3.3.20 \(\int (f x)^m (d+e x^2) (a+b x^2+c x^4)^3 \, dx\) [220]
Optimal. Leaf size=243 \[ \frac {a^3 d (f x)^{1+m}}{f (1+m)}+\frac {a^2 (3 b d+a e) (f x)^{3+m}}{f^3 (3+m)}+\frac {3 a \left (b^2 d+a c d+a b e\right ) (f x)^{5+m}}{f^5 (5+m)}+\frac {\left (b^3 d+6 a b c d+3 a b^2 e+3 a^2 c e\right ) (f x)^{7+m}}{f^7 (7+m)}+\frac {\left (3 b^2 c d+3 a c^2 d+b^3 e+6 a b c e\right ) (f x)^{9+m}}{f^9 (9+m)}+\frac {3 c \left (b c d+b^2 e+a c e\right ) (f x)^{11+m}}{f^{11} (11+m)}+\frac {c^2 (c d+3 b e) (f x)^{13+m}}{f^{13} (13+m)}+\frac {c^3 e (f x)^{15+m}}{f^{15} (15+m)} \]
[Out]
a^3*d*(f*x)^(1+m)/f/(1+m)+a^2*(a*e+3*b*d)*(f*x)^(3+m)/f^3/(3+m)+3*a*(a*b*e+a*c*d+b^2*d)*(f*x)^(5+m)/f^5/(5+m)+
(3*a^2*c*e+3*a*b^2*e+6*a*b*c*d+b^3*d)*(f*x)^(7+m)/f^7/(7+m)+(6*a*b*c*e+3*a*c^2*d+b^3*e+3*b^2*c*d)*(f*x)^(9+m)/
f^9/(9+m)+3*c*(a*c*e+b^2*e+b*c*d)*(f*x)^(11+m)/f^11/(11+m)+c^2*(3*b*e+c*d)*(f*x)^(13+m)/f^13/(13+m)+c^3*e*(f*x
)^(15+m)/f^15/(15+m)
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Rubi [A]
time = 0.11, antiderivative size = 243, 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^3 d (f x)^{m+1}}{f (m+1)}+\frac {(f x)^{m+7} \left (3 a^2 c e+3 a b^2 e+6 a b c d+b^3 d\right )}{f^7 (m+7)}+\frac {a^2 (f x)^{m+3} (a e+3 b d)}{f^3 (m+3)}+\frac {3 c (f x)^{m+11} \left (a c e+b^2 e+b c d\right )}{f^{11} (m+11)}+\frac {3 a (f x)^{m+5} \left (a b e+a c d+b^2 d\right )}{f^5 (m+5)}+\frac {(f x)^{m+9} \left (6 a b c e+3 a c^2 d+b^3 e+3 b^2 c d\right )}{f^9 (m+9)}+\frac {c^2 (f x)^{m+13} (3 b e+c d)}{f^{13} (m+13)}+\frac {c^3 e (f x)^{m+15}}{f^{15} (m+15)} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(f*x)^m*(d + e*x^2)*(a + b*x^2 + c*x^4)^3,x]
[Out]
(a^3*d*(f*x)^(1 + m))/(f*(1 + m)) + (a^2*(3*b*d + a*e)*(f*x)^(3 + m))/(f^3*(3 + m)) + (3*a*(b^2*d + a*c*d + a*
b*e)*(f*x)^(5 + m))/(f^5*(5 + m)) + ((b^3*d + 6*a*b*c*d + 3*a*b^2*e + 3*a^2*c*e)*(f*x)^(7 + m))/(f^7*(7 + m))
+ ((3*b^2*c*d + 3*a*c^2*d + b^3*e + 6*a*b*c*e)*(f*x)^(9 + m))/(f^9*(9 + m)) + (3*c*(b*c*d + b^2*e + a*c*e)*(f*
x)^(11 + m))/(f^11*(11 + m)) + (c^2*(c*d + 3*b*e)*(f*x)^(13 + m))/(f^13*(13 + m)) + (c^3*e*(f*x)^(15 + m))/(f^
15*(15 + 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 )^3 \, dx &=\int \left (a^3 d (f x)^m+\frac {a^2 (3 b d+a e) (f x)^{2+m}}{f^2}+\frac {3 a \left (b^2 d+a c d+a b e\right ) (f x)^{4+m}}{f^4}+\frac {\left (b^3 d+6 a b c d+3 a b^2 e+3 a^2 c e\right ) (f x)^{6+m}}{f^6}+\frac {\left (3 b^2 c d+3 a c^2 d+b^3 e+6 a b c e\right ) (f x)^{8+m}}{f^8}+\frac {3 c \left (b c d+b^2 e+a c e\right ) (f x)^{10+m}}{f^{10}}+\frac {c^2 (c d+3 b e) (f x)^{12+m}}{f^{12}}+\frac {c^3 e (f x)^{14+m}}{f^{14}}\right ) \, dx\\ &=\frac {a^3 d (f x)^{1+m}}{f (1+m)}+\frac {a^2 (3 b d+a e) (f x)^{3+m}}{f^3 (3+m)}+\frac {3 a \left (b^2 d+a c d+a b e\right ) (f x)^{5+m}}{f^5 (5+m)}+\frac {\left (b^3 d+6 a b c d+3 a b^2 e+3 a^2 c e\right ) (f x)^{7+m}}{f^7 (7+m)}+\frac {\left (3 b^2 c d+3 a c^2 d+b^3 e+6 a b c e\right ) (f x)^{9+m}}{f^9 (9+m)}+\frac {3 c \left (b c d+b^2 e+a c e\right ) (f x)^{11+m}}{f^{11} (11+m)}+\frac {c^2 (c d+3 b e) (f x)^{13+m}}{f^{13} (13+m)}+\frac {c^3 e (f x)^{15+m}}{f^{15} (15+m)}\\ \end {align*}
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Mathematica [A]
time = 0.77, size = 191, normalized size = 0.79 \begin {gather*} x (f x)^m \left (\frac {a^3 d}{1+m}+\frac {a^2 (3 b d+a e) x^2}{3+m}+\frac {3 a \left (b^2 d+a c d+a b e\right ) x^4}{5+m}+\frac {\left (b^3 d+6 a b c d+3 a b^2 e+3 a^2 c e\right ) x^6}{7+m}+\frac {\left (3 b^2 c d+3 a c^2 d+b^3 e+6 a b c e\right ) x^8}{9+m}+\frac {3 c \left (b c d+b^2 e+a c e\right ) x^{10}}{11+m}+\frac {c^2 (c d+3 b e) x^{12}}{13+m}+\frac {c^3 e x^{14}}{15+m}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(f*x)^m*(d + e*x^2)*(a + b*x^2 + c*x^4)^3,x]
[Out]
x*(f*x)^m*((a^3*d)/(1 + m) + (a^2*(3*b*d + a*e)*x^2)/(3 + m) + (3*a*(b^2*d + a*c*d + a*b*e)*x^4)/(5 + m) + ((b
^3*d + 6*a*b*c*d + 3*a*b^2*e + 3*a^2*c*e)*x^6)/(7 + m) + ((3*b^2*c*d + 3*a*c^2*d + b^3*e + 6*a*b*c*e)*x^8)/(9
+ m) + (3*c*(b*c*d + b^2*e + a*c*e)*x^10)/(11 + m) + (c^2*(c*d + 3*b*e)*x^12)/(13 + m) + (c^3*e*x^14)/(15 + m)
)
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1934\) vs.
