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

Optimal. Leaf size=71 \[ \frac{A b^2 x^{m+5}}{m+5}+\frac{b x^{m+7} (2 A c+b B)}{m+7}+\frac{c x^{m+9} (A c+2 b B)}{m+9}+\frac{B c^2 x^{m+11}}{m+11} \]

[Out]

(A*b^2*x^(5 + m))/(5 + m) + (b*(b*B + 2*A*c)*x^(7 + m))/(7 + m) + (c*(2*b*B + A*
c)*x^(9 + m))/(9 + m) + (B*c^2*x^(11 + m))/(11 + m)

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Rubi [A]  time = 0.140596, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{A b^2 x^{m+5}}{m+5}+\frac{b x^{m+7} (2 A c+b B)}{m+7}+\frac{c x^{m+9} (A c+2 b B)}{m+9}+\frac{B c^2 x^{m+11}}{m+11} \]

Antiderivative was successfully verified.

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

[Out]

(A*b^2*x^(5 + m))/(5 + m) + (b*(b*B + 2*A*c)*x^(7 + m))/(7 + m) + (c*(2*b*B + A*
c)*x^(9 + m))/(9 + m) + (B*c^2*x^(11 + m))/(11 + m)

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Rubi in Sympy [A]  time = 22.2144, size = 63, normalized size = 0.89 \[ \frac{A b^{2} x^{m + 5}}{m + 5} + \frac{B c^{2} x^{m + 11}}{m + 11} + \frac{b x^{m + 7} \left (2 A c + B b\right )}{m + 7} + \frac{c x^{m + 9} \left (A c + 2 B b\right )}{m + 9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*b**2*x**(m + 5)/(m + 5) + B*c**2*x**(m + 11)/(m + 11) + b*x**(m + 7)*(2*A*c +
B*b)/(m + 7) + c*x**(m + 9)*(A*c + 2*B*b)/(m + 9)

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Mathematica [A]  time = 0.0720595, size = 67, normalized size = 0.94 \[ x^m \left (\frac{A b^2 x^5}{m+5}+\frac{c x^9 (A c+2 b B)}{m+9}+\frac{b x^7 (2 A c+b B)}{m+7}+\frac{B c^2 x^{11}}{m+11}\right ) \]

Antiderivative was successfully verified.

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

[Out]

x^m*((A*b^2*x^5)/(5 + m) + (b*(b*B + 2*A*c)*x^7)/(7 + m) + (c*(2*b*B + A*c)*x^9)
/(9 + m) + (B*c^2*x^11)/(11 + m))

