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

Optimal. Leaf size=155 \[ \frac{a^2 d (f x)^{m+1}}{f (m+1)}+\frac{(f x)^{m+5} \left (2 a b e+2 a c d+b^2 d\right )}{f^5 (m+5)}+\frac{(f x)^{m+7} \left (2 a c e+b^2 e+2 b c d\right )}{f^7 (m+7)}+\frac{a (f x)^{m+3} (a e+2 b d)}{f^3 (m+3)}+\frac{c (f x)^{m+9} (2 b e+c d)}{f^9 (m+9)}+\frac{c^2 e (f x)^{m+11}}{f^{11} (m+11)} \]

[Out]

(a^2*d*(f*x)^(1 + m))/(f*(1 + m)) + (a*(2*b*d + a*e)*(f*x)^(3 + m))/(f^3*(3 + m)) + ((b^2*d + 2*a*c*d + 2*a*b*
e)*(f*x)^(5 + m))/(f^5*(5 + m)) + ((2*b*c*d + b^2*e + 2*a*c*e)*(f*x)^(7 + m))/(f^7*(7 + m)) + (c*(c*d + 2*b*e)
*(f*x)^(9 + m))/(f^9*(9 + m)) + (c^2*e*(f*x)^(11 + m))/(f^11*(11 + m))

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Rubi [A]  time = 0.0994351, antiderivative size = 155, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.037, Rules used = {1261} \[ \frac{a^2 d (f x)^{m+1}}{f (m+1)}+\frac{(f x)^{m+5} \left (2 a b e+2 a c d+b^2 d\right )}{f^5 (m+5)}+\frac{(f x)^{m+7} \left (2 a c e+b^2 e+2 b c d\right )}{f^7 (m+7)}+\frac{a (f x)^{m+3} (a e+2 b d)}{f^3 (m+3)}+\frac{c (f x)^{m+9} (2 b e+c d)}{f^9 (m+9)}+\frac{c^2 e (f x)^{m+11}}{f^{11} (m+11)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*d*(f*x)^(1 + m))/(f*(1 + m)) + (a*(2*b*d + a*e)*(f*x)^(3 + m))/(f^3*(3 + m)) + ((b^2*d + 2*a*c*d + 2*a*b*
e)*(f*x)^(5 + m))/(f^5*(5 + m)) + ((2*b*c*d + b^2*e + 2*a*c*e)*(f*x)^(7 + m))/(f^7*(7 + m)) + (c*(c*d + 2*b*e)
*(f*x)^(9 + m))/(f^9*(9 + m)) + (c^2*e*(f*x)^(11 + m))/(f^11*(11 + 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 )^2 \, dx &=\int \left (a^2 d (f x)^m+\frac{a (2 b d+a e) (f x)^{2+m}}{f^2}+\frac{\left (b^2 d+2 a c d+2 a b e\right ) (f x)^{4+m}}{f^4}+\frac{\left (2 b c d+b^2 e+2 a c e\right ) (f x)^{6+m}}{f^6}+\frac{c (c d+2 b e) (f x)^{8+m}}{f^8}+\frac{c^2 e (f x)^{10+m}}{f^{10}}\right ) \, dx\\ &=\frac{a^2 d (f x)^{1+m}}{f (1+m)}+\frac{a (2 b d+a e) (f x)^{3+m}}{f^3 (3+m)}+\frac{\left (b^2 d+2 a c d+2 a b e\right ) (f x)^{5+m}}{f^5 (5+m)}+\frac{\left (2 b c d+b^2 e+2 a c e\right ) (f x)^{7+m}}{f^7 (7+m)}+\frac{c (c d+2 b e) (f x)^{9+m}}{f^9 (9+m)}+\frac{c^2 e (f x)^{11+m}}{f^{11} (11+m)}\\ \end{align*}

