3.7.7 \(\int (d x)^m (a^2+2 a b x^2+b^2 x^4)^3 \, dx\)
Optimal. Leaf size=150 \[ \frac {a^6 (d x)^{m+1}}{d (m+1)}+\frac {6 a^5 b (d x)^{m+3}}{d^3 (m+3)}+\frac {15 a^4 b^2 (d x)^{m+5}}{d^5 (m+5)}+\frac {20 a^3 b^3 (d x)^{m+7}}{d^7 (m+7)}+\frac {15 a^2 b^4 (d x)^{m+9}}{d^9 (m+9)}+\frac {6 a b^5 (d x)^{m+11}}{d^{11} (m+11)}+\frac {b^6 (d x)^{m+13}}{d^{13} (m+13)} \]
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Rubi [A] time = 0.12, antiderivative size = 150, normalized size of antiderivative = 1.00,
number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used =
{28, 270} \begin {gather*} \frac {15 a^4 b^2 (d x)^{m+5}}{d^5 (m+5)}+\frac {20 a^3 b^3 (d x)^{m+7}}{d^7 (m+7)}+\frac {15 a^2 b^4 (d x)^{m+9}}{d^9 (m+9)}+\frac {6 a^5 b (d x)^{m+3}}{d^3 (m+3)}+\frac {a^6 (d x)^{m+1}}{d (m+1)}+\frac {6 a b^5 (d x)^{m+11}}{d^{11} (m+11)}+\frac {b^6 (d x)^{m+13}}{d^{13} (m+13)} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(d*x)^m*(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]
[Out]
(a^6*(d*x)^(1 + m))/(d*(1 + m)) + (6*a^5*b*(d*x)^(3 + m))/(d^3*(3 + m)) + (15*a^4*b^2*(d*x)^(5 + m))/(d^5*(5 +
m)) + (20*a^3*b^3*(d*x)^(7 + m))/(d^7*(7 + m)) + (15*a^2*b^4*(d*x)^(9 + m))/(d^9*(9 + m)) + (6*a*b^5*(d*x)^(1
1 + m))/(d^11*(11 + m)) + (b^6*(d*x)^(13 + m))/(d^13*(13 + m))
Rule 28
Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/c^p, Int[u*(b/2 + c*x^n)^(2*
p), x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Rule 270
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]
Rubi steps
\begin {align*} \int (d x)^m \left (a^2+2 a b x^2+b^2 x^4\right )^3 \, dx &=\frac {\int (d x)^m \left (a b+b^2 x^2\right )^6 \, dx}{b^6}\\ &=\frac {\int \left (a^6 b^6 (d x)^m+\frac {6 a^5 b^7 (d x)^{2+m}}{d^2}+\frac {15 a^4 b^8 (d x)^{4+m}}{d^4}+\frac {20 a^3 b^9 (d x)^{6+m}}{d^6}+\frac {15 a^2 b^{10} (d x)^{8+m}}{d^8}+\frac {6 a b^{11} (d x)^{10+m}}{d^{10}}+\frac {b^{12} (d x)^{12+m}}{d^{12}}\right ) \, dx}{b^6}\\ &=\frac {a^6 (d x)^{1+m}}{d (1+m)}+\frac {6 a^5 b (d x)^{3+m}}{d^3 (3+m)}+\frac {15 a^4 b^2 (d x)^{5+m}}{d^5 (5+m)}+\frac {20 a^3 b^3 (d x)^{7+m}}{d^7 (7+m)}+\frac {15 a^2 b^4 (d x)^{9+m}}{d^9 (9+m)}+\frac {6 a b^5 (d x)^{11+m}}{d^{11} (11+m)}+\frac {b^6 (d x)^{13+m}}{d^{13} (13+m)}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 105, normalized size = 0.70 \begin {gather*} x (d x)^m \left (\frac {a^6}{m+1}+\frac {6 a^5 b x^2}{m+3}+\frac {15 a^4 b^2 x^4}{m+5}+\frac {20 a^3 b^3 x^6}{m+7}+\frac {15 a^2 b^4 x^8}{m+9}+\frac {6 a b^5 x^{10}}{m+11}+\frac {b^6 x^{12}}{m+13}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(d*x)^m*(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]
[Out]
x*(d*x)^m*(a^6/(1 + m) + (6*a^5*b*x^2)/(3 + m) + (15*a^4*b^2*x^4)/(5 + m) + (20*a^3*b^3*x^6)/(7 + m) + (15*a^2
*b^4*x^8)/(9 + m) + (6*a*b^5*x^10)/(11 + m) + (b^6*x^12)/(13 + m))
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IntegrateAlgebraic [F] time = 1.07, size = 0, normalized size = 0.00 \begin {gather*} \int (d x)^m \left (a^2+2 a b x^2+b^2 x^4\right )^3 \, dx \end {gather*}
Verification is not applicable to the result.
