3.270 \(\int x^m (A+B x^2) (b x^2+c x^4)^2 \, dx\)

Optimal. Leaf size=71 \[ \frac {A b^2 x^{m+5}}{m+5}+\frac {b x^{m+7} (2 A c+b B)}{m+7}+\frac {c x^{m+9} (A c+2 b B)}{m+9}+\frac {B c^2 x^{m+11}}{m+11} \]

[Out]

A*b^2*x^(5+m)/(5+m)+b*(2*A*c+B*b)*x^(7+m)/(7+m)+c*(A*c+2*B*b)*x^(9+m)/(9+m)+B*c^2*x^(11+m)/(11+m)

________________________________________________________________________________________

Rubi [A]  time = 0.05, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1584, 448} \[ \frac {A b^2 x^{m+5}}{m+5}+\frac {b x^{m+7} (2 A c+b B)}{m+7}+\frac {c x^{m+9} (A c+2 b B)}{m+9}+\frac {B c^2 x^{m+11}}{m+11} \]

Antiderivative was successfully verified.

[In]

Int[x^m*(A + B*x^2)*(b*x^2 + c*x^4)^2,x]

[Out]

(A*b^2*x^(5 + m))/(5 + m) + (b*(b*B + 2*A*c)*x^(7 + m))/(7 + m) + (c*(2*b*B + A*c)*x^(9 + m))/(9 + m) + (B*c^2
*x^(11 + m))/(11 + m)

Rule 448

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[ExpandI
ntegrand[(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] &
& IGtQ[p, 0] && IGtQ[q, 0]

Rule 1584

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.07, size = 66, normalized size = 0.93 \[ x^{m+5} \left (\frac {A b^2}{m+5}+\frac {c x^4 (A c+2 b B)}{m+9}+\frac {b x^2 (2 A c+b B)}{m+7}+\frac {B c^2 x^6}{m+11}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^m*(A + B*x^2)*(b*x^2 + c*x^4)^2,x]

[Out]

x^(5 + m)*((A*b^2)/(5 + m) + (b*(b*B + 2*A*c)*x^2)/(7 + m) + (c*(2*b*B + A*c)*x^4)/(9 + m) + (B*c^2*x^6)/(11 +
m))

________________________________________________________________________________________

fricas [B]  time = 0.97, size = 217, normalized size = 3.06 \[ \frac {{\left ({\left (B c^{2} m^{3} + 21 \, B c^{2} m^{2} + 143 \, B c^{2} m + 315 \, B c^{2}\right )} x^{11} + {\left ({\left (2 \, B b c + A c^{2}\right )} m^{3} + 770 \, B b c + 385 \, A c^{2} + 23 \, {\left (2 \, B b c + A c^{2}\right )} m^{2} + 167 \, {\left (2 \, B b c + A c^{2}\right )} m\right )} x^{9} + {\left ({\left (B b^{2} + 2 \, A b c\right )} m^{3} + 495 \, B b^{2} + 990 \, A b c + 25 \, {\left (B b^{2} + 2 \, A b c\right )} m^{2} + 199 \, {\left (B b^{2} + 2 \, A b c\right )} m\right )} x^{7} + {\left (A b^{2} m^{3} + 27 \, A b^{2} m^{2} + 239 \, A b^{2} m + 693 \, A b^{2}\right )} x^{5}\right )} x^{m}}{m^{4} + 32 \, m^{3} + 374 \, m^{2} + 1888 \, m + 3465} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(B*x^2+A)*(c*x^4+b*x^2)^2,x, algorithm="fricas")

[Out]

((B*c^2*m^3 + 21*B*c^2*m^2 + 143*B*c^2*m + 315*B*c^2)*x^11 + ((2*B*b*c + A*c^2)*m^3 + 770*B*b*c + 385*A*c^2 +
23*(2*B*b*c + A*c^2)*m^2 + 167*(2*B*b*c + A*c^2)*m)*x^9 + ((B*b^2 + 2*A*b*c)*m^3 + 495*B*b^2 + 990*A*b*c + 25*
(B*b^2 + 2*A*b*c)*m^2 + 199*(B*b^2 + 2*A*b*c)*m)*x^7 + (A*b^2*m^3 + 27*A*b^2*m^2 + 239*A*b^2*m + 693*A*b^2)*x^
5)*x^m/(m^4 + 32*m^3 + 374*m^2 + 1888*m + 3465)

