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

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 = 0.0474501, 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, Rules used = {1261} \[ \frac{(f x)^{m+3} (a e+b d)}{f^3 (m+3)}+\frac{a d (f x)^{m+1}}{f (m+1)}+\frac{(f x)^{m+5} (b e+c d)}{f^5 (m+5)}+\frac{c e (f x)^{m+7}}{f^7 (m+7)} \]

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))

Rule 1261

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] &&
NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && IGtQ[q, -2]

Rubi steps

\begin{align*} \int (f x)^m \left (d+e x^2\right ) \left (a+b x^2+c x^4\right ) \, dx &=\int \left (a d (f x)^m+\frac{(b d+a e) (f x)^{2+m}}{f^2}+\frac{(c d+b e) (f x)^{4+m}}{f^4}+\frac{c e (f x)^{6+m}}{f^6}\right ) \, dx\\ &=\frac{a d (f x)^{1+m}}{f (1+m)}+\frac{(b d+a e) (f x)^{3+m}}{f^3 (3+m)}+\frac{(c d+b e) (f x)^{5+m}}{f^5 (5+m)}+\frac{c e (f x)^{7+m}}{f^7 (7+m)}\\ \end{align*}

Mathematica [A]  time = 0.0494136, size = 59, normalized size = 0.71 \[ x (f x)^m \left (\frac{x^2 (a e+b d)}{m+3}+\frac{a d}{m+1}+\frac{x^4 (b e+c d)}{m+5}+\frac{c e x^6}{m+7}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

x*(f*x)^m*((a*d)/(1 + m) + ((b*d + a*e)*x^2)/(3 + m) + ((c*d + b*e)*x^4)/(5 + m) + (c*e*x^6)/(7 + m))

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Maple [B]  time = 0.003, size = 221, normalized size = 2.7 \begin{align*}{\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 ) }} \end{align*}

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(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.66453, size = 414, normalized size = 4.99 \begin{align*} \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} \end{align*}

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, 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 = 2.0871, size = 1056, normalized size = 12.72 \begin{align*} \text{result too large to display} \end{align*}

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*lo
g(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 + 176*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 + 105) + 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 + 16*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 [B]  time = 1.10684, size = 473, normalized size = 5.7 \begin{align*} \frac{\left (f x\right )^{m} c m^{3} x^{7} e + 9 \, \left (f x\right )^{m} c m^{2} x^{7} e + \left (f x\right )^{m} c d m^{3} x^{5} + \left (f x\right )^{m} b m^{3} x^{5} e + 23 \, \left (f x\right )^{m} c m x^{7} e + 11 \, \left (f x\right )^{m} c d m^{2} x^{5} + 11 \, \left (f x\right )^{m} b m^{2} x^{5} e + 15 \, \left (f x\right )^{m} c x^{7} e + \left (f x\right )^{m} b d m^{3} x^{3} + 31 \, \left (f x\right )^{m} c d m x^{5} + \left (f x\right )^{m} a m^{3} x^{3} e + 31 \, \left (f x\right )^{m} b m x^{5} e + 13 \, \left (f x\right )^{m} b d m^{2} x^{3} + 21 \, \left (f x\right )^{m} c d x^{5} + 13 \, \left (f x\right )^{m} a m^{2} x^{3} e + 21 \, \left (f x\right )^{m} b x^{5} e + \left (f x\right )^{m} a d m^{3} x + 47 \, \left (f x\right )^{m} b d m x^{3} + 47 \, \left (f x\right )^{m} a m x^{3} e + 15 \, \left (f x\right )^{m} a d m^{2} x + 35 \, \left (f x\right )^{m} b d x^{3} + 35 \, \left (f x\right )^{m} a x^{3} e + 71 \, \left (f x\right )^{m} a d m x + 105 \, \left (f x\right )^{m} a d x}{m^{4} + 16 \, m^{3} + 86 \, m^{2} + 176 \, m + 105} \end{align*}

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, algorithm="giac")

[Out]

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