3.222 \(\int (f x)^m \left (d+e x^2\right ) \left (a+b x^2+c x^4\right ) \, dx\)

Optimal. Leaf size=83 \[ \frac{(f x)^{m+3} (a e+b d)}{f^3 (m+3)}+\frac{a d (f x)^{m+1}}{f (m+1)}+\frac{(f x)^{m+5} (b e+c d)}{f^5 (m+5)}+\frac{c e (f x)^{m+7}}{f^7 (m+7)} \]

[Out]

(a*d*(f*x)^(1 + m))/(f*(1 + m)) + ((b*d + a*e)*(f*x)^(3 + m))/(f^3*(3 + m)) + ((
c*d + b*e)*(f*x)^(5 + m))/(f^5*(5 + m)) + (c*e*(f*x)^(7 + m))/(f^7*(7 + m))

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Rubi [A]  time = 0.11259, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04 \[ \frac{(f x)^{m+3} (a e+b d)}{f^3 (m+3)}+\frac{a d (f x)^{m+1}}{f (m+1)}+\frac{(f x)^{m+5} (b e+c d)}{f^5 (m+5)}+\frac{c e (f x)^{m+7}}{f^7 (m+7)} \]

Antiderivative was successfully verified.

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

[Out]

(a*d*(f*x)^(1 + m))/(f*(1 + m)) + ((b*d + a*e)*(f*x)^(3 + m))/(f^3*(3 + m)) + ((
c*d + b*e)*(f*x)^(5 + m))/(f^5*(5 + m)) + (c*e*(f*x)^(7 + m))/(f^7*(7 + m))

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Rubi in Sympy [A]  time = 21.1202, size = 71, normalized size = 0.86 \[ \frac{a d \left (f x\right )^{m + 1}}{f \left (m + 1\right )} + \frac{c e \left (f x\right )^{m + 7}}{f^{7} \left (m + 7\right )} + \frac{\left (f x\right )^{m + 3} \left (a e + b d\right )}{f^{3} \left (m + 3\right )} + \frac{\left (f x\right )^{m + 5} \left (b e + c d\right )}{f^{5} \left (m + 5\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((f*x)**m*(e*x**2+d)*(c*x**4+b*x**2+a),x)

[Out]

a*d*(f*x)**(m + 1)/(f*(m + 1)) + c*e*(f*x)**(m + 7)/(f**7*(m + 7)) + (f*x)**(m +
3)*(a*e + b*d)/(f**3*(m + 3)) + (f*x)**(m + 5)*(b*e + c*d)/(f**5*(m + 5))

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Mathematica [A]  time = 0.0806152, size = 59, normalized size = 0.71 \[ (f x)^m \left (\frac{x^3 (a e+b d)}{m+3}+\frac{a d x}{m+1}+\frac{x^5 (b e+c d)}{m+5}+\frac{c e x^7}{m+7}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(f*x)^m*((a*d*x)/(1 + m) + ((b*d + a*e)*x^3)/(3 + m) + ((c*d + b*e)*x^5)/(5 + m)
+ (c*e*x^7)/(7 + m))

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Maple [B]  time = 0.006, size = 221, normalized size = 2.7 \[{\frac{ \left ( ce{m}^{3}{x}^{6}+9\,ce{m}^{2}{x}^{6}+be{m}^{3}{x}^{4}+cd{m}^{3}{x}^{4}+23\,cem{x}^{6}+11\,be{m}^{2}{x}^{4}+11\,cd{m}^{2}{x}^{4}+15\,ce{x}^{6}+ae{m}^{3}{x}^{2}+bd{m}^{3}{x}^{2}+31\,bem{x}^{4}+31\,cdm{x}^{4}+13\,ae{m}^{2}{x}^{2}+13\,bd{m}^{2}{x}^{2}+21\,be{x}^{4}+21\,cd{x}^{4}+ad{m}^{3}+47\,aem{x}^{2}+47\,bdm{x}^{2}+15\,ad{m}^{2}+35\,ae{x}^{2}+35\,bd{x}^{2}+71\,adm+105\,ad \right ) x \left ( fx \right ) ^{m}}{ \left ( 7+m \right ) \left ( 5+m \right ) \left ( 3+m \right ) \left ( 1+m \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((f*x)^m*(e*x^2+d)*(c*x^4+b*x^2+a),x)