[In]
IntegrateAlgebraic[(d*x)^m*(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]
[Out]
Defer[IntegrateAlgebraic][(d*x)^m*(a^2 + 2*a*b*x^2 + b^2*x^4)^3, x]
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fricas [B] time = 1.99, size = 507, normalized size = 3.38 \begin {gather*} \frac {{\left ({\left (b^{6} m^{6} + 36 \, b^{6} m^{5} + 505 \, b^{6} m^{4} + 3480 \, b^{6} m^{3} + 12139 \, b^{6} m^{2} + 19524 \, b^{6} m + 10395 \, b^{6}\right )} x^{13} + 6 \, {\left (a b^{5} m^{6} + 38 \, a b^{5} m^{5} + 555 \, a b^{5} m^{4} + 3940 \, a b^{5} m^{3} + 14039 \, a b^{5} m^{2} + 22902 \, a b^{5} m + 12285 \, a b^{5}\right )} x^{11} + 15 \, {\left (a^{2} b^{4} m^{6} + 40 \, a^{2} b^{4} m^{5} + 613 \, a^{2} b^{4} m^{4} + 4528 \, a^{2} b^{4} m^{3} + 16627 \, a^{2} b^{4} m^{2} + 27688 \, a^{2} b^{4} m + 15015 \, a^{2} b^{4}\right )} x^{9} + 20 \, {\left (a^{3} b^{3} m^{6} + 42 \, a^{3} b^{3} m^{5} + 679 \, a^{3} b^{3} m^{4} + 5292 \, a^{3} b^{3} m^{3} + 20335 \, a^{3} b^{3} m^{2} + 34986 \, a^{3} b^{3} m + 19305 \, a^{3} b^{3}\right )} x^{7} + 15 \, {\left (a^{4} b^{2} m^{6} + 44 \, a^{4} b^{2} m^{5} + 753 \, a^{4} b^{2} m^{4} + 6280 \, a^{4} b^{2} m^{3} + 25979 \, a^{4} b^{2} m^{2} + 47436 \, a^{4} b^{2} m + 27027 \, a^{4} b^{2}\right )} x^{5} + 6 \, {\left (a^{5} b m^{6} + 46 \, a^{5} b m^{5} + 835 \, a^{5} b m^{4} + 7540 \, a^{5} b m^{3} + 34759 \, a^{5} b m^{2} + 73054 \, a^{5} b m + 45045 \, a^{5} b\right )} x^{3} + {\left (a^{6} m^{6} + 48 \, a^{6} m^{5} + 925 \, a^{6} m^{4} + 9120 \, a^{6} m^{3} + 48259 \, a^{6} m^{2} + 129072 \, a^{6} m + 135135 \, a^{6}\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x)^m*(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="fricas")
[Out]
((b^6*m^6 + 36*b^6*m^5 + 505*b^6*m^4 + 3480*b^6*m^3 + 12139*b^6*m^2 + 19524*b^6*m + 10395*b^6)*x^13 + 6*(a*b^5
*m^6 + 38*a*b^5*m^5 + 555*a*b^5*m^4 + 3940*a*b^5*m^3 + 14039*a*b^5*m^2 + 22902*a*b^5*m + 12285*a*b^5)*x^11 + 1
5*(a^2*b^4*m^6 + 40*a^2*b^4*m^5 + 613*a^2*b^4*m^4 + 4528*a^2*b^4*m^3 + 16627*a^2*b^4*m^2 + 27688*a^2*b^4*m + 1
5015*a^2*b^4)*x^9 + 20*(a^3*b^3*m^6 + 42*a^3*b^3*m^5 + 679*a^3*b^3*m^4 + 5292*a^3*b^3*m^3 + 20335*a^3*b^3*m^2
+ 34986*a^3*b^3*m + 19305*a^3*b^3)*x^7 + 15*(a^4*b^2*m^6 + 44*a^4*b^2*m^5 + 753*a^4*b^2*m^4 + 6280*a^4*b^2*m^3
+ 25979*a^4*b^2*m^2 + 47436*a^4*b^2*m + 27027*a^4*b^2)*x^5 + 6*(a^5*b*m^6 + 46*a^5*b*m^5 + 835*a^5*b*m^4 + 75
40*a^5*b*m^3 + 34759*a^5*b*m^2 + 73054*a^5*b*m + 45045*a^5*b)*x^3 + (a^6*m^6 + 48*a^6*m^5 + 925*a^6*m^4 + 9120