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Maple [B]  time = 0.009, size = 262, normalized size = 3.7 \[{\frac{{x}^{5+m} \left ( B{c}^{2}{m}^{3}{x}^{6}+21\,B{c}^{2}{m}^{2}{x}^{6}+A{c}^{2}{m}^{3}{x}^{4}+2\,Bbc{m}^{3}{x}^{4}+143\,B{c}^{2}m{x}^{6}+23\,A{c}^{2}{m}^{2}{x}^{4}+46\,Bbc{m}^{2}{x}^{4}+315\,B{c}^{2}{x}^{6}+2\,Abc{m}^{3}{x}^{2}+167\,A{c}^{2}m{x}^{4}+B{b}^{2}{m}^{3}{x}^{2}+334\,Bbcm{x}^{4}+50\,Abc{m}^{2}{x}^{2}+385\,A{c}^{2}{x}^{4}+25\,B{b}^{2}{m}^{2}{x}^{2}+770\,B{x}^{4}bc+A{b}^{2}{m}^{3}+398\,Abcm{x}^{2}+199\,B{b}^{2}m{x}^{2}+27\,A{b}^{2}{m}^{2}+990\,Abc{x}^{2}+495\,B{b}^{2}{x}^{2}+239\,A{b}^{2}m+693\,{b}^{2}A \right ) }{ \left ( 11+m \right ) \left ( 9+m \right ) \left ( 7+m \right ) \left ( 5+m \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x^(5+m)*(B*c^2*m^3*x^6+21*B*c^2*m^2*x^6+A*c^2*m^3*x^4+2*B*b*c*m^3*x^4+143*B*c^2*
m*x^6+23*A*c^2*m^2*x^4+46*B*b*c*m^2*x^4+315*B*c^2*x^6+2*A*b*c*m^3*x^2+167*A*c^2*
m*x^4+B*b^2*m^3*x^2+334*B*b*c*m*x^4+50*A*b*c*m^2*x^2+385*A*c^2*x^4+25*B*b^2*m^2*
x^2+770*B*b*c*x^4+A*b^2*m^3+398*A*b*c*m*x^2+199*B*b^2*m*x^2+27*A*b^2*m^2+990*A*b
*c*x^2+495*B*b^2*x^2+239*A*b^2*m+693*A*b^2)/(11+m)/(9+m)/(7+m)/(5+m)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.226846, size = 293, normalized size = 4.13 \[ \frac{{\left ({\left (B c^{2} m^{3} + 21 \, B c^{2} m^{2} + 143 \, B c^{2} m + 315 \, B c^{2}\right )} x^{11} +{\left ({\left (2 \, B b c + A c^{2}\right )} m^{3} + 770 \, B b c + 385 \, A c^{2} + 23 \,{\left (2 \, B b c + A c^{2}\right )} m^{2} + 167 \,{\left (2 \, B b c + A c^{2}\right )} m\right )} x^{9} +{\left ({\left (B b^{2} + 2 \, A b c\right )} m^{3} + 495 \, B b^{2} + 990 \, A b c + 25 \,{\left (B b^{2} + 2 \, A b c\right )} m^{2} + 199 \,{\left (B b^{2} + 2 \, A b c\right )} m\right )} x^{7} +{\left (A b^{2} m^{3} + 27 \, A b^{2} m^{2} + 239 \, A b^{2} m + 693 \, A b^{2}\right )} x^{5}\right )} x^{m}}{m^{4} + 32 \, m^{3} + 374 \, m^{2} + 1888 \, m + 3465} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

((B*c^2*m^3 + 21*B*c^2*m^2 + 143*B*c^2*m + 315*B*c^2)*x^11 + ((2*B*b*c + A*c^2)*
m^3 + 770*B*b*c + 385*A*c^2 + 23*(2*B*b*c + A*c^2)*m^2 + 167*(2*B*b*c + A*c^2)*m
)*x^9 + ((B*b^2 + 2*A*b*c)*m^3 + 495*B*b^2 + 990*A*b*c + 25*(B*b^2 + 2*A*b*c)*m^
2 + 199*(B*b^2 + 2*A*b*c)*m)*x^7 + (A*b^2*m^3 + 27*A*b^2*m^2 + 239*A*b^2*m + 693
*A*b^2)*x^5)*x^m/(m^4 + 32*m^3 + 374*m^2 + 1888*m + 3465)

