3.41 \(\int \frac{\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^9} \, dx\)

Optimal. Leaf size=112 \[ -\frac{a^5 A}{8 x^8}-\frac{a^4 (a B+5 A b)}{6 x^6}-\frac{5 a^3 b (a B+2 A b)}{4 x^4}-\frac{5 a^2 b^2 (a B+A b)}{x^2}+\frac{1}{2} b^4 x^2 (5 a B+A b)+5 a b^3 \log (x) (2 a B+A b)+\frac{1}{4} b^5 B x^4 \]

[Out]

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

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Rubi [A]  time = 0.268635, antiderivative size = 112, 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 \[ -\frac{a^5 A}{8 x^8}-\frac{a^4 (a B+5 A b)}{6 x^6}-\frac{5 a^3 b (a B+2 A b)}{4 x^4}-\frac{5 a^2 b^2 (a B+A b)}{x^2}+\frac{1}{2} b^4 x^2 (5 a B+A b)+5 a b^3 \log (x) (2 a B+A b)+\frac{1}{4} b^5 B x^4 \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^2)^5*(A + B*x^2))/x^9,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**5*(B*x**2+A)/x**9,x)

[Out]

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

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Mathematica [A]  time = 0.103304, size = 116, normalized size = 1.04 \[ 5 a b^3 \log (x) (2 a B+A b)-\frac{a^5 \left (3 A+4 B x^2\right )+10 a^4 b x^2 \left (2 A+3 B x^2\right )+60 a^3 b^2 x^4 \left (A+2 B x^2\right )+120 a^2 A b^3 x^6-60 a b^4 B x^{10}-6 b^5 x^{10} \left (2 A+B x^2\right )}{24 x^8} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x^2)^5*(A + B*x^2))/x^9,x]

[Out]

-(120*a^2*A*b^3*x^6 - 60*a*b^4*B*x^10 - 6*b^5*x^10*(2*A + B*x^2) + 60*a^3*b^2*x^
4*(A + 2*B*x^2) + 10*a^4*b*x^2*(2*A + 3*B*x^2) + a^5*(3*A + 4*B*x^2))/(24*x^8) +
5*a*b^3*(A*b + 2*a*B)*Log[x]

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Maple [A]  time = 0.013, size = 124, normalized size = 1.1 \[{\frac{{b}^{5}B{x}^{4}}{4}}+{\frac{A{x}^{2}{b}^{5}}{2}}+{\frac{5\,B{x}^{2}a{b}^{4}}{2}}+5\,A\ln \left ( x \right ) a{b}^{4}+10\,B\ln \left ( x \right ){a}^{2}{b}^{3}-{\frac{5\,{a}^{4}bA}{6\,{x}^{6}}}-{\frac{{a}^{5}B}{6\,{x}^{6}}}-{\frac{5\,{a}^{3}{b}^{2}A}{2\,{x}^{4}}}-{\frac{5\,{a}^{4}bB}{4\,{x}^{4}}}-{\frac{A{a}^{5}}{8\,{x}^{8}}}-5\,{\frac{{a}^{2}{b}^{3}A}{{x}^{2}}}-5\,{\frac{{a}^{3}{b}^{2}B}{{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^5*(B*x^2+A)/x^9,x)

[Out]

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

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Maxima [A]  time = 1.35121, size = 166, normalized size = 1.48 \[ \frac{1}{4} \, B b^{5} x^{4} + \frac{1}{2} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{2} + \frac{5}{2} \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} \log \left (x^{2}\right ) - \frac{120 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + 3 \, A a^{5} + 30 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 4 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{24 \, x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^5/x^9,x, algorithm="maxima")

[Out]

1/4*B*b^5*x^4 + 1/2*(5*B*a*b^4 + A*b^5)*x^2 + 5/2*(2*B*a^2*b^3 + A*a*b^4)*log(x^
2) - 1/24*(120*(B*a^3*b^2 + A*a^2*b^3)*x^6 + 3*A*a^5 + 30*(B*a^4*b + 2*A*a^3*b^2
)*x^4 + 4*(B*a^5 + 5*A*a^4*b)*x^2)/x^8

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Fricas [A]  time = 0.223127, size = 166, normalized size = 1.48 \[ \frac{6 \, B b^{5} x^{12} + 12 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + 120 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} \log \left (x\right ) - 120 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} - 3 \, A a^{5} - 30 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} - 4 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{24 \, x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^5/x^9,x, algorithm="fricas")

[Out]

1/24*(6*B*b^5*x^12 + 12*(5*B*a*b^4 + A*b^5)*x^10 + 120*(2*B*a^2*b^3 + A*a*b^4)*x
^8*log(x) - 120*(B*a^3*b^2 + A*a^2*b^3)*x^6 - 3*A*a^5 - 30*(B*a^4*b + 2*A*a^3*b^
2)*x^4 - 4*(B*a^5 + 5*A*a^4*b)*x^2)/x^8

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Sympy [A]  time = 11.3661, size = 124, normalized size = 1.11 \[ \frac{B b^{5} x^{4}}{4} + 5 a b^{3} \left (A b + 2 B a\right ) \log{\left (x \right )} + x^{2} \left (\frac{A b^{5}}{2} + \frac{5 B a b^{4}}{2}\right ) - \frac{3 A a^{5} + x^{6} \left (120 A a^{2} b^{3} + 120 B a^{3} b^{2}\right ) + x^{4} \left (60 A a^{3} b^{2} + 30 B a^{4} b\right ) + x^{2} \left (20 A a^{4} b + 4 B a^{5}\right )}{24 x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**5*(B*x**2+A)/x**9,x)

[Out]

B*b**5*x**4/4 + 5*a*b**3*(A*b + 2*B*a)*log(x) + x**2*(A*b**5/2 + 5*B*a*b**4/2) -
(3*A*a**5 + x**6*(120*A*a**2*b**3 + 120*B*a**3*b**2) + x**4*(60*A*a**3*b**2 + 3
0*B*a**4*b) + x**2*(20*A*a**4*b + 4*B*a**5))/(24*x**8)

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GIAC/XCAS [A]  time = 0.226235, size = 203, normalized size = 1.81 \[ \frac{1}{4} \, B b^{5} x^{4} + \frac{5}{2} \, B a b^{4} x^{2} + \frac{1}{2} \, A b^{5} x^{2} + \frac{5}{2} \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )}{\rm ln}\left (x^{2}\right ) - \frac{250 \, B a^{2} b^{3} x^{8} + 125 \, A a b^{4} x^{8} + 120 \, B a^{3} b^{2} x^{6} + 120 \, A a^{2} b^{3} x^{6} + 30 \, B a^{4} b x^{4} + 60 \, A a^{3} b^{2} x^{4} + 4 \, B a^{5} x^{2} + 20 \, A a^{4} b x^{2} + 3 \, A a^{5}}{24 \, x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^5/x^9,x, algorithm="giac")

[Out]

1/4*B*b^5*x^4 + 5/2*B*a*b^4*x^2 + 1/2*A*b^5*x^2 + 5/2*(2*B*a^2*b^3 + A*a*b^4)*ln
(x^2) - 1/24*(250*B*a^2*b^3*x^8 + 125*A*a*b^4*x^8 + 120*B*a^3*b^2*x^6 + 120*A*a^
2*b^3*x^6 + 30*B*a^4*b*x^4 + 60*A*a^3*b^2*x^4 + 4*B*a^5*x^2 + 20*A*a^4*b*x^2 + 3
*A*a^5)/x^8