*a^6*m^3 + 48259*a^6*m^2 + 129072*a^6*m + 135135*a^6)*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 = 0.25, size = 847, normalized size = 5.65 \begin {gather*} \frac {\left (d x\right )^{m} b^{6} m^{6} x^{13} + 36 \, \left (d x\right )^{m} b^{6} m^{5} x^{13} + 6 \, \left (d x\right )^{m} a b^{5} m^{6} x^{11} + 505 \, \left (d x\right )^{m} b^{6} m^{4} x^{13} + 228 \, \left (d x\right )^{m} a b^{5} m^{5} x^{11} + 3480 \, \left (d x\right )^{m} b^{6} m^{3} x^{13} + 15 \, \left (d x\right )^{m} a^{2} b^{4} m^{6} x^{9} + 3330 \, \left (d x\right )^{m} a b^{5} m^{4} x^{11} + 12139 \, \left (d x\right )^{m} b^{6} m^{2} x^{13} + 600 \, \left (d x\right )^{m} a^{2} b^{4} m^{5} x^{9} + 23640 \, \left (d x\right )^{m} a b^{5} m^{3} x^{11} + 19524 \, \left (d x\right )^{m} b^{6} m x^{13} + 20 \, \left (d x\right )^{m} a^{3} b^{3} m^{6} x^{7} + 9195 \, \left (d x\right )^{m} a^{2} b^{4} m^{4} x^{9} + 84234 \, \left (d x\right )^{m} a b^{5} m^{2} x^{11} + 10395 \, \left (d x\right )^{m} b^{6} x^{13} + 840 \, \left (d x\right )^{m} a^{3} b^{3} m^{5} x^{7} + 67920 \, \left (d x\right )^{m} a^{2} b^{4} m^{3} x^{9} + 137412 \, \left (d x\right )^{m} a b^{5} m x^{11} + 15 \, \left (d x\right )^{m} a^{4} b^{2} m^{6} x^{5} + 13580 \, \left (d x\right )^{m} a^{3} b^{3} m^{4} x^{7} + 249405 \, \left (d x\right )^{m} a^{2} b^{4} m^{2} x^{9} + 73710 \, \left (d x\right )^{m} a b^{5} x^{11} + 660 \, \left (d x\right )^{m} a^{4} b^{2} m^{5} x^{5} + 105840 \, \left (d x\right )^{m} a^{3} b^{3} m^{3} x^{7} + 415320 \, \left (d x\right )^{m} a^{2} b^{4} m x^{9} + 6 \, \left (d x\right )^{m} a^{5} b m^{6} x^{3} + 11295 \, \left (d x\right )^{m} a^{4} b^{2} m^{4} x^{5} + 406700 \, \left (d x\right )^{m} a^{3} b^{3} m^{2} x^{7} + 225225 \, \left (d x\right )^{m} a^{2} b^{4} x^{9} + 276 \, \left (d x\right )^{m} a^{5} b m^{5} x^{3} + 94200 \, \left (d x\right )^{m} a^{4} b^{2} m^{3} x^{5} + 699720 \, \left (d x\right )^{m} a^{3} b^{3} m x^{7} + \left (d x\right )^{m} a^{6} m^{6} x + 5010 \, \left (d x\right )^{m} a^{5} b m^{4} x^{3} + 389685 \, \left (d x\right )^{m} a^{4} b^{2} m^{2} x^{5} + 386100 \, \left (d x\right )^{m} a^{3} b^{3} x^{7} + 48 \, \left (d x\right )^{m} a^{6} m^{5} x + 45240 \, \left (d x\right )^{m} a^{5} b m^{3} x^{3} + 711540 \, \left (d x\right )^{m} a^{4} b^{2} m x^{5} + 925 \, \left (d x\right )^{m} a^{6} m^{4} x + 208554 \, \left (d x\right )^{m} a^{5} b m^{2} x^{3} + 405405 \, \left (d x\right )^{m} a^{4} b^{2} x^{5} + 9120 \, \left (d x\right )^{m} a^{6} m^{3} x + 438324 \, \left (d x\right )^{m} a^{5} b m x^{3} + 48259 \, \left (d x\right )^{m} a^{6} m^{2} x + 270270 \, \left (d x\right )^{m} a^{5} b x^{3} + 129072 \, \left (d x\right )^{m} a^{6} m x + 135135 \, \left (d x\right )^{m} a^{6} x}{m^{7} + 49 \, m^{6} + 973 \, m^{5} + 10045 \, m^{4} + 57379 \, m^{3} + 177331 \, m^{2} + 264207 \, m + 135135} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x)^m*(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="giac")
[Out]
((d*x)^m*b^6*m^6*x^13 + 36*(d*x)^m*b^6*m^5*x^13 + 6*(d*x)^m*a*b^5*m^6*x^11 + 505*(d*x)^m*b^6*m^4*x^13 + 228*(d
