3.4.24 \(\int x^m (a+b x^2)^2 (c+d x^2) \, dx\)

Optimal. Leaf size=71 \[ \frac {a^2 c x^{m+1}}{m+1}+\frac {a x^{m+3} (a d+2 b c)}{m+3}+\frac {b x^{m+5} (2 a d+b c)}{m+5}+\frac {b^2 d x^{m+7}}{m+7} \]

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Rubi [A]  time = 0.04, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {448} \begin {gather*} \frac {a^2 c x^{m+1}}{m+1}+\frac {a x^{m+3} (a d+2 b c)}{m+3}+\frac {b x^{m+5} (2 a d+b c)}{m+5}+\frac {b^2 d x^{m+7}}{m+7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^m*(a + b*x^2)^2*(c + d*x^2),x]

[Out]

(a^2*c*x^(1 + m))/(1 + m) + (a*(2*b*c + a*d)*x^(3 + m))/(3 + m) + (b*(b*c + 2*a*d)*x^(5 + m))/(5 + m) + (b^2*d
*x^(7 + m))/(7 + 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]

Rubi steps

\begin {align*} \int x^m \left (a+b x^2\right )^2 \left (c+d x^2\right ) \, dx &=\int \left (a^2 c x^m+a (2 b c+a d) x^{2+m}+b (b c+2 a d) x^{4+m}+b^2 d x^{6+m}\right ) \, dx\\ &=\frac {a^2 c x^{1+m}}{1+m}+\frac {a (2 b c+a d) x^{3+m}}{3+m}+\frac {b (b c+2 a d) x^{5+m}}{5+m}+\frac {b^2 d x^{7+m}}{7+m}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 66, normalized size = 0.93 \begin {gather*} x^{m+1} \left (\frac {a^2 c}{m+1}+\frac {b x^4 (2 a d+b c)}{m+5}+\frac {a x^2 (a d+2 b c)}{m+3}+\frac {b^2 d x^6}{m+7}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x^(1 + m)*((a^2*c)/(1 + m) + (a*(2*b*c + a*d)*x^2)/(3 + m) + (b*(b*c + 2*a*d)*x^4)/(5 + m) + (b^2*d*x^6)/(7 +
m))

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IntegrateAlgebraic [F]  time = 0.04, size = 0, normalized size = 0.00 \begin {gather*} \int x^m \left (a+b x^2\right )^2 \left (c+d x^2\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x^m*(a + b*x^2)^2*(c + d*x^2),x]

[Out]

Defer[IntegrateAlgebraic][x^m*(a + b*x^2)^2*(c + d*x^2), x]

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fricas [B]  time = 1.03, size = 215, normalized size = 3.03 \begin {gather*} \frac {{\left ({\left (b^{2} d m^{3} + 9 \, b^{2} d m^{2} + 23 \, b^{2} d m + 15 \, b^{2} d\right )} x^{7} + {\left ({\left (b^{2} c + 2 \, a b d\right )} m^{3} + 21 \, b^{2} c + 42 \, a b d + 11 \, {\left (b^{2} c + 2 \, a b d\right )} m^{2} + 31 \, {\left (b^{2} c + 2 \, a b d\right )} m\right )} x^{5} + {\left ({\left (2 \, a b c + a^{2} d\right )} m^{3} + 70 \, a b c + 35 \, a^{2} d + 13 \, {\left (2 \, a b c + a^{2} d\right )} m^{2} + 47 \, {\left (2 \, a b c + a^{2} d\right )} m\right )} x^{3} + {\left (a^{2} c m^{3} + 15 \, a^{2} c m^{2} + 71 \, a^{2} c m + 105 \, a^{2} c\right )} x\right )} x^{m}}{m^{4} + 16 \, m^{3} + 86 \, m^{2} + 176 \, m + 105} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((b^2*d*m^3 + 9*b^2*d*m^2 + 23*b^2*d*m + 15*b^2*d)*x^7 + ((b^2*c + 2*a*b*d)*m^3 + 21*b^2*c + 42*a*b*d + 11*(b^
2*c + 2*a*b*d)*m^2 + 31*(b^2*c + 2*a*b*d)*m)*x^5 + ((2*a*b*c + a^2*d)*m^3 + 70*a*b*c + 35*a^2*d + 13*(2*a*b*c
+ a^2*d)*m^2 + 47*(2*a*b*c + a^2*d)*m)*x^3 + (a^2*c*m^3 + 15*a^2*c*m^2 + 71*a^2*c*m + 105*a^2*c)*x)*x^m/(m^4 +
16*m^3 + 86*m^2 + 176*m + 105)