________________________________________________________________________________________

giac [B]  time = 0.19, size = 340, normalized size = 4.79 \[ \frac {B c^{2} m^{3} x^{11} x^{m} + 21 \, B c^{2} m^{2} x^{11} x^{m} + 2 \, B b c m^{3} x^{9} x^{m} + A c^{2} m^{3} x^{9} x^{m} + 143 \, B c^{2} m x^{11} x^{m} + 46 \, B b c m^{2} x^{9} x^{m} + 23 \, A c^{2} m^{2} x^{9} x^{m} + 315 \, B c^{2} x^{11} x^{m} + B b^{2} m^{3} x^{7} x^{m} + 2 \, A b c m^{3} x^{7} x^{m} + 334 \, B b c m x^{9} x^{m} + 167 \, A c^{2} m x^{9} x^{m} + 25 \, B b^{2} m^{2} x^{7} x^{m} + 50 \, A b c m^{2} x^{7} x^{m} + 770 \, B b c x^{9} x^{m} + 385 \, A c^{2} x^{9} x^{m} + A b^{2} m^{3} x^{5} x^{m} + 199 \, B b^{2} m x^{7} x^{m} + 398 \, A b c m x^{7} x^{m} + 27 \, A b^{2} m^{2} x^{5} x^{m} + 495 \, B b^{2} x^{7} x^{m} + 990 \, A b c x^{7} x^{m} + 239 \, A b^{2} m x^{5} x^{m} + 693 \, A b^{2} x^{5} x^{m}}{m^{4} + 32 \, m^{3} + 374 \, m^{2} + 1888 \, m + 3465} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(B*x^2+A)*(c*x^4+b*x^2)^2,x, algorithm="giac")

[Out]

(B*c^2*m^3*x^11*x^m + 21*B*c^2*m^2*x^11*x^m + 2*B*b*c*m^3*x^9*x^m + A*c^2*m^3*x^9*x^m + 143*B*c^2*m*x^11*x^m +
46*B*b*c*m^2*x^9*x^m + 23*A*c^2*m^2*x^9*x^m + 315*B*c^2*x^11*x^m + B*b^2*m^3*x^7*x^m + 2*A*b*c*m^3*x^7*x^m +
334*B*b*c*m*x^9*x^m + 167*A*c^2*m*x^9*x^m + 25*B*b^2*m^2*x^7*x^m + 50*A*b*c*m^2*x^7*x^m + 770*B*b*c*x^9*x^m +
385*A*c^2*x^9*x^m + A*b^2*m^3*x^5*x^m + 199*B*b^2*m*x^7*x^m + 398*A*b*c*m*x^7*x^m + 27*A*b^2*m^2*x^5*x^m + 495
*B*b^2*x^7*x^m + 990*A*b*c*x^7*x^m + 239*A*b^2*m*x^5*x^m + 693*A*b^2*x^5*x^m)/(m^4 + 32*m^3 + 374*m^2 + 1888*m
+ 3465)

________________________________________________________________________________________

maple [B]  time = 0.05, size = 262, normalized size = 3.69 \[ \frac {\left (B \,c^{2} m^{3} x^{6}+21 B \,c^{2} m^{2} x^{6}+A \,c^{2} m^{3} x^{4}+2 B b c \,m^{3} x^{4}+143 B \,c^{2} m \,x^{6}+23 A \,c^{2} m^{2} x^{4}+46 B b c \,m^{2} x^{4}+315 B \,c^{2} x^{6}+2 A b c \,m^{3} x^{2}+167 A \,c^{2} m \,x^{4}+B \,b^{2} m^{3} x^{2}+334 B b c m \,x^{4}+50 A b c \,m^{2} x^{2}+385 A \,c^{2} x^{4}+25 B \,b^{2} m^{2} x^{2}+770 B b c \,x^{4}+A \,b^{2} m^{3}+398 A b c m \,x^{2}+199 B \,b^{2} m \,x^{2}+27 A \,b^{2} m^{2}+990 A b c \,x^{2}+495 B \,b^{2} x^{2}+239 A \,b^{2} m +693 A \,b^{2}\right ) x^{m +5}}{\left (m +11\right ) \left (m +9\right ) \left (m +7\right ) \left (m +5\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*(B*x^2+A)*(c*x^4+b*x^2)^2,x)