[Out]

x*(c*e*m^3*x^6+9*c*e*m^2*x^6+b*e*m^3*x^4+c*d*m^3*x^4+23*c*e*m*x^6+11*b*e*m^2*x^4
+11*c*d*m^2*x^4+15*c*e*x^6+a*e*m^3*x^2+b*d*m^3*x^2+31*b*e*m*x^4+31*c*d*m*x^4+13*
a*e*m^2*x^2+13*b*d*m^2*x^2+21*b*e*x^4+21*c*d*x^4+a*d*m^3+47*a*e*m*x^2+47*b*d*m*x
^2+15*a*d*m^2+35*a*e*x^2+35*b*d*x^2+71*a*d*m+105*a*d)*(f*x)^m/(7+m)/(5+m)/(3+m)/
(1+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 + a)*(e*x^2 + d)*(f*x)^m,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.268219, size = 231, normalized size = 2.78 \[ \frac{{\left ({\left (c e m^{3} + 9 \, c e m^{2} + 23 \, c e m + 15 \, c e\right )} x^{7} +{\left ({\left (c d + b e\right )} m^{3} + 11 \,{\left (c d + b e\right )} m^{2} + 21 \, c d + 21 \, b e + 31 \,{\left (c d + b e\right )} m\right )} x^{5} +{\left ({\left (b d + a e\right )} m^{3} + 13 \,{\left (b d + a e\right )} m^{2} + 35 \, b d + 35 \, a e + 47 \,{\left (b d + a e\right )} m\right )} x^{3} +{\left (a d m^{3} + 15 \, a d m^{2} + 71 \, a d m + 105 \, a d\right )} x\right )} \left (f x\right )^{m}}{m^{4} + 16 \, m^{3} + 86 \, m^{2} + 176 \, m + 105} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

((c*e*m^3 + 9*c*e*m^2 + 23*c*e*m + 15*c*e)*x^7 + ((c*d + b*e)*m^3 + 11*(c*d + b*
e)*m^2 + 21*c*d + 21*b*e + 31*(c*d + b*e)*m)*x^5 + ((b*d + a*e)*m^3 + 13*(b*d +
a*e)*m^2 + 35*b*d + 35*a*e + 47*(b*d + a*e)*m)*x^3 + (a*d*m^3 + 15*a*d*m^2 + 71*
a*d*m + 105*a*d)*x)*(f*x)^m/(m^4 + 16*m^3 + 86*m^2 + 176*m + 105)

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Sympy [A]  time = 5.42355, size = 1056, normalized size = 12.72 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x)**m*(e*x**2+d)*(c*x**4+b*x**2+a),x)

[Out]

Piecewise(((-a*d/(6*x**6) - a*e/(4*x**4) - b*d/(4*x**4) - b*e/(2*x**2) - c*d/(2*
x**2) + c*e*log(x))/f**7, Eq(m, -7)), ((-a*d/(4*x**4) - a*e/(2*x**2) - b*d/(2*x*
*2) + b*e*log(x) + c*d*log(x) + c*e*x**2/2)/f**5, Eq(m, -5)), ((-a*d/(2*x**2) +
a*e*log(x) + b*d*log(x) + b*e*x**2/2 + c*d*x**2/2 + c*e*x**4/4)/f**3, Eq(m, -3))
, ((a*d*log(x) + a*e*x**2/2 + b*d*x**2/2 + b*e*x**4/4 + c*d*x**4/4 + c*e*x**6/6)
/f, Eq(m, -1)), (a*d*f**m*m**3*x*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) +
15*a*d*f**m*m**2*x*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 71*a*d*f**m*
m*x*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 105*a*d*f**m*x*x**m/(m**4 +
16*m**3 + 86*m**2 + 176*m + 105) + a*e*f**m*m**3*x**3*x**m/(m**4 + 16*m**3 + 86*
m**2 + 176*m + 105) + 13*a*e*f**m*m**2*x**3*x**m/(m**4 + 16*m**3 + 86*m**2 + 176
*m + 105) + 47*a*e*f**m*m*x**3*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 3
5*a*e*f**m*x**3*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + b*d*f**m*m**3*x*
*3*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 13*b*d*f**m*m**2*x**3*x**m/(m
**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 47*b*d*f**m*m*x**3*x**m/(m**4 + 16*m**3
+ 86*m**2 + 176*m + 105) + 35*b*d*f**m*x**3*x**m/(m**4 + 16*m**3 + 86*m**2 + 17
6*m + 105) + b*e*f**m*m**3*x**5*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) +
11*b*e*f**m*m**2*x**5*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 31*b*e*f**
m*m*x**5*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 21*b*e*f**m*x**5*x**m/(
m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + c*d*f**m*m**3*x**5*x**m/(m**4 + 16*m**
3 + 86*m**2 + 176*m + 105) + 11*c*d*f**m*m**2*x**5*x**m/(m**4 + 16*m**3 + 86*m**
2 + 176*m + 105) + 31*c*d*f**m*m*x**5*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 1
05) + 21*c*d*f**m*x**5*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + c*e*f**m*
m**3*x**7*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 9*c*e*f**m*m**2*x**7*x
**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 23*c*e*f**m*m*x**7*x**m/(m**4 + 1
6*m**3 + 86*m**2 + 176*m + 105) + 15*c*e*f**m*x**7*x**m/(m**4 + 16*m**3 + 86*m**
2 + 176*m + 105), True))

