3.326 \(\int x^m (a+b x^2)^2 (c+d x^2)^2 \, dx\)
Optimal. Leaf size=109 \[ \frac{x^{m+5} \left (a^2 d^2+4 a b c d+b^2 c^2\right )}{m+5}+\frac{a^2 c^2 x^{m+1}}{m+1}+\frac{2 a c x^{m+3} (a d+b c)}{m+3}+\frac{2 b d x^{m+7} (a d+b c)}{m+7}+\frac{b^2 d^2 x^{m+9}}{m+9} \]
[Out]
(a^2*c^2*x^(1 + m))/(1 + m) + (2*a*c*(b*c + a*d)*x^(3 + m))/(3 + m) + ((b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^(5 +
m))/(5 + m) + (2*b*d*(b*c + a*d)*x^(7 + m))/(7 + m) + (b^2*d^2*x^(9 + m))/(9 + m)
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Rubi [A] time = 0.0622496, antiderivative size = 109, normalized size of antiderivative = 1.,
number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used =
{448} \[ \frac{x^{m+5} \left (a^2 d^2+4 a b c d+b^2 c^2\right )}{m+5}+\frac{a^2 c^2 x^{m+1}}{m+1}+\frac{2 a c x^{m+3} (a d+b c)}{m+3}+\frac{2 b d x^{m+7} (a d+b c)}{m+7}+\frac{b^2 d^2 x^{m+9}}{m+9} \]
Antiderivative was successfully verified.
[In]
Int[x^m*(a + b*x^2)^2*(c + d*x^2)^2,x]
[Out]
(a^2*c^2*x^(1 + m))/(1 + m) + (2*a*c*(b*c + a*d)*x^(3 + m))/(3 + m) + ((b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^(5 +
m))/(5 + m) + (2*b*d*(b*c + a*d)*x^(7 + m))/(7 + m) + (b^2*d^2*x^(9 + m))/(9 + 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 )^2 \, dx &=\int \left (a^2 c^2 x^m+2 a c (b c+a d) x^{2+m}+\left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^{4+m}+2 b d (b c+a d) x^{6+m}+b^2 d^2 x^{8+m}\right ) \, dx\\ &=\frac{a^2 c^2 x^{1+m}}{1+m}+\frac{2 a c (b c+a d) x^{3+m}}{3+m}+\frac{\left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^{5+m}}{5+m}+\frac{2 b d (b c+a d) x^{7+m}}{7+m}+\frac{b^2 d^2 x^{9+m}}{9+m}\\ \end{align*}
Mathematica [A] time = 0.0723732, size = 101, normalized size = 0.93 \[ x^m \left (\frac{x^5 \left (a^2 d^2+4 a b c d+b^2 c^2\right )}{m+5}+\frac{a^2 c^2 x}{m+1}+\frac{2 b d x^7 (a d+b c)}{m+7}+\frac{2 a c x^3 (a d+b c)}{m+3}+\frac{b^2 d^2 x^9}{m+9}\right ) \]
Antiderivative was successfully verified.
[In]
Integrate[x^m*(a + b*x^2)^2*(c + d*x^2)^2,x]
[Out]
x^m*((a^2*c^2*x)/(1 + m) + (2*a*c*(b*c + a*d)*x^3)/(3 + m) + ((b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^5)/(5 + m) + (
2*b*d*(b*c + a*d)*x^7)/(7 + m) + (b^2*d^2*x^9)/(9 + m))
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Maple [B] time = 0.007, size = 569, normalized size = 5.2 \begin{align*}{\frac{{x}^{1+m} \left ({b}^{2}{d}^{2}{m}^{4}{x}^{8}+16\,{b}^{2}{d}^{2}{m}^{3}{x}^{8}+2\,ab{d}^{2}{m}^{4}{x}^{6}+2\,{b}^{2}cd{m}^{4}{x}^{6}+86\,{b}^{2}{d}^{2}{m}^{2}{x}^{8}+36\,ab{d}^{2}{m}^{3}{x}^{6}+36\,{b}^{2}cd{m}^{3}{x}^{6}+176\,{b}^{2}{d}^{2}m{x}^{8}+{a}^{2}{d}^{2}{m}^{4}{x}^{4}+4\,abcd{m}^{4}{x}^{4}+208\,ab{d}^{2}{m}^{2}{x}^{6}+{b}^{2}{c}^{2}{m}^{4}{x}^{4}+208\,{b}^{2}cd{m}^{2}{x}^{6}+105\,{b}^{2}{d}^{2}{x}^{8}+20\,{a}^{2}{d}^{2}{m}^{3}{x}^{4}+80\,abcd{m}^{3}{x}^{4}+444\,ab{d}^{2}m{x}^{6}+20\,{b}^{2}{c}^{2}{m}^{3}{x}^{4}+444\,{b}^{2}cdm{x}^{6}+2\,{a}^{2}cd{m}^{4}{x}^{2}