Mathematica [A]  time = 0.148003, size = 117, normalized size = 0.75 \[ x (f x)^m \left (\frac{a^2 d}{m+1}+\frac{x^6 \left (2 a c e+b^2 e+2 b c d\right )}{m+7}+\frac{x^4 \left (2 a b e+2 a c d+b^2 d\right )}{m+5}+\frac{a x^2 (a e+2 b d)}{m+3}+\frac{c x^8 (2 b e+c d)}{m+9}+\frac{c^2 e x^{10}}{m+11}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

x*(f*x)^m*((a^2*d)/(1 + m) + (a*(2*b*d + a*e)*x^2)/(3 + m) + ((b^2*d + 2*a*c*d + 2*a*b*e)*x^4)/(5 + m) + ((2*b
*c*d + b^2*e + 2*a*c*e)*x^6)/(7 + m) + (c*(c*d + 2*b*e)*x^8)/(9 + m) + (c^2*e*x^10)/(11 + m))

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Maple [B]  time = 0.008, size = 783, normalized size = 5.1 \begin{align*}{\frac{ \left ({c}^{2}e{m}^{5}{x}^{10}+25\,{c}^{2}e{m}^{4}{x}^{10}+2\,bce{m}^{5}{x}^{8}+{c}^{2}d{m}^{5}{x}^{8}+230\,{c}^{2}e{m}^{3}{x}^{10}+54\,bce{m}^{4}{x}^{8}+27\,{c}^{2}d{m}^{4}{x}^{8}+950\,{c}^{2}e{m}^{2}{x}^{10}+2\,ace{m}^{5}{x}^{6}+{b}^{2}e{m}^{5}{x}^{6}+2\,bcd{m}^{5}{x}^{6}+524\,bce{m}^{3}{x}^{8}+262\,{c}^{2}d{m}^{3}{x}^{8}+1689\,{c}^{2}em{x}^{10}+58\,ace{m}^{4}{x}^{6}+29\,{b}^{2}e{m}^{4}{x}^{6}+58\,bcd{m}^{4}{x}^{6}+2244\,bce{m}^{2}{x}^{8}+1122\,{c}^{2}d{m}^{2}{x}^{8}+945\,e{c}^{2}{x}^{10}+2\,abe{m}^{5}{x}^{4}+2\,acd{m}^{5}{x}^{4}+604\,ace{m}^{3}{x}^{6}+{b}^{2}d{m}^{5}{x}^{4}+302\,{b}^{2}e{m}^{3}{x}^{6}+604\,bcd{m}^{3}{x}^{6}+4082\,bcem{x}^{8}+2041\,{c}^{2}dm{x}^{8}+62\,abe{m}^{4}{x}^{4}+62\,acd{m}^{4}{x}^{4}+2732\,ace{m}^{2}{x}^{6}+31\,{b}^{2}d{m}^{4}{x}^{4}+1366\,{b}^{2}e{m}^{2}{x}^{6}+2732\,bcd{m}^{2}{x}^{6}+2310\,bce{x}^{8}+1155\,{c}^{2}d{x}^{8}+{a}^{2}e{m}^{5}{x}^{2}+2\,abd{m}^{5}{x}^{2}+700\,abe{m}^{3}{x}^{4}+700\,acd{m}^{3}{x}^{4}+5154\,acem{x}^{6}+350\,{b}^{2}d{m}^{3}{x}^{4}+2577\,{b}^{2}em{x}^{6}+5154\,bcdm{x}^{6}+33\,{a}^{2}e{m}^{4}{x}^{2}+66\,abd{m}^{4}{x}^{2}+3460\,abe{m}^{2}{x}^{4}+3460\,acd{m}^{2}{x}^{4}+2970\,ace{x}^{6}+1730\,{b}^{2}d{m}^{2}{x}^{4}+1485\,{b}^{2}e{x}^{6}+2970\,bcd{x}^{6}+{a}^{2}d{m}^{5}+406\,{a}^{2}e{m}^{3}{x}^{2}+812\,abd{m}^{3}{x}^{2}+6978\,abem{x}^{4}+6978\,acdm{x}^{4}+3489\,{b}^{2}dm{x}^{4}+35\,{a}^{2}d{m}^{4}+2262\,{a}^{2}e{m}^{2}{x}^{2}+4524\,abd{m}^{2}{x}^{2}+4158\,abe{x}^{4}+4158\,acd{x}^{4}+2079\,{b}^{2}d{x}^{4}+470\,{a}^{2}d{m}^{3}+5353\,{a}^{2}em{x}^{2}+10706\,abdm{x}^{2}+3010\,{a}^{2}d{m}^{2}+3465\,{a}^{2}e{x}^{2}+6930\,abd{x}^{2}+9129\,{a}^{2}dm+10395\,d{a}^{2} \right ) x \left ( fx \right ) ^{m}}{ \left ( 11+m \right ) \left ( 9+m \right ) \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)^2,x)

