3.405 \(\int (c x)^m \left (b x^2+c x^4\right ) \, dx\)

Optimal. Leaf size=34 \[ \frac{b (c x)^{m+3}}{c^3 (m+3)}+\frac{(c x)^{m+5}}{c^4 (m+5)} \]

[Out]

(b*(c*x)^(3 + m))/(c^3*(3 + m)) + (c*x)^(5 + m)/(c^4*(5 + m))

_______________________________________________________________________________________

Rubi [A]  time = 0.0361338, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ \frac{b (c x)^{m+3}}{c^3 (m+3)}+\frac{(c x)^{m+5}}{c^4 (m+5)} \]

Antiderivative was successfully verified.

[In]  Int[(c*x)^m*(b*x^2 + c*x^4),x]

[Out]

(b*(c*x)^(3 + m))/(c^3*(3 + m)) + (c*x)^(5 + m)/(c^4*(5 + m))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 10.1091, size = 27, normalized size = 0.79 \[ \frac{b \left (c x\right )^{m + 3}}{c^{3} \left (m + 3\right )} + \frac{\left (c x\right )^{m + 5}}{c^{4} \left (m + 5\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b*(c*x)**(m + 3)/(c**3*(m + 3)) + (c*x)**(m + 5)/(c**4*(m + 5))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0275057, size = 27, normalized size = 0.79 \[ (c x)^m \left (\frac{b x^3}{m+3}+\frac{c x^5}{m+5}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(c*x)^m*((b*x^3)/(3 + m) + (c*x^5)/(5 + m))

_______________________________________________________________________________________

Maple [A]  time = 0.003, size = 39, normalized size = 1.2 \[{\frac{ \left ( cx \right ) ^{m} \left ( cm{x}^{2}+3\,c{x}^{2}+bm+5\,b \right ){x}^{3}}{ \left ( 5+m \right ) \left ( 3+m \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x)^m*(c*x^4+b*x^2),x)

[Out]

(c*x)^m*(c*m*x^2+3*c*x^2+b*m+5*b)*x^3/(5+m)/(3+m)

_______________________________________________________________________________________

Maxima [A]  time = 0.69625, size = 46, normalized size = 1.35 \[ \frac{c^{m + 1} x^{5} x^{m}}{m + 5} + \frac{b c^{m} x^{3} x^{m}}{m + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c^(m + 1)*x^5*x^m/(m + 5) + b*c^m*x^3*x^m/(m + 3)

_______________________________________________________________________________________

Fricas [A]  time = 0.271867, size = 53, normalized size = 1.56 \[ \frac{{\left ({\left (c m + 3 \, c\right )} x^{5} +{\left (b m + 5 \, b\right )} x^{3}\right )} \left (c x\right )^{m}}{m^{2} + 8 \, m + 15} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

((c*m + 3*c)*x^5 + (b*m + 5*b)*x^3)*(c*x)^m/(m^2 + 8*m + 15)

_______________________________________________________________________________________

Sympy [A]  time = 1.9643, size = 119, normalized size = 3.5 \[ \begin{cases} \frac{- \frac{b}{2 x^{2}} + c \log{\left (x \right )}}{c^{5}} & \text{for}\: m = -5 \\\frac{b \log{\left (x \right )} + \frac{c x^{2}}{2}}{c^{3}} & \text{for}\: m = -3 \\\frac{b c^{m} m x^{3} x^{m}}{m^{2} + 8 m + 15} + \frac{5 b c^{m} x^{3} x^{m}}{m^{2} + 8 m + 15} + \frac{c c^{m} m x^{5} x^{m}}{m^{2} + 8 m + 15} + \frac{3 c c^{m} x^{5} x^{m}}{m^{2} + 8 m + 15} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise(((-b/(2*x**2) + c*log(x))/c**5, Eq(m, -5)), ((b*log(x) + c*x**2/2)/c**
3, Eq(m, -3)), (b*c**m*m*x**3*x**m/(m**2 + 8*m + 15) + 5*b*c**m*x**3*x**m/(m**2
+ 8*m + 15) + c*c**m*m*x**5*x**m/(m**2 + 8*m + 15) + 3*c*c**m*x**5*x**m/(m**2 +
8*m + 15), True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.271009, size = 86, normalized size = 2.53 \[ \frac{c m x^{5} e^{\left (m{\rm ln}\left (c x\right )\right )} + 3 \, c x^{5} e^{\left (m{\rm ln}\left (c x\right )\right )} + b m x^{3} e^{\left (m{\rm ln}\left (c x\right )\right )} + 5 \, b x^{3} e^{\left (m{\rm ln}\left (c x\right )\right )}}{m^{2} + 8 \, m + 15} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(c*m*x^5*e^(m*ln(c*x)) + 3*c*x^5*e^(m*ln(c*x)) + b*m*x^3*e^(m*ln(c*x)) + 5*b*x^3
*e^(m*ln(c*x)))/(m^2 + 8*m + 15)