+130\,{a}^{2}{d}^{2}{m}^{2}{x}^{4}+2\,ab{c}^{2}{m}^{4}{x}^{2}+520\,abcd{m}^{2}{x}^{4}+270\,{x}^{6}ab{d}^{2}+130\,{b}^{2}{c}^{2}{m}^{2}{x}^{4}+270\,{x}^{6}{b}^{2}cd+44\,{a}^{2}cd{m}^{3}{x}^{2}+300\,{a}^{2}{d}^{2}m{x}^{4}+44\,ab{c}^{2}{m}^{3}{x}^{2}+1200\,abcdm{x}^{4}+300\,{b}^{2}{c}^{2}m{x}^{4}+{a}^{2}{c}^{2}{m}^{4}+328\,{a}^{2}cd{m}^{2}{x}^{2}+189\,{x}^{4}{a}^{2}{d}^{2}+328\,ab{c}^{2}{m}^{2}{x}^{2}+756\,{x}^{4}abcd+189\,{x}^{4}{b}^{2}{c}^{2}+24\,{a}^{2}{c}^{2}{m}^{3}+916\,{a}^{2}cdm{x}^{2}+916\,ab{c}^{2}m{x}^{2}+206\,{a}^{2}{c}^{2}{m}^{2}+630\,{x}^{2}{a}^{2}cd+630\,a{c}^{2}b{x}^{2}+744\,{a}^{2}{c}^{2}m+945\,{a}^{2}{c}^{2} \right ) }{ \left ( 9+m \right ) \left ( 7+m \right ) \left ( 5+m \right ) \left ( 3+m \right ) \left ( 1+m \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^m*(b*x^2+a)^2*(d*x^2+c)^2,x)
[Out]
x^(1+m)*(b^2*d^2*m^4*x^8+16*b^2*d^2*m^3*x^8+2*a*b*d^2*m^4*x^6+2*b^2*c*d*m^4*x^6+86*b^2*d^2*m^2*x^8+36*a*b*d^2*
m^3*x^6+36*b^2*c*d*m^3*x^6+176*b^2*d^2*m*x^8+a^2*d^2*m^4*x^4+4*a*b*c*d*m^4*x^4+208*a*b*d^2*m^2*x^6+b^2*c^2*m^4
*x^4+208*b^2*c*d*m^2*x^6+105*b^2*d^2*x^8+20*a^2*d^2*m^3*x^4+80*a*b*c*d*m^3*x^4+444*a*b*d^2*m*x^6+20*b^2*c^2*m^
3*x^4+444*b^2*c*d*m*x^6+2*a^2*c*d*m^4*x^2+130*a^2*d^2*m^2*x^4+2*a*b*c^2*m^4*x^2+520*a*b*c*d*m^2*x^4+270*a*b*d^
2*x^6+130*b^2*c^2*m^2*x^4+270*b^2*c*d*x^6+44*a^2*c*d*m^3*x^2+300*a^2*d^2*m*x^4+44*a*b*c^2*m^3*x^2+1200*a*b*c*d
*m*x^4+300*b^2*c^2*m*x^4+a^2*c^2*m^4+328*a^2*c*d*m^2*x^2+189*a^2*d^2*x^4+328*a*b*c^2*m^2*x^2+756*a*b*c*d*x^4+1
89*b^2*c^2*x^4+24*a^2*c^2*m^3+916*a^2*c*d*m*x^2+916*a*b*c^2*m*x^2+206*a^2*c^2*m^2+630*a^2*c*d*x^2+630*a*b*c^2*
x^2+744*a^2*c^2*m+945*a^2*c^2)/(9+m)/(7+m)/(5+m)/(3+m)/(1+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)^2*(d*x^2+c)^2,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [B] time = 0.857922, size = 984, normalized size = 9.03 \begin{align*} \frac{{\left ({\left (b^{2} d^{2} m^{4} + 16 \, b^{2} d^{2} m^{3} + 86 \, b^{2} d^{2} m^{2} + 176 \, b^{2} d^{2} m + 105 \, b^{2} d^{2}\right )} x^{9} + 2 \,{\left ({\left (b^{2} c d + a b d^{2}\right )} m^{4} + 135 \, b^{2} c d + 135 \, a b d^{2} + 18 \,{\left (b^{2} c d + a b d^{2}\right )} m^{3} + 104 \,{\left (b^{2} c d + a b d^{2}\right )} m^{2} + 222 \,{\left (b^{2} c d + a b d^{2}\right )} m\right )} x^{7} +{\left ({\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} m^{4} + 189 \, b^{2} c^{2} + 756 \, a b c d + 189 \, a^{2} d^{2} + 20 \,{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} m^{3} + 130 \,{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} m^{2} + 300 \,{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} m\right )} x^{5} + 2 \,{\left ({\left (a b c^{2} + a^{2} c d\right )} m^{4} + 315 \, a b c^{2} + 315 \, a^{2} c d + 22 \,{\left (a b c^{2} + a^{2} c d\right )} m^{3} + 164 \,{\left (a b c^{2} + a^{2} c d\right )} m^{2} + 458 \,{\left (a b c^{2} + a^{2} c d\right )} m\right )} x^{3} +{\left (a^{2} c^{2} m^{4} + 24 \, a^{2} c^{2} m^{3} + 206 \, a^{2} c^{2} m^{2} + 744 \, a^{2} c^{2} m + 945 \, a^{2} c^{2}\right )} x\right )} x^{m}}{m^{5} + 25 \, m^{4} + 230 \, m^{3} + 950 \, m^{2} + 1689 \, m + 945} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^m*(b*x^2+a)^2*(d*x^2+c)^2,x, algorithm="fricas")