[Out]

x*(c^2*e*m^5*x^10+25*c^2*e*m^4*x^10+2*b*c*e*m^5*x^8+c^2*d*m^5*x^8+230*c^2*e*m^3*x^10+54*b*c*e*m^4*x^8+27*c^2*d
*m^4*x^8+950*c^2*e*m^2*x^10+2*a*c*e*m^5*x^6+b^2*e*m^5*x^6+2*b*c*d*m^5*x^6+524*b*c*e*m^3*x^8+262*c^2*d*m^3*x^8+
1689*c^2*e*m*x^10+58*a*c*e*m^4*x^6+29*b^2*e*m^4*x^6+58*b*c*d*m^4*x^6+2244*b*c*e*m^2*x^8+1122*c^2*d*m^2*x^8+945
*c^2*e*x^10+2*a*b*e*m^5*x^4+2*a*c*d*m^5*x^4+604*a*c*e*m^3*x^6+b^2*d*m^5*x^4+302*b^2*e*m^3*x^6+604*b*c*d*m^3*x^
6+4082*b*c*e*m*x^8+2041*c^2*d*m*x^8+62*a*b*e*m^4*x^4+62*a*c*d*m^4*x^4+2732*a*c*e*m^2*x^6+31*b^2*d*m^4*x^4+1366
*b^2*e*m^2*x^6+2732*b*c*d*m^2*x^6+2310*b*c*e*x^8+1155*c^2*d*x^8+a^2*e*m^5*x^2+2*a*b*d*m^5*x^2+700*a*b*e*m^3*x^
4+700*a*c*d*m^3*x^4+5154*a*c*e*m*x^6+350*b^2*d*m^3*x^4+2577*b^2*e*m*x^6+5154*b*c*d*m*x^6+33*a^2*e*m^4*x^2+66*a
*b*d*m^4*x^2+3460*a*b*e*m^2*x^4+3460*a*c*d*m^2*x^4+2970*a*c*e*x^6+1730*b^2*d*m^2*x^4+1485*b^2*e*x^6+2970*b*c*d
*x^6+a^2*d*m^5+406*a^2*e*m^3*x^2+812*a*b*d*m^3*x^2+6978*a*b*e*m*x^4+6978*a*c*d*m*x^4+3489*b^2*d*m*x^4+35*a^2*d
*m^4+2262*a^2*e*m^2*x^2+4524*a*b*d*m^2*x^2+4158*a*b*e*x^4+4158*a*c*d*x^4+2079*b^2*d*x^4+470*a^2*d*m^3+5353*a^2
*e*m*x^2+10706*a*b*d*m*x^2+3010*a^2*d*m^2+3465*a^2*e*x^2+6930*a*b*d*x^2+9129*a^2*d*m+10395*a^2*d)*(f*x)^m/(11+
m)/(9+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)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.38458, size = 1416, normalized size = 9.14 \begin{align*} \frac{{\left ({\left (c^{2} e m^{5} + 25 \, c^{2} e m^{4} + 230 \, c^{2} e m^{3} + 950 \, c^{2} e m^{2} + 1689 \, c^{2} e m + 945 \, c^{2} e\right )} x^{11} +{\left ({\left (c^{2} d + 2 \, b c e\right )} m^{5} + 27 \,{\left (c^{2} d + 2 \, b c e\right )} m^{4} + 262 \,{\left (c^{2} d + 2 \, b c e\right )} m^{3} + 1155 \, c^{2} d + 2310 \, b c e + 1122 \,{\left (c^{2} d + 2 \, b c e\right )} m^{2} + 2041 \,{\left (c^{2} d + 2 \, b c e\right )} m\right )} x^{9} +{\left ({\left (2 \, b c d +{\left (b^{2} + 2 \, a c\right )} e\right )} m^{5} + 29 \,{\left (2 \, b c d +{\left (b^{2} + 2 \, a c\right )} e\right )} m^{4} + 302 \,{\left (2 \, b c d +{\left (b^{2} + 2 \, a c\right )} e\right )} m^{3} + 2970 \, b c d + 1366 \,{\left (2 \, b c d +{\left (b^{2} + 2 \, a c\right )} e\right )} m^{2} + 1485 \,{\left (b^{2} + 2 \, a c\right )} e + 2577 \,{\left (2 \, b c d +{\left (b^{2} + 2 \, a c\right )} e\right )} m\right )} x^{7} +{\left ({\left (2 \, a b e +{\left (b^{2} + 2 \, a c\right )} d\right )} m^{5} + 31 \,{\left (2 \, a b e +{\left (b^{2} + 2 \, a c\right )} d\right )} m^{4} + 350 \,{\left (2 \, a b e +{\left (b^{2} + 2 \, a c\right )} d\right )} m^{3} + 4158 \, a b e + 1730 \,{\left (2 \, a b e +{\left (b^{2} + 2 \, a c\right )} d\right )} m^{2} + 2079 \,{\left (b^{2} + 2 \, a c\right )} d + 3489 \,{\left (2 \, a b e +{\left (b^{2} + 2 \, a c\right )} d\right )} m\right )} x^{5} +{\left ({\left (2 \, a b d + a^{2} e\right )} m^{5} + 33 \,{\left (2 \, a b d + a^{2} e\right )} m^{4} + 406 \,{\left (2 \, a b d + a^{2} e\right )} m^{3} + 6930 \, a b d + 3465 \, a^{2} e + 2262 \,{\left (2 \, a b d + a^{2} e\right )} m^{2} + 5353 \,{\left (2 \, a b d + a^{2} e\right )} m\right )} x^{3} +{\left (a^{2} d m^{5} + 35 \, a^{2} d m^{4} + 470 \, a^{2} d m^{3} + 3010 \, a^{2} d m^{2} + 9129 \, a^{2} d m + 10395 \, a^{2} d\right )} x\right )} \left (f x\right )^{m}}{m^{6} + 36 \, m^{5} + 505 \, m^{4} + 3480 \, m^{3} + 12139 \, m^{2} + 19524 \, m + 10395} \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)^2,x, algorithm="fricas")

[Out]

