3.3.21 \(\int (f x)^m (d+e x^2) (a+b x^2+c x^4)^2 \, dx\) [221]

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

[Out]

a^2*d*(f*x)^(1+m)/f/(1+m)+a*(a*e+2*b*d)*(f*x)^(3+m)/f^3/(3+m)+(2*a*b*e+2*a*c*d+b^2*d)*(f*x)^(5+m)/f^5/(5+m)+(2
*a*c*e+b^2*e+2*b*c*d)*(f*x)^(7+m)/f^7/(7+m)+c*(2*b*e+c*d)*(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 = 0.07, antiderivative size = 155, normalized size of antiderivative = 1.00, 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 = {1275} \begin {gather*} \frac {a^2 d (f x)^{m+1}}{f (m+1)}+\frac {(f x)^{m+7} \left (2 a c e+b^2 e+2 b c d\right )}{f^7 (m+7)}+\frac {(f x)^{m+5} \left (2 a b e+2 a c d+b^2 d\right )}{f^5 (m+5)}+\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)} \end {gather*}

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 1275

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*}

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Mathematica [A]
time = 0.45, size = 117, normalized size = 0.75 \begin {gather*} x (f x)^m \left (\frac {a^2 d}{1+m}+\frac {a (2 b d+a e) x^2}{3+m}+\frac {\left (b^2 d+2 a c d+2 a b e\right ) x^4}{5+m}+\frac {\left (2 b c d+b^2 e+2 a c e\right ) x^6}{7+m}+\frac {c (c d+2 b e) x^8}{9+m}+\frac {c^2 e x^{10}}{11+m}\right ) \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. \(782\) vs. \(2(155)=310\).
time = 0.02, size = 783, normalized size = 5.05

