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

Optimal. Leaf size=54 \[ \frac{b^2 (c x)^{m+5}}{c^5 (m+5)}+\frac{2 b (c x)^{m+7}}{c^6 (m+7)}+\frac{(c x)^{m+9}}{c^7 (m+9)} \]

[Out]

(b^2*(c*x)^(5 + m))/(c^5*(5 + m)) + (2*b*(c*x)^(7 + m))/(c^6*(7 + m)) + (c*x)^(9
+ m)/(c^7*(9 + m))

_______________________________________________________________________________________

Rubi [A]  time = 0.0916882, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{b^2 (c x)^{m+5}}{c^5 (m+5)}+\frac{2 b (c x)^{m+7}}{c^6 (m+7)}+\frac{(c x)^{m+9}}{c^7 (m+9)} \]

Antiderivative was successfully verified.

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

[Out]

(b^2*(c*x)^(5 + m))/(c^5*(5 + m)) + (2*b*(c*x)^(7 + m))/(c^6*(7 + m)) + (c*x)^(9
+ m)/(c^7*(9 + m))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 18.2137, size = 46, normalized size = 0.85 \[ \frac{b^{2} \left (c x\right )^{m + 5}}{c^{5} \left (m + 5\right )} + \frac{2 b \left (c x\right )^{m + 7}}{c^{6} \left (m + 7\right )} + \frac{\left (c x\right )^{m + 9}}{c^{7} \left (m + 9\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b**2*(c*x)**(m + 5)/(c**5*(m + 5)) + 2*b*(c*x)**(m + 7)/(c**6*(m + 7)) + (c*x)**
(m + 9)/(c**7*(m + 9))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0333006, size = 43, normalized size = 0.8 \[ (c x)^m \left (\frac{b^2 x^5}{m+5}+\frac{2 b c x^7}{m+7}+\frac{c^2 x^9}{m+9}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(c*x)^m*((b^2*x^5)/(5 + m) + (2*b*c*x^7)/(7 + m) + (c^2*x^9)/(9 + m))

_______________________________________________________________________________________

Maple [A]  time = 0.008, size = 96, normalized size = 1.8 \[{\frac{ \left ( cx \right ) ^{m} \left ({c}^{2}{m}^{2}{x}^{4}+12\,{c}^{2}m{x}^{4}+2\,bc{m}^{2}{x}^{2}+35\,{c}^{2}{x}^{4}+28\,bcm{x}^{2}+{b}^{2}{m}^{2}+90\,bc{x}^{2}+16\,{b}^{2}m+63\,{b}^{2} \right ){x}^{5}}{ \left ( 9+m \right ) \left ( 7+m \right ) \left ( 5+m \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x)^m*(c*x^4+b*x^2)^2,x)

[Out]

(c*x)^m*(c^2*m^2*x^4+12*c^2*m*x^4+2*b*c*m^2*x^2+35*c^2*x^4+28*b*c*m*x^2+b^2*m^2+
90*b*c*x^2+16*b^2*m+63*b^2)*x^5/(9+m)/(7+m)/(5+m)

_______________________________________________________________________________________

Maxima [A]  time = 0.700437, size = 74, normalized size = 1.37 \[ \frac{c^{m + 2} x^{9} x^{m}}{m + 9} + \frac{2 \, b c^{m + 1} x^{7} x^{m}}{m + 7} + \frac{b^{2} c^{m} x^{5} x^{m}}{m + 5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c^(m + 2)*x^9*x^m/(m + 9) + 2*b*c^(m + 1)*x^7*x^m/(m + 7) + b^2*c^m*x^5*x^m/(m +
5)

_______________________________________________________________________________________

Fricas [A]  time = 0.271168, size = 120, normalized size = 2.22 \[ \frac{{\left ({\left (c^{2} m^{2} + 12 \, c^{2} m + 35 \, c^{2}\right )} x^{9} + 2 \,{\left (b c m^{2} + 14 \, b c m + 45 \, b c\right )} x^{7} +{\left (b^{2} m^{2} + 16 \, b^{2} m + 63 \, b^{2}\right )} x^{5}\right )} \left (c x\right )^{m}}{m^{3} + 21 \, m^{2} + 143 \, m + 315} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

((c^2*m^2 + 12*c^2*m + 35*c^2)*x^9 + 2*(b*c*m^2 + 14*b*c*m + 45*b*c)*x^7 + (b^2*
m^2 + 16*b^2*m + 63*b^2)*x^5)*(c*x)^m/(m^3 + 21*m^2 + 143*m + 315)