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GIAC/XCAS [A]  time = 0.276233, size = 537, normalized size = 6.47 \[ \frac{c m^{3} x^{7} e^{\left (m{\rm ln}\left (f x\right ) + 1\right )} + 9 \, c m^{2} x^{7} e^{\left (m{\rm ln}\left (f x\right ) + 1\right )} + c d m^{3} x^{5} e^{\left (m{\rm ln}\left (f x\right )\right )} + b m^{3} x^{5} e^{\left (m{\rm ln}\left (f x\right ) + 1\right )} + 23 \, c m x^{7} e^{\left (m{\rm ln}\left (f x\right ) + 1\right )} + 11 \, c d m^{2} x^{5} e^{\left (m{\rm ln}\left (f x\right )\right )} + 11 \, b m^{2} x^{5} e^{\left (m{\rm ln}\left (f x\right ) + 1\right )} + 15 \, c x^{7} e^{\left (m{\rm ln}\left (f x\right ) + 1\right )} + b d m^{3} x^{3} e^{\left (m{\rm ln}\left (f x\right )\right )} + 31 \, c d m x^{5} e^{\left (m{\rm ln}\left (f x\right )\right )} + a m^{3} x^{3} e^{\left (m{\rm ln}\left (f x\right ) + 1\right )} + 31 \, b m x^{5} e^{\left (m{\rm ln}\left (f x\right ) + 1\right )} + 13 \, b d m^{2} x^{3} e^{\left (m{\rm ln}\left (f x\right )\right )} + 21 \, c d x^{5} e^{\left (m{\rm ln}\left (f x\right )\right )} + 13 \, a m^{2} x^{3} e^{\left (m{\rm ln}\left (f x\right ) + 1\right )} + 21 \, b x^{5} e^{\left (m{\rm ln}\left (f x\right ) + 1\right )} + a d m^{3} x e^{\left (m{\rm ln}\left (f x\right )\right )} + 47 \, b d m x^{3} e^{\left (m{\rm ln}\left (f x\right )\right )} + 47 \, a m x^{3} e^{\left (m{\rm ln}\left (f x\right ) + 1\right )} + 15 \, a d m^{2} x e^{\left (m{\rm ln}\left (f x\right )\right )} + 35 \, b d x^{3} e^{\left (m{\rm ln}\left (f x\right )\right )} + 35 \, a x^{3} e^{\left (m{\rm ln}\left (f x\right ) + 1\right )} + 71 \, a d m x e^{\left (m{\rm ln}\left (f x\right )\right )} + 105 \, a d x e^{\left (m{\rm ln}\left (f x\right )\right )}}{m^{4} + 16 \, m^{3} + 86 \, m^{2} + 176 \, m + 105} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(c*m^3*x^7*e^(m*ln(f*x) + 1) + 9*c*m^2*x^7*e^(m*ln(f*x) + 1) + c*d*m^3*x^5*e^(m*
ln(f*x)) + b*m^3*x^5*e^(m*ln(f*x) + 1) + 23*c*m*x^7*e^(m*ln(f*x) + 1) + 11*c*d*m
^2*x^5*e^(m*ln(f*x)) + 11*b*m^2*x^5*e^(m*ln(f*x) + 1) + 15*c*x^7*e^(m*ln(f*x) +
1) + b*d*m^3*x^3*e^(m*ln(f*x)) + 31*c*d*m*x^5*e^(m*ln(f*x)) + a*m^3*x^3*e^(m*ln(
f*x) + 1) + 31*b*m*x^5*e^(m*ln(f*x) + 1) + 13*b*d*m^2*x^3*e^(m*ln(f*x)) + 21*c*d
*x^5*e^(m*ln(f*x)) + 13*a*m^2*x^3*e^(m*ln(f*x) + 1) + 21*b*x^5*e^(m*ln(f*x) + 1)
+ a*d*m^3*x*e^(m*ln(f*x)) + 47*b*d*m*x^3*e^(m*ln(f*x)) + 47*a*m*x^3*e^(m*ln(f*x
) + 1) + 15*a*d*m^2*x*e^(m*ln(f*x)) + 35*b*d*x^3*e^(m*ln(f*x)) + 35*a*x^3*e^(m*l
n(f*x) + 1) + 71*a*d*m*x*e^(m*ln(f*x)) + 105*a*d*x*e^(m*ln(f*x)))/(m^4 + 16*m^3
+ 86*m^2 + 176*m + 105)