3.1105 \(\int (d x)^m (a+b x^2+c x^4)^3 \, dx\)
Optimal. Leaf size=156 \[ \frac {a^3 (d x)^{m+1}}{d (m+1)}+\frac {3 a^2 b (d x)^{m+3}}{d^3 (m+3)}+\frac {3 c \left (a c+b^2\right ) (d x)^{m+9}}{d^9 (m+9)}+\frac {b \left (6 a c+b^2\right ) (d x)^{m+7}}{d^7 (m+7)}+\frac {3 a \left (a c+b^2\right ) (d x)^{m+5}}{d^5 (m+5)}+\frac {3 b c^2 (d x)^{m+11}}{d^{11} (m+11)}+\frac {c^3 (d x)^{m+13}}{d^{13} (m+13)} \]
[Out]
a^3*(d*x)^(1+m)/d/(1+m)+3*a^2*b*(d*x)^(3+m)/d^3/(3+m)+3*a*(a*c+b^2)*(d*x)^(5+m)/d^5/(5+m)+b*(6*a*c+b^2)*(d*x)^
(7+m)/d^7/(7+m)+3*c*(a*c+b^2)*(d*x)^(9+m)/d^9/(9+m)+3*b*c^2*(d*x)^(11+m)/d^11/(11+m)+c^3*(d*x)^(13+m)/d^13/(13
+m)
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Rubi [A] time = 0.10, antiderivative size = 156, normalized size of antiderivative = 1.00,
number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used =
{1108} \[ \frac {3 a^2 b (d x)^{m+3}}{d^3 (m+3)}+\frac {a^3 (d x)^{m+1}}{d (m+1)}+\frac {3 a \left (a c+b^2\right ) (d x)^{m+5}}{d^5 (m+5)}+\frac {b \left (6 a c+b^2\right ) (d x)^{m+7}}{d^7 (m+7)}+\frac {3 c \left (a c+b^2\right ) (d x)^{m+9}}{d^9 (m+9)}+\frac {3 b c^2 (d x)^{m+11}}{d^{11} (m+11)}+\frac {c^3 (d x)^{m+13}}{d^{13} (m+13)} \]
Antiderivative was successfully verified.
[In]
Int[(d*x)^m*(a + b*x^2 + c*x^4)^3,x]
[Out]
(a^3*(d*x)^(1 + m))/(d*(1 + m)) + (3*a^2*b*(d*x)^(3 + m))/(d^3*(3 + m)) + (3*a*(b^2 + a*c)*(d*x)^(5 + m))/(d^5
*(5 + m)) + (b*(b^2 + 6*a*c)*(d*x)^(7 + m))/(d^7*(7 + m)) + (3*c*(b^2 + a*c)*(d*x)^(9 + m))/(d^9*(9 + m)) + (3
*b*c^2*(d*x)^(11 + m))/(d^11*(11 + m)) + (c^3*(d*x)^(13 + m))/(d^13*(13 + m))
Rule 1108
Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d*x)^m*(a
+ b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[p, 0] && !IntegerQ[(m + 1)/2]
Rubi steps
\begin {align*} \int (d x)^m \left (a+b x^2+c x^4\right )^3 \, dx &=\int \left (a^3 (d x)^m+\frac {3 a^2 b (d x)^{2+m}}{d^2}+\frac {3 a \left (b^2+a c\right ) (d x)^{4+m}}{d^4}+\frac {b \left (b^2+6 a c\right ) (d x)^{6+m}}{d^6}+\frac {3 c \left (b^2+a c\right ) (d x)^{8+m}}{d^8}+\frac {3 b c^2 (d x)^{10+m}}{d^{10}}+\frac {c^3 (d x)^{12+m}}{d^{12}}\right ) \, dx\\ &=\frac {a^3 (d x)^{1+m}}{d (1+m)}+\frac {3 a^2 b (d x)^{3+m}}{d^3 (3+m)}+\frac {3 a \left (b^2+a c\right ) (d x)^{5+m}}{d^5 (5+m)}+\frac {b \left (b^2+6 a c\right ) (d x)^{7+m}}{d^7 (7+m)}+\frac {3 c \left (b^2+a c\right ) (d x)^{9+m}}{d^9 (9+m)}+\frac {3 b c^2 (d x)^{11+m}}{d^{11} (11+m)}+\frac {c^3 (d x)^{13+m}}{d^{13} (13+m)}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 111, normalized size = 0.71 \[ x (d x)^m \left (\frac {a^3}{m+1}+\frac {3 a^2 b x^2}{m+3}+\frac {3 c x^8 \left (a c+b^2\right )}{m+9}+\frac {b x^6 \left (6 a c+b^2\right )}{m+7}+\frac {3 a x^4 \left (a c+b^2\right )}{m+5}+\frac {3 b c^2 x^{10}}{m+11}+\frac {c^3 x^{12}}{m+13}\right ) \]
Antiderivative was successfully verified.