[Out]
((b^2*d^2*m^4 + 16*b^2*d^2*m^3 + 86*b^2*d^2*m^2 + 176*b^2*d^2*m + 105*b^2*d^2)*x^9 + 2*((b^2*c*d + a*b*d^2)*m^
4 + 135*b^2*c*d + 135*a*b*d^2 + 18*(b^2*c*d + a*b*d^2)*m^3 + 104*(b^2*c*d + a*b*d^2)*m^2 + 222*(b^2*c*d + a*b*
d^2)*m)*x^7 + ((b^2*c^2 + 4*a*b*c*d + a^2*d^2)*m^4 + 189*b^2*c^2 + 756*a*b*c*d + 189*a^2*d^2 + 20*(b^2*c^2 + 4
*a*b*c*d + a^2*d^2)*m^3 + 130*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*m^2 + 300*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*m)*x^5
+ 2*((a*b*c^2 + a^2*c*d)*m^4 + 315*a*b*c^2 + 315*a^2*c*d + 22*(a*b*c^2 + a^2*c*d)*m^3 + 164*(a*b*c^2 + a^2*c*
d)*m^2 + 458*(a*b*c^2 + a^2*c*d)*m)*x^3 + (a^2*c^2*m^4 + 24*a^2*c^2*m^3 + 206*a^2*c^2*m^2 + 744*a^2*c^2*m + 94
5*a^2*c^2)*x)*x^m/(m^5 + 25*m^4 + 230*m^3 + 950*m^2 + 1689*m + 945)
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Sympy [A] time = 3.0362, size = 2363, normalized size = 21.68 \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)**2*(d*x**2+c)**2,x)
[Out]
Piecewise((-a**2*c**2/(8*x**8) - a**2*c*d/(3*x**6) - a**2*d**2/(4*x**4) - a*b*c**2/(3*x**6) - a*b*c*d/x**4 - a
*b*d**2/x**2 - b**2*c**2/(4*x**4) - b**2*c*d/x**2 + b**2*d**2*log(x), Eq(m, -9)), (-a**2*c**2/(6*x**6) - a**2*
c*d/(2*x**4) - a**2*d**2/(2*x**2) - a*b*c**2/(2*x**4) - 2*a*b*c*d/x**2 + 2*a*b*d**2*log(x) - b**2*c**2/(2*x**2
) + 2*b**2*c*d*log(x) + b**2*d**2*x**2/2, Eq(m, -7)), (-a**2*c**2/(4*x**4) - a**2*c*d/x**2 + a**2*d**2*log(x)
- a*b*c**2/x**2 + 4*a*b*c*d*log(x) + a*b*d**2*x**2 + b**2*c**2*log(x) + b**2*c*d*x**2 + b**2*d**2*x**4/4, Eq(m
, -5)), (-a**2*c**2/(2*x**2) + 2*a**2*c*d*log(x) + a**2*d**2*x**2/2 + 2*a*b*c**2*log(x) + 2*a*b*c*d*x**2 + a*b
*d**2*x**4/2 + b**2*c**2*x**2/2 + b**2*c*d*x**4/2 + b**2*d**2*x**6/6, Eq(m, -3)), (a**2*c**2*log(x) + a**2*c*d
*x**2 + a**2*d**2*x**4/4 + a*b*c**2*x**2 + a*b*c*d*x**4 + a*b*d**2*x**6/3 + b**2*c**2*x**4/4 + b**2*c*d*x**6/3
+ b**2*d**2*x**8/8, Eq(m, -1)), (a**2*c**2*m**4*x*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945)
+ 24*a**2*c**2*m**3*x*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 206*a**2*c**2*m**2*x*x**m/(
m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 744*a**2*c**2*m*x*x**m/(m**5 + 25*m**4 + 230*m**3 + 950
*m**2 + 1689*m + 945) + 945*a**2*c**2*x*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 2*a**2*c*
d*m**4*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 44*a**2*c*d*m**3*x**3*x**m/(m**5 + 25
*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 328*a**2*c*d*m**2*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**
2 + 1689*m + 945) + 916*a**2*c*d*m*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 630*a**2*