method	result	size
gosper	\(\frac {x \left (c^{2} e \,m^{5} x^{10}+25 c^{2} e \,m^{4} x^{10}+2 b c e \,m^{5} x^{8}+c^{2} d \,m^{5} x^{8}+230 c^{2} e \,m^{3} x^{10}+54 b c e \,m^{4} x^{8}+27 c^{2} d \,m^{4} x^{8}+950 c^{2} e \,m^{2} x^{10}+2 a c e \,m^{5} x^{6}+b^{2} e \,m^{5} x^{6}+2 b c d \,m^{5} x^{6}+524 b c e \,m^{3} x^{8}+262 c^{2} d \,m^{3} x^{8}+1689 m \,x^{10} c^{2} e +58 a c e \,m^{4} x^{6}+29 b^{2} e \,m^{4} x^{6}+58 b c d \,m^{4} x^{6}+2244 b c e \,m^{2} x^{8}+1122 c^{2} d \,m^{2} x^{8}+945 c^{2} e \,x^{10}+2 a b e \,m^{5} x^{4}+2 a c d \,m^{5} x^{4}+604 a c e \,m^{3} x^{6}+b^{2} d \,m^{5} x^{4}+302 b^{2} e \,m^{3} x^{6}+604 b c d \,m^{3} x^{6}+4082 b c e \,x^{8} m +2041 c^{2} d \,x^{8} m +62 a b e \,m^{4} x^{4}+62 a c d \,m^{4} x^{4}+2732 a c e \,m^{2} x^{6}+31 b^{2} d \,m^{4} x^{4}+1366 b^{2} e \,m^{2} x^{6}+2732 b c d \,m^{2} x^{6}+2310 b c e \,x^{8}+1155 c^{2} d \,x^{8}+a^{2} e \,m^{5} x^{2}+2 a b d \,m^{5} x^{2}+700 a b e \,m^{3} x^{4}+700 a c d \,m^{3} x^{4}+5154 a c e \,x^{6} m +350 b^{2} d \,m^{3} x^{4}+2577 b^{2} e \,x^{6} m +5154 b c d \,x^{6} m +33 a^{2} e \,m^{4} x^{2}+66 a b d \,m^{4} x^{2}+3460 a b e \,m^{2} x^{4}+3460 a c d \,m^{2} x^{4}+2970 a c e \,x^{6}+1730 b^{2} d \,m^{2} x^{4}+1485 b^{2} e \,x^{6}+2970 b c d \,x^{6}+a^{2} d \,m^{5}+406 a^{2} e \,m^{3} x^{2}+812 a b d \,m^{3} x^{2}+6978 a b e \,x^{4} m +6978 a c d \,x^{4} m +3489 b^{2} d \,x^{4} m +35 a^{2} d \,m^{4}+2262 a^{2} e \,m^{2} x^{2}+4524 a b d \,m^{2} x^{2}+4158 a b e \,x^{4}+4158 a c d \,x^{4}+2079 b^{2} d \,x^{4}+470 a^{2} d \,m^{3}+5353 a^{2} e \,x^{2} m +10706 a b d \,x^{2} m +3010 a^{2} d \,m^{2}+3465 a^{2} e \,x^{2}+6930 a b d \,x^{2}+9129 d \,a^{2} m +10395 d \,a^{2}\right ) \left (f x \right )^{m}}{\left (11+m \right ) \left (9+m \right ) \left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right )}\)	\(783\)
risch	\(\frac {x \left (c^{2} e \,m^{5} x^{10}+25 c^{2} e \,m^{4} x^{10}+2 b c e \,m^{5} x^{8}+c^{2} d \,m^{5} x^{8}+230 c^{2} e \,m^{3} x^{10}+54 b c e \,m^{4} x^{8}+27 c^{2} d \,m^{4} x^{8}+950 c^{2} e \,m^{2} x^{10}+2 a c e \,m^{5} x^{6}+b^{2} e \,m^{5} x^{6}+2 b c d \,m^{5} x^{6}+524 b c e \,m^{3} x^{8}+262 c^{2} d \,m^{3} x^{8}+1689 m \,x^{10} c^{2} e +58 a c e \,m^{4} x^{6}+29 b^{2} e \,m^{4} x^{6}+58 b c d \,m^{4} x^{6}+2244 b c e \,m^{2} x^{8}+1122 c^{2} d \,m^{2} x^{8}+945 c^{2} e \,x^{10}+2 a b e \,m^{5} x^{4}+2 a c d \,m^{5} x^{4}+604 a c e \,m^{3} x^{6}+b^{2} d \,m^{5} x^{4}+302 b^{2} e \,m^{3} x^{6}+604 b c d \,m^{3} x^{6}+4082 b c e \,x^{8} m +2041 c^{2} d \,x^{8} m +62 a b e \,m^{4} x^{4}+62 a c d \,m^{4} x^{4}+2732 a c e \,m^{2} x^{6}+31 b^{2} d \,m^{4} x^{4}+1366 b^{2} e \,m^{2} x^{6}+2732 b c d \,m^{2} x^{6}+2310 b c e \,x^{8}+1155 c^{2} d \,x^{8}+a^{2} e \,m^{5} x^{2}+2 a b d \,m^{5} x^{2}+700 a b e \,m^{3} x^{4}+700 a c d \,m^{3} x^{4}+5154 a c e \,x^{6} m +350 b^{2} d \,m^{3} x^{4}+2577 b^{2} e \,x^{6} m +5154 b c d \,x^{6} m +33 a^{2} e \,m^{4} x^{2}+66 a b d \,m^{4} x^{2}+3460 a b e \,m^{2} x^{4}+3460 a c d \,m^{2} x^{4}+2970 a c e \,x^{6}+1730 b^{2} d \,m^{2} x^{4}+1485 b^{2} e \,x^{6}+2970 b c d \,x^{6}+a^{2} d \,m^{5}+406 a^{2} e \,m^{3} x^{2}+812 a b d \,m^{3} x^{2}+6978 a b e \,x^{4} m +6978 a c d \,x^{4} m +3489 b^{2} d \,x^{4} m +35 a^{2} d \,m^{4}+2262 a^{2} e \,m^{2} x^{2}+4524 a b d \,m^{2} x^{2}+4158 a b e \,x^{4}+4158 a c d \,x^{4}+2079 b^{2} d \,x^{4}+470 a^{2} d \,m^{3}+5353 a^{2} e \,x^{2} m +10706 a b d \,x^{2} m +3010 a^{2} d \,m^{2}+3465 a^{2} e \,x^{2}+6930 a b d \,x^{2}+9129 d \,a^{2} m +10395 d \,a^{2}\right ) \left (f x \right )^{m}}{\left (11+m \right ) \left (9+m \right ) \left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right )}\)	\(783\)

