3.72 \(\int x^m (a x+b x^3+c x^5)^2 \, dx\)

Optimal. Leaf size=76 \[ \frac {a^2 x^{m+3}}{m+3}+\frac {x^{m+7} \left (2 a c+b^2\right )}{m+7}+\frac {2 a b x^{m+5}}{m+5}+\frac {2 b c x^{m+9}}{m+9}+\frac {c^2 x^{m+11}}{m+11} \]

[Out]

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

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Rubi [A]  time = 0.05, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1585, 1108} \[ \frac {a^2 x^{m+3}}{m+3}+\frac {x^{m+7} \left (2 a c+b^2\right )}{m+7}+\frac {2 a b x^{m+5}}{m+5}+\frac {2 b c x^{m+9}}{m+9}+\frac {c^2 x^{m+11}}{m+11} \]

Antiderivative was successfully verified.

[In]

Int[x^m*(a*x + b*x^3 + c*x^5)^2,x]

[Out]

(a^2*x^(3 + m))/(3 + m) + (2*a*b*x^(5 + m))/(5 + m) + ((b^2 + 2*a*c)*x^(7 + m))/(7 + m) + (2*b*c*x^(9 + m))/(9
+ m) + (c^2*x^(11 + m))/(11 + 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]

Rule 1585

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(m +
n*p)*(a + b*x^(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, m, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] &
& PosQ[r - p]

Rubi steps

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

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Mathematica [A]  time = 0.07, size = 69, normalized size = 0.91 \[ x^{m+3} \left (\frac {a^2}{m+3}+\frac {x^4 \left (2 a c+b^2\right )}{m+7}+\frac {2 a b x^2}{m+5}+\frac {2 b c x^6}{m+9}+\frac {c^2 x^8}{m+11}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^m*(a*x + b*x^3 + c*x^5)^2,x]

[Out]

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

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fricas [B]  time = 0.67, size = 241, normalized size = 3.17 \[ \frac {{\left ({\left (c^{2} m^{4} + 24 \, c^{2} m^{3} + 206 \, c^{2} m^{2} + 744 \, c^{2} m + 945 \, c^{2}\right )} x^{11} + 2 \, {\left (b c m^{4} + 26 \, b c m^{3} + 236 \, b c m^{2} + 886 \, b c m + 1155 \, b c\right )} x^{9} + {\left ({\left (b^{2} + 2 \, a c\right )} m^{4} + 28 \, {\left (b^{2} + 2 \, a c\right )} m^{3} + 274 \, {\left (b^{2} + 2 \, a c\right )} m^{2} + 1485 \, b^{2} + 2970 \, a c + 1092 \, {\left (b^{2} + 2 \, a c\right )} m\right )} x^{7} + 2 \, {\left (a b m^{4} + 30 \, a b m^{3} + 320 \, a b m^{2} + 1410 \, a b m + 2079 \, a b\right )} x^{5} + {\left (a^{2} m^{4} + 32 \, a^{2} m^{3} + 374 \, a^{2} m^{2} + 1888 \, a^{2} m + 3465 \, a^{2}\right )} x^{3}\right )} x^{m}}{m^{5} + 35 \, m^{4} + 470 \, m^{3} + 3010 \, m^{2} + 9129 \, m + 10395} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(c*x^5+b*x^3+a*x)^2,x, algorithm="fricas")

[Out]

((c^2*m^4 + 24*c^2*m^3 + 206*c^2*m^2 + 744*c^2*m + 945*c^2)*x^11 + 2*(b*c*m^4 + 26*b*c*m^3 + 236*b*c*m^2 + 886
*b*c*m + 1155*b*c)*x^9 + ((b^2 + 2*a*c)*m^4 + 28*(b^2 + 2*a*c)*m^3 + 274*(b^2 + 2*a*c)*m^2 + 1485*b^2 + 2970*a
*c + 1092*(b^2 + 2*a*c)*m)*x^7 + 2*(a*b*m^4 + 30*a*b*m^3 + 320*a*b*m^2 + 1410*a*b*m + 2079*a*b)*x^5 + (a^2*m^4
+ 32*a^2*m^3 + 374*a^2*m^2 + 1888*a^2*m + 3465*a^2)*x^3)*x^m/(m^5 + 35*m^4 + 470*m^3 + 3010*m^2 + 9129*m + 10
395)

