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)+(a*e+b*d)*(f*x)^(3+m)/f^3/(3+m)+(b*e+c*d)*(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 = 0.05, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, 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*}

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Mathematica [A]  time = 0.05, 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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fricas [B]  time = 0.91, size = 171, normalized size = 2.06 \[ \frac {{\left ({\left (c e m^{3} + 9 \, c e m^{2} + 23 \, c e m + 15 \, c e\right )} x^{7} + {\left ({\left (c d + b e\right )} m^{3} + 11 \, {\left (c d + b e\right )} m^{2} + 21 \, c d + 21 \, b e + 31 \, {\left (c d + b e\right )} m\right )} x^{5} + {\left ({\left (b d + a e\right )} m^{3} + 13 \, {\left (b d + a e\right )} m^{2} + 35 \, b d + 35 \, a e + 47 \, {\left (b d + a e\right )} m\right )} x^{3} + {\left (a d m^{3} + 15 \, a d m^{2} + 71 \, a d m + 105 \, a d\right )} x\right )} \left (f x\right )^{m}}{m^{4} + 16 \, m^{3} + 86 \, m^{2} + 176 \, m + 105} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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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giac [B]  time = 0.42, size = 350, normalized size = 4.22 \[ \frac {\left (f x\right )^{m} c m^{3} x^{7} e + 9 \, \left (f x\right )^{m} c m^{2} x^{7} e + \left (f x\right )^{m} c d m^{3} x^{5} + \left (f x\right )^{m} b m^{3} x^{5} e + 23 \, \left (f x\right )^{m} c m x^{7} e + 11 \, \left (f x\right )^{m} c d m^{2} x^{5} + 11 \, \left (f x\right )^{m} b m^{2} x^{5} e + 15 \, \left (f x\right )^{m} c x^{7} e + \left (f x\right )^{m} b d m^{3} x^{3} + 31 \, \left (f x\right )^{m} c d m x^{5} + \left (f x\right )^{m} a m^{3} x^{3} e + 31 \, \left (f x\right )^{m} b m x^{5} e + 13 \, \left (f x\right )^{m} b d m^{2} x^{3} + 21 \, \left (f x\right )^{m} c d x^{5} + 13 \, \left (f x\right )^{m} a m^{2} x^{3} e + 21 \, \left (f x\right )^{m} b x^{5} e + \left (f x\right )^{m} a d m^{3} x + 47 \, \left (f x\right )^{m} b d m x^{3} + 47 \, \left (f x\right )^{m} a m x^{3} e + 15 \, \left (f x\right )^{m} a d m^{2} x + 35 \, \left (f x\right )^{m} b d x^{3} + 35 \, \left (f x\right )^{m} a x^{3} e + 71 \, \left (f x\right )^{m} a d m x + 105 \, \left (f x\right )^{m} a d x}{m^{4} + 16 \, m^{3} + 86 \, m^{2} + 176 \, m + 105} \]

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)

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maple [B]  time = 0.00, size = 221, normalized size = 2.66 \[ \frac {\left (c e \,m^{3} x^{6}+9 c e \,m^{2} x^{6}+b e \,m^{3} x^{4}+c d \,m^{3} x^{4}+23 c e m \,x^{6}+11 b e \,m^{2} x^{4}+11 c d \,m^{2} x^{4}+15 c e \,x^{6}+a e \,m^{3} x^{2}+b d \,m^{3} x^{2}+31 b e m \,x^{4}+31 c d m \,x^{4}+13 a e \,m^{2} x^{2}+13 b d \,m^{2} x^{2}+21 b e \,x^{4}+21 c d \,x^{4}+a d \,m^{3}+47 a e m \,x^{2}+47 b d m \,x^{2}+15 a d \,m^{2}+35 a e \,x^{2}+35 b d \,x^{2}+71 a d m +105 a d \right ) x \left (f x \right )^{m}}{\left (m +7\right ) \left (m +5\right ) \left (m +3\right ) \left (m +1\right )} \]