c*d*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + a**2*d**2*m**4*x**5*x**m/(m**5 + 25*m**4
+ 230*m**3 + 950*m**2 + 1689*m + 945) + 20*a**2*d**2*m**3*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1
689*m + 945) + 130*a**2*d**2*m**2*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 300*a**2*d
**2*m*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 189*a**2*d**2*x**5*x**m/(m**5 + 25*m**
4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 2*a*b*c**2*m**4*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 16
89*m + 945) + 44*a*b*c**2*m**3*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 328*a*b*c**2*
m**2*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 916*a*b*c**2*m*x**3*x**m/(m**5 + 25*m**
4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 630*a*b*c**2*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*
m + 945) + 4*a*b*c*d*m**4*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 80*a*b*c*d*m**3*x*
*5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 520*a*b*c*d*m**2*x**5*x**m/(m**5 + 25*m**4 + 2
30*m**3 + 950*m**2 + 1689*m + 945) + 1200*a*b*c*d*m*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m +
945) + 756*a*b*c*d*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 2*a*b*d**2*m**4*x**7*x**
m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 36*a*b*d**2*m**3*x**7*x**m/(m**5 + 25*m**4 + 230*m**
3 + 950*m**2 + 1689*m + 945) + 208*a*b*d**2*m**2*x**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 94
5) + 444*a*b*d**2*m*x**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 270*a*b*d**2*x**7*x**m/(
m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + b**2*c**2*m**4*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 9
50*m**2 + 1689*m + 945) + 20*b**2*c**2*m**3*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) +
130*b**2*c**2*m**2*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 300*b**2*c**2*m*x**5*x**m
/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 189*b**2*c**2*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 +
950*m**2 + 1689*m + 945) + 2*b**2*c*d*m**4*x**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 3
6*b**2*c*d*m**3*x**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 208*b**2*c*d*m**2*x**7*x**m/
(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 444*b**2*c*d*m*x**7*x**m/(m**5 + 25*m**4 + 230*m**3 +
950*m**2 + 1689*m + 945) + 270*b**2*c*d*x**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + b**2
*d**2*m**4*x**9*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 16*b**2*d**2*m**3*x**9*x**m/(m**5
+ 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 86*b**2*d**2*m**2*x**9*x**m/(m**5 + 25*m**4 + 230*m**3 + 95
0*m**2 + 1689*m + 945) + 176*b**2*d**2*m*x**9*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 105
*b**2*d**2*x**9*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945), True))