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giac [B]  time = 0.43, size = 332, normalized size = 4.68 \begin {gather*} \frac {b^{2} d m^{3} x^{7} x^{m} + 9 \, b^{2} d m^{2} x^{7} x^{m} + b^{2} c m^{3} x^{5} x^{m} + 2 \, a b d m^{3} x^{5} x^{m} + 23 \, b^{2} d m x^{7} x^{m} + 11 \, b^{2} c m^{2} x^{5} x^{m} + 22 \, a b d m^{2} x^{5} x^{m} + 15 \, b^{2} d x^{7} x^{m} + 2 \, a b c m^{3} x^{3} x^{m} + a^{2} d m^{3} x^{3} x^{m} + 31 \, b^{2} c m x^{5} x^{m} + 62 \, a b d m x^{5} x^{m} + 26 \, a b c m^{2} x^{3} x^{m} + 13 \, a^{2} d m^{2} x^{3} x^{m} + 21 \, b^{2} c x^{5} x^{m} + 42 \, a b d x^{5} x^{m} + a^{2} c m^{3} x x^{m} + 94 \, a b c m x^{3} x^{m} + 47 \, a^{2} d m x^{3} x^{m} + 15 \, a^{2} c m^{2} x x^{m} + 70 \, a b c x^{3} x^{m} + 35 \, a^{2} d x^{3} x^{m} + 71 \, a^{2} c m x x^{m} + 105 \, a^{2} c x x^{m}}{m^{4} + 16 \, m^{3} + 86 \, m^{2} + 176 \, m + 105} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b^2*d*m^3*x^7*x^m + 9*b^2*d*m^2*x^7*x^m + b^2*c*m^3*x^5*x^m + 2*a*b*d*m^3*x^5*x^m + 23*b^2*d*m*x^7*x^m + 11*b
^2*c*m^2*x^5*x^m + 22*a*b*d*m^2*x^5*x^m + 15*b^2*d*x^7*x^m + 2*a*b*c*m^3*x^3*x^m + a^2*d*m^3*x^3*x^m + 31*b^2*
c*m*x^5*x^m + 62*a*b*d*m*x^5*x^m + 26*a*b*c*m^2*x^3*x^m + 13*a^2*d*m^2*x^3*x^m + 21*b^2*c*x^5*x^m + 42*a*b*d*x
^5*x^m + a^2*c*m^3*x*x^m + 94*a*b*c*m*x^3*x^m + 47*a^2*d*m*x^3*x^m + 15*a^2*c*m^2*x*x^m + 70*a*b*c*x^3*x^m + 3
5*a^2*d*x^3*x^m + 71*a^2*c*m*x*x^m + 105*a^2*c*x*x^m)/(m^4 + 16*m^3 + 86*m^2 + 176*m + 105)

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maple [B]  time = 0.01, size = 262, normalized size = 3.69 \begin {gather*} \frac {\left (b^{2} d \,m^{3} x^{6}+9 b^{2} d \,m^{2} x^{6}+2 a b d \,m^{3} x^{4}+b^{2} c \,m^{3} x^{4}+23 b^{2} d m \,x^{6}+22 a b d \,m^{2} x^{4}+11 b^{2} c \,m^{2} x^{4}+15 b^{2} d \,x^{6}+a^{2} d \,m^{3} x^{2}+2 a b c \,m^{3} x^{2}+62 a b d m \,x^{4}+31 b^{2} c m \,x^{4}+13 a^{2} d \,m^{2} x^{2}+26 a b c \,m^{2} x^{2}+42 a b d \,x^{4}+21 b^{2} c \,x^{4}+a^{2} c \,m^{3}+47 a^{2} d m \,x^{2}+94 a b c m \,x^{2}+15 a^{2} c \,m^{2}+35 a^{2} d \,x^{2}+70 a b c \,x^{2}+71 a^{2} c m +105 a^{2} c \right ) x^{m +1}}{\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(x^m*(b*x^2+a)^2*(d*x^2+c),x)