[In]
Integrate[(d*x)^m*(a + b*x^2 + c*x^4)^3,x]
[Out]
x*(d*x)^m*(a^3/(1 + m) + (3*a^2*b*x^2)/(3 + m) + (3*a*(b^2 + a*c)*x^4)/(5 + m) + (b*(b^2 + 6*a*c)*x^6)/(7 + m)
+ (3*c*(b^2 + a*c)*x^8)/(9 + m) + (3*b*c^2*x^10)/(11 + m) + (c^3*x^12)/(13 + m))
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fricas [B] time = 0.69, size = 594, normalized size = 3.81 \[ \frac {{\left ({\left (c^{3} m^{6} + 36 \, c^{3} m^{5} + 505 \, c^{3} m^{4} + 3480 \, c^{3} m^{3} + 12139 \, c^{3} m^{2} + 19524 \, c^{3} m + 10395 \, c^{3}\right )} x^{13} + 3 \, {\left (b c^{2} m^{6} + 38 \, b c^{2} m^{5} + 555 \, b c^{2} m^{4} + 3940 \, b c^{2} m^{3} + 14039 \, b c^{2} m^{2} + 22902 \, b c^{2} m + 12285 \, b c^{2}\right )} x^{11} + 3 \, {\left ({\left (b^{2} c + a c^{2}\right )} m^{6} + 40 \, {\left (b^{2} c + a c^{2}\right )} m^{5} + 613 \, {\left (b^{2} c + a c^{2}\right )} m^{4} + 4528 \, {\left (b^{2} c + a c^{2}\right )} m^{3} + 15015 \, b^{2} c + 15015 \, a c^{2} + 16627 \, {\left (b^{2} c + a c^{2}\right )} m^{2} + 27688 \, {\left (b^{2} c + a c^{2}\right )} m\right )} x^{9} + {\left ({\left (b^{3} + 6 \, a b c\right )} m^{6} + 42 \, {\left (b^{3} + 6 \, a b c\right )} m^{5} + 679 \, {\left (b^{3} + 6 \, a b c\right )} m^{4} + 5292 \, {\left (b^{3} + 6 \, a b c\right )} m^{3} + 19305 \, b^{3} + 115830 \, a b c + 20335 \, {\left (b^{3} + 6 \, a b c\right )} m^{2} + 34986 \, {\left (b^{3} + 6 \, a b c\right )} m\right )} x^{7} + 3 \, {\left ({\left (a b^{2} + a^{2} c\right )} m^{6} + 44 \, {\left (a b^{2} + a^{2} c\right )} m^{5} + 753 \, {\left (a b^{2} + a^{2} c\right )} m^{4} + 6280 \, {\left (a b^{2} + a^{2} c\right )} m^{3} + 27027 \, a b^{2} + 27027 \, a^{2} c + 25979 \, {\left (a b^{2} + a^{2} c\right )} m^{2} + 47436 \, {\left (a b^{2} + a^{2} c\right )} m\right )} x^{5} + 3 \, {\left (a^{2} b m^{6} + 46 \, a^{2} b m^{5} + 835 \, a^{2} b m^{4} + 7540 \, a^{2} b m^{3} + 34759 \, a^{2} b m^{2} + 73054 \, a^{2} b m + 45045 \, a^{2} b\right )} x^{3} + {\left (a^{3} m^{6} + 48 \, a^{3} m^{5} + 925 \, a^{3} m^{4} + 9120 \, a^{3} m^{3} + 48259 \, a^{3} m^{2} + 129072 \, a^{3} m + 135135 \, a^{3}\right )} x\right )} \left (d x\right )^{m}}{m^{7} + 49 \, m^{6} + 973 \, m^{5} + 10045 \, m^{4} + 57379 \, m^{3} + 177331 \, m^{2} + 264207 \, m + 135135} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x)^m*(c*x^4+b*x^2+a)^3,x, algorithm="fricas")
[Out]
((c^3*m^6 + 36*c^3*m^5 + 505*c^3*m^4 + 3480*c^3*m^3 + 12139*c^3*m^2 + 19524*c^3*m + 10395*c^3)*x^13 + 3*(b*c^2
*m^6 + 38*b*c^2*m^5 + 555*b*c^2*m^4 + 3940*b*c^2*m^3 + 14039*b*c^2*m^2 + 22902*b*c^2*m + 12285*b*c^2)*x^11 + 3
*((b^2*c + a*c^2)*m^6 + 40*(b^2*c + a*c^2)*m^5 + 613*(b^2*c + a*c^2)*m^4 + 4528*(b^2*c + a*c^2)*m^3 + 15015*b^
2*c + 15015*a*c^2 + 16627*(b^2*c + a*c^2)*m^2 + 27688*(b^2*c + a*c^2)*m)*x^9 + ((b^3 + 6*a*b*c)*m^6 + 42*(b^3
+ 6*a*b*c)*m^5 + 679*(b^3 + 6*a*b*c)*m^4 + 5292*(b^3 + 6*a*b*c)*m^3 + 19305*b^3 + 115830*a*b*c + 20335*(b^3 +
6*a*b*c)*m^2 + 34986*(b^3 + 6*a*b*c)*m)*x^7 + 3*((a*b^2 + a^2*c)*m^6 + 44*(a*b^2 + a^2*c)*m^5 + 753*(a*b^2 + a
^2*c)*m^4 + 6280*(a*b^2 + a^2*c)*m^3 + 27027*a*b^2 + 27027*a^2*c + 25979*(a*b^2 + a^2*c)*m^2 + 47436*(a*b^2 +