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,method=_RETURNVERBOSE)

[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 [A]
time = 0.29, size = 248, normalized size = 1.60 \begin {gather*} \frac {c^{2} f^{m} x^{11} e^{\left (m \log \left (x\right ) + 1\right )}}{m + 11} + \frac {c^{2} d f^{m} x^{9} x^{m}}{m + 9} + \frac {2 \, b c f^{m} x^{9} e^{\left (m \log \left (x\right ) + 1\right )}}{m + 9} + \frac {2 \, b c d f^{m} x^{7} x^{m}}{m + 7} + \frac {b^{2} f^{m} x^{7} e^{\left (m \log \left (x\right ) + 1\right )}}{m + 7} + \frac {2 \, a c f^{m} x^{7} e^{\left (m \log \left (x\right ) + 1\right )}}{m + 7} + \frac {b^{2} d f^{m} x^{5} x^{m}}{m + 5} + \frac {2 \, a c d f^{m} x^{5} x^{m}}{m + 5} + \frac {2 \, a b f^{m} x^{5} e^{\left (m \log \left (x\right ) + 1\right )}}{m + 5} + \frac {2 \, a b d f^{m} x^{3} x^{m}}{m + 3} + \frac {a^{2} f^{m} x^{3} e^{\left (m \log \left (x\right ) + 1\right )}}{m + 3} + \frac {\left (f x\right )^{m + 1} a^{2} d}{f {\left (m + 1\right )}} \end {gather*}

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]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 578 vs. \(2 (161) = 322\).
time = 0.41, size = 578, normalized size = 3.73 \begin {gather*} \frac {{\left ({\left (c^{2} d m^{5} + 27 \, c^{2} d m^{4} + 262 \, c^{2} d m^{3} + 1122 \, c^{2} d m^{2} + 2041 \, c^{2} d m + 1155 \, c^{2} d\right )} x^{9} + 2 \, {\left (b c d m^{5} + 29 \, b c d m^{4} + 302 \, b c d m^{3} + 1366 \, b c d m^{2} + 2577 \, b c d m + 1485 \, b c d\right )} x^{7} + {\left ({\left (b^{2} + 2 \, a c\right )} d m^{5} + 31 \, {\left (b^{2} + 2 \, a c\right )} d m^{4} + 350 \, {\left (b^{2} + 2 \, a c\right )} d m^{3} + 1730 \, {\left (b^{2} + 2 \, a c\right )} d m^{2} + 3489 \, {\left (b^{2} + 2 \, a c\right )} d m + 2079 \, {\left (b^{2} + 2 \, a c\right )} d\right )} x^{5} + 2 \, {\left (a b d m^{5} + 33 \, a b d m^{4} + 406 \, a b d m^{3} + 2262 \, a b d m^{2} + 5353 \, a b d m + 3465 \, a b d\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 + {\left ({\left (c^{2} m^{5} + 25 \, c^{2} m^{4} + 230 \, c^{2} m^{3} + 950 \, c^{2} m^{2} + 1689 \, c^{2} m + 945 \, c^{2}\right )} x^{11} + 2 \, {\left (b c m^{5} + 27 \, b c m^{4} + 262 \, b c m^{3} + 1122 \, b c m^{2} + 2041 \, b c m + 1155 \, b c\right )} x^{9} + {\left ({\left (b^{2} + 2 \, a c\right )} m^{5} + 29 \, {\left (b^{2} + 2 \, a c\right )} m^{4} + 302 \, {\left (b^{2} + 2 \, a c\right )} m^{3} + 1366 \, {\left (b^{2} + 2 \, a c\right )} m^{2} + 1485 \, b^{2} + 2970 \, a c + 2577 \, {\left (b^{2} + 2 \, a c\right )} m\right )} x^{7} + 2 \, {\left (a b m^{5} + 31 \, a b m^{4} + 350 \, a b m^{3} + 1730 \, a b m^{2} + 3489 \, a b m + 2079 \, a b\right )} x^{5} + {\left (a^{2} m^{5} + 33 \, a^{2} m^{4} + 406 \, a^{2} m^{3} + 2262 \, a^{2} m^{2} + 5353 \, a^{2} m + 3465 \, a^{2}\right )} x^{3}\right )} e\right )} \left (f x\right )^{m}}{m^{6} + 36 \, m^{5} + 505 \, m^{4} + 3480 \, m^{3} + 12139 \, m^{2} + 19524 \, m + 10395} \end {gather*}