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Sympy [A]  time = 6.37574, size = 1051, normalized size = 14.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-A*b**2/(6*x**6) - A*b*c/(2*x**4) - A*c**2/(2*x**2) - B*b**2/(4*x**4)
- B*b*c/x**2 + B*c**2*log(x), Eq(m, -11)), (-A*b**2/(4*x**4) - A*b*c/x**2 + A*c
**2*log(x) - B*b**2/(2*x**2) + 2*B*b*c*log(x) + B*c**2*x**2/2, Eq(m, -9)), (-A*b
**2/(2*x**2) + 2*A*b*c*log(x) + A*c**2*x**2/2 + B*b**2*log(x) + B*b*c*x**2 + B*c
**2*x**4/4, Eq(m, -7)), (A*b**2*log(x) + A*b*c*x**2 + A*c**2*x**4/4 + B*b**2*x**
2/2 + B*b*c*x**4/2 + B*c**2*x**6/6, Eq(m, -5)), (A*b**2*m**3*x**5*x**m/(m**4 + 3
2*m**3 + 374*m**2 + 1888*m + 3465) + 27*A*b**2*m**2*x**5*x**m/(m**4 + 32*m**3 +
374*m**2 + 1888*m + 3465) + 239*A*b**2*m*x**5*x**m/(m**4 + 32*m**3 + 374*m**2 +
1888*m + 3465) + 693*A*b**2*x**5*x**m/(m**4 + 32*m**3 + 374*m**2 + 1888*m + 3465
) + 2*A*b*c*m**3*x**7*x**m/(m**4 + 32*m**3 + 374*m**2 + 1888*m + 3465) + 50*A*b*
c*m**2*x**7*x**m/(m**4 + 32*m**3 + 374*m**2 + 1888*m + 3465) + 398*A*b*c*m*x**7*
x**m/(m**4 + 32*m**3 + 374*m**2 + 1888*m + 3465) + 990*A*b*c*x**7*x**m/(m**4 + 3
2*m**3 + 374*m**2 + 1888*m + 3465) + A*c**2*m**3*x**9*x**m/(m**4 + 32*m**3 + 374
*m**2 + 1888*m + 3465) + 23*A*c**2*m**2*x**9*x**m/(m**4 + 32*m**3 + 374*m**2 + 1
888*m + 3465) + 167*A*c**2*m*x**9*x**m/(m**4 + 32*m**3 + 374*m**2 + 1888*m + 346
5) + 385*A*c**2*x**9*x**m/(m**4 + 32*m**3 + 374*m**2 + 1888*m + 3465) + B*b**2*m
**3*x**7*x**m/(m**4 + 32*m**3 + 374*m**2 + 1888*m + 3465) + 25*B*b**2*m**2*x**7*
x**m/(m**4 + 32*m**3 + 374*m**2 + 1888*m + 3465) + 199*B*b**2*m*x**7*x**m/(m**4
+ 32*m**3 + 374*m**2 + 1888*m + 3465) + 495*B*b**2*x**7*x**m/(m**4 + 32*m**3 + 3
74*m**2 + 1888*m + 3465) + 2*B*b*c*m**3*x**9*x**m/(m**4 + 32*m**3 + 374*m**2 + 1
888*m + 3465) + 46*B*b*c*m**2*x**9*x**m/(m**4 + 32*m**3 + 374*m**2 + 1888*m + 34
65) + 334*B*b*c*m*x**9*x**m/(m**4 + 32*m**3 + 374*m**2 + 1888*m + 3465) + 770*B*
b*c*x**9*x**m/(m**4 + 32*m**3 + 374*m**2 + 1888*m + 3465) + B*c**2*m**3*x**11*x*
*m/(m**4 + 32*m**3 + 374*m**2 + 1888*m + 3465) + 21*B*c**2*m**2*x**11*x**m/(m**4
+ 32*m**3 + 374*m**2 + 1888*m + 3465) + 143*B*c**2*m*x**11*x**m/(m**4 + 32*m**3
+ 374*m**2 + 1888*m + 3465) + 315*B*c**2*x**11*x**m/(m**4 + 32*m**3 + 374*m**2
+ 1888*m + 3465), True))

