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) + ((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)
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Rubi [A] time = 0.0458617, antiderivative size = 76, normalized size of antiderivative = 1.,
number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, 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 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]
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 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*}
Mathematica [A] time = 0.0787956, 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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Maple [B] time = 0.006, size = 300, normalized size = 4. \begin{align*}{\frac{{x}^{3+m} \left ({c}^{2}{m}^{4}{x}^{8}+24\,{c}^{2}{m}^{3}{x}^{8}+2\,bc{m}^{4}{x}^{6}+206\,{c}^{2}{m}^{2}{x}^{8}+52\,bc{m}^{3}{x}^{6}+744\,{c}^{2}m{x}^{8}+2\,ac{m}^{4}{x}^{4}+{b}^{2}{m}^{4}{x}^{4}+472\,bc{m}^{2}{x}^{6}+945\,{c}^{2}{x}^{8}+56\,ac{m}^{3}{x}^{4}+28\,{b}^{2}{m}^{3}{x}^{4}+1772\,bcm{x}^{6}+2\,ab{m}^{4}{x}^{2}+548\,ac{m}^{2}{x}^{4}+274\,{b}^{2}{m}^{2}{x}^{4}+2310\,bc{x}^{6}+60\,ab{m}^{3}{x}^{2}+2184\,acm{x}^{4}+1092\,{b}^{2}m{x}^{4}+{a}^{2}{m}^{4}+640\,ab{m}^{2}{x}^{2}+2970\,ac{x}^{4}+1485\,{b}^{2}{x}^{4}+32\,{a}^{2}{m}^{3}+2820\,abm{x}^{2}+374\,{a}^{2}{m}^{2}+4158\,ab{x}^{2}+1888\,{a}^{2}m+3465\,{a}^{2} \right ) }{ \left ( 11+m \right ) \left ( 9+m \right ) \left ( 7+m \right ) \left ( 5+m \right ) \left ( 3+m \right ) }} \end{align*}
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^(3+m)*(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)/(11+m)/(9+m)/(7+m)/(
5+m)/(3+m)
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
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]
Exception raised: ValueError
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Fricas [B] time = 1.4098, size = 602, normalized size = 7.92 \begin{align*} \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} \end{align*}
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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Sympy [A] time = 3.99035, size = 1377, normalized size = 18.12 \begin{align*} \text{result too large to display} \end{align*}
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))
________________________________________________________________________________________
Giac [B] time = 1.1273, size = 539, normalized size = 7.09 \begin{align*} \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} \end{align*}
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)