a^2*c)*m)*x^5 + 3*(a^2*b*m^6 + 46*a^2*b*m^5 + 835*a^2*b*m^4 + 7540*a^2*b*m^3 + 34759*a^2*b*m^2 + 73054*a^2*b*m
+ 45045*a^2*b)*x^3 + (a^3*m^6 + 48*a^3*m^5 + 925*a^3*m^4 + 9120*a^3*m^3 + 48259*a^3*m^2 + 129072*a^3*m + 1351
35*a^3)*x)*(d*x)^m/(m^7 + 49*m^6 + 973*m^5 + 10045*m^4 + 57379*m^3 + 177331*m^2 + 264207*m + 135135)
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giac [B] time = 0.23, size = 1132, normalized size = 7.26 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x)^m*(c*x^4+b*x^2+a)^3,x, algorithm="giac")
[Out]
((d*x)^m*c^3*m^6*x^13 + 36*(d*x)^m*c^3*m^5*x^13 + 3*(d*x)^m*b*c^2*m^6*x^11 + 505*(d*x)^m*c^3*m^4*x^13 + 114*(d
*x)^m*b*c^2*m^5*x^11 + 3480*(d*x)^m*c^3*m^3*x^13 + 3*(d*x)^m*b^2*c*m^6*x^9 + 3*(d*x)^m*a*c^2*m^6*x^9 + 1665*(d
*x)^m*b*c^2*m^4*x^11 + 12139*(d*x)^m*c^3*m^2*x^13 + 120*(d*x)^m*b^2*c*m^5*x^9 + 120*(d*x)^m*a*c^2*m^5*x^9 + 11
820*(d*x)^m*b*c^2*m^3*x^11 + 19524*(d*x)^m*c^3*m*x^13 + (d*x)^m*b^3*m^6*x^7 + 6*(d*x)^m*a*b*c*m^6*x^7 + 1839*(
d*x)^m*b^2*c*m^4*x^9 + 1839*(d*x)^m*a*c^2*m^4*x^9 + 42117*(d*x)^m*b*c^2*m^2*x^11 + 10395*(d*x)^m*c^3*x^13 + 42
*(d*x)^m*b^3*m^5*x^7 + 252*(d*x)^m*a*b*c*m^5*x^7 + 13584*(d*x)^m*b^2*c*m^3*x^9 + 13584*(d*x)^m*a*c^2*m^3*x^9 +
68706*(d*x)^m*b*c^2*m*x^11 + 3*(d*x)^m*a*b^2*m^6*x^5 + 3*(d*x)^m*a^2*c*m^6*x^5 + 679*(d*x)^m*b^3*m^4*x^7 + 40
74*(d*x)^m*a*b*c*m^4*x^7 + 49881*(d*x)^m*b^2*c*m^2*x^9 + 49881*(d*x)^m*a*c^2*m^2*x^9 + 36855*(d*x)^m*b*c^2*x^1
1 + 132*(d*x)^m*a*b^2*m^5*x^5 + 132*(d*x)^m*a^2*c*m^5*x^5 + 5292*(d*x)^m*b^3*m^3*x^7 + 31752*(d*x)^m*a*b*c*m^3
*x^7 + 83064*(d*x)^m*b^2*c*m*x^9 + 83064*(d*x)^m*a*c^2*m*x^9 + 3*(d*x)^m*a^2*b*m^6*x^3 + 2259*(d*x)^m*a*b^2*m^
4*x^5 + 2259*(d*x)^m*a^2*c*m^4*x^5 + 20335*(d*x)^m*b^3*m^2*x^7 + 122010*(d*x)^m*a*b*c*m^2*x^7 + 45045*(d*x)^m*
b^2*c*x^9 + 45045*(d*x)^m*a*c^2*x^9 + 138*(d*x)^m*a^2*b*m^5*x^3 + 18840*(d*x)^m*a*b^2*m^3*x^5 + 18840*(d*x)^m*
a^2*c*m^3*x^5 + 34986*(d*x)^m*b^3*m*x^7 + 209916*(d*x)^m*a*b*c*m*x^7 + (d*x)^m*a^3*m^6*x + 2505*(d*x)^m*a^2*b*
m^4*x^3 + 77937*(d*x)^m*a*b^2*m^2*x^5 + 77937*(d*x)^m*a^2*c*m^2*x^5 + 19305*(d*x)^m*b^3*x^7 + 115830*(d*x)^m*a
*b*c*x^7 + 48*(d*x)^m*a^3*m^5*x + 22620*(d*x)^m*a^2*b*m^3*x^3 + 142308*(d*x)^m*a*b^2*m*x^5 + 142308*(d*x)^m*a^
2*c*m*x^5 + 925*(d*x)^m*a^3*m^4*x + 104277*(d*x)^m*a^2*b*m^2*x^3 + 81081*(d*x)^m*a*b^2*x^5 + 81081*(d*x)^m*a^2
*c*x^5 + 9120*(d*x)^m*a^3*m^3*x + 219162*(d*x)^m*a^2*b*m*x^3 + 48259*(d*x)^m*a^3*m^2*x + 135135*(d*x)^m*a^2*b*
x^3 + 129072*(d*x)^m*a^3*m*x + 135135*(d*x)^m*a^3*x)/(m^7 + 49*m^6 + 973*m^5 + 10045*m^4 + 57379*m^3 + 177331*
m^2 + 264207*m + 135135)
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maple [B] time = 0.01, size = 782, normalized size = 5.01 \[ \frac {\left (c^{3} m^{6} x^{12}+36 c^{3} m^{5} x^{12}+3 b \,c^{2} m^{6} x^{10}+505 c^{3} m^{4} x^{12}+114 b \,c^{2} m^{5} x^{10}+3480 c^{3} m^{3} x^{12}+3 a \,c^{2} m^{6} x^{8}+3 b^{2} c \,m^{6} x^{8}+1665 b \,c^{2} m^{4} x^{10}+12139 c^{3} m^{2} x^{12}+120 a \,c^{2} m^{5} x^{8}+120 b^{2} c \,m^{5} x^{8}+11820 b \,c^{2} m^{3} x^{10}+19524 c^{3} m \,x^{12}+6 a b c \,m^{6} x^{6}+1839 a \,c^{2} m^{4} x^{8}+b^{3} m^{6} x^{6}+1839 b^{2} c \,m^{4} x^{8}+42117 b \,c^{2} m^{2} x^{10}+10395 c^{3} x^{12}+252 a b c \,m^{5} x^{6}+13584 a \,c^{2} m^{3} x^{8}+42 b^{3} m^{5} x^{6}+13584 b^{2} c \,m^{3} x^{8}+68706 b \,c^{2} m \,x^{10}+3 a^{2} c \,m^{6} x^{4}+3 a \,b^{2} m^{6} x^{4}+4074 a b c \,m^{4} x^{6}+49881 a \,c^{2} m^{2} x^{8}+679 b^{3} m^{4} x^{6}+49881 b^{2} c \,m^{2} x^{8}+36855 b \,c^{2} x^{10}+132 a^{2} c \,m^{5} x^{4}+132 a \,b^{2} m^{5} x^{4}+31752 a b c \,m^{3} x^{6}+83064 a \,c^{2} m \,x^{8}+5292 b^{3} m^{3} x^{6}+83064 b^{2} c m \,x^{8}+3 a^{2} b \,m^{6} x^{2}+2259 a^{2} c \,m^{4} x^{4}+2259 a \,b^{2} m^{4} x^{4}+122010 a b c \,m^{2} x^{6}+45045 a \,c^{2} x^{8}+20335 b^{3} m^{2} x^{6}+45045 b^{2} c \,x^{8}+138 a^{2} b \,m^{5} x^{2}+18840 a^{2} c \,m^{3} x^{4}+18840 a \,b^{2} m^{3} x^{4}+209916 a b c m \,x^{6}+34986 b^{3} m \,x^{6}+a^{3} m^{6}+2505 a^{2} b \,m^{4} x^{2}+77937 a^{2} c \,m^{2} x^{4}+77937 a \,b^{2} m^{2} x^{4}+115830 a b c \,x^{6}+19305 b^{3} x^{6}+48 a^{3} m^{5}+22620 a^{2} b \,m^{3} x^{2}+142308 a^{2} c m \,x^{4}+142308 a \,b^{2} m \,x^{4}+925 a^{3} m^{4}+104277 a^{2} b \,m^{2} x^{2}+81081 a^{2} c \,x^{4}+81081 a \,b^{2} x^{4}+9120 a^{3} m^{3}+219162 a^{2} b m \,x^{2}+48259 a^{3} m^{2}+135135 a^{2} b \,x^{2}+129072 a^{3} m +135135 a^{3}\right ) x \left (d x \right )^{m}}{\left (m +13\right ) \left (m +11\right ) \left (m +9\right ) \left (m +7\right ) \left (m +5\right ) \left (m +3\right ) \left (m +1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((d*x)^m*(c*x^4+b*x^2+a)^3,x)
[Out]
x*(c^3*m^6*x^12+36*c^3*m^5*x^12+3*b*c^2*m^6*x^10+505*c^3*m^4*x^12+114*b*c^2*m^5*x^10+3480*c^3*m^3*x^12+3*a*c^2
*m^6*x^8+3*b^2*c*m^6*x^8+1665*b*c^2*m^4*x^10+12139*c^3*m^2*x^12+120*a*c^2*m^5*x^8+120*b^2*c*m^5*x^8+11820*b*c^
2*m^3*x^10+19524*c^3*m*x^12+6*a*b*c*m^6*x^6+1839*a*c^2*m^4*x^8+b^3*m^6*x^6+1839*b^2*c*m^4*x^8+42117*b*c^2*m^2*
x^10+10395*c^3*x^12+252*a*b*c*m^5*x^6+13584*a*c^2*m^3*x^8+42*b^3*m^5*x^6+13584*b^2*c*m^3*x^8+68706*b*c^2*m*x^1
0+3*a^2*c*m^6*x^4+3*a*b^2*m^6*x^4+4074*a*b*c*m^4*x^6+49881*a*c^2*m^2*x^8+679*b^3*m^4*x^6+49881*b^2*c*m^2*x^8+3
6855*b*c^2*x^10+132*a^2*c*m^5*x^4+132*a*b^2*m^5*x^4+31752*a*b*c*m^3*x^6+83064*a*c^2*m*x^8+5292*b^3*m^3*x^6+830
64*b^2*c*m*x^8+3*a^2*b*m^6*x^2+2259*a^2*c*m^4*x^4+2259*a*b^2*m^4*x^4+122010*a*b*c*m^2*x^6+45045*a*c^2*x^8+2033
5*b^3*m^2*x^6+45045*b^2*c*x^8+138*a^2*b*m^5*x^2+18840*a^2*c*m^3*x^4+18840*a*b^2*m^3*x^4+209916*a*b*c*m*x^6+349
86*b^3*m*x^6+a^3*m^6+2505*a^2*b*m^4*x^2+77937*a^2*c*m^2*x^4+77937*a*b^2*m^2*x^4+115830*a*b*c*x^6+19305*b^3*x^6
+48*a^3*m^5+22620*a^2*b*m^3*x^2+142308*a^2*c*m*x^4+142308*a*b^2*m*x^4+925*a^3*m^4+104277*a^2*b*m^2*x^2+81081*a
^2*c*x^4+81081*a*b^2*x^4+9120*a^3*m^3+219162*a^2*b*m*x^2+48259*a^3*m^2+135135*a^2*b*x^2+129072*a^3*m+135135*a^
3)*(d*x)^m/(m+13)/(m+11)/(m+9)/(m+7)/(m+5)/(m+3)/(m+1)
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maxima [A] time = 1.31, size = 195, normalized size = 1.25 \[ \frac {c^{3} d^{m} x^{13} x^{m}}{m + 13} + \frac {3 \, b c^{2} d^{m} x^{11} x^{m}}{m + 11} + \frac {3 \, b^{2} c d^{m} x^{9} x^{m}}{m + 9} + \frac {3 \, a c^{2} d^{m} x^{9} x^{m}}{m + 9} + \frac {b^{3} d^{m} x^{7} x^{m}}{m + 7} + \frac {6 \, a b c d^{m} x^{7} x^{m}}{m + 7} + \frac {3 \, a b^{2} d^{m} x^{5} x^{m}}{m + 5} + \frac {3 \, a^{2} c d^{m} x^{5} x^{m}}{m + 5} + \frac {3 \, a^{2} b d^{m} x^{3} x^{m}}{m + 3} + \frac {\left (d x\right )^{m + 1} a^{3}}{d {\left (m + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x)^m*(c*x^4+b*x^2+a)^3,x, algorithm="maxima")