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giac [B]  time = 0.52, size = 399, normalized size = 5.25 \[ \frac {c^{2} m^{4} x^{11} x^{m} + 24 \, c^{2} m^{3} x^{11} x^{m} + 2 \, b c m^{4} x^{9} x^{m} + 206 \, c^{2} m^{2} x^{11} x^{m} + 52 \, b c m^{3} x^{9} x^{m} + 744 \, c^{2} m x^{11} x^{m} + b^{2} m^{4} x^{7} x^{m} + 2 \, a c m^{4} x^{7} x^{m} + 472 \, b c m^{2} x^{9} x^{m} + 945 \, c^{2} x^{11} x^{m} + 28 \, b^{2} m^{3} x^{7} x^{m} + 56 \, a c m^{3} x^{7} x^{m} + 1772 \, b c m x^{9} x^{m} + 2 \, a b m^{4} x^{5} x^{m} + 274 \, b^{2} m^{2} x^{7} x^{m} + 548 \, a c m^{2} x^{7} x^{m} + 2310 \, b c x^{9} x^{m} + 60 \, a b m^{3} x^{5} x^{m} + 1092 \, b^{2} m x^{7} x^{m} + 2184 \, a c m x^{7} x^{m} + a^{2} m^{4} x^{3} x^{m} + 640 \, a b m^{2} x^{5} x^{m} + 1485 \, b^{2} x^{7} x^{m} + 2970 \, a c x^{7} x^{m} + 32 \, a^{2} m^{3} x^{3} x^{m} + 2820 \, a b m x^{5} x^{m} + 374 \, a^{2} m^{2} x^{3} x^{m} + 4158 \, a b x^{5} x^{m} + 1888 \, a^{2} m x^{3} x^{m} + 3465 \, a^{2} x^{3} x^{m}}{m^{5} + 35 \, m^{4} + 470 \, m^{3} + 3010 \, m^{2} + 9129 \, m + 10395} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(c*x^5+b*x^3+a*x)^2,x, algorithm="giac")

[Out]

(c^2*m^4*x^11*x^m + 24*c^2*m^3*x^11*x^m + 2*b*c*m^4*x^9*x^m + 206*c^2*m^2*x^11*x^m + 52*b*c*m^3*x^9*x^m + 744*
c^2*m*x^11*x^m + b^2*m^4*x^7*x^m + 2*a*c*m^4*x^7*x^m + 472*b*c*m^2*x^9*x^m + 945*c^2*x^11*x^m + 28*b^2*m^3*x^7
*x^m + 56*a*c*m^3*x^7*x^m + 1772*b*c*m*x^9*x^m + 2*a*b*m^4*x^5*x^m + 274*b^2*m^2*x^7*x^m + 548*a*c*m^2*x^7*x^m
+ 2310*b*c*x^9*x^m + 60*a*b*m^3*x^5*x^m + 1092*b^2*m*x^7*x^m + 2184*a*c*m*x^7*x^m + a^2*m^4*x^3*x^m + 640*a*b
*m^2*x^5*x^m + 1485*b^2*x^7*x^m + 2970*a*c*x^7*x^m + 32*a^2*m^3*x^3*x^m + 2820*a*b*m*x^5*x^m + 374*a^2*m^2*x^3
*x^m + 4158*a*b*x^5*x^m + 1888*a^2*m*x^3*x^m + 3465*a^2*x^3*x^m)/(m^5 + 35*m^4 + 470*m^3 + 3010*m^2 + 9129*m +
10395)

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maple [B]  time = 0.00, size = 300, normalized size = 3.95 \[ \frac {\left (c^{2} m^{4} x^{8}+24 c^{2} m^{3} x^{8}+2 b c \,m^{4} x^{6}+206 c^{2} m^{2} x^{8}+52 b c \,m^{3} x^{6}+744 c^{2} m \,x^{8}+2 a c \,m^{4} x^{4}+b^{2} m^{4} x^{4}+472 b c \,m^{2} x^{6}+945 c^{2} x^{8}+56 a c \,m^{3} x^{4}+28 b^{2} m^{3} x^{4}+1772 b c m \,x^{6}+2 a b \,m^{4} x^{2}+548 a c \,m^{2} x^{4}+274 b^{2} m^{2} x^{4}+2310 b c \,x^{6}+60 a b \,m^{3} x^{2}+2184 a c m \,x^{4}+1092 b^{2} m \,x^{4}+a^{2} m^{4}+640 a b \,m^{2} x^{2}+2970 a c \,x^{4}+1485 b^{2} x^{4}+32 a^{2} m^{3}+2820 a b m \,x^{2}+374 a^{2} m^{2}+4158 a b \,x^{2}+1888 a^{2} m +3465 a^{2}\right ) x^{m +3}}{\left (m +11\right ) \left (m +9\right ) \left (m +7\right ) \left (m +5\right ) \left (m +3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*(c*x^5+b*x^3+a*x)^2,x)