\(2(243)=486\).
time = 0.03, size = 1935, normalized size = 7.96
| | |
| method |
result |
size |
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| gosper |
\(\text {Expression too large to display}\) |
\(1935\) |
| risch |
\(\text {Expression too large to display}\) |
\(1935\) |
| | |
|
|
|
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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)^3,x,method=_RETURNVERBOSE)
[Out]
x*(c^3*e*m^7*x^14+49*c^3*e*m^6*x^14+3*b*c^2*e*m^7*x^12+c^3*d*m^7*x^12+973*c^3*e*m^5*x^14+153*b*c^2*e*m^6*x^12+
51*c^3*d*m^6*x^12+10045*c^3*e*m^4*x^14+3*a*c^2*e*m^7*x^10+3*b^2*c*e*m^7*x^10+3*b*c^2*d*m^7*x^10+3135*b*c^2*e*m
^5*x^12+1045*c^3*d*m^5*x^12+57379*c^3*e*m^3*x^14+159*a*c^2*e*m^6*x^10+159*b^2*c*e*m^6*x^10+159*b*c^2*d*m^6*x^1
0+33165*b*c^2*e*m^4*x^12+11055*c^3*d*m^4*x^12+177331*c^3*e*m^2*x^14+6*a*b*c*e*m^7*x^8+3*a*c^2*d*m^7*x^8+3375*a
*c^2*e*m^5*x^10+b^3*e*m^7*x^8+3*b^2*c*d*m^7*x^8+3375*b^2*c*e*m^5*x^10+3375*b*c^2*d*m^5*x^10+193017*b*c^2*e*m^3
*x^12+64339*c^3*d*m^3*x^12+264207*c^3*e*m*x^14+330*a*b*c*e*m^6*x^8+165*a*c^2*d*m^6*x^8+36795*a*c^2*e*m^4*x^10+
55*b^3*e*m^6*x^8+165*b^2*c*d*m^6*x^8+36795*b^2*c*e*m^4*x^10+36795*b*c^2*d*m^4*x^10+604827*b*c^2*e*m^2*x^12+201
609*c^3*d*m^2*x^12+135135*c^3*e*x^14+3*a^2*c*e*m^7*x^6+3*a*b^2*e*m^7*x^6+6*a*b*c*d*m^7*x^6+7278*a*b*c*e*m^5*x^
8+3639*a*c^2*d*m^5*x^8+219417*a*c^2*e*m^3*x^10+b^3*d*m^7*x^6+1213*b^3*e*m^5*x^8+3639*b^2*c*d*m^5*x^8+219417*b^
2*c*e*m^3*x^10+219417*b*c^2*d*m^3*x^10+909765*b*c^2*e*m*x^12+303255*c^3*d*m*x^12+171*a^2*c*e*m^6*x^6+171*a*b^2
*e*m^6*x^6+342*a*b*c*d*m^6*x^6+82338*a*b*c*e*m^4*x^8+41169*a*c^2*d*m^4*x^8+700461*a*c^2*e*m^2*x^10+57*b^3*d*m^
6*x^6+13723*b^3*e*m^4*x^8+41169*b^2*c*d*m^4*x^8+700461*b^2*c*e*m^2*x^10+700461*b*c^2*d*m^2*x^10+467775*b*c^2*e
*x^12+155925*c^3*d*x^12+3*a^2*b*e*m^7*x^4+3*a^2*c*d*m^7*x^4+3927*a^2*c*e*m^5*x^6+3*a*b^2*d*m^7*x^4+3927*a*b^2*
e*m^5*x^6+7854*a*b*c*d*m^5*x^6+507282*a*b*c*e*m^3*x^8+253641*a*c^2*d*m^3*x^8+1067445*a*c^2*e*m*x^10+1309*b^3*d
*m^5*x^6+84547*b^3*e*m^3*x^8+253641*b^2*c*d*m^3*x^8+1067445*b^2*c*e*m*x^10+1067445*b*c^2*d*m*x^10+177*a^2*b*e*
m^6*x^4+177*a^2*c*d*m^6*x^4+46431*a^2*c*e*m^4*x^6+177*a*b^2*d*m^6*x^4+46431*a*b^2*e*m^4*x^6+92862*a*b*c*d*m^4*
x^6+1662558*a*b*c*e*m^2*x^8+831279*a*c^2*d*m^2*x^8+552825*a*c^2*e*x^10+15477*b^3*d*m^4*x^6+277093*b^3*e*m^2*x^
8+831279*b^2*c*d*m^2*x^8+552825*b^2*c*e*x^10+552825*b*c^2*d*x^10+a^3*e*m^7*x^2+3*a^2*b*d*m^7*x^2+4239*a^2*b*e*
m^5*x^4+4239*a^2*c*d*m^5*x^4+299145*a^2*c*e*m^3*x^6+4239*a*b^2*d*m^5*x^4+299145*a*b^2*e*m^3*x^6+598290*a*b*c*d
*m^3*x^6+2582010*a*b*c*e*m*x^8+1291005*a*c^2*d*m*x^8+99715*b^3*d*m^3*x^6+430335*b^3*e*m*x^8+1291005*b^2*c*d*m*
x^8+61*a^3*e*m^6*x^2+183*a^2*b*d*m^6*x^2+52725*a^2*b*e*m^4*x^4+52725*a^2*c*d*m^4*x^4+1020033*a^2*c*e*m^2*x^6+5
2725*a*b^2*d*m^4*x^4+1020033*a*b^2*e*m^2*x^6+2040066*a*b*c*d*m^2*x^6+1351350*a*b*c*e*x^8+675675*a*c^2*d*x^8+34
0011*b^3*d*m^2*x^6+225225*b^3*e*x^8+675675*b^2*c*d*x^8+a^3*d*m^7+1525*a^3*e*m^5*x^2+4575*a^2*b*d*m^5*x^2+36053
7*a^2*b*e*m^3*x^4+360537*a^2*c*d*m^3*x^4+1632285*a^2*c*e*m*x^6+360537*a*b^2*d*m^3*x^4+1632285*a*b^2*e*m*x^6+32
64570*a*b*c*d*m*x^6+544095*b^3*d*m*x^6+63*a^3*d*m^6+20065*a^3*e*m^4*x^2+60195*a^2*b*d*m^4*x^2+1311363*a^2*b*e*
m^2*x^4+1311363*a^2*c*d*m^2*x^4+868725*a^2*c*e*x^6+1311363*a*b^2*d*m^2*x^4+868725*a*b^2*e*x^6+1737450*a*b*c*d*
x^6+289575*b^3*d*x^6+1645*a^3*d*m^5+147859*a^3*e*m^3*x^2+443577*a^2*b*d*m^3*x^2+2215701*a^2*b*e*m*x^4+2215701*
a^2*c*d*m*x^4+2215701*a*b^2*d*m*x^4+22995*a^3*d*m^4+594439*a^3*e*m^2*x^2+1783317*a^2*b*d*m^2*x^2+1216215*a^2*b
*e*x^4+1216215*a^2*c*d*x^4+1216215*a*b^2*d*x^4+185059*a^3*d*m^3+1140855*a^3*e*m*x^2+3422565*a^2*b*d*m*x^2+8529
57*a^3*d*m^2+675675*a^3*e*x^2+2027025*a^2*b*d*x^2+2071215*a^3*d*m+2027025*a^3*d)*(f*x)^m/(1+m)/(3+m)/(5+m)/(7+
m)/(9+m)/(11+m)/(13+m)/(15+m)
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Maxima [A]