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*d*m^5 + 27*c^2*d*m^4 + 262*c^2*d*m^3 + 1122*c^2*d*m^2 + 2041*c^2*d*m + 1155*c^2*d)*x^9 + 2*(b*c*d*m^5 +
29*b*c*d*m^4 + 302*b*c*d*m^3 + 1366*b*c*d*m^2 + 2577*b*c*d*m + 1485*b*c*d)*x^7 + ((b^2 + 2*a*c)*d*m^5 + 31*(b^
2 + 2*a*c)*d*m^4 + 350*(b^2 + 2*a*c)*d*m^3 + 1730*(b^2 + 2*a*c)*d*m^2 + 3489*(b^2 + 2*a*c)*d*m + 2079*(b^2 + 2
*a*c)*d)*x^5 + 2*(a*b*d*m^5 + 33*a*b*d*m^4 + 406*a*b*d*m^3 + 2262*a*b*d*m^2 + 5353*a*b*d*m + 3465*a*b*d)*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 + ((c^2*m^5 + 25*c
^2*m^4 + 230*c^2*m^3 + 950*c^2*m^2 + 1689*c^2*m + 945*c^2)*x^11 + 2*(b*c*m^5 + 27*b*c*m^4 + 262*b*c*m^3 + 1122
*b*c*m^2 + 2041*b*c*m + 1155*b*c)*x^9 + ((b^2 + 2*a*c)*m^5 + 29*(b^2 + 2*a*c)*m^4 + 302*(b^2 + 2*a*c)*m^3 + 13
66*(b^2 + 2*a*c)*m^2 + 1485*b^2 + 2970*a*c + 2577*(b^2 + 2*a*c)*m)*x^7 + 2*(a*b*m^5 + 31*a*b*m^4 + 350*a*b*m^3
+ 1730*a*b*m^2 + 3489*a*b*m + 2079*a*b)*x^5 + (a^2*m^5 + 33*a^2*m^4 + 406*a^2*m^3 + 2262*a^2*m^2 + 5353*a^2*m
+ 3465*a^2)*x^3)*e)*(f*x)^m/(m^6 + 36*m^5 + 505*m^4 + 3480*m^3 + 12139*m^2 + 19524*m + 10395)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 4068 vs. \(2 (146) = 292\).
time = 0.86, size = 4068, normalized size = 26.25 \begin {gather*} \text {Too large to display} \end {gather*}

