**2)/2

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Mathematica [A]  time = 0.120187, size = 118, normalized size = 1.3 \[ b^5 B \log (x)-\frac{2 a^5 \left (5 A+6 B x^2\right )+15 a^4 b x^2 \left (4 A+5 B x^2\right )+50 a^3 b^2 x^4 \left (3 A+4 B x^2\right )+100 a^2 b^3 x^6 \left (2 A+3 B x^2\right )+150 a b^4 x^8 \left (A+2 B x^2\right )+60 A b^5 x^{10}}{120 x^{12}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x^2)^5*(A + B*x^2))/x^13,x]

[Out]

-(60*A*b^5*x^10 + 150*a*b^4*x^8*(A + 2*B*x^2) + 100*a^2*b^3*x^6*(2*A + 3*B*x^2)
+ 50*a^3*b^2*x^4*(3*A + 4*B*x^2) + 15*a^4*b*x^2*(4*A + 5*B*x^2) + 2*a^5*(5*A + 6
*B*x^2))/(120*x^12) + b^5*B*Log[x]

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Maple [A]  time = 0.011, size = 124, normalized size = 1.4 \[ -{\frac{A{a}^{5}}{12\,{x}^{12}}}+{b}^{5}B\ln \left ( x \right ) -{\frac{5\,{a}^{2}{b}^{3}A}{3\,{x}^{6}}}-{\frac{5\,{a}^{3}{b}^{2}B}{3\,{x}^{6}}}-{\frac{5\,a{b}^{4}A}{4\,{x}^{4}}}-{\frac{5\,{a}^{2}{b}^{3}B}{2\,{x}^{4}}}-{\frac{{a}^{4}bA}{2\,{x}^{10}}}-{\frac{{a}^{5}B}{10\,{x}^{10}}}-{\frac{5\,{a}^{3}{b}^{2}A}{4\,{x}^{8}}}-{\frac{5\,{a}^{4}bB}{8\,{x}^{8}}}-{\frac{{b}^{5}A}{2\,{x}^{2}}}-{\frac{5\,a{b}^{4}B}{2\,{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^5*(B*x^2+A)/x^13,x)

[Out]

-1/12*A*a^5/x^12+b^5*B*ln(x)-5/3*a^2*b^3/x^6*A-5/3*a^3*b^2*B/x^6-5/4*a*b^4/x^4*A
-5/2*a^2*b^3*B/x^4-1/2*a^4/x^10*A*b-1/10*a^5*B/x^10-5/4*a^3*b^2/x^8*A-5/8*a^4*b*
B/x^8-1/2*b^5/x^2*A-5/2*a*b^4*B/x^2

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Maxima [A]  time = 1.3423, size = 166, normalized size = 1.82 \[ \frac{1}{2} \, B b^{5} \log \left (x^{2}\right ) - \frac{60 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + 150 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 200 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + 10 \, A a^{5} + 75 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 12 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{120 \, x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^5/x^13,x, algorithm="maxima")

[Out]

1/2*B*b^5*log(x^2) - 1/120*(60*(5*B*a*b^4 + A*b^5)*x^10 + 150*(2*B*a^2*b^3 + A*a
*b^4)*x^8 + 200*(B*a^3*b^2 + A*a^2*b^3)*x^6 + 10*A*a^5 + 75*(B*a^4*b + 2*A*a^3*b
^2)*x^4 + 12*(B*a^5 + 5*A*a^4*b)*x^2)/x^12

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Fricas [A]  time = 0.246158, size = 166, normalized size = 1.82 \[ \frac{120 \, B b^{5} x^{12} \log \left (x\right ) - 60 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} - 150 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} - 200 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} - 10 \, A a^{5} - 75 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} - 12 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{120 \, x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^5/x^13,x, algorithm="fricas")

[Out]

1/120*(120*B*b^5*x^12*log(x) - 60*(5*B*a*b^4 + A*b^5)*x^10 - 150*(2*B*a^2*b^3 +
A*a*b^4)*x^8 - 200*(B*a^3*b^2 + A*a^2*b^3)*x^6 - 10*A*a^5 - 75*(B*a^4*b + 2*A*a^
3*b^2)*x^4 - 12*(B*a^5 + 5*A*a^4*b)*x^2)/x^12

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Sympy [A]  time = 43.0222, size = 124, normalized size = 1.36 \[ B b^{5} \log{\left (x \right )} - \frac{10 A a^{5} + x^{10} \left (60 A b^{5} + 300 B a b^{4}\right ) + x^{8} \left (150 A a b^{4} + 300 B a^{2} b^{3}\right ) + x^{6} \left (200 A a^{2} b^{3} + 200 B a^{3} b^{2}\right ) + x^{4} \left (150 A a^{3} b^{2} + 75 B a^{4} b\right ) + x^{2} \left (60 A a^{4} b + 12 B a^{5}\right )}{120 x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**5*(B*x**2+A)/x**13,x)

[Out]

B*b**5*log(x) - (10*A*a**5 + x**10*(60*A*b**5 + 300*B*a*b**4) + x**8*(150*A*a*b*
*4 + 300*B*a**2*b**3) + x**6*(200*A*a**2*b**3 + 200*B*a**3*b**2) + x**4*(150*A*a
**3*b**2 + 75*B*a**4*b) + x**2*(60*A*a**4*b + 12*B*a**5))/(120*x**12)

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GIAC/XCAS [A]  time = 0.228281, size = 186, normalized size = 2.04 \[ \frac{1}{2} \, B b^{5}{\rm ln}\left (x^{2}\right ) - \frac{147 \, B b^{5} x^{12} + 300 \, B a b^{4} x^{10} + 60 \, A b^{5} x^{10} + 300 \, B a^{2} b^{3} x^{8} + 150 \, A a b^{4} x^{8} + 200 \, B a^{3} b^{2} x^{6} + 200 \, A a^{2} b^{3} x^{6} + 75 \, B a^{4} b x^{4} + 150 \, A a^{3} b^{2} x^{4} + 12 \, B a^{5} x^{2} + 60 \, A a^{4} b x^{2} + 10 \, A a^{5}}{120 \, x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^5/x^13,x, algorithm="giac")

[Out]

1/2*B*b^5*ln(x^2) - 1/120*(147*B*b^5*x^12 + 300*B*a*b^4*x^10 + 60*A*b^5*x^10 + 3
00*B*a^2*b^3*x^8 + 150*A*a*b^4*x^8 + 200*B*a^3*b^2*x^6 + 200*A*a^2*b^3*x^6 + 75*
B*a^4*b*x^4 + 150*A*a^3*b^2*x^4 + 12*B*a^5*x^2 + 60*A*a^4*b*x^2 + 10*A*a^5)/x^12