3.36 \(\int \frac{\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^4} \, dx\)

Optimal. Leaf size=108 \[ -\frac{a^5 A}{3 x^3}-\frac{a^4 (a B+5 A b)}{x}+5 a^3 b x (a B+2 A b)+\frac{10}{3} a^2 b^2 x^3 (a B+A b)+\frac{1}{7} b^4 x^7 (5 a B+A b)+a b^3 x^5 (2 a B+A b)+\frac{1}{9} b^5 B x^9 \]

[Out]

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

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Rubi [A]  time = 0.191129, 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 \[ -\frac{a^5 A}{3 x^3}-\frac{a^4 (a B+5 A b)}{x}+5 a^3 b x (a B+2 A b)+\frac{10}{3} a^2 b^2 x^3 (a B+A b)+\frac{1}{7} b^4 x^7 (5 a B+A b)+a b^3 x^5 (2 a B+A b)+\frac{1}{9} b^5 B x^9 \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^2)^5*(A + B*x^2))/x^4,x]

[Out]

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

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Rubi in Sympy [A]  time = 27.0943, size = 104, normalized size = 0.96 \[ - \frac{A a^{5}}{3 x^{3}} + \frac{B b^{5} x^{9}}{9} - \frac{a^{4} \left (5 A b + B a\right )}{x} + 5 a^{3} b x \left (2 A b + B a\right ) + \frac{10 a^{2} b^{2} x^{3} \left (A b + B a\right )}{3} + a b^{3} x^{5} \left (A b + 2 B a\right ) + \frac{b^{4} x^{7} \left (A b + 5 B a\right )}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**5*(B*x**2+A)/x**4,x)

[Out]

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

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Mathematica [A]  time = 0.062461, size = 110, normalized size = 1.02 \[ -\frac{a^5 A}{3 x^3}+5 a^3 b x (a B+2 A b)+\frac{10}{3} a^2 b^2 x^3 (a B+A b)+\frac{a^5 (-B)-5 a^4 A b}{x}+\frac{1}{7} b^4 x^7 (5 a B+A b)+a b^3 x^5 (2 a B+A b)+\frac{1}{9} b^5 B x^9 \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x^2)^5*(A + B*x^2))/x^4,x]

[Out]

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

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Maple [A]  time = 0.008, size = 118, normalized size = 1.1 \[{\frac{{b}^{5}B{x}^{9}}{9}}+{\frac{A{x}^{7}{b}^{5}}{7}}+{\frac{5\,B{x}^{7}a{b}^{4}}{7}}+A{x}^{5}a{b}^{4}+2\,B{x}^{5}{a}^{2}{b}^{3}+{\frac{10\,A{x}^{3}{a}^{2}{b}^{3}}{3}}+{\frac{10\,B{x}^{3}{a}^{3}{b}^{2}}{3}}+10\,Ax{a}^{3}{b}^{2}+5\,Bx{a}^{4}b-{\frac{A{a}^{5}}{3\,{x}^{3}}}-{\frac{{a}^{4} \left ( 5\,Ab+Ba \right ) }{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^5*(B*x^2+A)/x^4,x)

[Out]

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

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Maxima [A]  time = 1.34786, size = 159, normalized size = 1.47 \[ \frac{1}{9} \, B b^{5} x^{9} + \frac{1}{7} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{7} +{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{5} + \frac{10}{3} \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 5 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x - \frac{A a^{5} + 3 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^5/x^4,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.226258, size = 163, normalized size = 1.51 \[ \frac{7 \, B b^{5} x^{12} + 9 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + 63 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 210 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} - 21 \, A a^{5} + 315 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} - 63 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{63 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^5/x^4,x, algorithm="fricas")

[Out]

1/63*(7*B*b^5*x^12 + 9*(5*B*a*b^4 + A*b^5)*x^10 + 63*(2*B*a^2*b^3 + A*a*b^4)*x^8
+ 210*(B*a^3*b^2 + A*a^2*b^3)*x^6 - 21*A*a^5 + 315*(B*a^4*b + 2*A*a^3*b^2)*x^4
- 63*(B*a^5 + 5*A*a^4*b)*x^2)/x^3

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Sympy [A]  time = 2.02657, size = 126, normalized size = 1.17 \[ \frac{B b^{5} x^{9}}{9} + x^{7} \left (\frac{A b^{5}}{7} + \frac{5 B a b^{4}}{7}\right ) + x^{5} \left (A a b^{4} + 2 B a^{2} b^{3}\right ) + x^{3} \left (\frac{10 A a^{2} b^{3}}{3} + \frac{10 B a^{3} b^{2}}{3}\right ) + x \left (10 A a^{3} b^{2} + 5 B a^{4} b\right ) - \frac{A a^{5} + x^{2} \left (15 A a^{4} b + 3 B a^{5}\right )}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**5*(B*x**2+A)/x**4,x)

[Out]

B*b**5*x**9/9 + x**7*(A*b**5/7 + 5*B*a*b**4/7) + x**5*(A*a*b**4 + 2*B*a**2*b**3)
+ x**3*(10*A*a**2*b**3/3 + 10*B*a**3*b**2/3) + x*(10*A*a**3*b**2 + 5*B*a**4*b)
- (A*a**5 + x**2*(15*A*a**4*b + 3*B*a**5))/(3*x**3)

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GIAC/XCAS [A]  time = 0.244244, size = 165, normalized size = 1.53 \[ \frac{1}{9} \, B b^{5} x^{9} + \frac{5}{7} \, B a b^{4} x^{7} + \frac{1}{7} \, A b^{5} x^{7} + 2 \, B a^{2} b^{3} x^{5} + A a b^{4} x^{5} + \frac{10}{3} \, B a^{3} b^{2} x^{3} + \frac{10}{3} \, A a^{2} b^{3} x^{3} + 5 \, B a^{4} b x + 10 \, A a^{3} b^{2} x - \frac{3 \, B a^{5} x^{2} + 15 \, A a^{4} b x^{2} + A a^{5}}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^5/x^4,x, algorithm="giac")

[Out]

1/9*B*b^5*x^9 + 5/7*B*a*b^4*x^7 + 1/7*A*b^5*x^7 + 2*B*a^2*b^3*x^5 + A*a*b^4*x^5
+ 10/3*B*a^3*b^2*x^3 + 10/3*A*a^2*b^3*x^3 + 5*B*a^4*b*x + 10*A*a^3*b^2*x - 1/3*(
3*B*a^5*x^2 + 15*A*a^4*b*x^2 + A*a^5)/x^3