[Out]

x^(m+5)*(B*c^2*m^3*x^6+21*B*c^2*m^2*x^6+A*c^2*m^3*x^4+2*B*b*c*m^3*x^4+143*B*c^2*m*x^6+23*A*c^2*m^2*x^4+46*B*b*
c*m^2*x^4+315*B*c^2*x^6+2*A*b*c*m^3*x^2+167*A*c^2*m*x^4+B*b^2*m^3*x^2+334*B*b*c*m*x^4+50*A*b*c*m^2*x^2+385*A*c
^2*x^4+25*B*b^2*m^2*x^2+770*B*b*c*x^4+A*b^2*m^3+398*A*b*c*m*x^2+199*B*b^2*m*x^2+27*A*b^2*m^2+990*A*b*c*x^2+495
*B*b^2*x^2+239*A*b^2*m+693*A*b^2)/(m+11)/(m+9)/(m+7)/(m+5)

________________________________________________________________________________________

maxima [A]  time = 1.37, size = 91, normalized size = 1.28 \[ \frac {B c^{2} x^{m + 11}}{m + 11} + \frac {2 \, B b c x^{m + 9}}{m + 9} + \frac {A c^{2} x^{m + 9}}{m + 9} + \frac {B b^{2} x^{m + 7}}{m + 7} + \frac {2 \, A b c x^{m + 7}}{m + 7} + \frac {A b^{2} x^{m + 5}}{m + 5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(B*x^2+A)*(c*x^4+b*x^2)^2,x, algorithm="maxima")

[Out]

B*c^2*x^(m + 11)/(m + 11) + 2*B*b*c*x^(m + 9)/(m + 9) + A*c^2*x^(m + 9)/(m + 9) + B*b^2*x^(m + 7)/(m + 7) + 2*
A*b*c*x^(m + 7)/(m + 7) + A*b^2*x^(m + 5)/(m + 5)

________________________________________________________________________________________

mupad [B]  time = 0.34, size = 179, normalized size = 2.52 \[ x^m\,\left (\frac {A\,b^2\,x^5\,\left (m^3+27\,m^2+239\,m+693\right )}{m^4+32\,m^3+374\,m^2+1888\,m+3465}+\frac {B\,c^2\,x^{11}\,\left (m^3+21\,m^2+143\,m+315\right )}{m^4+32\,m^3+374\,m^2+1888\,m+3465}+\frac {b\,x^7\,\left (2\,A\,c+B\,b\right )\,\left (m^3+25\,m^2+199\,m+495\right )}{m^4+32\,m^3+374\,m^2+1888\,m+3465}+\frac {c\,x^9\,\left (A\,c+2\,B\,b\right )\,\left (m^3+23\,m^2+167\,m+385\right )}{m^4+32\,m^3+374\,m^2+1888\,m+3465}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*(A + B*x^2)*(b*x^2 + c*x^4)^2,x)

[Out]

x^m*((A*b^2*x^5*(239*m + 27*m^2 + m^3 + 693))/(1888*m + 374*m^2 + 32*m^3 + m^4 + 3465) + (B*c^2*x^11*(143*m +
21*m^2 + m^3 + 315))/(1888*m + 374*m^2 + 32*m^3 + m^4 + 3465) + (b*x^7*(2*A*c + B*b)*(199*m + 25*m^2 + m^3 + 4
95))/(1888*m + 374*m^2 + 32*m^3 + m^4 + 3465) + (c*x^9*(A*c + 2*B*b)*(167*m + 23*m^2 + m^3 + 385))/(1888*m + 3
74*m^2 + 32*m^3 + m^4 + 3465))

