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*(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)

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Rubi [A]  time = 0.0519871, antiderivative size = 71, normalized size of antiderivative = 1., 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 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]

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]

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 = 0.0571561, 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))

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Maple [B]  time = 0.006, size = 262, normalized size = 3.7 \begin{align*}{\frac{{x}^{5+m} \left ( B{c}^{2}{m}^{3}{x}^{6}+21\,B{c}^{2}{m}^{2}{x}^{6}+A{c}^{2}{m}^{3}{x}^{4}+2\,Bbc{m}^{3}{x}^{4}+143\,B{c}^{2}m{x}^{6}+23\,A{c}^{2}{m}^{2}{x}^{4}+46\,Bbc{m}^{2}{x}^{4}+315\,B{c}^{2}{x}^{6}+2\,Abc{m}^{3}{x}^{2}+167\,A{c}^{2}m{x}^{4}+B{b}^{2}{m}^{3}{x}^{2}+334\,Bbcm{x}^{4}+50\,Abc{m}^{2}{x}^{2}+385\,A{c}^{2}{x}^{4}+25\,B{b}^{2}{m}^{2}{x}^{2}+770\,B{x}^{4}bc+A{b}^{2}{m}^{3}+398\,Abcm{x}^{2}+199\,B{b}^{2}m{x}^{2}+27\,A{b}^{2}{m}^{2}+990\,Abc{x}^{2}+495\,B{x}^{2}{b}^{2}+239\,A{b}^{2}m+693\,A{b}^{2} \right ) }{ \left ( 11+m \right ) \left ( 9+m \right ) \left ( 7+m \right ) \left ( 5+m \right ) }} \end{align*}

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^(5+m)*(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)/(11+m)/(9+m)/(7+m)/(5+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*(B*x^2+A)*(c*x^4+b*x^2)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.30448, size = 512, normalized size = 7.21 \begin{align*} \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} \end{align*}

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)

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Sympy [A]  time = 5.04163, size = 1051, normalized size = 14.8 \begin{align*} \text{result too large to display} \end{align*}

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))

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Giac [B]  time = 1.34679, size = 459, normalized size = 6.46 \begin{align*} \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} \end{align*}

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)