time = 0.31, size = 438, normalized size = 1.80 \begin {gather*} \frac {c^{3} f^{m} x^{15} e^{\left (m \log \left (x\right ) + 1\right )}}{m + 15} + \frac {c^{3} d f^{m} x^{13} x^{m}}{m + 13} + \frac {3 \, b c^{2} f^{m} x^{13} e^{\left (m \log \left (x\right ) + 1\right )}}{m + 13} + \frac {3 \, b c^{2} d f^{m} x^{11} x^{m}}{m + 11} + \frac {3 \, b^{2} c f^{m} x^{11} e^{\left (m \log \left (x\right ) + 1\right )}}{m + 11} + \frac {3 \, a c^{2} f^{m} x^{11} e^{\left (m \log \left (x\right ) + 1\right )}}{m + 11} + \frac {3 \, b^{2} c d f^{m} x^{9} x^{m}}{m + 9} + \frac {3 \, a c^{2} d f^{m} x^{9} x^{m}}{m + 9} + \frac {b^{3} f^{m} x^{9} e^{\left (m \log \left (x\right ) + 1\right )}}{m + 9} + \frac {6 \, a b c f^{m} x^{9} e^{\left (m \log \left (x\right ) + 1\right )}}{m + 9} + \frac {b^{3} d f^{m} x^{7} x^{m}}{m + 7} + \frac {6 \, a b c d f^{m} x^{7} x^{m}}{m + 7} + \frac {3 \, a b^{2} f^{m} x^{7} e^{\left (m \log \left (x\right ) + 1\right )}}{m + 7} + \frac {3 \, a^{2} c f^{m} x^{7} e^{\left (m \log \left (x\right ) + 1\right )}}{m + 7} + \frac {3 \, a b^{2} d f^{m} x^{5} x^{m}}{m + 5} + \frac {3 \, a^{2} c d f^{m} x^{5} x^{m}}{m + 5} + \frac {3 \, a^{2} b f^{m} x^{5} e^{\left (m \log \left (x\right ) + 1\right )}}{m + 5} + \frac {3 \, a^{2} b d f^{m} x^{3} x^{m}}{m + 3} + \frac {a^{3} f^{m} x^{3} e^{\left (m \log \left (x\right ) + 1\right )}}{m + 3} + \frac {\left (f x\right )^{m + 1} a^{3} 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)^3,x, algorithm="maxima")
[Out]
c^3*f^m*x^15*e^(m*log(x) + 1)/(m + 15) + c^3*d*f^m*x^13*x^m/(m + 13) + 3*b*c^2*f^m*x^13*e^(m*log(x) + 1)/(m +
13) + 3*b*c^2*d*f^m*x^11*x^m/(m + 11) + 3*b^2*c*f^m*x^11*e^(m*log(x) + 1)/(m + 11) + 3*a*c^2*f^m*x^11*e^(m*log
(x) + 1)/(m + 11) + 3*b^2*c*d*f^m*x^9*x^m/(m + 9) + 3*a*c^2*d*f^m*x^9*x^m/(m + 9) + b^3*f^m*x^9*e^(m*log(x) +
1)/(m + 9) + 6*a*b*c*f^m*x^9*e^(m*log(x) + 1)/(m + 9) + b^3*d*f^m*x^7*x^m/(m + 7) + 6*a*b*c*d*f^m*x^7*x^m/(m +
7) + 3*a*b^2*f^m*x^7*e^(m*log(x) + 1)/(m + 7) + 3*a^2*c*f^m*x^7*e^(m*log(x) + 1)/(m + 7) + 3*a*b^2*d*f^m*x^5*
x^m/(m + 5) + 3*a^2*c*d*f^m*x^5*x^m/(m + 5) + 3*a^2*b*f^m*x^5*e^(m*log(x) + 1)/(m + 5) + 3*a^2*b*d*f^m*x^3*x^m
/(m + 3) + a^3*f^m*x^3*e^(m*log(x) + 1)/(m + 3) + (f*x)^(m + 1)*a^3*d/(f*(m + 1))
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1376 vs.
\(2 (253) = 506\).
time = 0.39, size = 1376, normalized size = 5.66 \begin {gather*} \frac {{\left ({\left (c^{3} d m^{7} + 51 \, c^{3} d m^{6} + 1045 \, c^{3} d m^{5} + 11055 \, c^{3} d m^{4} + 64339 \, c^{3} d m^{3} + 201609 \, c^{3} d m^{2} + 303255 \, c^{3} d m + 155925 \, c^{3} d\right )} x^{13} + 3 \, {\left (b c^{2} d m^{7} + 53 \, b c^{2} d m^{6} + 1125 \, b c^{2} d m^{5} + 12265 \, b c^{2} d m^{4} + 73139 \, b c^{2} d m^{3} + 233487 \, b c^{2} d m^{2} + 355815 \, b c^{2} d m + 184275 \, b c^{2} d\right )} x^{11} + 3 \, {\left ({\left (b^{2} c + a c^{2}\right )} d m^{7} + 55 \, {\left (b^{2} c + a c^{2}\right )} d m^{6} + 1213 \, {\left (b^{2} c + a c^{2}\right )} d m^{5} + 13723 \, {\left (b^{2} c + a c^{2}\right )} d m^{4} + 84547 \, {\left (b^{2} c + a c^{2}\right )} d m^{3} + 277093 \, {\left (b^{2} c + a c^{2}\right )} d m^{2} + 430335 \, {\left (b^{2} c + a c^{2}\right )} d m + 225225 \, {\left (b^{2} c + a c^{2}\right )} d\right )} x^{9} + {\left ({\left (b^{3} + 6 \, a b c\right )} d m^{7} + 57 \, {\left (b^{3} + 6 \, a b c\right )} d m^{6} + 1309 \, {\left (b^{3} + 6 \, a b c\right )} d m^{5} + 15477 \, {\left (b^{3} + 6 \, a b c\right )} d m^{4} + 99715 \, {\left (b^{3} + 6 \, a b c\right )} d m^{3} + 340011 \, {\left (b^{3} + 6 \, a b c\right )} d m^{2} + 544095 \, {\left (b^{3} + 6 \, a b c\right )} d m + 289575 \, {\left (b^{3} + 6 \, a b c\right )} d\right )} x^{7} + 3 \, {\left ({\left (a b^{2} + a^{2} c\right )} d m^{7} + 59 \, {\left (a b^{2} + a^{2} c\right )} d m^{6} + 1413 \, {\left (a b^{2} + a^{2} c\right )} d m^{5} + 17575 \, {\left (a b^{2} + a^{2} c\right )} d m^{4} + 120179 \, {\left (a b^{2} + a^{2} c\right )} d m^{3} + 437121 \, {\left (a b^{2} + a^{2} c\right )} d m^{2} + 738567 \, {\left (a b^{2} + a^{2} c\right )} d m + 405405 \, {\left (a b^{2} + a^{2} c\right )} d\right )} x^{5} + 3 \, {\left (a^{2} b d m^{7} + 61 \, a^{2} b d m^{6} + 1525 \, a^{2} b d m^{5} + 20065 \, a^{2} b d m^{4} + 147859 \, a^{2} b d m^{3} + 594439 \, a^{2} b d m^{2} + 1140855 \, a^{2} b d m + 675675 \, a^{2} b d\right )} x^{3} + {\left (a^{3} d m^{7} + 63 \, a^{3} d m^{6} + 1645 \, a^{3} d m^{5} + 22995 \, a^{3} d m^{4} + 185059 \, a^{3} d m^{3} + 852957 \, a^{3} d m^{2} + 2071215 \, a^{3} d m + 2027025 \, a^{3} d\right )} x + {\left ({\left (c^{3} m^{7} + 49 \, c^{3} m^{6} + 973 \, c^{3} m^{5} + 10045 \, c^{3} m^{4} + 57379 \, c^{3} m^{3} + 177331 \, c^{3} m^{2} + 264207 \, c^{3} m + 135135 \, c^{3}\right )} x^{15} + 3 \, {\left (b c^{2} m^{7} + 51 \, b c^{2} m^{6} + 1045 \, b c^{2} m^{5} + 11055 \, b c^{2} m^{4} + 64339 \, b c^{2} m^{3} + 201609 \, b c^{2} m^{2} + 303255 \, b c^{2} m + 155925 \, b c^{2}\right )} x^{13} + 3 \, {\left ({\left (b^{2} c + a c^{2}\right )} m^{7} + 53 \, {\left (b^{2} c + a c^{2}\right )} m^{6} + 1125 \, {\left (b^{2} c + a c^{2}\right )} m^{5} + 12265 \, {\left (b^{2} c + a c^{2}\right )} m^{4} + 73139 \, {\left (b^{2} c + a c^{2}\right )} m^{3} + 184275 \, b^{2} c + 184275 \, a c^{2} + 233487 \, {\left (b^{2} c + a c^{2}\right )} m^{2} + 355815 \, {\left (b^{2} c + a c^{2}\right )} m\right )} x^{11} + {\left ({\left (b^{3} + 6 \, a b c\right )} m^{7} + 55 \, {\left (b^{3} + 6 \, a b c\right )} m^{6} + 1213 \, {\left (b^{3} + 6 \, a b c\right )} m^{5} + 13723 \, {\left (b^{3} + 6 \, a b c\right )} m^{4} + 84547 \, {\left (b^{3} + 6 \, a b c\right )} m^{3} + 225225 \, b^{3} + 1351350 \, a b c + 277093 \, {\left (b^{3} + 6 \, a b c\right )} m^{2} + 430335 \, {\left (b^{3} + 6 \, a b c\right )} m\right )} x^{9} + 3 \, {\left ({\left (a b^{2} + a^{2} c\right )} m^{7} + 57 \, {\left (a b^{2} + a^{2} c\right )} m^{6} + 1309 \, {\left (a b^{2} + a^{2} c\right )} m^{5} + 15477 \, {\left (a b^{2} + a^{2} c\right )} m^{4} + 99715 \, {\left (a b^{2} + a^{2} c\right )} m^{3} + 289575 \, a b^{2} + 289575 \, a^{2} c + 340011 \, {\left (a b^{2} + a^{2} c\right )} m^{2} + 544095 \, {\left (a b^{2} + a^{2} c\right )} m\right )} x^{7} + 3 \, {\left (a^{2} b m^{7} + 59 \, a^{2} b m^{6} + 1413 \, a^{2} b m^{5} + 17575 \, a^{2} b m^{4} + 120179 \, a^{2} b m^{3} + 437121 \, a^{2} b m^{2} + 738567 \, a^{2} b m + 405405 \, a^{2} b\right )} x^{5} + {\left (a^{3} m^{7} + 61 \, a^{3} m^{6} + 1525 \, a^{3} m^{5} + 20065 \, a^{3} m^{4} + 147859 \, a^{3} m^{3} + 594439 \, a^{3} m^{2} + 1140855 \, a^{3} m + 675675 \, a^{3}\right )} x^{3}\right )} e\right )} \left (f x\right )^{m}}{m^{8} + 64 \, m^{7} + 1708 \, m^{6} + 24640 \, m^{5} + 208054 \, m^{4} + 1038016 \, m^{3} + 2924172 \, m^{2} + 4098240 \, m + 2027025} \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)^3,x, algorithm="fricas")
[Out]
((c^3*d*m^7 + 51*c^3*d*m^6 + 1045*c^3*d*m^5 + 11055*c^3*d*m^4 + 64339*c^3*d*m^3 + 201609*c^3*d*m^2 + 303255*c^
3*d*m + 155925*c^3*d)*x^13 + 3*(b*c^2*d*m^7 + 53*b*c^2*d*m^6 + 1125*b*c^2*d*m^5 + 12265*b*c^2*d*m^4 + 73139*b*
c^2*d*m^3 + 233487*b*c^2*d*m^2 + 355815*b*c^2*d*m + 184275*b*c^2*d)*x^11 + 3*((b^2*c + a*c^2)*d*m^7 + 55*(b^2*
c + a*c^2)*d*m^6 + 1213*(b^2*c + a*c^2)*d*m^5 + 13723*(b^2*c + a*c^2)*d*m^4 + 84547*(b^2*c + a*c^2)*d*m^3 + 27
7093*(b^2*c + a*c^2)*d*m^2 + 430335*(b^2*c + a*c^2)*d*m + 225225*(b^2*c + a*c^2)*d)*x^9 + ((b^3 + 6*a*b*c)*d*m
^7 + 57*(b^3 + 6*a*b*c)*d*m^6 + 1309*(b^3 + 6*a*b*c)*d*m^5 + 15477*(b^3 + 6*a*b*c)*d*m^4 + 99715*(b^3 + 6*a*b*
c)*d*m^3 + 340011*(b^3 + 6*a*b*c)*d*m^2 + 544095*(b^3 + 6*a*b*c)*d*m + 289575*(b^3 + 6*a*b*c)*d)*x^7 + 3*((a*b
^2 + a^2*c)*d*m^7 + 59*(a*b^2 + a^2*c)*d*m^6 + 1413*(a*b^2 + a^2*c)*d*m^5 + 17575*(a*b^2 + a^2*c)*d*m^4 + 1201
79*(a*b^2 + a^2*c)*d*m^3 + 437121*(a*b^2 + a^2*c)*d*m^2 + 738567*(a*b^2 + a^2*c)*d*m + 405405*(a*b^2 + a^2*c)*
d)*x^5 + 3*(a^2*b*d*m^7 + 61*a^2*b*d*m^6 + 1525*a^2*b*d*m^5 + 20065*a^2*b*d*m^4 + 147859*a^2*b*d*m^3 + 594439*
a^2*b*d*m^2 + 1140855*a^2*b*d*m + 675675*a^2*b*d)*x^3 + (a^3*d*m^7 + 63*a^3*d*m^6 + 1645*a^3*d*m^5 + 22995*a^3
*d*m^4 + 185059*a^3*d*m^3 + 852957*a^3*d*m^2 + 2071215*a^3*d*m + 2027025*a^3*d)*x + ((c^3*m^7 + 49*c^3*m^6 + 9
73*c^3*m^5 + 10045*c^3*m^4 + 57379*c^3*m^3 + 177331*c^3*m^2 + 264207*c^3*m + 135135*c^3)*x^15 + 3*(b*c^2*m^7 +
51*b*c^2*m^6 + 1045*b*c^2*m^5 + 11055*b*c^2*m^4 + 64339*b*c^2*m^3 + 201609*b*c^2*m^2 + 303255*b*c^2*m + 15592
5*b*c^2)*x^13 + 3*((b^2*c + a*c^2)*m^7 + 53*(b^2*c + a*c^2)*m^6 + 1125*(b^2*c + a*c^2)*m^5 + 12265*(b^2*c + a*
c^2)*m^4 + 73139*(b^2*c + a*c^2)*m^3 + 184275*b^2*c + 184275*a*c^2 + 233487*(b^2*c + a*c^2)*m^2 + 355815*(b^2*
c + a*c^2)*m)*x^11 + ((b^3 + 6*a*b*c)*m^7 + 55*(b^3 + 6*a*b*c)*m^6 + 1213*(b^3 + 6*a*b*c)*m^5 + 13723*(b^3 + 6
*a*b*c)*m^4 + 84547*(b^3 + 6*a*b*c)*m^3 + 225225*b^3 + 1351350*a*b*c + 277093*(b^3 + 6*a*b*c)*m^2 + 430335*(b^
3 + 6*a*b*c)*m)*x^9 + 3*((a*b^2 + a^2*c)*m^7 + 57*(a*b^2 + a^2*c)*m^6 + 1309*(a*b^2 + a^2*c)*m^5 + 15477*(a*b^
2 + a^2*c)*m^4 + 99715*(a*b^2 + a^2*c)*m^3 + 289575*a*b^2 + 289575*a^2*c + 340011*(a*b^2 + a^2*c)*m^2 + 544095
*(a*b^2 + a^2*c)*m)*x^7 + 3*(a^2*b*m^7 + 59*a^2*b*m^6 + 1413*a^2*b*m^5 + 17575*a^2*b*m^4 + 120179*a^2*b*m^3 +
437121*a^2*b*m^2 + 738567*a^2*b*m + 405405*a^2*b)*x^5 + (a^3*m^7 + 61*a^3*m^6 + 1525*a^3*m^5 + 20065*a^3*m^4 +
147859*a^3*m^3 + 594439*a^3*m^2 + 1140855*a^3*m + 675675*a^3)*x^3)*e)*(f*x)^m/(m^8 + 64*m^7 + 1708*m^6 + 2464
0*m^5 + 208054*m^4 + 1038016*m^3 + 2924172*m^2 + 4098240*m + 2027025)
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 11266 vs.