*x)^m*a*b^5*m^5*x^11 + 3480*(d*x)^m*b^6*m^3*x^13 + 15*(d*x)^m*a^2*b^4*m^6*x^9 + 3330*(d*x)^m*a*b^5*m^4*x^11 +
12139*(d*x)^m*b^6*m^2*x^13 + 600*(d*x)^m*a^2*b^4*m^5*x^9 + 23640*(d*x)^m*a*b^5*m^3*x^11 + 19524*(d*x)^m*b^6*m*
x^13 + 20*(d*x)^m*a^3*b^3*m^6*x^7 + 9195*(d*x)^m*a^2*b^4*m^4*x^9 + 84234*(d*x)^m*a*b^5*m^2*x^11 + 10395*(d*x)^
m*b^6*x^13 + 840*(d*x)^m*a^3*b^3*m^5*x^7 + 67920*(d*x)^m*a^2*b^4*m^3*x^9 + 137412*(d*x)^m*a*b^5*m*x^11 + 15*(d
*x)^m*a^4*b^2*m^6*x^5 + 13580*(d*x)^m*a^3*b^3*m^4*x^7 + 249405*(d*x)^m*a^2*b^4*m^2*x^9 + 73710*(d*x)^m*a*b^5*x
^11 + 660*(d*x)^m*a^4*b^2*m^5*x^5 + 105840*(d*x)^m*a^3*b^3*m^3*x^7 + 415320*(d*x)^m*a^2*b^4*m*x^9 + 6*(d*x)^m*
a^5*b*m^6*x^3 + 11295*(d*x)^m*a^4*b^2*m^4*x^5 + 406700*(d*x)^m*a^3*b^3*m^2*x^7 + 225225*(d*x)^m*a^2*b^4*x^9 +
276*(d*x)^m*a^5*b*m^5*x^3 + 94200*(d*x)^m*a^4*b^2*m^3*x^5 + 699720*(d*x)^m*a^3*b^3*m*x^7 + (d*x)^m*a^6*m^6*x +
5010*(d*x)^m*a^5*b*m^4*x^3 + 389685*(d*x)^m*a^4*b^2*m^2*x^5 + 386100*(d*x)^m*a^3*b^3*x^7 + 48*(d*x)^m*a^6*m^5
*x + 45240*(d*x)^m*a^5*b*m^3*x^3 + 711540*(d*x)^m*a^4*b^2*m*x^5 + 925*(d*x)^m*a^6*m^4*x + 208554*(d*x)^m*a^5*b
*m^2*x^3 + 405405*(d*x)^m*a^4*b^2*x^5 + 9120*(d*x)^m*a^6*m^3*x + 438324*(d*x)^m*a^5*b*m*x^3 + 48259*(d*x)^m*a^
6*m^2*x + 270270*(d*x)^m*a^5*b*x^3 + 129072*(d*x)^m*a^6*m*x + 135135*(d*x)^m*a^6*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 = 602, normalized size = 4.01 \begin {gather*} \frac {\left (b^{6} m^{6} x^{12}+36 b^{6} m^{5} x^{12}+6 a \,b^{5} m^{6} x^{10}+505 b^{6} m^{4} x^{12}+228 a \,b^{5} m^{5} x^{10}+3480 b^{6} m^{3} x^{12}+15 a^{2} b^{4} m^{6} x^{8}+3330 a \,b^{5} m^{4} x^{10}+12139 b^{6} m^{2} x^{12}+600 a^{2} b^{4} m^{5} x^{8}+23640 a \,b^{5} m^{3} x^{10}+19524 b^{6} m \,x^{12}+20 a^{3} b^{3} m^{6} x^{6}+9195 a^{2} b^{4} m^{4} x^{8}+84234 a \,b^{5} m^{2} x^{10}+10395 b^{6} x^{12}+840 a^{3} b^{3} m^{5} x^{6}+67920 a^{2} b^{4} m^{3} x^{8}+137412 a \,b^{5} m \,x^{10}+15 a^{4} b^{2} m^{6} x^{4}+13580 a^{3} b^{3} m^{4} x^{6}+249405 a^{2} b^{4} m^{2} x^{8}+73710 a \,b^{5} x^{10}+660 a^{4} b^{2} m^{5} x^{4}+105840 a^{3} b^{3} m^{3} x^{6}+415320 a^{2} b^{4} m \,x^{8}+6 a^{5} b \,m^{6} x^{2}+11295 a^{4} b^{2} m^{4} x^{4}+406700 a^{3} b^{3} m^{2} x^{6}+225225 a^{2} b^{4} x^{8}+276 a^{5} b \,m^{5} x^{2}+94200 a^{4} b^{2} m^{3} x^{4}+699720 a^{3} b^{3} m \,x^{6}+a^{6} m^{6}+5010 a^{5} b \,m^{4} x^{2}+389685 a^{4} b^{2} m^{2} x^{4}+386100 a^{3} b^{3} x^{6}+48 a^{6} m^{5}+45240 a^{5} b \,m^{3} x^{2}+711540 a^{4} b^{2} m \,x^{4}+925 a^{6} m^{4}+208554 a^{5} b \,m^{2} x^{2}+405405 a^{4} b^{2} x^{4}+9120 a^{6} m^{3}+438324 a^{5} b m \,x^{2}+48259 a^{6} m^{2}+270270 a^{5} b \,x^{2}+129072 a^{6} m +135135 a^{6}\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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((d*x)^m*(b^2*x^4+2*a*b*x^2+a^2)^3,x)
[Out]