[Out]
c^3*d^m*x^13*x^m/(m + 13) + 3*b*c^2*d^m*x^11*x^m/(m + 11) + 3*b^2*c*d^m*x^9*x^m/(m + 9) + 3*a*c^2*d^m*x^9*x^m/
(m + 9) + b^3*d^m*x^7*x^m/(m + 7) + 6*a*b*c*d^m*x^7*x^m/(m + 7) + 3*a*b^2*d^m*x^5*x^m/(m + 5) + 3*a^2*c*d^m*x^
5*x^m/(m + 5) + 3*a^2*b*d^m*x^3*x^m/(m + 3) + (d*x)^(m + 1)*a^3/(d*(m + 1))
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mupad [B] time = 4.83, size = 546, normalized size = 3.50 \[ \frac {a^3\,x\,{\left (d\,x\right )}^m\,\left (m^6+48\,m^5+925\,m^4+9120\,m^3+48259\,m^2+129072\,m+135135\right )}{m^7+49\,m^6+973\,m^5+10045\,m^4+57379\,m^3+177331\,m^2+264207\,m+135135}+\frac {c^3\,x^{13}\,{\left (d\,x\right )}^m\,\left (m^6+36\,m^5+505\,m^4+3480\,m^3+12139\,m^2+19524\,m+10395\right )}{m^7+49\,m^6+973\,m^5+10045\,m^4+57379\,m^3+177331\,m^2+264207\,m+135135}+\frac {3\,a^2\,b\,x^3\,{\left (d\,x\right )}^m\,\left (m^6+46\,m^5+835\,m^4+7540\,m^3+34759\,m^2+73054\,m+45045\right )}{m^7+49\,m^6+973\,m^5+10045\,m^4+57379\,m^3+177331\,m^2+264207\,m+135135}+\frac {3\,b\,c^2\,x^{11}\,{\left (d\,x\right )}^m\,\left (m^6+38\,m^5+555\,m^4+3940\,m^3+14039\,m^2+22902\,m+12285\right )}{m^7+49\,m^6+973\,m^5+10045\,m^4+57379\,m^3+177331\,m^2+264207\,m+135135}+\frac {3\,a\,x^5\,{\left (d\,x\right )}^m\,\left (b^2+a\,c\right )\,\left (m^6+44\,m^5+753\,m^4+6280\,m^3+25979\,m^2+47436\,m+27027\right )}{m^7+49\,m^6+973\,m^5+10045\,m^4+57379\,m^3+177331\,m^2+264207\,m+135135}+\frac {b\,x^7\,{\left (d\,x\right )}^m\,\left (b^2+6\,a\,c\right )\,\left (m^6+42\,m^5+679\,m^4+5292\,m^3+20335\,m^2+34986\,m+19305\right )}{m^7+49\,m^6+973\,m^5+10045\,m^4+57379\,m^3+177331\,m^2+264207\,m+135135}+\frac {3\,c\,x^9\,{\left (d\,x\right )}^m\,\left (b^2+a\,c\right )\,\left (m^6+40\,m^5+613\,m^4+4528\,m^3+16627\,m^2+27688\,m+15015\right )}{m^7+49\,m^6+973\,m^5+10045\,m^4+57379\,m^3+177331\,m^2+264207\,m+135135} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((d*x)^m*(a + b*x^2 + c*x^4)^3,x)
[Out]
(a^3*x*(d*x)^m*(129072*m + 48259*m^2 + 9120*m^3 + 925*m^4 + 48*m^5 + m^6 + 135135))/(264207*m + 177331*m^2 + 5
7379*m^3 + 10045*m^4 + 973*m^5 + 49*m^6 + m^7 + 135135) + (c^3*x^13*(d*x)^m*(19524*m + 12139*m^2 + 3480*m^3 +
505*m^4 + 36*m^5 + m^6 + 10395))/(264207*m + 177331*m^2 + 57379*m^3 + 10045*m^4 + 973*m^5 + 49*m^6 + m^7 + 135
135) + (3*a^2*b*x^3*(d*x)^m*(73054*m + 34759*m^2 + 7540*m^3 + 835*m^4 + 46*m^5 + m^6 + 45045))/(264207*m + 177
331*m^2 + 57379*m^3 + 10045*m^4 + 973*m^5 + 49*m^6 + m^7 + 135135) + (3*b*c^2*x^11*(d*x)^m*(22902*m + 14039*m^
2 + 3940*m^3 + 555*m^4 + 38*m^5 + m^6 + 12285))/(264207*m + 177331*m^2 + 57379*m^3 + 10045*m^4 + 973*m^5 + 49*
m^6 + m^7 + 135135) + (3*a*x^5*(d*x)^m*(a*c + b^2)*(47436*m + 25979*m^2 + 6280*m^3 + 753*m^4 + 44*m^5 + m^6 +
27027))/(264207*m + 177331*m^2 + 57379*m^3 + 10045*m^4 + 973*m^5 + 49*m^6 + m^7 + 135135) + (b*x^7*(d*x)^m*(6*