\(2 (238) = 476\).
time = 1.62, size = 11266, normalized size = 46.36 \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)**3,x)
[Out]
Piecewise(((-a**3*d/(14*x**14) - a**3*e/(12*x**12) - a**2*b*d/(4*x**12) - 3*a**2*b*e/(10*x**10) - 3*a**2*c*d/(
10*x**10) - 3*a**2*c*e/(8*x**8) - 3*a*b**2*d/(10*x**10) - 3*a*b**2*e/(8*x**8) - 3*a*b*c*d/(4*x**8) - a*b*c*e/x
**6 - a*c**2*d/(2*x**6) - 3*a*c**2*e/(4*x**4) - b**3*d/(8*x**8) - b**3*e/(6*x**6) - b**2*c*d/(2*x**6) - 3*b**2
*c*e/(4*x**4) - 3*b*c**2*d/(4*x**4) - 3*b*c**2*e/(2*x**2) - c**3*d/(2*x**2) + c**3*e*log(x))/f**15, Eq(m, -15)
), ((-a**3*d/(12*x**12) - a**3*e/(10*x**10) - 3*a**2*b*d/(10*x**10) - 3*a**2*b*e/(8*x**8) - 3*a**2*c*d/(8*x**8
) - a**2*c*e/(2*x**6) - 3*a*b**2*d/(8*x**8) - a*b**2*e/(2*x**6) - a*b*c*d/x**6 - 3*a*b*c*e/(2*x**4) - 3*a*c**2
*d/(4*x**4) - 3*a*c**2*e/(2*x**2) - b**3*d/(6*x**6) - b**3*e/(4*x**4) - 3*b**2*c*d/(4*x**4) - 3*b**2*c*e/(2*x*
*2) - 3*b*c**2*d/(2*x**2) + 3*b*c**2*e*log(x) + c**3*d*log(x) + c**3*e*x**2/2)/f**13, Eq(m, -13)), ((-a**3*d/(
10*x**10) - a**3*e/(8*x**8) - 3*a**2*b*d/(8*x**8) - a**2*b*e/(2*x**6) - a**2*c*d/(2*x**6) - 3*a**2*c*e/(4*x**4
) - a*b**2*d/(2*x**6) - 3*a*b**2*e/(4*x**4) - 3*a*b*c*d/(2*x**4) - 3*a*b*c*e/x**2 - 3*a*c**2*d/(2*x**2) + 3*a*
c**2*e*log(x) - b**3*d/(4*x**4) - b**3*e/(2*x**2) - 3*b**2*c*d/(2*x**2) + 3*b**2*c*e*log(x) + 3*b*c**2*d*log(x
) + 3*b*c**2*e*x**2/2 + c**3*d*x**2/2 + c**3*e*x**4/4)/f**11, Eq(m, -11)), ((-a**3*d/(8*x**8) - a**3*e/(6*x**6
) - a**2*b*d/(2*x**6) - 3*a**2*b*e/(4*x**4) - 3*a**2*c*d/(4*x**4) - 3*a**2*c*e/(2*x**2) - 3*a*b**2*d/(4*x**4)
- 3*a*b**2*e/(2*x**2) - 3*a*b*c*d/x**2 + 6*a*b*c*e*log(x) + 3*a*c**2*d*log(x) + 3*a*c**2*e*x**2/2 - b**3*d/(2*
x**2) + b**3*e*log(x) + 3*b**2*c*d*log(x) + 3*b**2*c*e*x**2/2 + 3*b*c**2*d*x**2/2 + 3*b*c**2*e*x**4/4 + c**3*d
*x**4/4 + c**3*e*x**6/6)/f**9, Eq(m, -9)), ((-a**3*d/(6*x**6) - a**3*e/(4*x**4) - 3*a**2*b*d/(4*x**4) - 3*a**2
*b*e/(2*x**2) - 3*a**2*c*d/(2*x**2) + 3*a**2*c*e*log(x) - 3*a*b**2*d/(2*x**2) + 3*a*b**2*e*log(x) + 6*a*b*c*d*
log(x) + 3*a*b*c*e*x**2 + 3*a*c**2*d*x**2/2 + 3*a*c**2*e*x**4/4 + b**3*d*log(x) + b**3*e*x**2/2 + 3*b**2*c*d*x
**2/2 + 3*b**2*c*e*x**4/4 + 3*b*c**2*d*x**4/4 + b*c**2*e*x**6/2 + c**3*d*x**6/6 + c**3*e*x**8/8)/f**7, Eq(m, -
7)), ((-a**3*d/(4*x**4) - a**3*e/(2*x**2) - 3*a**2*b*d/(2*x**2) + 3*a**2*b*e*log(x) + 3*a**2*c*d*log(x) + 3*a*
*2*c*e*x**2/2 + 3*a*b**2*d*log(x) + 3*a*b**2*e*x**2/2 + 3*a*b*c*d*x**2 + 3*a*b*c*e*x**4/2 + 3*a*c**2*d*x**4/4
+ a*c**2*e*x**6/2 + b**3*d*x**2/2 + b**3*e*x**4/4 + 3*b**2*c*d*x**4/4 + b**2*c*e*x**6/2 + b*c**2*d*x**6/2 + 3*
b*c**2*e*x**8/8 + c**3*d*x**8/8 + c**3*e*x**10/10)/f**5, Eq(m, -5)), ((-a**3*d/(2*x**2) + a**3*e*log(x) + 3*a*
*2*b*d*log(x) + 3*a**2*b*e*x**2/2 + 3*a**2*c*d*x**2/2 + 3*a**2*c*e*x**4/4 + 3*a*b**2*d*x**2/2 + 3*a*b**2*e*x**
4/4 + 3*a*b*c*d*x**4/2 + a*b*c*e*x**6 + a*c**2*d*x**6/2 + 3*a*c**2*e*x**8/8 + b**3*d*x**4/4 + b**3*e*x**6/6 +
b**2*c*d*x**6/2 + 3*b**2*c*e*x**8/8 + 3*b*c**2*d*x**8/8 + 3*b*c**2*e*x**10/10 + c**3*d*x**10/10 + c**3*e*x**12
/12)/f**3, Eq(m, -3)), ((a**3*d*log(x) + a**3*e*x**2/2 + 3*a**2*b*d*x**2/2 + 3*a**2*b*e*x**4/4 + 3*a**2*c*d*x*
*4/4 + a**2*c*e*x**6/2 + 3*a*b**2*d*x**4/4 + a*b**2*e*x**6/2 + a*b*c*d*x**6 + 3*a*b*c*e*x**8/4 + 3*a*c**2*d*x*
*8/8 + 3*a*c**2*e*x**10/10 + b**3*d*x**6/6 + b**3*e*x**8/8 + 3*b**2*c*d*x**8/8 + 3*b**2*c*e*x**10/10 + 3*b*c**
2*d*x**10/10 + b*c**2*e*x**12/4 + c**3*d*x**12/12 + c**3*e*x**14/14)/f, Eq(m, -1)), (a**3*d*m**7*x*(f*x)**m/(m
**8 + 64*m**7 + 1708*m**6 + 24640*m**5 + 208054*m**4 + 1038016*m**3 + 2924172*m**2 + 4098240*m + 2027025) + 63
*a**3*d*m**6*x*(f*x)**m/(m**8 + 64*m**7 + 1708*m**6 + 24640*m**5 + 208054*m**4 + 1038016*m**3 + 2924172*m**2 +