[Out]

x^(m+1)*(b^2*d*m^3*x^6+9*b^2*d*m^2*x^6+2*a*b*d*m^3*x^4+b^2*c*m^3*x^4+23*b^2*d*m*x^6+22*a*b*d*m^2*x^4+11*b^2*c*
m^2*x^4+15*b^2*d*x^6+a^2*d*m^3*x^2+2*a*b*c*m^3*x^2+62*a*b*d*m*x^4+31*b^2*c*m*x^4+13*a^2*d*m^2*x^2+26*a*b*c*m^2
*x^2+42*a*b*d*x^4+21*b^2*c*x^4+a^2*c*m^3+47*a^2*d*m*x^2+94*a*b*c*m*x^2+15*a^2*c*m^2+35*a^2*d*x^2+70*a*b*c*x^2+
71*a^2*c*m+105*a^2*c)/(m+7)/(m+5)/(m+3)/(m+1)

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maxima [A]  time = 0.95, size = 91, normalized size = 1.28 \begin {gather*} \frac {b^{2} d x^{m + 7}}{m + 7} + \frac {b^{2} c x^{m + 5}}{m + 5} + \frac {2 \, a b d x^{m + 5}}{m + 5} + \frac {2 \, a b c x^{m + 3}}{m + 3} + \frac {a^{2} d x^{m + 3}}{m + 3} + \frac {a^{2} c x^{m + 1}}{m + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^2*d*x^(m + 7)/(m + 7) + b^2*c*x^(m + 5)/(m + 5) + 2*a*b*d*x^(m + 5)/(m + 5) + 2*a*b*c*x^(m + 3)/(m + 3) + a^
2*d*x^(m + 3)/(m + 3) + a^2*c*x^(m + 1)/(m + 1)

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mupad [B]  time = 0.36, size = 177, normalized size = 2.49 \begin {gather*} x^m\,\left (\frac {a\,x^3\,\left (a\,d+2\,b\,c\right )\,\left (m^3+13\,m^2+47\,m+35\right )}{m^4+16\,m^3+86\,m^2+176\,m+105}+\frac {b\,x^5\,\left (2\,a\,d+b\,c\right )\,\left (m^3+11\,m^2+31\,m+21\right )}{m^4+16\,m^3+86\,m^2+176\,m+105}+\frac {b^2\,d\,x^7\,\left (m^3+9\,m^2+23\,m+15\right )}{m^4+16\,m^3+86\,m^2+176\,m+105}+\frac {a^2\,c\,x\,\left (m^3+15\,m^2+71\,m+105\right )}{m^4+16\,m^3+86\,m^2+176\,m+105}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*(a + b*x^2)^2*(c + d*x^2),x)

[Out]

x^m*((a*x^3*(a*d + 2*b*c)*(47*m + 13*m^2 + m^3 + 35))/(176*m + 86*m^2 + 16*m^3 + m^4 + 105) + (b*x^5*(2*a*d +
b*c)*(31*m + 11*m^2 + m^3 + 21))/(176*m + 86*m^2 + 16*m^3 + m^4 + 105) + (b^2*d*x^7*(23*m + 9*m^2 + m^3 + 15))
/(176*m + 86*m^2 + 16*m^3 + m^4 + 105) + (a^2*c*x*(71*m + 15*m^2 + m^3 + 105))/(176*m + 86*m^2 + 16*m^3 + m^4
+ 105))