a*c + b^2)*(34986*m + 20335*m^2 + 5292*m^3 + 679*m^4 + 42*m^5 + m^6 + 19305))/(264207*m + 177331*m^2 + 57379*m
^3 + 10045*m^4 + 973*m^5 + 49*m^6 + m^7 + 135135) + (3*c*x^9*(d*x)^m*(a*c + b^2)*(27688*m + 16627*m^2 + 4528*m
^3 + 613*m^4 + 40*m^5 + m^6 + 15015))/(264207*m + 177331*m^2 + 57379*m^3 + 10045*m^4 + 973*m^5 + 49*m^6 + m^7
+ 135135)
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sympy [A] time = 7.27, size = 4451, normalized size = 28.53 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x)**m*(c*x**4+b*x**2+a)**3,x)
[Out]
Piecewise(((-a**3/(12*x**12) - 3*a**2*b/(10*x**10) - 3*a**2*c/(8*x**8) - 3*a*b**2/(8*x**8) - a*b*c/x**6 - 3*a*
c**2/(4*x**4) - b**3/(6*x**6) - 3*b**2*c/(4*x**4) - 3*b*c**2/(2*x**2) + c**3*log(x))/d**13, Eq(m, -13)), ((-a*
*3/(10*x**10) - 3*a**2*b/(8*x**8) - a**2*c/(2*x**6) - a*b**2/(2*x**6) - 3*a*b*c/(2*x**4) - 3*a*c**2/(2*x**2) -
b**3/(4*x**4) - 3*b**2*c/(2*x**2) + 3*b*c**2*log(x) + c**3*x**2/2)/d**11, Eq(m, -11)), ((-a**3/(8*x**8) - a**
2*b/(2*x**6) - 3*a**2*c/(4*x**4) - 3*a*b**2/(4*x**4) - 3*a*b*c/x**2 + 3*a*c**2*log(x) - b**3/(2*x**2) + 3*b**2
*c*log(x) + 3*b*c**2*x**2/2 + c**3*x**4/4)/d**9, Eq(m, -9)), ((-a**3/(6*x**6) - 3*a**2*b/(4*x**4) - 3*a**2*c/(
2*x**2) - 3*a*b**2/(2*x**2) + 6*a*b*c*log(x) + 3*a*c**2*x**2/2 + b**3*log(x) + 3*b**2*c*x**2/2 + 3*b*c**2*x**4
/4 + c**3*x**6/6)/d**7, Eq(m, -7)), ((-a**3/(4*x**4) - 3*a**2*b/(2*x**2) + 3*a**2*c*log(x) + 3*a*b**2*log(x) +
3*a*b*c*x**2 + 3*a*c**2*x**4/4 + b**3*x**2/2 + 3*b**2*c*x**4/4 + b*c**2*x**6/2 + c**3*x**8/8)/d**5, Eq(m, -5)
), ((-a**3/(2*x**2) + 3*a**2*b*log(x) + 3*a**2*c*x**2/2 + 3*a*b**2*x**2/2 + 3*a*b*c*x**4/2 + a*c**2*x**6/2 + b
**3*x**4/4 + b**2*c*x**6/2 + 3*b*c**2*x**8/8 + c**3*x**10/10)/d**3, Eq(m, -3)), ((a**3*log(x) + 3*a**2*b*x**2/
2 + 3*a**2*c*x**4/4 + 3*a*b**2*x**4/4 + a*b*c*x**6 + 3*a*c**2*x**8/8 + b**3*x**6/6 + 3*b**2*c*x**8/8 + 3*b*c**
2*x**10/10 + c**3*x**12/12)/d, Eq(m, -1)), (a**3*d**m*m**6*x*x**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57
379*m**3 + 177331*m**2 + 264207*m + 135135) + 48*a**3*d**m*m**5*x*x**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4
+ 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 925*a**3*d**m*m**4*x*x**m/(m**7 + 49*m**6 + 973*m**5 + 1004
5*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 9120*a**3*d**m*m**3*x*x**m/(m**7 + 49*m**6 + 973*m**5
+ 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 48259*a**3*d**m*m**2*x*x**m/(m**7 + 49*m**6 +
973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 129072*a**3*d**m*m*x*x**m/(m**7 + 49*m
**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 135135*a**3*d**m*x*x**m/(m**7 +
49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 3*a**2*b*d**m*m**6*x**3*x**m
/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 138*a**2*b*d**m*m**
5*x**3*x**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 2505*a**
2*b*d**m*m**4*x**3*x**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135
) + 22620*a**2*b*d**m*m**3*x**3*x**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 2642
07*m + 135135) + 104277*a**2*b*d**m*m**2*x**3*x**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 1773