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*m**5*x*(f*x)**m/(m**6 + 36*m
**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 35*a**2*d*m**4*x*(f*x)**m/(m**6 + 36*m**5 + 505*m
**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 470*a**2*d*m**3*x*(f*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480
*m**3 + 12139*m**2 + 19524*m + 10395) + 3010*a**2*d*m**2*x*(f*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 1
2139*m**2 + 19524*m + 10395) + 9129*a**2*d*m*x*(f*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 +
19524*m + 10395) + 10395*a**2*d*x*(f*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 103
95) + a**2*e*m**5*x**3*(f*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 33*a*
*2*e*m**4*x**3*(f*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 406*a**2*e*m*
*3*x**3*(f*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 2262*a**2*e*m**2*x**
3*(f*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 5353*a**2*e*m*x**3*(f*x)**
m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 3465*a**2*e*x**3*(f*x)**m/(m**6 + 3
6*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 2*a*b*d*m**5*x**3*(f*x)**m/(m**6 + 36*m**5 + 5
05*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 66*a*b*d*m**4*x**3*(f*x)**m/(m**6 + 36*m**5 + 505*m**4 +
3480*m**3 + 12139*m**2 + 19524*m + 10395) + 812*a*b*d*m**3*x**3*(f*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m*
*3 + 12139*m**2 + 19524*m + 10395) + 4524*a*b*d*m**2*x**3*(f*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12
139*m**2 + 19524*m + 10395) + 10706*a*b*d*m*x**3*(f*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2
+ 19524*m + 10395) + 6930*a*b*d*x**3*(f*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m +
10395) + 2*a*b*e*m**5*x**5*(f*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 6
2*a*b*e*m**4*x**5*(f*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 700*a*b*e*
m**3*x**5*(f*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 3460*a*b*e*m**2*x*
*5*(f*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 6978*a*b*e*m*x**5*(f*x)**
m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 4158*a*b*e*x**5*(f*x)**m/(m**6 + 36
*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 2*a*c*d*m**5*x**5*(f*x)**m/(m**6 + 36*m**5 + 50
5*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 62*a*c*d*m**4*x**5*(f*x)**m/(m**6 + 36*m**5 + 505*m**4 +
3480*m**3 + 12139*m**2 + 19524*m + 10395) + 700*a*c*d*m**3*x**5*(f*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**
3 + 12139*m**2 + 19524*m + 10395) + 3460*a*c*d*m**2*x**5*(f*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 121
39*m**2 + 19524*m + 10395) + 6978*a*c*d*m*x**5*(f*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 +
19524*m + 10395) + 4158*a*c*d*x**5*(f*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10
395) + 2*a*c*e*m**5*x**7*(f*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 58*
a*c*e*m**4*x**7*(f*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 604*a*c*e*m*
*3*x**7*(f*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 2732*a*c*e*m**2*x**7
*(f*x)**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 5154*a*c*e*m*x**7*(f*x)**m/
(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 2970*a*c*e*x**7*(f*x)**m/(m**6 + 36*m
**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + b**2*d*m**5*x**5*(f*x)**m/(m**6 + 36*m**5 + 505*m
**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395)...