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sympy [A]  time = 1.76, size = 1044, normalized size = 14.70 \begin {gather*} \begin {cases} - \frac {a^{2} c}{6 x^{6}} - \frac {a^{2} d}{4 x^{4}} - \frac {a b c}{2 x^{4}} - \frac {a b d}{x^{2}} - \frac {b^{2} c}{2 x^{2}} + b^{2} d \log {\relax (x )} & \text {for}\: m = -7 \\- \frac {a^{2} c}{4 x^{4}} - \frac {a^{2} d}{2 x^{2}} - \frac {a b c}{x^{2}} + 2 a b d \log {\relax (x )} + b^{2} c \log {\relax (x )} + \frac {b^{2} d x^{2}}{2} & \text {for}\: m = -5 \\- \frac {a^{2} c}{2 x^{2}} + a^{2} d \log {\relax (x )} + 2 a b c \log {\relax (x )} + a b d x^{2} + \frac {b^{2} c x^{2}}{2} + \frac {b^{2} d x^{4}}{4} & \text {for}\: m = -3 \\a^{2} c \log {\relax (x )} + \frac {a^{2} d x^{2}}{2} + a b c x^{2} + \frac {a b d x^{4}}{2} + \frac {b^{2} c x^{4}}{4} + \frac {b^{2} d x^{6}}{6} & \text {for}\: m = -1 \\\frac {a^{2} c m^{3} x x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {15 a^{2} c m^{2} x x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {71 a^{2} c m x x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {105 a^{2} c x x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {a^{2} d m^{3} x^{3} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {13 a^{2} d m^{2} x^{3} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {47 a^{2} d m x^{3} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {35 a^{2} d x^{3} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {2 a b c m^{3} x^{3} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {26 a b c m^{2} x^{3} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {94 a b c m x^{3} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {70 a b c x^{3} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {2 a b d m^{3} x^{5} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {22 a b d m^{2} x^{5} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {62 a b d m x^{5} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {42 a b d x^{5} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {b^{2} c m^{3} x^{5} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {11 b^{2} c m^{2} x^{5} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {31 b^{2} c m x^{5} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {21 b^{2} c x^{5} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {b^{2} d m^{3} x^{7} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {9 b^{2} d m^{2} x^{7} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {23 b^{2} d m x^{7} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {15 b^{2} d x^{7} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-a**2*c/(6*x**6) - a**2*d/(4*x**4) - a*b*c/(2*x**4) - a*b*d/x**2 - b**2*c/(2*x**2) + b**2*d*log(x),
Eq(m, -7)), (-a**2*c/(4*x**4) - a**2*d/(2*x**2) - a*b*c/x**2 + 2*a*b*d*log(x) + b**2*c*log(x) + b**2*d*x**2/2
, Eq(m, -5)), (-a**2*c/(2*x**2) + a**2*d*log(x) + 2*a*b*c*log(x) + a*b*d*x**2 + b**2*c*x**2/2 + b**2*d*x**4/4,
Eq(m, -3)), (a**2*c*log(x) + a**2*d*x**2/2 + a*b*c*x**2 + a*b*d*x**4/2 + b**2*c*x**4/4 + b**2*d*x**6/6, Eq(m,
-1)), (a**2*c*m**3*x*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 15*a**2*c*m**2*x*x**m/(m**4 + 16*m**3 +
86*m**2 + 176*m + 105) + 71*a**2*c*m*x*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 105*a**2*c*x*x**m/(m**4
+ 16*m**3 + 86*m**2 + 176*m + 105) + a**2*d*m**3*x**3*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 13*a**2
*d*m**2*x**3*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 47*a**2*d*m*x**3*x**m/(m**4 + 16*m**3 + 86*m**2 +
176*m + 105) + 35*a**2*d*x**3*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 2*a*b*c*m**3*x**3*x**m/(m**4 +
16*m**3 + 86*m**2 + 176*m + 105) + 26*a*b*c*m**2*x**3*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 94*a*b*c
*m*x**3*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 70*a*b*c*x**3*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m +
105) + 2*a*b*d*m**3*x**5*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 22*a*b*d*m**2*x**5*x**m/(m**4 + 16*m
**3 + 86*m**2 + 176*m + 105) + 62*a*b*d*m*x**5*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 42*a*b*d*x**5*x
**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + b**2*c*m**3*x**5*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105)
+ 11*b**2*c*m**2*x**5*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 31*b**2*c*m*x**5*x**m/(m**4 + 16*m**3 +
86*m**2 + 176*m + 105) + 21*b**2*c*x**5*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + b**2*d*m**3*x**7*x**m/
(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 9*b**2*d*m**2*x**7*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) +
23*b**2*d*m*x**7*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 15*b**2*d*x**7*x**m/(m**4 + 16*m**3 + 86*m**2
+ 176*m + 105), True))

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