3.20 \(\int \frac{\left (a+b x^2\right )^2 \left (A+B x^2\right )}{x^7} \, dx\)

Optimal. Leaf size=51 \[ -\frac{a^2 A}{6 x^6}-\frac{a (a B+2 A b)}{4 x^4}-\frac{b (2 a B+A b)}{2 x^2}+b^2 B \log (x) \]

[Out]

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

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Rubi [A]  time = 0.0988146, antiderivative size = 51, 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 \[ -\frac{a^2 A}{6 x^6}-\frac{a (a B+2 A b)}{4 x^4}-\frac{b (2 a B+A b)}{2 x^2}+b^2 B \log (x) \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^2)^2*(A + B*x^2))/x^7,x]

[Out]

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

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Rubi in Sympy [A]  time = 15.6329, size = 51, normalized size = 1. \[ - \frac{A a^{2}}{6 x^{6}} + \frac{B b^{2} \log{\left (x^{2} \right )}}{2} - \frac{a \left (2 A b + B a\right )}{4 x^{4}} - \frac{b \left (A b + 2 B a\right )}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**2*(B*x**2+A)/x**7,x)

[Out]

-A*a**2/(6*x**6) + B*b**2*log(x**2)/2 - a*(2*A*b + B*a)/(4*x**4) - b*(A*b + 2*B*
a)/(2*x**2)

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Mathematica [A]  time = 0.0482106, size = 54, normalized size = 1.06 \[ b^2 B \log (x)-\frac{a^2 \left (2 A+3 B x^2\right )+6 a b x^2 \left (A+2 B x^2\right )+6 A b^2 x^4}{12 x^6} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x^2)^2*(A + B*x^2))/x^7,x]

[Out]

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

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Maple [A]  time = 0.009, size = 52, normalized size = 1. \[{b}^{2}B\ln \left ( x \right ) -{\frac{A{a}^{2}}{6\,{x}^{6}}}-{\frac{abA}{2\,{x}^{4}}}-{\frac{{a}^{2}B}{4\,{x}^{4}}}-{\frac{{b}^{2}A}{2\,{x}^{2}}}-{\frac{abB}{{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^2*(B*x^2+A)/x^7,x)

[Out]

b^2*B*ln(x)-1/6*a^2*A/x^6-1/2*a/x^4*A*b-1/4*a^2/x^4*B-1/2*b^2/x^2*A-b/x^2*B*a

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Maxima [A]  time = 1.34602, size = 74, normalized size = 1.45 \[ \frac{1}{2} \, B b^{2} \log \left (x^{2}\right ) - \frac{6 \,{\left (2 \, B a b + A b^{2}\right )} x^{4} + 2 \, A a^{2} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} x^{2}}{12 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^2/x^7,x, algorithm="maxima")

[Out]

1/2*B*b^2*log(x^2) - 1/12*(6*(2*B*a*b + A*b^2)*x^4 + 2*A*a^2 + 3*(B*a^2 + 2*A*a*
b)*x^2)/x^6

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Fricas [A]  time = 0.238589, size = 74, normalized size = 1.45 \[ \frac{12 \, B b^{2} x^{6} \log \left (x\right ) - 6 \,{\left (2 \, B a b + A b^{2}\right )} x^{4} - 2 \, A a^{2} - 3 \,{\left (B a^{2} + 2 \, A a b\right )} x^{2}}{12 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^2/x^7,x, algorithm="fricas")

[Out]

1/12*(12*B*b^2*x^6*log(x) - 6*(2*B*a*b + A*b^2)*x^4 - 2*A*a^2 - 3*(B*a^2 + 2*A*a
*b)*x^2)/x^6

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Sympy [A]  time = 4.57595, size = 53, normalized size = 1.04 \[ B b^{2} \log{\left (x \right )} - \frac{2 A a^{2} + x^{4} \left (6 A b^{2} + 12 B a b\right ) + x^{2} \left (6 A a b + 3 B a^{2}\right )}{12 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**2*(B*x**2+A)/x**7,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.229301, size = 89, normalized size = 1.75 \[ \frac{1}{2} \, B b^{2}{\rm ln}\left (x^{2}\right ) - \frac{11 \, B b^{2} x^{6} + 12 \, B a b x^{4} + 6 \, A b^{2} x^{4} + 3 \, B a^{2} x^{2} + 6 \, A a b x^{2} + 2 \, A a^{2}}{12 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^2/x^7,x, algorithm="giac")

[Out]

1/2*B*b^2*ln(x^2) - 1/12*(11*B*b^2*x^6 + 12*B*a*b*x^4 + 6*A*b^2*x^4 + 3*B*a^2*x^
2 + 6*A*a*b*x^2 + 2*A*a^2)/x^6