________________________________________________________________________________________

sympy [A]  time = 4.42, size = 1051, normalized size = 14.80 \[ \begin {cases} - \frac {A b^{2}}{6 x^{6}} - \frac {A b c}{2 x^{4}} - \frac {A c^{2}}{2 x^{2}} - \frac {B b^{2}}{4 x^{4}} - \frac {B b c}{x^{2}} + B c^{2} \log {\relax (x )} & \text {for}\: m = -11 \\- \frac {A b^{2}}{4 x^{4}} - \frac {A b c}{x^{2}} + A c^{2} \log {\relax (x )} - \frac {B b^{2}}{2 x^{2}} + 2 B b c \log {\relax (x )} + \frac {B c^{2} x^{2}}{2} & \text {for}\: m = -9 \\- \frac {A b^{2}}{2 x^{2}} + 2 A b c \log {\relax (x )} + \frac {A c^{2} x^{2}}{2} + B b^{2} \log {\relax (x )} + B b c x^{2} + \frac {B c^{2} x^{4}}{4} & \text {for}\: m = -7 \\A b^{2} \log {\relax (x )} + A b c x^{2} + \frac {A c^{2} x^{4}}{4} + \frac {B b^{2} x^{2}}{2} + \frac {B b c x^{4}}{2} + \frac {B c^{2} x^{6}}{6} & \text {for}\: m = -5 \\\frac {A b^{2} m^{3} x^{5} x^{m}}{m^{4} + 32 m^{3} + 374 m^{2} + 1888 m + 3465} + \frac {27 A b^{2} m^{2} x^{5} x^{m}}{m^{4} + 32 m^{3} + 374 m^{2} + 1888 m + 3465} + \frac {239 A b^{2} m x^{5} x^{m}}{m^{4} + 32 m^{3} + 374 m^{2} + 1888 m + 3465} + \frac {693 A b^{2} x^{5} x^{m}}{m^{4} + 32 m^{3} + 374 m^{2} + 1888 m + 3465} + \frac {2 A b c m^{3} x^{7} x^{m}}{m^{4} + 32 m^{3} + 374 m^{2} + 1888 m + 3465} + \frac {50 A b c m^{2} x^{7} x^{m}}{m^{4} + 32 m^{3} + 374 m^{2} + 1888 m + 3465} + \frac {398 A b c m x^{7} x^{m}}{m^{4} + 32 m^{3} + 374 m^{2} + 1888 m + 3465} + \frac {990 A b c x^{7} x^{m}}{m^{4} + 32 m^{3} + 374 m^{2} + 1888 m + 3465} + \frac {A c^{2} m^{3} x^{9} x^{m}}{m^{4} + 32 m^{3} + 374 m^{2} + 1888 m + 3465} + \frac {23 A c^{2} m^{2} x^{9} x^{m}}{m^{4} + 32 m^{3} + 374 m^{2} + 1888 m + 3465} + \frac {167 A c^{2} m x^{9} x^{m}}{m^{4} + 32 m^{3} + 374 m^{2} + 1888 m + 3465} + \frac {385 A c^{2} x^{9} x^{m}}{m^{4} + 32 m^{3} + 374 m^{2} + 1888 m + 3465} + \frac {B b^{2} m^{3} x^{7} x^{m}}{m^{4} + 32 m^{3} + 374 m^{2} + 1888 m + 3465} + \frac {25 B b^{2} m^{2} x^{7} x^{m}}{m^{4} + 32 m^{3} + 374 m^{2} + 1888 m + 3465} + \frac {199 B b^{2} m x^{7} x^{m}}{m^{4} + 32 m^{3} + 374 m^{2} + 1888 m + 3465} + \frac {495 B b^{2} x^{7} x^{m}}{m^{4} + 32 m^{3} + 374 m^{2} + 1888 m + 3465} + \frac {2 B b c m^{3} x^{9} x^{m}}{m^{4} + 32 m^{3} + 374 m^{2} + 1888 m + 3465} + \frac {46 B b c m^{2} x^{9} x^{m}}{m^{4} + 32 m^{3} + 374 m^{2} + 1888 m + 3465} + \frac {334 B b c m x^{9} x^{m}}{m^{4} + 32 m^{3} + 374 m^{2} + 1888 m + 3465} + \frac {770 B b c x^{9} x^{m}}{m^{4} + 32 m^{3} + 374 m^{2} + 1888 m + 3465} + \frac {B c^{2} m^{3} x^{11} x^{m}}{m^{4} + 32 m^{3} + 374 m^{2} + 1888 m + 3465} + \frac {21 B c^{2} m^{2} x^{11} x^{m}}{m^{4} + 32 m^{3} + 374 m^{2} + 1888 m + 3465} + \frac {143 B c^{2} m x^{11} x^{m}}{m^{4} + 32 m^{3} + 374 m^{2} + 1888 m + 3465} + \frac {315 B c^{2} x^{11} x^{m}}{m^{4} + 32 m^{3} + 374 m^{2} + 1888 m + 3465} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**m*(B*x**2+A)*(c*x**4+b*x**2)**2,x)

