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

Optimal. Leaf size=108 \[ -\frac{a^5 A}{x}+a^4 x (a B+5 A b)+\frac{5}{3} a^3 b x^3 (a B+2 A b)+2 a^2 b^2 x^5 (a B+A b)+\frac{1}{9} b^4 x^9 (5 a B+A b)+\frac{5}{7} a b^3 x^7 (2 a B+A b)+\frac{1}{11} b^5 B x^{11} \]

[Out]

-((a^5*A)/x) + a^4*(5*A*b + a*B)*x + (5*a^3*b*(2*A*b + a*B)*x^3)/3 + 2*a^2*b^2*(
A*b + a*B)*x^5 + (5*a*b^3*(A*b + 2*a*B)*x^7)/7 + (b^4*(A*b + 5*a*B)*x^9)/9 + (b^
5*B*x^11)/11

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Rubi [A]  time = 0.185105, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ -\frac{a^5 A}{x}+a^4 x (a B+5 A b)+\frac{5}{3} a^3 b x^3 (a B+2 A b)+2 a^2 b^2 x^5 (a B+A b)+\frac{1}{9} b^4 x^9 (5 a B+A b)+\frac{5}{7} a b^3 x^7 (2 a B+A b)+\frac{1}{11} b^5 B x^{11} \]

Antiderivative was successfully verified.

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

[Out]

-((a^5*A)/x) + a^4*(5*A*b + a*B)*x + (5*a^3*b*(2*A*b + a*B)*x^3)/3 + 2*a^2*b^2*(
A*b + a*B)*x^5 + (5*a*b^3*(A*b + 2*a*B)*x^7)/7 + (b^4*(A*b + 5*a*B)*x^9)/9 + (b^
5*B*x^11)/11

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{A a^{5}}{x} + \frac{B b^{5} x^{11}}{11} + \frac{5 a^{3} b x^{3} \left (2 A b + B a\right )}{3} + 2 a^{2} b^{2} x^{5} \left (A b + B a\right ) + \frac{5 a b^{3} x^{7} \left (A b + 2 B a\right )}{7} + \frac{b^{4} x^{9} \left (A b + 5 B a\right )}{9} + \frac{a^{4} \left (5 A b + B a\right ) \int B\, dx}{B} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A*a**5/x + B*b**5*x**11/11 + 5*a**3*b*x**3*(2*A*b + B*a)/3 + 2*a**2*b**2*x**5*(
A*b + B*a) + 5*a*b**3*x**7*(A*b + 2*B*a)/7 + b**4*x**9*(A*b + 5*B*a)/9 + a**4*(5
*A*b + B*a)*Integral(B, x)/B

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Mathematica [A]  time = 0.0541114, size = 108, normalized size = 1. \[ -\frac{a^5 A}{x}+a^4 x (a B+5 A b)+\frac{5}{3} a^3 b x^3 (a B+2 A b)+2 a^2 b^2 x^5 (a B+A b)+\frac{1}{9} b^4 x^9 (5 a B+A b)+\frac{5}{7} a b^3 x^7 (2 a B+A b)+\frac{1}{11} b^5 B x^{11} \]

Antiderivative was successfully verified.

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

[Out]

-((a^5*A)/x) + a^4*(5*A*b + a*B)*x + (5*a^3*b*(2*A*b + a*B)*x^3)/3 + 2*a^2*b^2*(
A*b + a*B)*x^5 + (5*a*b^3*(A*b + 2*a*B)*x^7)/7 + (b^4*(A*b + 5*a*B)*x^9)/9 + (b^
5*B*x^11)/11

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Maple [A]  time = 0.006, size = 121, normalized size = 1.1 \[{\frac{{b}^{5}B{x}^{11}}{11}}+{\frac{A{x}^{9}{b}^{5}}{9}}+{\frac{5\,B{x}^{9}a{b}^{4}}{9}}+{\frac{5\,A{x}^{7}a{b}^{4}}{7}}+{\frac{10\,B{x}^{7}{a}^{2}{b}^{3}}{7}}+2\,A{x}^{5}{a}^{2}{b}^{3}+2\,B{x}^{5}{a}^{3}{b}^{2}+{\frac{10\,A{x}^{3}{a}^{3}{b}^{2}}{3}}+{\frac{5\,B{x}^{3}{a}^{4}b}{3}}+5\,Ax{a}^{4}b+Bx{a}^{5}-{\frac{A{a}^{5}}{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/11*b^5*B*x^11+1/9*A*x^9*b^5+5/9*B*x^9*a*b^4+5/7*A*x^7*a*b^4+10/7*B*x^7*a^2*b^3
+2*A*x^5*a^2*b^3+2*B*x^5*a^3*b^2+10/3*A*x^3*a^3*b^2+5/3*B*x^3*a^4*b+5*A*x*a^4*b+
B*x*a^5-a^5*A/x

