3.327 \(\int x^m \left (a+b x^2\right )^2 \left (c+d x^2\right ) \, dx\)

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

[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)

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Rubi [A]  time = 0.11054, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ \frac{a^2 c x^{m+1}}{m+1}+\frac{a x^{m+3} (a d+2 b c)}{m+3}+\frac{b x^{m+5} (2 a d+b c)}{m+5}+\frac{b^2 d x^{m+7}}{m+7} \]

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)

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Rubi in Sympy [A]  time = 15.8642, size = 63, normalized size = 0.89 \[ \frac{a^{2} c x^{m + 1}}{m + 1} + \frac{a x^{m + 3} \left (a d + 2 b c\right )}{m + 3} + \frac{b^{2} d x^{m + 7}}{m + 7} + \frac{b x^{m + 5} \left (2 a d + b c\right )}{m + 5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**m*(b*x**2+a)**2*(d*x**2+c),x)

[Out]

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

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Mathematica [A]  time = 0.0816094, size = 65, normalized size = 0.92 \[ x^m \left (\frac{a^2 c x}{m+1}+\frac{b x^5 (2 a d+b c)}{m+5}+\frac{a x^3 (a d+2 b c)}{m+3}+\frac{b^2 d x^7}{m+7}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.008, size = 262, normalized size = 3.7 \[{\frac{{x}^{1+m} \left ({b}^{2}d{m}^{3}{x}^{6}+9\,{b}^{2}d{m}^{2}{x}^{6}+2\,abd{m}^{3}{x}^{4}+{b}^{2}c{m}^{3}{x}^{4}+23\,{b}^{2}dm{x}^{6}+22\,abd{m}^{2}{x}^{4}+11\,{b}^{2}c{m}^{2}{x}^{4}+15\,{b}^{2}d{x}^{6}+{a}^{2}d{m}^{3}{x}^{2}+2\,abc{m}^{3}{x}^{2}+62\,abdm{x}^{4}+31\,{b}^{2}cm{x}^{4}+13\,{a}^{2}d{m}^{2}{x}^{2}+26\,abc{m}^{2}{x}^{2}+42\,{x}^{4}abd+21\,{b}^{2}c{x}^{4}+{a}^{2}c{m}^{3}+47\,{a}^{2}dm{x}^{2}+94\,abcm{x}^{2}+15\,{a}^{2}c{m}^{2}+35\,{x}^{2}{a}^{2}d+70\,abc{x}^{2}+71\,{a}^{2}cm+105\,{a}^{2}c \right ) }{ \left ( 7+m \right ) \left ( 5+m \right ) \left ( 3+m \right ) \left ( 1+m \right ) }} \]

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^(1+m)*(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)/(7+m)/(5+m)/(3+m)/(1+m)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.240005, size = 290, normalized size = 4.08 \[ \frac{{\left ({\left (b^{2} d m^{3} + 9 \, b^{2} d m^{2} + 23 \, b^{2} d m + 15 \, b^{2} d\right )} x^{7} +{\left ({\left (b^{2} c + 2 \, a b d\right )} m^{3} + 21 \, b^{2} c + 42 \, a b d + 11 \,{\left (b^{2} c + 2 \, a b d\right )} m^{2} + 31 \,{\left (b^{2} c + 2 \, a b d\right )} m\right )} x^{5} +{\left ({\left (2 \, a b c + a^{2} d\right )} m^{3} + 70 \, a b c + 35 \, a^{2} d + 13 \,{\left (2 \, a b c + a^{2} d\right )} m^{2} + 47 \,{\left (2 \, a b c + a^{2} d\right )} m\right )} x^{3} +{\left (a^{2} c m^{3} + 15 \, a^{2} c m^{2} + 71 \, a^{2} c m + 105 \, a^{2} c\right )} x\right )} x^{m}}{m^{4} + 16 \, m^{3} + 86 \, m^{2} + 176 \, m + 105} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*(d*x^2 + c)*x^m,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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Sympy [A]  time = 5.47498, size = 1044, normalized size = 14.7 \[ \text{result too large to display} \]

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), Tru
e))