31*m**2 + 264207*m + 135135) + 219162*a**2*b*d**m*m*x**3*x**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*
m**3 + 177331*m**2 + 264207*m + 135135) + 135135*a**2*b*d**m*x**3*x**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4
+ 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 3*a**2*c*d**m*m**6*x**5*x**m/(m**7 + 49*m**6 + 973*m**5 + 1
0045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 132*a**2*c*d**m*m**5*x**5*x**m/(m**7 + 49*m**6 + 9
73*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 2259*a**2*c*d**m*m**4*x**5*x**m/(m**7 +
49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 18840*a**2*c*d**m*m**3*x**5
*x**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 77937*a**2*c*d
**m*m**2*x**5*x**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 1
42308*a**2*c*d**m*m*x**5*x**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m +
135135) + 81081*a**2*c*d**m*x**5*x**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264
207*m + 135135) + 3*a*b**2*d**m*m**6*x**5*x**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m
**2 + 264207*m + 135135) + 132*a*b**2*d**m*m**5*x**5*x**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3
+ 177331*m**2 + 264207*m + 135135) + 2259*a*b**2*d**m*m**4*x**5*x**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4
+ 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 18840*a*b**2*d**m*m**3*x**5*x**m/(m**7 + 49*m**6 + 973*m**5
+ 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 77937*a*b**2*d**m*m**2*x**5*x**m/(m**7 + 49*m**
6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 142308*a*b**2*d**m*m*x**5*x**m/(m*
*7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 81081*a*b**2*d**m*x**5*
x**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 6*a*b*c*d**m*m*
*6*x**7*x**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 252*a*b
*c*d**m*m**5*x**7*x**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135)
+ 4074*a*b*c*d**m*m**4*x**7*x**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*
m + 135135) + 31752*a*b*c*d**m*m**3*x**7*x**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m*
*2 + 264207*m + 135135) + 122010*a*b*c*d**m*m**2*x**7*x**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**
3 + 177331*m**2 + 264207*m + 135135) + 209916*a*b*c*d**m*m*x**7*x**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 +
57379*m**3 + 177331*m**2 + 264207*m + 135135) + 115830*a*b*c*d**m*x**7*x**m/(m**7 + 49*m**6 + 973*m**5 + 1004
5*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 3*a*c**2*d**m*m**6*x**9*x**m/(m**7 + 49*m**6 + 973*m*
*5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 120*a*c**2*d**m*m**5*x**9*x**m/(m**7 + 49*m*
*6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 1839*a*c**2*d**m*m**4*x**9*x**m/(
m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 13584*a*c**2*d**m*m**
3*x**9*x**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 49881*a*
c**2*d**m*m**2*x**9*x**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 13513
5) + 83064*a*c**2*d**m*m*x**9*x**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207
