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

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

[Out]

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

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Rubi [A]  time = 0.277694, 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 \[ -\frac{a^5 A}{4 x^4}-\frac{a^4 (a B+5 A b)}{2 x^2}+5 a^3 b \log (x) (a B+2 A b)+5 a^2 b^2 x^2 (a B+A b)+\frac{1}{6} b^4 x^6 (5 a B+A b)+\frac{5}{4} a b^3 x^4 (2 a B+A b)+\frac{1}{8} b^5 B x^8 \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0665043, size = 112, normalized size = 1. \[ -\frac{a^5 A}{4 x^4}-\frac{a^4 (a B+5 A b)}{2 x^2}+5 a^3 b \log (x) (a B+2 A b)+5 a^2 b^2 x^2 (a B+A b)+\frac{1}{6} b^4 x^6 (5 a B+A b)+\frac{5}{4} a b^3 x^4 (2 a B+A b)+\frac{1}{8} b^5 B x^8 \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 124, normalized size = 1.1 \[{\frac{{b}^{5}B{x}^{8}}{8}}+{\frac{A{x}^{6}{b}^{5}}{6}}+{\frac{5\,B{x}^{6}a{b}^{4}}{6}}+{\frac{5\,A{x}^{4}a{b}^{4}}{4}}+{\frac{5\,B{x}^{4}{a}^{2}{b}^{3}}{2}}+5\,A{x}^{2}{a}^{2}{b}^{3}+5\,B{x}^{2}{a}^{3}{b}^{2}+10\,A\ln \left ( x \right ){a}^{3}{b}^{2}+5\,B\ln \left ( x \right ){a}^{4}b-{\frac{A{a}^{5}}{4\,{x}^{4}}}-{\frac{5\,{a}^{4}bA}{2\,{x}^{2}}}-{\frac{{a}^{5}B}{2\,{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.36597, size = 165, normalized size = 1.47 \[ \frac{1}{8} \, B b^{5} x^{8} + \frac{1}{6} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{6} + \frac{5}{4} \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 5 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{2} + \frac{5}{2} \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} \log \left (x^{2}\right ) - \frac{A a^{5} + 2 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{4 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.22511, size = 166, normalized size = 1.48 \[ \frac{3 \, B b^{5} x^{12} + 4 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + 30 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 120 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} - 6 \, A a^{5} + 120 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} \log \left (x\right ) - 12 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{24 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 3.27077, size = 126, normalized size = 1.12 \[ \frac{B b^{5} x^{8}}{8} + 5 a^{3} b \left (2 A b + B a\right ) \log{\left (x \right )} + x^{6} \left (\frac{A b^{5}}{6} + \frac{5 B a b^{4}}{6}\right ) + x^{4} \left (\frac{5 A a b^{4}}{4} + \frac{5 B a^{2} b^{3}}{2}\right ) + x^{2} \left (5 A a^{2} b^{3} + 5 B a^{3} b^{2}\right ) - \frac{A a^{5} + x^{2} \left (10 A a^{4} b + 2 B a^{5}\right )}{4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.234827, size = 201, normalized size = 1.79 \[ \frac{1}{8} \, B b^{5} x^{8} + \frac{5}{6} \, B a b^{4} x^{6} + \frac{1}{6} \, A b^{5} x^{6} + \frac{5}{2} \, B a^{2} b^{3} x^{4} + \frac{5}{4} \, A a b^{4} x^{4} + 5 \, B a^{3} b^{2} x^{2} + 5 \, A a^{2} b^{3} x^{2} + \frac{5}{2} \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )}{\rm ln}\left (x^{2}\right ) - \frac{15 \, B a^{4} b x^{4} + 30 \, A a^{3} b^{2} x^{4} + 2 \, B a^{5} x^{2} + 10 \, A a^{4} b x^{2} + A a^{5}}{4 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*B*b^5*x^8 + 5/6*B*a*b^4*x^6 + 1/6*A*b^5*x^6 + 5/2*B*a^2*b^3*x^4 + 5/4*A*a*b^
4*x^4 + 5*B*a^3*b^2*x^2 + 5*A*a^2*b^3*x^2 + 5/2*(B*a^4*b + 2*A*a^3*b^2)*ln(x^2)
- 1/4*(15*B*a^4*b*x^4 + 30*A*a^3*b^2*x^4 + 2*B*a^5*x^2 + 10*A*a^4*b*x^2 + A*a^5)
/x^4