(d*x)^m*(b^6*m^6*x^12+36*b^6*m^5*x^12+6*a*b^5*m^6*x^10+505*b^6*m^4*x^12+228*a*b^5*m^5*x^10+3480*b^6*m^3*x^12+1
5*a^2*b^4*m^6*x^8+3330*a*b^5*m^4*x^10+12139*b^6*m^2*x^12+600*a^2*b^4*m^5*x^8+23640*a*b^5*m^3*x^10+19524*b^6*m*
x^12+20*a^3*b^3*m^6*x^6+9195*a^2*b^4*m^4*x^8+84234*a*b^5*m^2*x^10+10395*b^6*x^12+840*a^3*b^3*m^5*x^6+67920*a^2
*b^4*m^3*x^8+137412*a*b^5*m*x^10+15*a^4*b^2*m^6*x^4+13580*a^3*b^3*m^4*x^6+249405*a^2*b^4*m^2*x^8+73710*a*b^5*x
^10+660*a^4*b^2*m^5*x^4+105840*a^3*b^3*m^3*x^6+415320*a^2*b^4*m*x^8+6*a^5*b*m^6*x^2+11295*a^4*b^2*m^4*x^4+4067
00*a^3*b^3*m^2*x^6+225225*a^2*b^4*x^8+276*a^5*b*m^5*x^2+94200*a^4*b^2*m^3*x^4+699720*a^3*b^3*m*x^6+a^6*m^6+501
0*a^5*b*m^4*x^2+389685*a^4*b^2*m^2*x^4+386100*a^3*b^3*x^6+48*a^6*m^5+45240*a^5*b*m^3*x^2+711540*a^4*b^2*m*x^4+
925*a^6*m^4+208554*a^5*b*m^2*x^2+405405*a^4*b^2*x^4+9120*a^6*m^3+438324*a^5*b*m*x^2+48259*a^6*m^2+270270*a^5*b
*x^2+129072*a^6*m+135135*a^6)*x/(m+13)/(m+11)/(m+9)/(m+7)/(m+5)/(m+3)/(m+1)
________________________________________________________________________________________
maxima [A] time = 1.54, size = 144, normalized size = 0.96 \begin {gather*} \frac {b^{6} d^{m} x^{13} x^{m}}{m + 13} + \frac {6 \, a b^{5} d^{m} x^{11} x^{m}}{m + 11} + \frac {15 \, a^{2} b^{4} d^{m} x^{9} x^{m}}{m + 9} + \frac {20 \, a^{3} b^{3} d^{m} x^{7} x^{m}}{m + 7} + \frac {15 \, a^{4} b^{2} d^{m} x^{5} x^{m}}{m + 5} + \frac {6 \, a^{5} b d^{m} x^{3} x^{m}}{m + 3} + \frac {\left (d x\right )^{m + 1} a^{6}}{d {\left (m + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x)^m*(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="maxima")
[Out]
b^6*d^m*x^13*x^m/(m + 13) + 6*a*b^5*d^m*x^11*x^m/(m + 11) + 15*a^2*b^4*d^m*x^9*x^m/(m + 9) + 20*a^3*b^3*d^m*x^
7*x^m/(m + 7) + 15*a^4*b^2*d^m*x^5*x^m/(m + 5) + 6*a^5*b*d^m*x^3*x^m/(m + 3) + (d*x)^(m + 1)*a^6/(d*(m + 1))
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mupad [B] time = 4.58, size = 540, normalized size = 3.60 \begin {gather*} \frac {a^6\,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 {b^6\,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 {6\,a\,b^5\,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 {6\,a^5\,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 {15\,a^2\,b^4\,x^9\,{\left (d\,x\right )}^m\,\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}+\frac {20\,a^3\,b^3\,x^7\,{\left (d\,x\right )}^m\,\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 {15\,a^4\,b^2\,x^5\,{\left (d\,x\right )}^m\,\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((d*x)^m*(a^2 + b^2*x^4 + 2*a*b*x^2)^3,x)
[Out]
(a^6*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) + (b^6*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) + (6*a*b^5*x^11*(d*x)^m*(22902*m + 14039*m^2 + 3940*m^3 + 555*m^4 + 38*m^5 + m^6 + 12285))/(264207*m + 17