4098240*m + 2027025) + 1645*a**3*d*m**5*x*(f*x)**m/(m**8 + 64*m**7 + 1708*m**6 + 24640*m**5 + 208054*m**4 + 1
038016*m**3 + 2924172*m**2 + 4098240*m + 2027025) + 22995*a**3*d*m**4*x*(f*x)**m/(m**8 + 64*m**7 + 1708*m**6 +
24640*m**5 + 208054*m**4 + 1038016*m**3 + 2924172*m**2 + 4098240*m + 2027025) + 185059*a**3*d*m**3*x*(f*x)**m
/(m**8 + 64*m**7 + 1708*m**6 + 24640*m**5 + 208054*m**4 + 1038016*m**3 + 2924172*m**2 + 4098240*m + 2027025) +
852957*a**3*d*m**2*x*(f*x)**m/(m**8 + 64*m**7 + 1708*m**6 + 24640*m**5 + 208054*m**4 + 1038016*m**3 + 2924172
*m**2 + 4098240*m + 2027025) + 2071215*a**3*d*m*x*(f*x)**m/(m**8 + 64*m**7 + 1708*m**6 + 24640*m**5 + 208054*m
**4 + 1038016*m**3 + 2924172*m**2 + 4098240*m + 2027025) + 2027025*a**3*d*x*(f*x)**m/(m**8 + 64*m**7 + 1708*m*
*6 + 24640*m**5 + 208054*m**4 + 1038016*m**3 + 2924172*m**2 + 4098240*m + 2027025) + a**3*e*m**7*x**3*(f*x)**m
/(m**8 + 64*m**7 + 1708*m**6 + 24640*m**5 + 208054*m**4 + 1038016*m**3 + 2924172*m**2 + 4098240*m + 2027025) +
61*a**3*e*m**6*x**3*(f*x)**m/(m**8 + 64*m**7 + 1708*m**6 + 24640*m**5 + 208054*m**4 + 1038016*m**3 + 2924172*
m**2 + 4098240*m + 2027025) + 1525*a**3*e*m**5*x**3*(f*x)**m/(m**8 + 64*m**7 + 1708*m**6 + 24640*m**5 + 208054
*m**4 + 1038016*m**3 + 2924172*m**2 + 4098240*m + 2027025) + 20065*a**3*e*m**4*x**3*(f*x)**m/(m**8 + 64*m**7 +
1708*m**6 + 24640*m**5 + 208054*m**4 + 1038016*m**3 + 2924172*m**2 + 4098240*m + 2027025) + 147859*a**3*e*m**
3*x**3*(f*x)**m/(m**8 + 64*m**7 + 1708*m**6 + 2...
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 2816 vs.
\(2 (253) = 506\).
time = 5.87, size = 2816, normalized size = 11.59 \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)^3,x, algorithm="giac")
[Out]
((f*x)^m*c^3*m^7*x^15*e + 49*(f*x)^m*c^3*m^6*x^15*e + (f*x)^m*c^3*d*m^7*x^13 + 3*(f*x)^m*b*c^2*m^7*x^13*e + 97
3*(f*x)^m*c^3*m^5*x^15*e + 51*(f*x)^m*c^3*d*m^6*x^13 + 153*(f*x)^m*b*c^2*m^6*x^13*e + 10045*(f*x)^m*c^3*m^4*x^
15*e + 3*(f*x)^m*b*c^2*d*m^7*x^11 + 1045*(f*x)^m*c^3*d*m^5*x^13 + 3*(f*x)^m*b^2*c*m^7*x^11*e + 3*(f*x)^m*a*c^2
*m^7*x^11*e + 3135*(f*x)^m*b*c^2*m^5*x^13*e + 57379*(f*x)^m*c^3*m^3*x^15*e + 159*(f*x)^m*b*c^2*d*m^6*x^11 + 11
055*(f*x)^m*c^3*d*m^4*x^13 + 159*(f*x)^m*b^2*c*m^6*x^11*e + 159*(f*x)^m*a*c^2*m^6*x^11*e + 33165*(f*x)^m*b*c^2
*m^4*x^13*e + 177331*(f*x)^m*c^3*m^2*x^15*e + 3*(f*x)^m*b^2*c*d*m^7*x^9 + 3*(f*x)^m*a*c^2*d*m^7*x^9 + 3375*(f*
x)^m*b*c^2*d*m^5*x^11 + 64339*(f*x)^m*c^3*d*m^3*x^13 + (f*x)^m*b^3*m^7*x^9*e + 6*(f*x)^m*a*b*c*m^7*x^9*e + 337
5*(f*x)^m*b^2*c*m^5*x^11*e + 3375*(f*x)^m*a*c^2*m^5*x^11*e + 193017*(f*x)^m*b*c^2*m^3*x^13*e + 264207*(f*x)^m*
c^3*m*x^15*e + 165*(f*x)^m*b^2*c*d*m^6*x^9 + 165*(f*x)^m*a*c^2*d*m^6*x^9 + 36795*(f*x)^m*b*c^2*d*m^4*x^11 + 20
1609*(f*x)^m*c^3*d*m^2*x^13 + 55*(f*x)^m*b^3*m^6*x^9*e + 330*(f*x)^m*a*b*c*m^6*x^9*e + 36795*(f*x)^m*b^2*c*m^4
*x^11*e + 36795*(f*x)^m*a*c^2*m^4*x^11*e + 604827*(f*x)^m*b*c^2*m^2*x^13*e + 135135*(f*x)^m*c^3*x^15*e + (f*x)
^m*b^3*d*m^7*x^7 + 6*(f*x)^m*a*b*c*d*m^7*x^7 + 3639*(f*x)^m*b^2*c*d*m^5*x^9 + 3639*(f*x)^m*a*c^2*d*m^5*x^9 + 2
19417*(f*x)^m*b*c^2*d*m^3*x^11 + 303255*(f*x)^m*c^3*d*m*x^13 + 3*(f*x)^m*a*b^2*m^7*x^7*e + 3*(f*x)^m*a^2*c*m^7
*x^7*e + 1213*(f*x)^m*b^3*m^5*x^9*e + 7278*(f*x)^m*a*b*c*m^5*x^9*e + 219417*(f*x)^m*b^2*c*m^3*x^11*e + 219417*
(f*x)^m*a*c^2*m^3*x^11*e + 909765*(f*x)^m*b*c^2*m*x^13*e + 57*(f*x)^m*b^3*d*m^6*x^7 + 342*(f*x)^m*a*b*c*d*m^6*
x^7 + 41169*(f*x)^m*b^2*c*d*m^4*x^9 + 41169*(f*x)^m*a*c^2*d*m^4*x^9 + 700461*(f*x)^m*b*c^2*d*m^2*x^11 + 155925
*(f*x)^m*c^3*d*x^13 + 171*(f*x)^m*a*b^2*m^6*x^7*e + 171*(f*x)^m*a^2*c*m^6*x^7*e + 13723*(f*x)^m*b^3*m^4*x^9*e