_______________________________________________________________________________________

Sympy [A]  time = 7.01663, size = 352, normalized size = 6.52 \[ \begin{cases} \frac{- \frac{b^{2}}{4 x^{4}} - \frac{b c}{x^{2}} + c^{2} \log{\left (x \right )}}{c^{9}} & \text{for}\: m = -9 \\\frac{- \frac{b^{2}}{2 x^{2}} + 2 b c \log{\left (x \right )} + \frac{c^{2} x^{2}}{2}}{c^{7}} & \text{for}\: m = -7 \\\frac{b^{2} \log{\left (x \right )} + b c x^{2} + \frac{c^{2} x^{4}}{4}}{c^{5}} & \text{for}\: m = -5 \\\frac{b^{2} c^{m} m^{2} x^{5} x^{m}}{m^{3} + 21 m^{2} + 143 m + 315} + \frac{16 b^{2} c^{m} m x^{5} x^{m}}{m^{3} + 21 m^{2} + 143 m + 315} + \frac{63 b^{2} c^{m} x^{5} x^{m}}{m^{3} + 21 m^{2} + 143 m + 315} + \frac{2 b c c^{m} m^{2} x^{7} x^{m}}{m^{3} + 21 m^{2} + 143 m + 315} + \frac{28 b c c^{m} m x^{7} x^{m}}{m^{3} + 21 m^{2} + 143 m + 315} + \frac{90 b c c^{m} x^{7} x^{m}}{m^{3} + 21 m^{2} + 143 m + 315} + \frac{c^{2} c^{m} m^{2} x^{9} x^{m}}{m^{3} + 21 m^{2} + 143 m + 315} + \frac{12 c^{2} c^{m} m x^{9} x^{m}}{m^{3} + 21 m^{2} + 143 m + 315} + \frac{35 c^{2} c^{m} x^{9} x^{m}}{m^{3} + 21 m^{2} + 143 m + 315} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise(((-b**2/(4*x**4) - b*c/x**2 + c**2*log(x))/c**9, Eq(m, -9)), ((-b**2/(
2*x**2) + 2*b*c*log(x) + c**2*x**2/2)/c**7, Eq(m, -7)), ((b**2*log(x) + b*c*x**2
+ c**2*x**4/4)/c**5, Eq(m, -5)), (b**2*c**m*m**2*x**5*x**m/(m**3 + 21*m**2 + 14
3*m + 315) + 16*b**2*c**m*m*x**5*x**m/(m**3 + 21*m**2 + 143*m + 315) + 63*b**2*c
**m*x**5*x**m/(m**3 + 21*m**2 + 143*m + 315) + 2*b*c*c**m*m**2*x**7*x**m/(m**3 +
21*m**2 + 143*m + 315) + 28*b*c*c**m*m*x**7*x**m/(m**3 + 21*m**2 + 143*m + 315)
+ 90*b*c*c**m*x**7*x**m/(m**3 + 21*m**2 + 143*m + 315) + c**2*c**m*m**2*x**9*x*
*m/(m**3 + 21*m**2 + 143*m + 315) + 12*c**2*c**m*m*x**9*x**m/(m**3 + 21*m**2 + 1
43*m + 315) + 35*c**2*c**m*x**9*x**m/(m**3 + 21*m**2 + 143*m + 315), True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.27412, size = 215, normalized size = 3.98 \[ \frac{c^{2} m^{2} x^{9} e^{\left (m{\rm ln}\left (c x\right )\right )} + 12 \, c^{2} m x^{9} e^{\left (m{\rm ln}\left (c x\right )\right )} + 2 \, b c m^{2} x^{7} e^{\left (m{\rm ln}\left (c x\right )\right )} + 35 \, c^{2} x^{9} e^{\left (m{\rm ln}\left (c x\right )\right )} + 28 \, b c m x^{7} e^{\left (m{\rm ln}\left (c x\right )\right )} + b^{2} m^{2} x^{5} e^{\left (m{\rm ln}\left (c x\right )\right )} + 90 \, b c x^{7} e^{\left (m{\rm ln}\left (c x\right )\right )} + 16 \, b^{2} m x^{5} e^{\left (m{\rm ln}\left (c x\right )\right )} + 63 \, b^{2} x^{5} e^{\left (m{\rm ln}\left (c x\right )\right )}}{m^{3} + 21 \, m^{2} + 143 \, m + 315} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(c^2*m^2*x^9*e^(m*ln(c*x)) + 12*c^2*m*x^9*e^(m*ln(c*x)) + 2*b*c*m^2*x^7*e^(m*ln(
c*x)) + 35*c^2*x^9*e^(m*ln(c*x)) + 28*b*c*m*x^7*e^(m*ln(c*x)) + b^2*m^2*x^5*e^(m
*ln(c*x)) + 90*b*c*x^7*e^(m*ln(c*x)) + 16*b^2*m*x^5*e^(m*ln(c*x)) + 63*b^2*x^5*e
^(m*ln(c*x)))/(m^3 + 21*m^2 + 143*m + 315)