7331*m^2 + 57379*m^3 + 10045*m^4 + 973*m^5 + 49*m^6 + m^7 + 135135) + (6*a^5*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 + 177331*m^2 + 57379*m^3 + 10045*m^4 + 973*m^5 + 49*
m^6 + m^7 + 135135) + (15*a^2*b^4*x^9*(d*x)^m*(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) + (20*a^3*b^3*x^7*(d*x)^m
*(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) + (15*a^4*b^2*x^5*(d*x)^m*(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)
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sympy [A] time = 7.61, size = 3188, normalized size = 21.25
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x)**m*(b**2*x**4+2*a*b*x**2+a**2)**3,x)
[Out]
Piecewise(((-a**6/(12*x**12) - 3*a**5*b/(5*x**10) - 15*a**4*b**2/(8*x**8) - 10*a**3*b**3/(3*x**6) - 15*a**2*b*
*4/(4*x**4) - 3*a*b**5/x**2 + b**6*log(x))/d**13, Eq(m, -13)), ((-a**6/(10*x**10) - 3*a**5*b/(4*x**8) - 5*a**4
*b**2/(2*x**6) - 5*a**3*b**3/x**4 - 15*a**2*b**4/(2*x**2) + 6*a*b**5*log(x) + b**6*x**2/2)/d**11, Eq(m, -11)),
((-a**6/(8*x**8) - a**5*b/x**6 - 15*a**4*b**2/(4*x**4) - 10*a**3*b**3/x**2 + 15*a**2*b**4*log(x) + 3*a*b**5*x
**2 + b**6*x**4/4)/d**9, Eq(m, -9)), ((-a**6/(6*x**6) - 3*a**5*b/(2*x**4) - 15*a**4*b**2/(2*x**2) + 20*a**3*b*
*3*log(x) + 15*a**2*b**4*x**2/2 + 3*a*b**5*x**4/2 + b**6*x**6/6)/d**7, Eq(m, -7)), ((-a**6/(4*x**4) - 3*a**5*b
/x**2 + 15*a**4*b**2*log(x) + 10*a**3*b**3*x**2 + 15*a**2*b**4*x**4/4 + a*b**5*x**6 + b**6*x**8/8)/d**5, Eq(m,
-5)), ((-a**6/(2*x**2) + 6*a**5*b*log(x) + 15*a**4*b**2*x**2/2 + 5*a**3*b**3*x**4 + 5*a**2*b**4*x**6/2 + 3*a*
b**5*x**8/4 + b**6*x**10/10)/d**3, Eq(m, -3)), ((a**6*log(x) + 3*a**5*b*x**2 + 15*a**4*b**2*x**4/4 + 10*a**3*b
**3*x**6/3 + 15*a**2*b**4*x**8/8 + 3*a*b**5*x**10/5 + b**6*x**12/12)/d, Eq(m, -1)), (a**6*d**m*m**6*x*x**m/(m*
*7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 48*a**6*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**6*d**m*m**4
*x*x**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 9120*a**6*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**6*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**6*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 + 1
35135) + 135135*a**6*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) + 6*a**5*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) + 276*a**5*b*d**m*m**5*x**3*x**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 17
7331*m**2 + 264207*m + 135135) + 5010*a**5*b*d**m*m**4*x**3*x**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 573
79*m**3 + 177331*m**2 + 264207*m + 135135) + 45240*a**5*b*d**m*m**3*x**3*x**m/(m**7 + 49*m**6 + 973*m**5 + 100
45*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 208554*a**5*b*d**m*m**2*x**3*x**m/(m**7 + 49*m**6 +
973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 438324*a**5*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) + 270270*a**5*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) + 15*a**4*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) + 660*a