[Out]

Piecewise((-A*b**2/(6*x**6) - A*b*c/(2*x**4) - A*c**2/(2*x**2) - B*b**2/(4*x**4) - B*b*c/x**2 + B*c**2*log(x),
Eq(m, -11)), (-A*b**2/(4*x**4) - A*b*c/x**2 + A*c**2*log(x) - B*b**2/(2*x**2) + 2*B*b*c*log(x) + B*c**2*x**2/
2, Eq(m, -9)), (-A*b**2/(2*x**2) + 2*A*b*c*log(x) + A*c**2*x**2/2 + B*b**2*log(x) + B*b*c*x**2 + B*c**2*x**4/4
, Eq(m, -7)), (A*b**2*log(x) + A*b*c*x**2 + A*c**2*x**4/4 + B*b**2*x**2/2 + B*b*c*x**4/2 + B*c**2*x**6/6, Eq(m
, -5)), (A*b**2*m**3*x**5*x**m/(m**4 + 32*m**3 + 374*m**2 + 1888*m + 3465) + 27*A*b**2*m**2*x**5*x**m/(m**4 +
32*m**3 + 374*m**2 + 1888*m + 3465) + 239*A*b**2*m*x**5*x**m/(m**4 + 32*m**3 + 374*m**2 + 1888*m + 3465) + 693
*A*b**2*x**5*x**m/(m**4 + 32*m**3 + 374*m**2 + 1888*m + 3465) + 2*A*b*c*m**3*x**7*x**m/(m**4 + 32*m**3 + 374*m
**2 + 1888*m + 3465) + 50*A*b*c*m**2*x**7*x**m/(m**4 + 32*m**3 + 374*m**2 + 1888*m + 3465) + 398*A*b*c*m*x**7*
x**m/(m**4 + 32*m**3 + 374*m**2 + 1888*m + 3465) + 990*A*b*c*x**7*x**m/(m**4 + 32*m**3 + 374*m**2 + 1888*m + 3
465) + A*c**2*m**3*x**9*x**m/(m**4 + 32*m**3 + 374*m**2 + 1888*m + 3465) + 23*A*c**2*m**2*x**9*x**m/(m**4 + 32
*m**3 + 374*m**2 + 1888*m + 3465) + 167*A*c**2*m*x**9*x**m/(m**4 + 32*m**3 + 374*m**2 + 1888*m + 3465) + 385*A
*c**2*x**9*x**m/(m**4 + 32*m**3 + 374*m**2 + 1888*m + 3465) + B*b**2*m**3*x**7*x**m/(m**4 + 32*m**3 + 374*m**2
+ 1888*m + 3465) + 25*B*b**2*m**2*x**7*x**m/(m**4 + 32*m**3 + 374*m**2 + 1888*m + 3465) + 199*B*b**2*m*x**7*x
**m/(m**4 + 32*m**3 + 374*m**2 + 1888*m + 3465) + 495*B*b**2*x**7*x**m/(m**4 + 32*m**3 + 374*m**2 + 1888*m + 3
465) + 2*B*b*c*m**3*x**9*x**m/(m**4 + 32*m**3 + 374*m**2 + 1888*m + 3465) + 46*B*b*c*m**2*x**9*x**m/(m**4 + 32
*m**3 + 374*m**2 + 1888*m + 3465) + 334*B*b*c*m*x**9*x**m/(m**4 + 32*m**3 + 374*m**2 + 1888*m + 3465) + 770*B*
b*c*x**9*x**m/(m**4 + 32*m**3 + 374*m**2 + 1888*m + 3465) + B*c**2*m**3*x**11*x**m/(m**4 + 32*m**3 + 374*m**2
+ 1888*m + 3465) + 21*B*c**2*m**2*x**11*x**m/(m**4 + 32*m**3 + 374*m**2 + 1888*m + 3465) + 143*B*c**2*m*x**11*
x**m/(m**4 + 32*m**3 + 374*m**2 + 1888*m + 3465) + 315*B*c**2*x**11*x**m/(m**4 + 32*m**3 + 374*m**2 + 1888*m +
3465), True))

________________________________________________________________________________________