((c^2*e*m^5 + 25*c^2*e*m^4 + 230*c^2*e*m^3 + 950*c^2*e*m^2 + 1689*c^2*e*m + 945*c^2*e)*x^11 + ((c^2*d + 2*b*c*
e)*m^5 + 27*(c^2*d + 2*b*c*e)*m^4 + 262*(c^2*d + 2*b*c*e)*m^3 + 1155*c^2*d + 2310*b*c*e + 1122*(c^2*d + 2*b*c*
e)*m^2 + 2041*(c^2*d + 2*b*c*e)*m)*x^9 + ((2*b*c*d + (b^2 + 2*a*c)*e)*m^5 + 29*(2*b*c*d + (b^2 + 2*a*c)*e)*m^4
+ 302*(2*b*c*d + (b^2 + 2*a*c)*e)*m^3 + 2970*b*c*d + 1366*(2*b*c*d + (b^2 + 2*a*c)*e)*m^2 + 1485*(b^2 + 2*a*c
)*e + 2577*(2*b*c*d + (b^2 + 2*a*c)*e)*m)*x^7 + ((2*a*b*e + (b^2 + 2*a*c)*d)*m^5 + 31*(2*a*b*e + (b^2 + 2*a*c)
*d)*m^4 + 350*(2*a*b*e + (b^2 + 2*a*c)*d)*m^3 + 4158*a*b*e + 1730*(2*a*b*e + (b^2 + 2*a*c)*d)*m^2 + 2079*(b^2
+ 2*a*c)*d + 3489*(2*a*b*e + (b^2 + 2*a*c)*d)*m)*x^5 + ((2*a*b*d + a^2*e)*m^5 + 33*(2*a*b*d + a^2*e)*m^4 + 406
*(2*a*b*d + a^2*e)*m^3 + 6930*a*b*d + 3465*a^2*e + 2262*(2*a*b*d + a^2*e)*m^2 + 5353*(2*a*b*d + a^2*e)*m)*x^3
+ (a^2*d*m^5 + 35*a^2*d*m^4 + 470*a^2*d*m^3 + 3010*a^2*d*m^2 + 9129*a^2*d*m + 10395*a^2*d)*x)*(f*x)^m/(m^6 + 3
6*m^5 + 505*m^4 + 3480*m^3 + 12139*m^2 + 19524*m + 10395)

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Sympy [A]  time = 6.13589, size = 4190, normalized size = 27.03 \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)**2,x)

[Out]