+ 82338*(f*x)^m*a*b*c*m^4*x^9*e + 700461*(f*x)^m*b^2*c*m^2*x^11*e + 700461*(f*x)^m*a*c^2*m^2*x^11*e + 467775*(
f*x)^m*b*c^2*x^13*e + 3*(f*x)^m*a*b^2*d*m^7*x^5 + 3*(f*x)^m*a^2*c*d*m^7*x^5 + 1309*(f*x)^m*b^3*d*m^5*x^7 + 785
4*(f*x)^m*a*b*c*d*m^5*x^7 + 253641*(f*x)^m*b^2*c*d*m^3*x^9 + 253641*(f*x)^m*a*c^2*d*m^3*x^9 + 1067445*(f*x)^m*
b*c^2*d*m*x^11 + 3*(f*x)^m*a^2*b*m^7*x^5*e + 3927*(f*x)^m*a*b^2*m^5*x^7*e + 3927*(f*x)^m*a^2*c*m^5*x^7*e + 845
47*(f*x)^m*b^3*m^3*x^9*e + 507282*(f*x)^m*a*b*c*m^3*x^9*e + 1067445*(f*x)^m*b^2*c*m*x^11*e + 1067445*(f*x)^m*a
*c^2*m*x^11*e + 177*(f*x)^m*a*b^2*d*m^6*x^5 + 177*(f*x)^m*a^2*c*d*m^6*x^5 + 15477*(f*x)^m*b^3*d*m^4*x^7 + 9286
2*(f*x)^m*a*b*c*d*m^4*x^7 + 831279*(f*x)^m*b^2*c*d*m^2*x^9 + 831279*(f*x)^m*a*c^2*d*m^2*x^9 + 552825*(f*x)^m*b
*c^2*d*x^11 + 177*(f*x)^m*a^2*b*m^6*x^5*e + 46431*(f*x)^m*a*b^2*m^4*x^7*e + 46431*(f*x)^m*a^2*c*m^4*x^7*e + 27
7093*(f*x)^m*b^3*m^2*x^9*e + 1662558*(f*x)^m*a*b*c*m^2*x^9*e + 552825*(f*x)^m*b^2*c*x^11*e + 552825*(f*x)^m*a*
c^2*x^11*e + 3*(f*x)^m*a^2*b*d*m^7*x^3 + 4239*(f*x)^m*a*b^2*d*m^5*x^5 + 4239*(f*x)^m*a^2*c*d*m^5*x^5 + 99715*(
f*x)^m*b^3*d*m^3*x^7 + 598290*(f*x)^m*a*b*c*d*m^3*x^7 + 1291005*(f*x)^m*b^2*c*d*m*x^9 + 1291005*(f*x)^m*a*c^2*
d*m*x^9 + (f*x)^m*a^3*m^7*x^3*e + 4239*(f*x)^m*a^2*b*m^5*x^5*e + 299145*(f*x)^m*a*b^2*m^3*x^7*e + 299145*(f*x)
^m*a^2*c*m^3*x^7*e + 430335*(f*x)^m*b^3*m*x^9*e + 2582010*(f*x)^m*a*b*c*m*x^9*e + 183*(f*x)^m*a^2*b*d*m^6*x^3
+ 52725*(f*x)^m*a*b^2*d*m^4*x^5 + 52725*(f*x)^m*a^2*c*d*m^4*x^5 + 340011*(f*x)^m*b^3*d*m^2*x^7 + 2040066*(f*x)
^m*a*b*c*d*m^2*x^7 + 675675*(f*x)^m*b^2*c*d*x^9 + 675675*(f*x)^m*a*c^2*d*x^9 + 61*(f*x)^m*a^3*m^6*x^3*e + 5272
5*(f*x)^m*a^2*b*m^4*x^5*e + 1020033*(f*x)^m*a*b^2*m^2*x^7*e + 1020033*(f*x)^m*a^2*c*m^2*x^7*e + 225225*(f*x)^m
*b^3*x^9*e + 1351350*(f*x)^m*a*b*c*x^9*e + (f*x)^m*a^3*d*m^7*x + 4575*(f*x)^m*a^2*b*d*m^5*x^3 + 360537*(f*x)^m
*a*b^2*d*m^3*x^5 + 360537*(f*x)^m*a^2*c*d*m^3*x^5 + 544095*(f*x)^m*b^3*d*m*x^7 + 3264570*(f*x)^m*a*b*c*d*m*x^7
+ 1525*(f*x)^m*a^3*m^5*x^3*e + 360537*(f*x)^m*a^2*b*m^3*x^5*e + 1632285*(f*x)^m*a*b^2*m*x^7*e + 1632285*(f*x)
^m*a^2*c*m*x^7*e + 63*(f*x)^m*a^3*d*m^6*x + 60195*(f*x)^m*a^2*b*d*m^4*x^3 + 1311363*(f*x)^m*a*b^2*d*m^2*x^5 +
1311363*(f*x)^m*a^2*c*d*m^2*x^5 + 289575*(f*x)^m*b^3*d*x^7 + 1737450*(f*x)^m*a*b*c*d*x^7 + 20065*(f*x)^m*a^3*m
^4*x^3*e + 1311363*(f*x)^m*a^2*b*m^2*x^5*e + 868725*(f*x)^m*a*b^2*x^7*e + 868725*(f*x)^m*a^2*c*x^7*e + 1645*(f
*x)^m*a^3*d*m^5*x + 443577*(f*x)^m*a^2*b*d*m^3*x^3 + 2215701*(f*x)^m*a*b^2*d*m*x^5 + 2215701*(f*x)^m*a^2*c*d*m
*x^5 + 147859*(f*x)^m*a^3*m^3*x^3*e + 2215701*(f*x)^m*a^2*b*m*x^5*e + 22995*(f*x)^m*a^3*d*m^4*x + 1783317*(f*x
)^m*a^2*b*d*m^2*x^3 + 1216215*(f*x)^m*a*b^2*d*x^5 + 1216215*(f*x)^m*a^2*c*d*x^5 + 594439*(f*x)^m*a^3*m^2*x^3*e
+ 1216215*(f*x)^m*a^2*b*x^5*e + 185059*(f*x)^m*a^3*d*m^3*x + 3422565*(f*x)^m*a^2*b*d*m*x^3 + 1140855*(f*x)^m*
a^3*m*x^3*e + 852957*(f*x)^m*a^3*d*m^2*x + 2027025*(f*x)^m*a^2*b*d*x^3 + 675675*(f*x)^m*a^3*x^3*e + 2071215*(f
*x)^m*a^3*d*m*x + 2027025*(f*x)^m*a^3*d*x)/(m^8 + 64*m^7 + 1708*m^6 + 24640*m^5 + 208054*m^4 + 1038016*m^3 + 2
924172*m^2 + 4098240*m + 2027025)
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Mupad [B]