*m + 135135) + 45045*a*c**2*d**m*x**9*x**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2
+ 264207*m + 135135) + b**3*d**m*m**6*x**7*x**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*
m**2 + 264207*m + 135135) + 42*b**3*d**m*m**5*x**7*x**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 +
177331*m**2 + 264207*m + 135135) + 679*b**3*d**m*m**4*x**7*x**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 573
79*m**3 + 177331*m**2 + 264207*m + 135135) + 5292*b**3*d**m*m**3*x**7*x**m/(m**7 + 49*m**6 + 973*m**5 + 10045*
m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 20335*b**3*d**m*m**2*x**7*x**m/(m**7 + 49*m**6 + 973*m*
*5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 34986*b**3*d**m*m*x**7*x**m/(m**7 + 49*m**6
+ 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 19305*b**3*d**m*x**7*x**m/(m**7 + 49
*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 3*b**2*c*d**m*m**6*x**9*x**m/(
m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 120*b**2*c*d**m*m**5*
x**9*x**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 1839*b**2*
c*d**m*m**4*x**9*x**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135)
+ 13584*b**2*c*d**m*m**3*x**9*x**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207
*m + 135135) + 49881*b**2*c*d**m*m**2*x**9*x**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*
m**2 + 264207*m + 135135) + 83064*b**2*c*d**m*m*x**9*x**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3
+ 177331*m**2 + 264207*m + 135135) + 45045*b**2*c*d**m*x**9*x**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57
379*m**3 + 177331*m**2 + 264207*m + 135135) + 3*b*c**2*d**m*m**6*x**11*x**m/(m**7 + 49*m**6 + 973*m**5 + 10045
*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 114*b*c**2*d**m*m**5*x**11*x**m/(m**7 + 49*m**6 + 973*
m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 1665*b*c**2*d**m*m**4*x**11*x**m/(m**7 + 4
9*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 11820*b*c**2*d**m*m**3*x**11*
x**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 42117*b*c**2*d*
*m*m**2*x**11*x**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 6
8706*b*c**2*d**m*m*x**11*x**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m +
135135) + 36855*b*c**2*d**m*x**11*x**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 26
4207*m + 135135) + c**3*d**m*m**6*x**13*x**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**
2 + 264207*m + 135135) + 36*c**3*d**m*m**5*x**13*x**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 1
77331*m**2 + 264207*m + 135135) + 505*c**3*d**m*m**4*x**13*x**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 5737
9*m**3 + 177331*m**2 + 264207*m + 135135) + 3480*c**3*d**m*m**3*x**13*x**m/(m**7 + 49*m**6 + 973*m**5 + 10045*
m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 12139*c**3*d**m*m**2*x**13*x**m/(m**7 + 49*m**6 + 973*m
**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 19524*c**3*d**m*m*x**13*x**m/(m**7 + 49*m**
6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 10395*c**3*d**m*x**13*x**m/(m**7 +
49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135), True))
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