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GIAC/XCAS [A]  time = 0.263233, size = 513, normalized size = 7.23 \[ \frac{b^{2} d m^{3} x^{7} e^{\left (m{\rm ln}\left (x\right )\right )} + 9 \, b^{2} d m^{2} x^{7} e^{\left (m{\rm ln}\left (x\right )\right )} + b^{2} c m^{3} x^{5} e^{\left (m{\rm ln}\left (x\right )\right )} + 2 \, a b d m^{3} x^{5} e^{\left (m{\rm ln}\left (x\right )\right )} + 23 \, b^{2} d m x^{7} e^{\left (m{\rm ln}\left (x\right )\right )} + 11 \, b^{2} c m^{2} x^{5} e^{\left (m{\rm ln}\left (x\right )\right )} + 22 \, a b d m^{2} x^{5} e^{\left (m{\rm ln}\left (x\right )\right )} + 15 \, b^{2} d x^{7} e^{\left (m{\rm ln}\left (x\right )\right )} + 2 \, a b c m^{3} x^{3} e^{\left (m{\rm ln}\left (x\right )\right )} + a^{2} d m^{3} x^{3} e^{\left (m{\rm ln}\left (x\right )\right )} + 31 \, b^{2} c m x^{5} e^{\left (m{\rm ln}\left (x\right )\right )} + 62 \, a b d m x^{5} e^{\left (m{\rm ln}\left (x\right )\right )} + 26 \, a b c m^{2} x^{3} e^{\left (m{\rm ln}\left (x\right )\right )} + 13 \, a^{2} d m^{2} x^{3} e^{\left (m{\rm ln}\left (x\right )\right )} + 21 \, b^{2} c x^{5} e^{\left (m{\rm ln}\left (x\right )\right )} + 42 \, a b d x^{5} e^{\left (m{\rm ln}\left (x\right )\right )} + a^{2} c m^{3} x e^{\left (m{\rm ln}\left (x\right )\right )} + 94 \, a b c m x^{3} e^{\left (m{\rm ln}\left (x\right )\right )} + 47 \, a^{2} d m x^{3} e^{\left (m{\rm ln}\left (x\right )\right )} + 15 \, a^{2} c m^{2} x e^{\left (m{\rm ln}\left (x\right )\right )} + 70 \, a b c x^{3} e^{\left (m{\rm ln}\left (x\right )\right )} + 35 \, a^{2} d x^{3} e^{\left (m{\rm ln}\left (x\right )\right )} + 71 \, a^{2} c m x e^{\left (m{\rm ln}\left (x\right )\right )} + 105 \, a^{2} c x e^{\left (m{\rm ln}\left (x\right )\right )}}{m^{4} + 16 \, m^{3} + 86 \, m^{2} + 176 \, m + 105} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(b^2*d*m^3*x^7*e^(m*ln(x)) + 9*b^2*d*m^2*x^7*e^(m*ln(x)) + b^2*c*m^3*x^5*e^(m*ln
(x)) + 2*a*b*d*m^3*x^5*e^(m*ln(x)) + 23*b^2*d*m*x^7*e^(m*ln(x)) + 11*b^2*c*m^2*x
^5*e^(m*ln(x)) + 22*a*b*d*m^2*x^5*e^(m*ln(x)) + 15*b^2*d*x^7*e^(m*ln(x)) + 2*a*b
*c*m^3*x^3*e^(m*ln(x)) + a^2*d*m^3*x^3*e^(m*ln(x)) + 31*b^2*c*m*x^5*e^(m*ln(x))
+ 62*a*b*d*m*x^5*e^(m*ln(x)) + 26*a*b*c*m^2*x^3*e^(m*ln(x)) + 13*a^2*d*m^2*x^3*e
^(m*ln(x)) + 21*b^2*c*x^5*e^(m*ln(x)) + 42*a*b*d*x^5*e^(m*ln(x)) + a^2*c*m^3*x*e
^(m*ln(x)) + 94*a*b*c*m*x^3*e^(m*ln(x)) + 47*a^2*d*m*x^3*e^(m*ln(x)) + 15*a^2*c*
m^2*x*e^(m*ln(x)) + 70*a*b*c*x^3*e^(m*ln(x)) + 35*a^2*d*x^3*e^(m*ln(x)) + 71*a^2
*c*m*x*e^(m*ln(x)) + 105*a^2*c*x*e^(m*ln(x)))/(m^4 + 16*m^3 + 86*m^2 + 176*m + 1
05)