3.19 \(\int \frac{\left (A+B x^2\right ) \left (b x^2+c x^4\right )^2}{x^7} \, dx\)

Optimal. Leaf size=51 \[ -\frac{A b^2}{2 x^2}+\frac{1}{2} c x^2 (A c+2 b B)+b \log (x) (2 A c+b B)+\frac{1}{4} B c^2 x^4 \]

[Out]

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

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Rubi [A]  time = 0.152335, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{A b^2}{2 x^2}+\frac{1}{2} c x^2 (A c+2 b B)+b \log (x) (2 A c+b B)+\frac{1}{4} B c^2 x^4 \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x^2)*(b*x^2 + c*x^4)^2)/x^7,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x**2+A)*(c*x**4+b*x**2)**2/x**7,x)

[Out]

-A*b**2/(2*x**2) + B*c**2*Integral(x, (x, x**2))/2 + b*(2*A*c + B*b)*log(x**2)/2
+ c*(A*c + 2*B*b)*Integral(A, (x, x**2))/(2*A)

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Mathematica [A]  time = 0.0544585, size = 49, normalized size = 0.96 \[ \frac{1}{4} \left (-\frac{2 A b^2}{x^2}+2 c x^2 (A c+2 b B)+4 b \log (x) (2 A c+b B)+B c^2 x^4\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^2)/x^7,x]

[Out]

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

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Maple [A]  time = 0.009, size = 50, normalized size = 1. \[{\frac{B{c}^{2}{x}^{4}}{4}}+{\frac{A{x}^{2}{c}^{2}}{2}}+B{x}^{2}bc+2\,A\ln \left ( x \right ) bc+B{b}^{2}\ln \left ( x \right ) -{\frac{{b}^{2}A}{2\,{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^2+A)*(c*x^4+b*x^2)^2/x^7,x)

[Out]

1/4*B*c^2*x^4+1/2*A*x^2*c^2+B*x^2*b*c+2*A*ln(x)*b*c+B*b^2*ln(x)-1/2*A*b^2/x^2

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Maxima [A]  time = 1.37847, size = 70, normalized size = 1.37 \[ \frac{1}{4} \, B c^{2} x^{4} + \frac{1}{2} \,{\left (2 \, B b c + A c^{2}\right )} x^{2} + \frac{1}{2} \,{\left (B b^{2} + 2 \, A b c\right )} \log \left (x^{2}\right ) - \frac{A b^{2}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2)^2*(B*x^2 + A)/x^7,x, algorithm="maxima")

[Out]

1/4*B*c^2*x^4 + 1/2*(2*B*b*c + A*c^2)*x^2 + 1/2*(B*b^2 + 2*A*b*c)*log(x^2) - 1/2
*A*b^2/x^2

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Fricas [A]  time = 0.213112, size = 73, normalized size = 1.43 \[ \frac{B c^{2} x^{6} + 2 \,{\left (2 \, B b c + A c^{2}\right )} x^{4} + 4 \,{\left (B b^{2} + 2 \, A b c\right )} x^{2} \log \left (x\right ) - 2 \, A b^{2}}{4 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2)^2*(B*x^2 + A)/x^7,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.745577, size = 48, normalized size = 0.94 \[ - \frac{A b^{2}}{2 x^{2}} + \frac{B c^{2} x^{4}}{4} + b \left (2 A c + B b\right ) \log{\left (x \right )} + x^{2} \left (\frac{A c^{2}}{2} + B b c\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x**2+A)*(c*x**4+b*x**2)**2/x**7,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.208525, size = 95, normalized size = 1.86 \[ \frac{1}{4} \, B c^{2} x^{4} + B b c x^{2} + \frac{1}{2} \, A c^{2} x^{2} + \frac{1}{2} \,{\left (B b^{2} + 2 \, A b c\right )}{\rm ln}\left (x^{2}\right ) - \frac{B b^{2} x^{2} + 2 \, A b c x^{2} + A b^{2}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2)^2*(B*x^2 + A)/x^7,x, algorithm="giac")

[Out]

1/4*B*c^2*x^4 + B*b*c*x^2 + 1/2*A*c^2*x^2 + 1/2*(B*b^2 + 2*A*b*c)*ln(x^2) - 1/2*
(B*b^2*x^2 + 2*A*b*c*x^2 + A*b^2)/x^2