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1178 vs. \(2 (161) = 322\).
time = 3.83, size = 1178, normalized size = 7.60 \begin {gather*} \frac {\left (f x\right )^{m} c^{2} m^{5} x^{11} e + 25 \, \left (f x\right )^{m} c^{2} m^{4} x^{11} e + \left (f x\right )^{m} c^{2} d m^{5} x^{9} + 2 \, \left (f x\right )^{m} b c m^{5} x^{9} e + 230 \, \left (f x\right )^{m} c^{2} m^{3} x^{11} e + 27 \, \left (f x\right )^{m} c^{2} d m^{4} x^{9} + 54 \, \left (f x\right )^{m} b c m^{4} x^{9} e + 950 \, \left (f x\right )^{m} c^{2} m^{2} x^{11} e + 2 \, \left (f x\right )^{m} b c d m^{5} x^{7} + 262 \, \left (f x\right )^{m} c^{2} d m^{3} x^{9} + \left (f x\right )^{m} b^{2} m^{5} x^{7} e + 2 \, \left (f x\right )^{m} a c m^{5} x^{7} e + 524 \, \left (f x\right )^{m} b c m^{3} x^{9} e + 1689 \, \left (f x\right )^{m} c^{2} m x^{11} e + 58 \, \left (f x\right )^{m} b c d m^{4} x^{7} + 1122 \, \left (f x\right )^{m} c^{2} d m^{2} x^{9} + 29 \, \left (f x\right )^{m} b^{2} m^{4} x^{7} e + 58 \, \left (f x\right )^{m} a c m^{4} x^{7} e + 2244 \, \left (f x\right )^{m} b c m^{2} x^{9} e + 945 \, \left (f x\right )^{m} c^{2} x^{11} e + \left (f x\right )^{m} b^{2} d m^{5} x^{5} + 2 \, \left (f x\right )^{m} a c d m^{5} x^{5} + 604 \, \left (f x\right )^{m} b c d m^{3} x^{7} + 2041 \, \left (f x\right )^{m} c^{2} d m x^{9} + 2 \, \left (f x\right )^{m} a b m^{5} x^{5} e + 302 \, \left (f x\right )^{m} b^{2} m^{3} x^{7} e + 604 \, \left (f x\right )^{m} a c m^{3} x^{7} e + 4082 \, \left (f x\right )^{m} b c m x^{9} e + 31 \, \left (f x\right )^{m} b^{2} d m^{4} x^{5} + 62 \, \left (f x\right )^{m} a c d m^{4} x^{5} + 2732 \, \left (f x\right )^{m} b c d m^{2} x^{7} + 1155 \, \left (f x\right )^{m} c^{2} d x^{9} + 62 \, \left (f x\right )^{m} a b m^{4} x^{5} e + 1366 \, \left (f x\right )^{m} b^{2} m^{2} x^{7} e + 2732 \, \left (f x\right )^{m} a c m^{2} x^{7} e + 2310 \, \left (f x\right )^{m} b c x^{9} e + 2 \, \left (f x\right )^{m} a b d m^{5} x^{3} + 350 \, \left (f x\right )^{m} b^{2} d m^{3} x^{5} + 700 \, \left (f x\right )^{m} a c d m^{3} x^{5} + 5154 \, \left (f x\right )^{m} b c d m x^{7} + \left (f x\right )^{m} a^{2} m^{5} x^{3} e + 700 \, \left (f x\right )^{m} a b m^{3} x^{5} e + 2577 \, \left (f x\right )^{m} b^{2} m x^{7} e + 5154 \, \left (f x\right )^{m} a c m x^{7} e + 66 \, \left (f x\right )^{m} a b d m^{4} x^{3} + 1730 \, \left (f x\right )^{m} b^{2} d m^{2} x^{5} + 3460 \, \left (f x\right )^{m} a c d m^{2} x^{5} + 2970 \, \left (f x\right )^{m} b c d x^{7} + 33 \, \left (f x\right )^{m} a^{2} m^{4} x^{3} e + 3460 \, \left (f x\right )^{m} a b m^{2} x^{5} e + 1485 \, \left (f x\right )^{m} b^{2} x^{7} e + 2970 \, \left (f x\right )^{m} a c x^{7} e + \left (f x\right )^{m} a^{2} d m^{5} x + 812 \, \left (f x\right )^{m} a b d m^{3} x^{3} + 3489 \, \left (f x\right )^{m} b^{2} d m x^{5} + 6978 \, \left (f x\right )^{m} a c d m x^{5} + 406 \, \left (f x\right )^{m} a^{2} m^{3} x^{3} e + 6978 \, \left (f x\right )^{m} a b m x^{5} e + 35 \, \left (f x\right )^{m} a^{2} d m^{4} x + 4524 \, \left (f x\right )^{m} a b d m^{2} x^{3} + 2079 \, \left (f x\right )^{m} b^{2} d x^{5} + 4158 \, \left (f x\right )^{m} a c d x^{5} + 2262 \, \left (f x\right )^{m} a^{2} m^{2} x^{3} e + 4158 \, \left (f x\right )^{m} a b x^{5} e + 470 \, \left (f x\right )^{m} a^{2} d m^{3} x + 10706 \, \left (f x\right )^{m} a b d m x^{3} + 5353 \, \left (f x\right )^{m} a^{2} m x^{3} e + 3010 \, \left (f x\right )^{m} a^{2} d m^{2} x + 6930 \, \left (f x\right )^{m} a b d x^{3} + 3465 \, \left (f x\right )^{m} a^{2} x^{3} e + 9129 \, \left (f x\right )^{m} a^{2} d m x + 10395 \, \left (f x\right )^{m} a^{2} d x}{m^{6} + 36 \, m^{5} + 505 \, m^{4} + 3480 \, m^{3} + 12139 \, m^{2} + 19524 \, m + 10395} \end {gather*}