[Out]

x^(m+3)*(c^2*m^4*x^8+24*c^2*m^3*x^8+2*b*c*m^4*x^6+206*c^2*m^2*x^8+52*b*c*m^3*x^6+744*c^2*m*x^8+2*a*c*m^4*x^4+b
^2*m^4*x^4+472*b*c*m^2*x^6+945*c^2*x^8+56*a*c*m^3*x^4+28*b^2*m^3*x^4+1772*b*c*m*x^6+2*a*b*m^4*x^2+548*a*c*m^2*
x^4+274*b^2*m^2*x^4+2310*b*c*x^6+60*a*b*m^3*x^2+2184*a*c*m*x^4+1092*b^2*m*x^4+a^2*m^4+640*a*b*m^2*x^2+2970*a*c
*x^4+1485*b^2*x^4+32*a^2*m^3+2820*a*b*m*x^2+374*a^2*m^2+4158*a*b*x^2+1888*a^2*m+3465*a^2)/(m+11)/(m+9)/(m+7)/(
m+5)/(m+3)

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maxima [A]  time = 0.44, size = 85, normalized size = 1.12 \[ \frac {c^{2} x^{m + 11}}{m + 11} + \frac {2 \, b c x^{m + 9}}{m + 9} + \frac {b^{2} x^{m + 7}}{m + 7} + \frac {2 \, a c x^{m + 7}}{m + 7} + \frac {2 \, a b x^{m + 5}}{m + 5} + \frac {a^{2} x^{m + 3}}{m + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(c*x^5+b*x^3+a*x)^2,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 2.19, size = 271, normalized size = 3.57 \[ \frac {a^2\,x^m\,x^3\,\left (m^4+32\,m^3+374\,m^2+1888\,m+3465\right )}{m^5+35\,m^4+470\,m^3+3010\,m^2+9129\,m+10395}+\frac {c^2\,x^m\,x^{11}\,\left (m^4+24\,m^3+206\,m^2+744\,m+945\right )}{m^5+35\,m^4+470\,m^3+3010\,m^2+9129\,m+10395}+\frac {x^m\,x^7\,\left (b^2+2\,a\,c\right )\,\left (m^4+28\,m^3+274\,m^2+1092\,m+1485\right )}{m^5+35\,m^4+470\,m^3+3010\,m^2+9129\,m+10395}+\frac {2\,a\,b\,x^m\,x^5\,\left (m^4+30\,m^3+320\,m^2+1410\,m+2079\right )}{m^5+35\,m^4+470\,m^3+3010\,m^2+9129\,m+10395}+\frac {2\,b\,c\,x^m\,x^9\,\left (m^4+26\,m^3+236\,m^2+886\,m+1155\right )}{m^5+35\,m^4+470\,m^3+3010\,m^2+9129\,m+10395} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*(a*x + b*x^3 + c*x^5)^2,x)

[Out]

(a^2*x^m*x^3*(1888*m + 374*m^2 + 32*m^3 + m^4 + 3465))/(9129*m + 3010*m^2 + 470*m^3 + 35*m^4 + m^5 + 10395) +
(c^2*x^m*x^11*(744*m + 206*m^2 + 24*m^3 + m^4 + 945))/(9129*m + 3010*m^2 + 470*m^3 + 35*m^4 + m^5 + 10395) + (
x^m*x^7*(2*a*c + b^2)*(1092*m + 274*m^2 + 28*m^3 + m^4 + 1485))/(9129*m + 3010*m^2 + 470*m^3 + 35*m^4 + m^5 +
10395) + (2*a*b*x^m*x^5*(1410*m + 320*m^2 + 30*m^3 + m^4 + 2079))/(9129*m + 3010*m^2 + 470*m^3 + 35*m^4 + m^5
+ 10395) + (2*b*c*x^m*x^9*(886*m + 236*m^2 + 26*m^3 + m^4 + 1155))/(9129*m + 3010*m^2 + 470*m^3 + 35*m^4 + m^5
+ 10395)

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sympy [A]  time = 4.29, size = 1377, normalized size = 18.12 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**m*(c*x**5+b*x**3+a*x)**2,x)

[Out]