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GIAC/XCAS [A]  time = 0.221147, size = 524, normalized size = 7.38 \[ \frac{B c^{2} m^{3} x^{11} e^{\left (m{\rm ln}\left (x\right )\right )} + 21 \, B c^{2} m^{2} x^{11} e^{\left (m{\rm ln}\left (x\right )\right )} + 2 \, B b c m^{3} x^{9} e^{\left (m{\rm ln}\left (x\right )\right )} + A c^{2} m^{3} x^{9} e^{\left (m{\rm ln}\left (x\right )\right )} + 143 \, B c^{2} m x^{11} e^{\left (m{\rm ln}\left (x\right )\right )} + 46 \, B b c m^{2} x^{9} e^{\left (m{\rm ln}\left (x\right )\right )} + 23 \, A c^{2} m^{2} x^{9} e^{\left (m{\rm ln}\left (x\right )\right )} + 315 \, B c^{2} x^{11} e^{\left (m{\rm ln}\left (x\right )\right )} + B b^{2} m^{3} x^{7} e^{\left (m{\rm ln}\left (x\right )\right )} + 2 \, A b c m^{3} x^{7} e^{\left (m{\rm ln}\left (x\right )\right )} + 334 \, B b c m x^{9} e^{\left (m{\rm ln}\left (x\right )\right )} + 167 \, A c^{2} m x^{9} e^{\left (m{\rm ln}\left (x\right )\right )} + 25 \, B b^{2} m^{2} x^{7} e^{\left (m{\rm ln}\left (x\right )\right )} + 50 \, A b c m^{2} x^{7} e^{\left (m{\rm ln}\left (x\right )\right )} + 770 \, B b c x^{9} e^{\left (m{\rm ln}\left (x\right )\right )} + 385 \, A c^{2} x^{9} e^{\left (m{\rm ln}\left (x\right )\right )} + A b^{2} m^{3} x^{5} e^{\left (m{\rm ln}\left (x\right )\right )} + 199 \, B b^{2} m x^{7} e^{\left (m{\rm ln}\left (x\right )\right )} + 398 \, A b c m x^{7} e^{\left (m{\rm ln}\left (x\right )\right )} + 27 \, A b^{2} m^{2} x^{5} e^{\left (m{\rm ln}\left (x\right )\right )} + 495 \, B b^{2} x^{7} e^{\left (m{\rm ln}\left (x\right )\right )} + 990 \, A b c x^{7} e^{\left (m{\rm ln}\left (x\right )\right )} + 239 \, A b^{2} m x^{5} e^{\left (m{\rm ln}\left (x\right )\right )} + 693 \, A b^{2} x^{5} e^{\left (m{\rm ln}\left (x\right )\right )}}{m^{4} + 32 \, m^{3} + 374 \, m^{2} + 1888 \, m + 3465} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(B*c^2*m^3*x^11*e^(m*ln(x)) + 21*B*c^2*m^2*x^11*e^(m*ln(x)) + 2*B*b*c*m^3*x^9*e^
(m*ln(x)) + A*c^2*m^3*x^9*e^(m*ln(x)) + 143*B*c^2*m*x^11*e^(m*ln(x)) + 46*B*b*c*
m^2*x^9*e^(m*ln(x)) + 23*A*c^2*m^2*x^9*e^(m*ln(x)) + 315*B*c^2*x^11*e^(m*ln(x))
+ B*b^2*m^3*x^7*e^(m*ln(x)) + 2*A*b*c*m^3*x^7*e^(m*ln(x)) + 334*B*b*c*m*x^9*e^(m
*ln(x)) + 167*A*c^2*m*x^9*e^(m*ln(x)) + 25*B*b^2*m^2*x^7*e^(m*ln(x)) + 50*A*b*c*
m^2*x^7*e^(m*ln(x)) + 770*B*b*c*x^9*e^(m*ln(x)) + 385*A*c^2*x^9*e^(m*ln(x)) + A*
b^2*m^3*x^5*e^(m*ln(x)) + 199*B*b^2*m*x^7*e^(m*ln(x)) + 398*A*b*c*m*x^7*e^(m*ln(
x)) + 27*A*b^2*m^2*x^5*e^(m*ln(x)) + 495*B*b^2*x^7*e^(m*ln(x)) + 990*A*b*c*x^7*e
^(m*ln(x)) + 239*A*b^2*m*x^5*e^(m*ln(x)) + 693*A*b^2*x^5*e^(m*ln(x)))/(m^4 + 32*
m^3 + 374*m^2 + 1888*m + 3465)