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Giac [B] time = 1.19144, size = 949, normalized size = 8.71 \begin{align*} \frac{b^{2} d^{2} m^{4} x^{9} x^{m} + 16 \, b^{2} d^{2} m^{3} x^{9} x^{m} + 2 \, b^{2} c d m^{4} x^{7} x^{m} + 2 \, a b d^{2} m^{4} x^{7} x^{m} + 86 \, b^{2} d^{2} m^{2} x^{9} x^{m} + 36 \, b^{2} c d m^{3} x^{7} x^{m} + 36 \, a b d^{2} m^{3} x^{7} x^{m} + 176 \, b^{2} d^{2} m x^{9} x^{m} + b^{2} c^{2} m^{4} x^{5} x^{m} + 4 \, a b c d m^{4} x^{5} x^{m} + a^{2} d^{2} m^{4} x^{5} x^{m} + 208 \, b^{2} c d m^{2} x^{7} x^{m} + 208 \, a b d^{2} m^{2} x^{7} x^{m} + 105 \, b^{2} d^{2} x^{9} x^{m} + 20 \, b^{2} c^{2} m^{3} x^{5} x^{m} + 80 \, a b c d m^{3} x^{5} x^{m} + 20 \, a^{2} d^{2} m^{3} x^{5} x^{m} + 444 \, b^{2} c d m x^{7} x^{m} + 444 \, a b d^{2} m x^{7} x^{m} + 2 \, a b c^{2} m^{4} x^{3} x^{m} + 2 \, a^{2} c d m^{4} x^{3} x^{m} + 130 \, b^{2} c^{2} m^{2} x^{5} x^{m} + 520 \, a b c d m^{2} x^{5} x^{m} + 130 \, a^{2} d^{2} m^{2} x^{5} x^{m} + 270 \, b^{2} c d x^{7} x^{m} + 270 \, a b d^{2} x^{7} x^{m} + 44 \, a b c^{2} m^{3} x^{3} x^{m} + 44 \, a^{2} c d m^{3} x^{3} x^{m} + 300 \, b^{2} c^{2} m x^{5} x^{m} + 1200 \, a b c d m x^{5} x^{m} + 300 \, a^{2} d^{2} m x^{5} x^{m} + a^{2} c^{2} m^{4} x x^{m} + 328 \, a b c^{2} m^{2} x^{3} x^{m} + 328 \, a^{2} c d m^{2} x^{3} x^{m} + 189 \, b^{2} c^{2} x^{5} x^{m} + 756 \, a b c d x^{5} x^{m} + 189 \, a^{2} d^{2} x^{5} x^{m} + 24 \, a^{2} c^{2} m^{3} x x^{m} + 916 \, a b c^{2} m x^{3} x^{m} + 916 \, a^{2} c d m x^{3} x^{m} + 206 \, a^{2} c^{2} m^{2} x x^{m} + 630 \, a b c^{2} x^{3} x^{m} + 630 \, a^{2} c d x^{3} x^{m} + 744 \, a^{2} c^{2} m x x^{m} + 945 \, a^{2} c^{2} x x^{m}}{m^{5} + 25 \, m^{4} + 230 \, m^{3} + 950 \, m^{2} + 1689 \, m + 945} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^m*(b*x^2+a)^2*(d*x^2+c)^2,x, algorithm="giac")
[Out]
(b^2*d^2*m^4*x^9*x^m + 16*b^2*d^2*m^3*x^9*x^m + 2*b^2*c*d*m^4*x^7*x^m + 2*a*b*d^2*m^4*x^7*x^m + 86*b^2*d^2*m^2
*x^9*x^m + 36*b^2*c*d*m^3*x^7*x^m + 36*a*b*d^2*m^3*x^7*x^m + 176*b^2*d^2*m*x^9*x^m + b^2*c^2*m^4*x^5*x^m + 4*a
*b*c*d*m^4*x^5*x^m + a^2*d^2*m^4*x^5*x^m + 208*b^2*c*d*m^2*x^7*x^m + 208*a*b*d^2*m^2*x^7*x^m + 105*b^2*d^2*x^9
*x^m + 20*b^2*c^2*m^3*x^5*x^m + 80*a*b*c*d*m^3*x^5*x^m + 20*a^2*d^2*m^3*x^5*x^m + 444*b^2*c*d*m*x^7*x^m + 444*
a*b*d^2*m*x^7*x^m + 2*a*b*c^2*m^4*x^3*x^m + 2*a^2*c*d*m^4*x^3*x^m + 130*b^2*c^2*m^2*x^5*x^m + 520*a*b*c*d*m^2*
x^5*x^m + 130*a^2*d^2*m^2*x^5*x^m + 270*b^2*c*d*x^7*x^m + 270*a*b*d^2*x^7*x^m + 44*a*b*c^2*m^3*x^3*x^m + 44*a^
2*c*d*m^3*x^3*x^m + 300*b^2*c^2*m*x^5*x^m + 1200*a*b*c*d*m*x^5*x^m + 300*a^2*d^2*m*x^5*x^m + a^2*c^2*m^4*x*x^m
+ 328*a*b*c^2*m^2*x^3*x^m + 328*a^2*c*d*m^2*x^3*x^m + 189*b^2*c^2*x^5*x^m + 756*a*b*c*d*x^5*x^m + 189*a^2*d^2
*x^5*x^m + 24*a^2*c^2*m^3*x*x^m + 916*a*b*c^2*m*x^3*x^m + 916*a^2*c*d*m*x^3*x^m + 206*a^2*c^2*m^2*x*x^m + 630*
a*b*c^2*x^3*x^m + 630*a^2*c*d*x^3*x^m + 744*a^2*c^2*m*x*x^m + 945*a^2*c^2*x*x^m)/(m^5 + 25*m^4 + 230*m^3 + 950
*m^2 + 1689*m + 945)