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)

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Mupad [B]
time = 0.60, size = 429, normalized size = 2.77 \begin {gather*} \frac {x^5\,{\left (f\,x\right )}^m\,\left (d\,b^2+2\,a\,e\,b+2\,a\,c\,d\right )\,\left (m^5+31\,m^4+350\,m^3+1730\,m^2+3489\,m+2079\right )}{m^6+36\,m^5+505\,m^4+3480\,m^3+12139\,m^2+19524\,m+10395}+\frac {x^7\,{\left (f\,x\right )}^m\,\left (e\,b^2+2\,c\,d\,b+2\,a\,c\,e\right )\,\left (m^5+29\,m^4+302\,m^3+1366\,m^2+2577\,m+1485\right )}{m^6+36\,m^5+505\,m^4+3480\,m^3+12139\,m^2+19524\,m+10395}+\frac {a^2\,d\,x\,{\left (f\,x\right )}^m\,\left (m^5+35\,m^4+470\,m^3+3010\,m^2+9129\,m+10395\right )}{m^6+36\,m^5+505\,m^4+3480\,m^3+12139\,m^2+19524\,m+10395}+\frac {a\,x^3\,{\left (f\,x\right )}^m\,\left (a\,e+2\,b\,d\right )\,\left (m^5+33\,m^4+406\,m^3+2262\,m^2+5353\,m+3465\right )}{m^6+36\,m^5+505\,m^4+3480\,m^3+12139\,m^2+19524\,m+10395}+\frac {c\,x^9\,{\left (f\,x\right )}^m\,\left (2\,b\,e+c\,d\right )\,\left (m^5+27\,m^4+262\,m^3+1122\,m^2+2041\,m+1155\right )}{m^6+36\,m^5+505\,m^4+3480\,m^3+12139\,m^2+19524\,m+10395}+\frac {c^2\,e\,x^{11}\,{\left (f\,x\right )}^m\,\left (m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945\right )}{m^6+36\,m^5+505\,m^4+3480\,m^3+12139\,m^2+19524\,m+10395} \end {gather*}

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)^2,x)

[Out]

(x^5*(f*x)^m*(b^2*d + 2*a*b*e + 2*a*c*d)*(3489*m + 1730*m^2 + 350*m^3 + 31*m^4 + m^5 + 2079))/(19524*m + 12139
*m^2 + 3480*m^3 + 505*m^4 + 36*m^5 + m^6 + 10395) + (x^7*(f*x)^m*(b^2*e + 2*a*c*e + 2*b*c*d)*(2577*m + 1366*m^
2 + 302*m^3 + 29*m^4 + m^5 + 1485))/(19524*m + 12139*m^2 + 3480*m^3 + 505*m^4 + 36*m^5 + m^6 + 10395) + (a^2*d
*x*(f*x)^m*(9129*m + 3010*m^2 + 470*m^3 + 35*m^4 + m^5 + 10395))/(19524*m + 12139*m^2 + 3480*m^3 + 505*m^4 + 3
6*m^5 + m^6 + 10395) + (a*x^3*(f*x)^m*(a*e + 2*b*d)*(5353*m + 2262*m^2 + 406*m^3 + 33*m^4 + m^5 + 3465))/(1952
4*m + 12139*m^2 + 3480*m^3 + 505*m^4 + 36*m^5 + m^6 + 10395) + (c*x^9*(f*x)^m*(2*b*e + c*d)*(2041*m + 1122*m^2
+ 262*m^3 + 27*m^4 + m^5 + 1155))/(19524*m + 12139*m^2 + 3480*m^3 + 505*m^4 + 36*m^5 + m^6 + 10395) + (c^2*e*
x^11*(f*x)^m*(1689*m + 950*m^2 + 230*m^3 + 25*m^4 + m^5 + 945))/(19524*m + 12139*m^2 + 3480*m^3 + 505*m^4 + 36
*m^5 + m^6 + 10395)

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