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Maxima [A]  time = 1.34748, size = 157, normalized size = 1.45 \[ \frac{1}{11} \, B b^{5} x^{11} + \frac{1}{9} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{9} + \frac{5}{7} \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{7} + 2 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{5} - \frac{A a^{5}}{x} + \frac{5}{3} \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{3} +{\left (B a^{5} + 5 \, A a^{4} b\right )} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/11*B*b^5*x^11 + 1/9*(5*B*a*b^4 + A*b^5)*x^9 + 5/7*(2*B*a^2*b^3 + A*a*b^4)*x^7
+ 2*(B*a^3*b^2 + A*a^2*b^3)*x^5 - A*a^5/x + 5/3*(B*a^4*b + 2*A*a^3*b^2)*x^3 + (B
*a^5 + 5*A*a^4*b)*x

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Fricas [A]  time = 0.22893, size = 163, normalized size = 1.51 \[ \frac{63 \, B b^{5} x^{12} + 77 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + 495 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 1386 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} - 693 \, A a^{5} + 1155 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 693 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{693 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/693*(63*B*b^5*x^12 + 77*(5*B*a*b^4 + A*b^5)*x^10 + 495*(2*B*a^2*b^3 + A*a*b^4)
*x^8 + 1386*(B*a^3*b^2 + A*a^2*b^3)*x^6 - 693*A*a^5 + 1155*(B*a^4*b + 2*A*a^3*b^
2)*x^4 + 693*(B*a^5 + 5*A*a^4*b)*x^2)/x

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Sympy [A]  time = 1.56495, size = 126, normalized size = 1.17 \[ - \frac{A a^{5}}{x} + \frac{B b^{5} x^{11}}{11} + x^{9} \left (\frac{A b^{5}}{9} + \frac{5 B a b^{4}}{9}\right ) + x^{7} \left (\frac{5 A a b^{4}}{7} + \frac{10 B a^{2} b^{3}}{7}\right ) + x^{5} \left (2 A a^{2} b^{3} + 2 B a^{3} b^{2}\right ) + x^{3} \left (\frac{10 A a^{3} b^{2}}{3} + \frac{5 B a^{4} b}{3}\right ) + x \left (5 A a^{4} b + B a^{5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A*a**5/x + B*b**5*x**11/11 + x**9*(A*b**5/9 + 5*B*a*b**4/9) + x**7*(5*A*a*b**4/
7 + 10*B*a**2*b**3/7) + x**5*(2*A*a**2*b**3 + 2*B*a**3*b**2) + x**3*(10*A*a**3*b
**2/3 + 5*B*a**4*b/3) + x*(5*A*a**4*b + B*a**5)

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GIAC/XCAS [A]  time = 0.226972, size = 162, normalized size = 1.5 \[ \frac{1}{11} \, B b^{5} x^{11} + \frac{5}{9} \, B a b^{4} x^{9} + \frac{1}{9} \, A b^{5} x^{9} + \frac{10}{7} \, B a^{2} b^{3} x^{7} + \frac{5}{7} \, A a b^{4} x^{7} + 2 \, B a^{3} b^{2} x^{5} + 2 \, A a^{2} b^{3} x^{5} + \frac{5}{3} \, B a^{4} b x^{3} + \frac{10}{3} \, A a^{3} b^{2} x^{3} + B a^{5} x + 5 \, A a^{4} b x - \frac{A a^{5}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/11*B*b^5*x^11 + 5/9*B*a*b^4*x^9 + 1/9*A*b^5*x^9 + 10/7*B*a^2*b^3*x^7 + 5/7*A*a
*b^4*x^7 + 2*B*a^3*b^2*x^5 + 2*A*a^2*b^3*x^5 + 5/3*B*a^4*b*x^3 + 10/3*A*a^3*b^2*
x^3 + B*a^5*x + 5*A*a^4*b*x - A*a^5/x