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/(m+7)/(m+5)/(m+3)/(m+1)

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maxima [A]  time = 1.06, size = 104, normalized size = 1.25 \[ \frac {c e f^{m} x^{7} x^{m}}{m + 7} + \frac {c d f^{m} x^{5} x^{m}}{m + 5} + \frac {b e f^{m} x^{5} x^{m}}{m + 5} + \frac {b d f^{m} x^{3} x^{m}}{m + 3} + \frac {a e f^{m} x^{3} x^{m}}{m + 3} + \frac {\left (f x\right )^{m + 1} a d}{f {\left (m + 1\right )}} \]

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]

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

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mupad [B]  time = 0.34, size = 171, normalized size = 2.06 \[ {\left (f\,x\right )}^m\,\left (\frac {x^3\,\left (a\,e+b\,d\right )\,\left (m^3+13\,m^2+47\,m+35\right )}{m^4+16\,m^3+86\,m^2+176\,m+105}+\frac {x^5\,\left (b\,e+c\,d\right )\,\left (m^3+11\,m^2+31\,m+21\right )}{m^4+16\,m^3+86\,m^2+176\,m+105}+\frac {a\,d\,x\,\left (m^3+15\,m^2+71\,m+105\right )}{m^4+16\,m^3+86\,m^2+176\,m+105}+\frac {c\,e\,x^7\,\left (m^3+9\,m^2+23\,m+15\right )}{m^4+16\,m^3+86\,m^2+176\,m+105}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 1.86, size = 1056, normalized size = 12.72 \[ \begin {cases} \frac {- \frac {a d}{6 x^{6}} - \frac {a e}{4 x^{4}} - \frac {b d}{4 x^{4}} - \frac {b e}{2 x^{2}} - \frac {c d}{2 x^{2}} + c e \log {\relax (x )}}{f^{7}} & \text {for}\: m = -7 \\\frac {- \frac {a d}{4 x^{4}} - \frac {a e}{2 x^{2}} - \frac {b d}{2 x^{2}} + b e \log {\relax (x )} + c d \log {\relax (x )} + \frac {c e x^{2}}{2}}{f^{5}} & \text {for}\: m = -5 \\\frac {- \frac {a d}{2 x^{2}} + a e \log {\relax (x )} + b d \log {\relax (x )} + \frac {b e x^{2}}{2} + \frac {c d x^{2}}{2} + \frac {c e x^{4}}{4}}{f^{3}} & \text {for}\: m = -3 \\\frac {a d \log {\relax (x )} + \frac {a e x^{2}}{2} + \frac {b d x^{2}}{2} + \frac {b e x^{4}}{4} + \frac {c d x^{4}}{4} + \frac {c e x^{6}}{6}}{f} & \text {for}\: m = -1 \\\frac {a d f^{m} m^{3} x x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {15 a d f^{m} m^{2} x x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {71 a d f^{m} m x x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {105 a d f^{m} x x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {a e f^{m} m^{3} x^{3} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {13 a e f^{m} m^{2} x^{3} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {47 a e f^{m} m x^{3} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {35 a e f^{m} x^{3} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {b d f^{m} m^{3} x^{3} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {13 b d f^{m} m^{2} x^{3} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {47 b d f^{m} m x^{3} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {35 b d f^{m} x^{3} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {b e f^{m} m^{3} x^{5} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {11 b e f^{m} m^{2} x^{5} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {31 b e f^{m} m x^{5} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {21 b e f^{m} x^{5} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {c d f^{m} m^{3} x^{5} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {11 c d f^{m} m^{2} x^{5} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {31 c d f^{m} m x^{5} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {21 c d f^{m} x^{5} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {c e f^{m} m^{3} x^{7} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {9 c e f^{m} m^{2} x^{7} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {23 c e f^{m} m x^{7} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {15 c e f^{m} x^{7} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} & \text {otherwise} \end {cases} \]

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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