3.177 \(\int \frac{x^7}{b x^2+c x^4} \, dx\)

Optimal. Leaf size=40 \[ \frac{b^2 \log \left (b+c x^2\right )}{2 c^3}-\frac{b x^2}{2 c^2}+\frac{x^4}{4 c} \]

[Out]

-(b*x^2)/(2*c^2) + x^4/(4*c) + (b^2*Log[b + c*x^2])/(2*c^3)

_______________________________________________________________________________________

Rubi [A]  time = 0.0743586, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{b^2 \log \left (b+c x^2\right )}{2 c^3}-\frac{b x^2}{2 c^2}+\frac{x^4}{4 c} \]

Antiderivative was successfully verified.

[In]  Int[x^7/(b*x^2 + c*x^4),x]

[Out]

-(b*x^2)/(2*c^2) + x^4/(4*c) + (b^2*Log[b + c*x^2])/(2*c^3)

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{b^{2} \log{\left (b + c x^{2} \right )}}{2 c^{3}} + \frac{\int ^{x^{2}} x\, dx}{2 c} - \frac{\int ^{x^{2}} b\, dx}{2 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b**2*log(b + c*x**2)/(2*c**3) + Integral(x, (x, x**2))/(2*c) - Integral(b, (x, x
**2))/(2*c**2)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0093579, size = 40, normalized size = 1. \[ \frac{b^2 \log \left (b+c x^2\right )}{2 c^3}-\frac{b x^2}{2 c^2}+\frac{x^4}{4 c} \]

Antiderivative was successfully verified.

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

[Out]

-(b*x^2)/(2*c^2) + x^4/(4*c) + (b^2*Log[b + c*x^2])/(2*c^3)

_______________________________________________________________________________________

Maple [A]  time = 0.004, size = 35, normalized size = 0.9 \[ -{\frac{b{x}^{2}}{2\,{c}^{2}}}+{\frac{{x}^{4}}{4\,c}}+{\frac{{b}^{2}\ln \left ( c{x}^{2}+b \right ) }{2\,{c}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7/(c*x^4+b*x^2),x)

[Out]

-1/2*b*x^2/c^2+1/4*x^4/c+1/2*b^2*ln(c*x^2+b)/c^3

_______________________________________________________________________________________

Maxima [A]  time = 0.688344, size = 46, normalized size = 1.15 \[ \frac{b^{2} \log \left (c x^{2} + b\right )}{2 \, c^{3}} + \frac{c x^{4} - 2 \, b x^{2}}{4 \, c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*b^2*log(c*x^2 + b)/c^3 + 1/4*(c*x^4 - 2*b*x^2)/c^2

_______________________________________________________________________________________

Fricas [A]  time = 0.251709, size = 45, normalized size = 1.12 \[ \frac{c^{2} x^{4} - 2 \, b c x^{2} + 2 \, b^{2} \log \left (c x^{2} + b\right )}{4 \, c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(c^2*x^4 - 2*b*c*x^2 + 2*b^2*log(c*x^2 + b))/c^3

_______________________________________________________________________________________

Sympy [A]  time = 1.2139, size = 32, normalized size = 0.8 \[ \frac{b^{2} \log{\left (b + c x^{2} \right )}}{2 c^{3}} - \frac{b x^{2}}{2 c^{2}} + \frac{x^{4}}{4 c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b**2*log(b + c*x**2)/(2*c**3) - b*x**2/(2*c**2) + x**4/(4*c)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.270056, size = 47, normalized size = 1.18 \[ \frac{b^{2}{\rm ln}\left ({\left | c x^{2} + b \right |}\right )}{2 \, c^{3}} + \frac{c x^{4} - 2 \, b x^{2}}{4 \, c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*b^2*ln(abs(c*x^2 + b))/c^3 + 1/4*(c*x^4 - 2*b*x^2)/c^2