Piecewise(((-a**2*d/(10*x**10) - a**2*e/(8*x**8) - a*b*d/(4*x**8) - a*b*e/(3*x**6) - a*c*d/(3*x**6) - a*c*e/(2
*x**4) - b**2*d/(6*x**6) - b**2*e/(4*x**4) - b*c*d/(2*x**4) - b*c*e/x**2 - c**2*d/(2*x**2) + c**2*e*log(x))/f*
*11, Eq(m, -11)), ((-a**2*d/(8*x**8) - a**2*e/(6*x**6) - a*b*d/(3*x**6) - a*b*e/(2*x**4) - a*c*d/(2*x**4) - a*
c*e/x**2 - b**2*d/(4*x**4) - b**2*e/(2*x**2) - b*c*d/x**2 + 2*b*c*e*log(x) + c**2*d*log(x) + c**2*e*x**2/2)/f*
*9, Eq(m, -9)), ((-a**2*d/(6*x**6) - a**2*e/(4*x**4) - a*b*d/(2*x**4) - a*b*e/x**2 - a*c*d/x**2 + 2*a*c*e*log(
x) - b**2*d/(2*x**2) + b**2*e*log(x) + 2*b*c*d*log(x) + b*c*e*x**2 + c**2*d*x**2/2 + c**2*e*x**4/4)/f**7, Eq(m
, -7)), ((-a**2*d/(4*x**4) - a**2*e/(2*x**2) - a*b*d/x**2 + 2*a*b*e*log(x) + 2*a*c*d*log(x) + a*c*e*x**2 + b**
2*d*log(x) + b**2*e*x**2/2 + b*c*d*x**2 + b*c*e*x**4/2 + c**2*d*x**4/4 + c**2*e*x**6/6)/f**5, Eq(m, -5)), ((-a
**2*d/(2*x**2) + a**2*e*log(x) + 2*a*b*d*log(x) + a*b*e*x**2 + a*c*d*x**2 + a*c*e*x**4/2 + b**2*d*x**2/2 + b**
2*e*x**4/4 + b*c*d*x**4/2 + b*c*e*x**6/3 + c**2*d*x**6/6 + c**2*e*x**8/8)/f**3, Eq(m, -3)), ((a**2*d*log(x) +
a**2*e*x**2/2 + a*b*d*x**2 + a*b*e*x**4/2 + a*c*d*x**4/2 + a*c*e*x**6/3 + b**2*d*x**4/4 + b**2*e*x**6/6 + b*c*
d*x**6/3 + b*c*e*x**8/4 + c**2*d*x**8/8 + c**2*e*x**10/10)/f, Eq(m, -1)), (a**2*d*f**m*m**5*x*x**m/(m**6 + 36*
m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 35*a**2*d*f**m*m**4*x*x**m/(m**6 + 36*m**5 + 505
*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 470*a**2*d*f**m*m**3*x*x**m/(m**6 + 36*m**5 + 505*m**4 + 3
480*m**3 + 12139*m**2 + 19524*m + 10395) + 3010*a**2*d*f**m*m**2*x*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3
+ 12139*m**2 + 19524*m + 10395) + 9129*a**2*d*f**m*m*x*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m*
*2 + 19524*m + 10395) + 10395*a**2*d*f**m*x*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m
+ 10395) + a**2*e*f**m*m**5*x**3*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395)
+ 33*a**2*e*f**m*m**4*x**3*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 406*a
**2*e*f**m*m**3*x**3*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 2262*a**2*e
*f**m*m**2*x**3*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 5353*a**2*e*f**m
*m*x**3*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 3465*a**2*e*f**m*x**3*x*
*m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 2*a*b*d*f**m*m**5*x**3*x**m/(m**6
+ 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 66*a*b*d*f**m*m**4*x**3*x**m/(m**6 + 36*m**
5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 812*a*b*d*f**m*m**3*x**3*x**m/(m**6 + 36*m**5 + 505
*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 4524*a*b*d*f**m*m**2*x**3*x**m/(m**6 + 36*m**5 + 505*m**4
+ 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 10706*a*b*d*f**m*m*x**3*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m
**3 + 12139*m**2 + 19524*m + 10395) + 6930*a*b*d*f**m*x**3*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139
*m**2 + 19524*m + 10395) + 2*a*b*e*f**m*m**5*x**5*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 1
9524*m + 10395) + 62*a*b*e*f**m*m**4*x**5*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m +
10395) + 700*a*b*e*f**m*m**3*x**5*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395)
+ 3460*a*b*e*f**m*m**2*x**5*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 697
8*a*b*e*f**m*m*x**5*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 4158*a*b*e*f
**m*x**5*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 2*a*c*d*f**m*m**5*x**5*
x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 62*a*c*d*f**m*m**4*x**5*x**m/(m*
*6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 700*a*c*d*f**m*m**3*x**5*x**m/(m**6 + 36
*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 3460*a*c*d*f**m*m**2*x**5*x**m/(m**6 + 36*m**5
+ 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 6978*a*c*d*f**m*m*x**5*x**m/(m**6 + 36*m**5 + 505*m**
4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 4158*a*c*d*f**m*x**5*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m*