**4*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 + 1
35135) + 11295*a**4*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) + 94200*a**4*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) + 389685*a**4*b**2*d**m*m**2*x**5*x**m/(m**7 + 49*m**6 + 973*m**5 + 100
45*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 711540*a**4*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) + 405405*a**4*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) + 20*a**3*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) + 840*a**3*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) +
13580*a**3*b**3*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 + 2642
07*m + 135135) + 105840*a**3*b**3*d**m*m**3*x**7*x**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 1
77331*m**2 + 264207*m + 135135) + 406700*a**3*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) + 699720*a**3*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) + 386100*a**3*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) + 15*a**2*b**4*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) + 600*a**2*b**4*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) + 9195*a
**2*b**4*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 + 1
35135) + 67920*a**2*b**4*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) + 249405*a**2*b**4*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) + 415320*a**2*b**4*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) + 225225*a**2*b**4*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) + 6*a*b**5*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) + 228*a*b**5*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) + 3330*a*b**5*d**m*m**
4*x**11*x**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 23640*a
*b**5*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 + 135
135) + 84234*a*b**5*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) + 137412*a*b**5*d**m*m*x**11*x**m/(m**7 + 49*m**6 + 973*m**5 + 10045*m**4 + 57379*m**3 + 17
7331*m**2 + 264207*m + 135135) + 73710*a*b**5*d**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) + b**6*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*b**6*d**m*m**5*x**13*x**m/(m**7 + 49*m**6 + 973*m**5 + 1004
5*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 505*b**6*d**m*m**4*x**13*x**m/(m**7 + 49*m**6 + 973*m
**5 + 10045*m**4 + 57379*m**3 + 177331*m**2 + 264207*m + 135135) + 3480*b**6*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*b**6*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*b**6*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*b**6*
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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