time = 1.06, size = 769, normalized size = 3.16 \begin {gather*} \frac {x^7\,{\left (f\,x\right )}^m\,\left (3\,c\,e\,a^2+3\,e\,a\,b^2+6\,c\,d\,a\,b+d\,b^3\right )\,\left (m^7+57\,m^6+1309\,m^5+15477\,m^4+99715\,m^3+340011\,m^2+544095\,m+289575\right )}{m^8+64\,m^7+1708\,m^6+24640\,m^5+208054\,m^4+1038016\,m^3+2924172\,m^2+4098240\,m+2027025}+\frac {x^9\,{\left (f\,x\right )}^m\,\left (e\,b^3+3\,d\,b^2\,c+6\,a\,e\,b\,c+3\,a\,d\,c^2\right )\,\left (m^7+55\,m^6+1213\,m^5+13723\,m^4+84547\,m^3+277093\,m^2+430335\,m+225225\right )}{m^8+64\,m^7+1708\,m^6+24640\,m^5+208054\,m^4+1038016\,m^3+2924172\,m^2+4098240\,m+2027025}+\frac {a^3\,d\,x\,{\left (f\,x\right )}^m\,\left (m^7+63\,m^6+1645\,m^5+22995\,m^4+185059\,m^3+852957\,m^2+2071215\,m+2027025\right )}{m^8+64\,m^7+1708\,m^6+24640\,m^5+208054\,m^4+1038016\,m^3+2924172\,m^2+4098240\,m+2027025}+\frac {c^3\,e\,x^{15}\,{\left (f\,x\right )}^m\,\left (m^7+49\,m^6+973\,m^5+10045\,m^4+57379\,m^3+177331\,m^2+264207\,m+135135\right )}{m^8+64\,m^7+1708\,m^6+24640\,m^5+208054\,m^4+1038016\,m^3+2924172\,m^2+4098240\,m+2027025}+\frac {3\,a\,x^5\,{\left (f\,x\right )}^m\,\left (d\,b^2+a\,e\,b+a\,c\,d\right )\,\left (m^7+59\,m^6+1413\,m^5+17575\,m^4+120179\,m^3+437121\,m^2+738567\,m+405405\right )}{m^8+64\,m^7+1708\,m^6+24640\,m^5+208054\,m^4+1038016\,m^3+2924172\,m^2+4098240\,m+2027025}+\frac {3\,c\,x^{11}\,{\left (f\,x\right )}^m\,\left (e\,b^2+c\,d\,b+a\,c\,e\right )\,\left (m^7+53\,m^6+1125\,m^5+12265\,m^4+73139\,m^3+233487\,m^2+355815\,m+184275\right )}{m^8+64\,m^7+1708\,m^6+24640\,m^5+208054\,m^4+1038016\,m^3+2924172\,m^2+4098240\,m+2027025}+\frac {a^2\,x^3\,{\left (f\,x\right )}^m\,\left (a\,e+3\,b\,d\right )\,\left (m^7+61\,m^6+1525\,m^5+20065\,m^4+147859\,m^3+594439\,m^2+1140855\,m+675675\right )}{m^8+64\,m^7+1708\,m^6+24640\,m^5+208054\,m^4+1038016\,m^3+2924172\,m^2+4098240\,m+2027025}+\frac {c^2\,x^{13}\,{\left (f\,x\right )}^m\,\left (3\,b\,e+c\,d\right )\,\left (m^7+51\,m^6+1045\,m^5+11055\,m^4+64339\,m^3+201609\,m^2+303255\,m+155925\right )}{m^8+64\,m^7+1708\,m^6+24640\,m^5+208054\,m^4+1038016\,m^3+2924172\,m^2+4098240\,m+2027025} \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)^3,x)
[Out]
(x^7*(f*x)^m*(b^3*d + 3*a*b^2*e + 3*a^2*c*e + 6*a*b*c*d)*(544095*m + 340011*m^2 + 99715*m^3 + 15477*m^4 + 1309
*m^5 + 57*m^6 + m^7 + 289575))/(4098240*m + 2924172*m^2 + 1038016*m^3 + 208054*m^4 + 24640*m^5 + 1708*m^6 + 64
*m^7 + m^8 + 2027025) + (x^9*(f*x)^m*(b^3*e + 3*a*c^2*d + 3*b^2*c*d + 6*a*b*c*e)*(430335*m + 277093*m^2 + 8454
7*m^3 + 13723*m^4 + 1213*m^5 + 55*m^6 + m^7 + 225225))/(4098240*m + 2924172*m^2 + 1038016*m^3 + 208054*m^4 + 2
4640*m^5 + 1708*m^6 + 64*m^7 + m^8 + 2027025) + (a^3*d*x*(f*x)^m*(2071215*m + 852957*m^2 + 185059*m^3 + 22995*
m^4 + 1645*m^5 + 63*m^6 + m^7 + 2027025))/(4098240*m + 2924172*m^2 + 1038016*m^3 + 208054*m^4 + 24640*m^5 + 17
08*m^6 + 64*m^7 + m^8 + 2027025) + (c^3*e*x^15*(f*x)^m*(264207*m + 177331*m^2 + 57379*m^3 + 10045*m^4 + 973*m^
5 + 49*m^6 + m^7 + 135135))/(4098240*m + 2924172*m^2 + 1038016*m^3 + 208054*m^4 + 24640*m^5 + 1708*m^6 + 64*m^
7 + m^8 + 2027025) + (3*a*x^5*(f*x)^m*(b^2*d + a*b*e + a*c*d)*(738567*m + 437121*m^2 + 120179*m^3 + 17575*m^4
+ 1413*m^5 + 59*m^6 + m^7 + 405405))/(4098240*m + 2924172*m^2 + 1038016*m^3 + 208054*m^4 + 24640*m^5 + 1708*m^
6 + 64*m^7 + m^8 + 2027025) + (3*c*x^11*(f*x)^m*(b^2*e + a*c*e + b*c*d)*(355815*m + 233487*m^2 + 73139*m^3 + 1
2265*m^4 + 1125*m^5 + 53*m^6 + m^7 + 184275))/(4098240*m + 2924172*m^2 + 1038016*m^3 + 208054*m^4 + 24640*m^5
+ 1708*m^6 + 64*m^7 + m^8 + 2027025) + (a^2*x^3*(f*x)^m*(a*e + 3*b*d)*(1140855*m + 594439*m^2 + 147859*m^3 + 2
0065*m^4 + 1525*m^5 + 61*m^6 + m^7 + 675675))/(4098240*m + 2924172*m^2 + 1038016*m^3 + 208054*m^4 + 24640*m^5
+ 1708*m^6 + 64*m^7 + m^8 + 2027025) + (c^2*x^13*(f*x)^m*(3*b*e + c*d)*(303255*m + 201609*m^2 + 64339*m^3 + 11
055*m^4 + 1045*m^5 + 51*m^6 + m^7 + 155925))/(4098240*m + 2924172*m^2 + 1038016*m^3 + 208054*m^4 + 24640*m^5 +
1708*m^6 + 64*m^7 + m^8 + 2027025)
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