*3 + 12139*m**2 + 19524*m + 10395) + 2*a*c*e*f**m*m**5*x**7*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 1213
9*m**2 + 19524*m + 10395) + 58*a*c*e*f**m*m**4*x**7*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 +
19524*m + 10395) + 604*a*c*e*f**m*m**3*x**7*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*
m + 10395) + 2732*a*c*e*f**m*m**2*x**7*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10
395) + 5154*a*c*e*f**m*m*x**7*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 29
70*a*c*e*f**m*x**7*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + b**2*d*f**m*m
**5*x**5*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 31*b**2*d*f**m*m**4*x**
5*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 350*b**2*d*f**m*m**3*x**5*x**m
/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 1730*b**2*d*f**m*m**2*x**5*x**m/(m**
6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 3489*b**2*d*f**m*m*x**5*x**m/(m**6 + 36*m
**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 2079*b**2*d*f**m*x**5*x**m/(m**6 + 36*m**5 + 505*
m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + b**2*e*f**m*m**5*x**7*x**m/(m**6 + 36*m**5 + 505*m**4 + 348
0*m**3 + 12139*m**2 + 19524*m + 10395) + 29*b**2*e*f**m*m**4*x**7*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3
+ 12139*m**2 + 19524*m + 10395) + 302*b**2*e*f**m*m**3*x**7*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 1213
9*m**2 + 19524*m + 10395) + 1366*b**2*e*f**m*m**2*x**7*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**
2 + 19524*m + 10395) + 2577*b**2*e*f**m*m*x**7*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 1952
4*m + 10395) + 1485*b**2*e*f**m*x**7*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 1039
5) + 2*b*c*d*f**m*m**5*x**7*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 58*b
*c*d*f**m*m**4*x**7*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 604*b*c*d*f*
*m*m**3*x**7*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 2732*b*c*d*f**m*m**
2*x**7*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 5154*b*c*d*f**m*m*x**7*x*
*m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 2970*b*c*d*f**m*x**7*x**m/(m**6 +
36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 2*b*c*e*f**m*m**5*x**9*x**m/(m**6 + 36*m**5 +
505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 54*b*c*e*f**m*m**4*x**9*x**m/(m**6 + 36*m**5 + 505*m**
4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 524*b*c*e*f**m*m**3*x**9*x**m/(m**6 + 36*m**5 + 505*m**4 + 348
0*m**3 + 12139*m**2 + 19524*m + 10395) + 2244*b*c*e*f**m*m**2*x**9*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3
+ 12139*m**2 + 19524*m + 10395) + 4082*b*c*e*f**m*m*x**9*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*
m**2 + 19524*m + 10395) + 2310*b*c*e*f**m*x**9*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 1952
4*m + 10395) + c**2*d*f**m*m**5*x**9*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 1039
5) + 27*c**2*d*f**m*m**4*x**9*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 26
2*c**2*d*f**m*m**3*x**9*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 1122*c**
2*d*f**m*m**2*x**9*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 2041*c**2*d*f
**m*m*x**9*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 1155*c**2*d*f**m*x**9
*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + c**2*e*f**m*m**5*x**11*x**m/(m*
*6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 25*c**2*e*f**m*m**4*x**11*x**m/(m**6 + 3
6*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 230*c**2*e*f**m*m**3*x**11*x**m/(m**6 + 36*m**
5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 950*c**2*e*f**m*m**2*x**11*x**m/(m**6 + 36*m**5 + 5
05*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 1689*c**2*e*f**m*m*x**11*x**m/(m**6 + 36*m**5 + 505*m**4
+ 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 945*c**2*e*f**m*x**11*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m*
*3 + 12139*m**2 + 19524*m + 10395), True))