Piecewise((-a**2/(8*x**8) - a*b/(3*x**6) - a*c/(2*x**4) - b**2/(4*x**4) - b*c/x**2 + c**2*log(x), Eq(m, -11)),
(-a**2/(6*x**6) - a*b/(2*x**4) - a*c/x**2 - b**2/(2*x**2) + 2*b*c*log(x) + c**2*x**2/2, Eq(m, -9)), (-a**2/(4
*x**4) - a*b/x**2 + 2*a*c*log(x) + b**2*log(x) + b*c*x**2 + c**2*x**4/4, Eq(m, -7)), (-a**2/(2*x**2) + 2*a*b*l
og(x) + a*c*x**2 + b**2*x**2/2 + b*c*x**4/2 + c**2*x**6/6, Eq(m, -5)), (a**2*log(x) + a*b*x**2 + a*c*x**4/2 +
b**2*x**4/4 + b*c*x**6/3 + c**2*x**8/8, Eq(m, -3)), (a**2*m**4*x**3*x**m/(m**5 + 35*m**4 + 470*m**3 + 3010*m**
2 + 9129*m + 10395) + 32*a**2*m**3*x**3*x**m/(m**5 + 35*m**4 + 470*m**3 + 3010*m**2 + 9129*m + 10395) + 374*a*
*2*m**2*x**3*x**m/(m**5 + 35*m**4 + 470*m**3 + 3010*m**2 + 9129*m + 10395) + 1888*a**2*m*x**3*x**m/(m**5 + 35*
m**4 + 470*m**3 + 3010*m**2 + 9129*m + 10395) + 3465*a**2*x**3*x**m/(m**5 + 35*m**4 + 470*m**3 + 3010*m**2 + 9
129*m + 10395) + 2*a*b*m**4*x**5*x**m/(m**5 + 35*m**4 + 470*m**3 + 3010*m**2 + 9129*m + 10395) + 60*a*b*m**3*x
**5*x**m/(m**5 + 35*m**4 + 470*m**3 + 3010*m**2 + 9129*m + 10395) + 640*a*b*m**2*x**5*x**m/(m**5 + 35*m**4 + 4
70*m**3 + 3010*m**2 + 9129*m + 10395) + 2820*a*b*m*x**5*x**m/(m**5 + 35*m**4 + 470*m**3 + 3010*m**2 + 9129*m +
10395) + 4158*a*b*x**5*x**m/(m**5 + 35*m**4 + 470*m**3 + 3010*m**2 + 9129*m + 10395) + 2*a*c*m**4*x**7*x**m/(
m**5 + 35*m**4 + 470*m**3 + 3010*m**2 + 9129*m + 10395) + 56*a*c*m**3*x**7*x**m/(m**5 + 35*m**4 + 470*m**3 + 3
010*m**2 + 9129*m + 10395) + 548*a*c*m**2*x**7*x**m/(m**5 + 35*m**4 + 470*m**3 + 3010*m**2 + 9129*m + 10395) +
2184*a*c*m*x**7*x**m/(m**5 + 35*m**4 + 470*m**3 + 3010*m**2 + 9129*m + 10395) + 2970*a*c*x**7*x**m/(m**5 + 35
*m**4 + 470*m**3 + 3010*m**2 + 9129*m + 10395) + b**2*m**4*x**7*x**m/(m**5 + 35*m**4 + 470*m**3 + 3010*m**2 +
9129*m + 10395) + 28*b**2*m**3*x**7*x**m/(m**5 + 35*m**4 + 470*m**3 + 3010*m**2 + 9129*m + 10395) + 274*b**2*m
**2*x**7*x**m/(m**5 + 35*m**4 + 470*m**3 + 3010*m**2 + 9129*m + 10395) + 1092*b**2*m*x**7*x**m/(m**5 + 35*m**4
+ 470*m**3 + 3010*m**2 + 9129*m + 10395) + 1485*b**2*x**7*x**m/(m**5 + 35*m**4 + 470*m**3 + 3010*m**2 + 9129*
m + 10395) + 2*b*c*m**4*x**9*x**m/(m**5 + 35*m**4 + 470*m**3 + 3010*m**2 + 9129*m + 10395) + 52*b*c*m**3*x**9*
x**m/(m**5 + 35*m**4 + 470*m**3 + 3010*m**2 + 9129*m + 10395) + 472*b*c*m**2*x**9*x**m/(m**5 + 35*m**4 + 470*m
**3 + 3010*m**2 + 9129*m + 10395) + 1772*b*c*m*x**9*x**m/(m**5 + 35*m**4 + 470*m**3 + 3010*m**2 + 9129*m + 103
95) + 2310*b*c*x**9*x**m/(m**5 + 35*m**4 + 470*m**3 + 3010*m**2 + 9129*m + 10395) + c**2*m**4*x**11*x**m/(m**5
+ 35*m**4 + 470*m**3 + 3010*m**2 + 9129*m + 10395) + 24*c**2*m**3*x**11*x**m/(m**5 + 35*m**4 + 470*m**3 + 301
0*m**2 + 9129*m + 10395) + 206*c**2*m**2*x**11*x**m/(m**5 + 35*m**4 + 470*m**3 + 3010*m**2 + 9129*m + 10395) +
744*c**2*m*x**11*x**m/(m**5 + 35*m**4 + 470*m**3 + 3010*m**2 + 9129*m + 10395) + 945*c**2*x**11*x**m/(m**5 +
35*m**4 + 470*m**3 + 3010*m**2 + 9129*m + 10395), True))

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