________________________________________________________________________________________

Giac [B]  time = 1.13304, size = 1590, normalized size = 10.26 \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)^2,x, algorithm="giac")

[Out]

((f*x)^m*c^2*m^5*x^11*e + 25*(f*x)^m*c^2*m^4*x^11*e + (f*x)^m*c^2*d*m^5*x^9 + 2*(f*x)^m*b*c*m^5*x^9*e + 230*(f
*x)^m*c^2*m^3*x^11*e + 27*(f*x)^m*c^2*d*m^4*x^9 + 54*(f*x)^m*b*c*m^4*x^9*e + 950*(f*x)^m*c^2*m^2*x^11*e + 2*(f
*x)^m*b*c*d*m^5*x^7 + 262*(f*x)^m*c^2*d*m^3*x^9 + (f*x)^m*b^2*m^5*x^7*e + 2*(f*x)^m*a*c*m^5*x^7*e + 524*(f*x)^
m*b*c*m^3*x^9*e + 1689*(f*x)^m*c^2*m*x^11*e + 58*(f*x)^m*b*c*d*m^4*x^7 + 1122*(f*x)^m*c^2*d*m^2*x^9 + 29*(f*x)
^m*b^2*m^4*x^7*e + 58*(f*x)^m*a*c*m^4*x^7*e + 2244*(f*x)^m*b*c*m^2*x^9*e + 945*(f*x)^m*c^2*x^11*e + (f*x)^m*b^
2*d*m^5*x^5 + 2*(f*x)^m*a*c*d*m^5*x^5 + 604*(f*x)^m*b*c*d*m^3*x^7 + 2041*(f*x)^m*c^2*d*m*x^9 + 2*(f*x)^m*a*b*m
^5*x^5*e + 302*(f*x)^m*b^2*m^3*x^7*e + 604*(f*x)^m*a*c*m^3*x^7*e + 4082*(f*x)^m*b*c*m*x^9*e + 31*(f*x)^m*b^2*d
*m^4*x^5 + 62*(f*x)^m*a*c*d*m^4*x^5 + 2732*(f*x)^m*b*c*d*m^2*x^7 + 1155*(f*x)^m*c^2*d*x^9 + 62*(f*x)^m*a*b*m^4
*x^5*e + 1366*(f*x)^m*b^2*m^2*x^7*e + 2732*(f*x)^m*a*c*m^2*x^7*e + 2310*(f*x)^m*b*c*x^9*e + 2*(f*x)^m*a*b*d*m^
5*x^3 + 350*(f*x)^m*b^2*d*m^3*x^5 + 700*(f*x)^m*a*c*d*m^3*x^5 + 5154*(f*x)^m*b*c*d*m*x^7 + (f*x)^m*a^2*m^5*x^3
*e + 700*(f*x)^m*a*b*m^3*x^5*e + 2577*(f*x)^m*b^2*m*x^7*e + 5154*(f*x)^m*a*c*m*x^7*e + 66*(f*x)^m*a*b*d*m^4*x^
3 + 1730*(f*x)^m*b^2*d*m^2*x^5 + 3460*(f*x)^m*a*c*d*m^2*x^5 + 2970*(f*x)^m*b*c*d*x^7 + 33*(f*x)^m*a^2*m^4*x^3*
e + 3460*(f*x)^m*a*b*m^2*x^5*e + 1485*(f*x)^m*b^2*x^7*e + 2970*(f*x)^m*a*c*x^7*e + (f*x)^m*a^2*d*m^5*x + 812*(
f*x)^m*a*b*d*m^3*x^3 + 3489*(f*x)^m*b^2*d*m*x^5 + 6978*(f*x)^m*a*c*d*m*x^5 + 406*(f*x)^m*a^2*m^3*x^3*e + 6978*
(f*x)^m*a*b*m*x^5*e + 35*(f*x)^m*a^2*d*m^4*x + 4524*(f*x)^m*a*b*d*m^2*x^3 + 2079*(f*x)^m*b^2*d*x^5 + 4158*(f*x
)^m*a*c*d*x^5 + 2262*(f*x)^m*a^2*m^2*x^3*e + 4158*(f*x)^m*a*b*x^5*e + 470*(f*x)^m*a^2*d*m^3*x + 10706*(f*x)^m*
a*b*d*m*x^3 + 5353*(f*x)^m*a^2*m*x^3*e + 3010*(f*x)^m*a^2*d*m^2*x + 6930*(f*x)^m*a*b*d*x^3 + 3465*(f*x)^m*a^2*
x^3*e + 9129*(f*x)^m*a^2*d*m*x + 10395*(f*x)^m*a^2*d*x)/(m^6 + 36*m^5 + 505*m^4 + 3480*m^3 + 12139*m^2 + 19524
*m + 10395)