3.1106 \(\int (d x)^m (a+b x^2+c x^4)^2 \, dx\)

Optimal. Leaf size=101 \[ \frac{a^2 (d x)^{m+1}}{d (m+1)}+\frac{\left (2 a c+b^2\right ) (d x)^{m+5}}{d^5 (m+5)}+\frac{2 a b (d x)^{m+3}}{d^3 (m+3)}+\frac{2 b c (d x)^{m+7}}{d^7 (m+7)}+\frac{c^2 (d x)^{m+9}}{d^9 (m+9)} \]

[Out]

(a^2*(d*x)^(1 + m))/(d*(1 + m)) + (2*a*b*(d*x)^(3 + m))/(d^3*(3 + m)) + ((b^2 + 2*a*c)*(d*x)^(5 + m))/(d^5*(5
+ m)) + (2*b*c*(d*x)^(7 + m))/(d^7*(7 + m)) + (c^2*(d*x)^(9 + m))/(d^9*(9 + m))

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Rubi [A]  time = 0.0528217, antiderivative size = 101, 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, Rules used = {1108} \[ \frac{a^2 (d x)^{m+1}}{d (m+1)}+\frac{\left (2 a c+b^2\right ) (d x)^{m+5}}{d^5 (m+5)}+\frac{2 a b (d x)^{m+3}}{d^3 (m+3)}+\frac{2 b c (d x)^{m+7}}{d^7 (m+7)}+\frac{c^2 (d x)^{m+9}}{d^9 (m+9)} \]

Antiderivative was successfully verified.

[In]

Int[(d*x)^m*(a + b*x^2 + c*x^4)^2,x]

[Out]

(a^2*(d*x)^(1 + m))/(d*(1 + m)) + (2*a*b*(d*x)^(3 + m))/(d^3*(3 + m)) + ((b^2 + 2*a*c)*(d*x)^(5 + m))/(d^5*(5
+ m)) + (2*b*c*(d*x)^(7 + m))/(d^7*(7 + m)) + (c^2*(d*x)^(9 + m))/(d^9*(9 + m))

Rule 1108

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d*x)^m*(a
+ b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[p, 0] &&  !IntegerQ[(m + 1)/2]

Rubi steps

\begin{align*} \int (d x)^m \left (a+b x^2+c x^4\right )^2 \, dx &=\int \left (a^2 (d x)^m+\frac{2 a b (d x)^{2+m}}{d^2}+\frac{\left (b^2+2 a c\right ) (d x)^{4+m}}{d^4}+\frac{2 b c (d x)^{6+m}}{d^6}+\frac{c^2 (d x)^{8+m}}{d^8}\right ) \, dx\\ &=\frac{a^2 (d x)^{1+m}}{d (1+m)}+\frac{2 a b (d x)^{3+m}}{d^3 (3+m)}+\frac{\left (b^2+2 a c\right ) (d x)^{5+m}}{d^5 (5+m)}+\frac{2 b c (d x)^{7+m}}{d^7 (7+m)}+\frac{c^2 (d x)^{9+m}}{d^9 (9+m)}\\ \end{align*}

Mathematica [A]  time = 0.0526927, size = 70, normalized size = 0.69 \[ x (d x)^m \left (\frac{a^2}{m+1}+\frac{x^4 \left (2 a c+b^2\right )}{m+5}+\frac{2 a b x^2}{m+3}+\frac{2 b c x^6}{m+7}+\frac{c^2 x^8}{m+9}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(d*x)^m*(a + b*x^2 + c*x^4)^2,x]

[Out]

x*(d*x)^m*(a^2/(1 + m) + (2*a*b*x^2)/(3 + m) + ((b^2 + 2*a*c)*x^4)/(5 + m) + (2*b*c*x^6)/(7 + m) + (c^2*x^8)/(
9 + m))

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Maple [B]  time = 0.048, size = 301, normalized size = 3. \begin{align*}{\frac{ \left ({c}^{2}{m}^{4}{x}^{8}+16\,{c}^{2}{m}^{3}{x}^{8}+2\,bc{m}^{4}{x}^{6}+86\,{c}^{2}{m}^{2}{x}^{8}+36\,bc{m}^{3}{x}^{6}+176\,{c}^{2}m{x}^{8}+2\,ac{m}^{4}{x}^{4}+{b}^{2}{m}^{4}{x}^{4}+208\,bc{m}^{2}{x}^{6}+105\,{c}^{2}{x}^{8}+40\,ac{m}^{3}{x}^{4}+20\,{b}^{2}{m}^{3}{x}^{4}+444\,bcm{x}^{6}+2\,ab{m}^{4}{x}^{2}+260\,ac{m}^{2}{x}^{4}+130\,{b}^{2}{m}^{2}{x}^{4}+270\,bc{x}^{6}+44\,ab{m}^{3}{x}^{2}+600\,acm{x}^{4}+300\,{b}^{2}m{x}^{4}+{a}^{2}{m}^{4}+328\,ab{m}^{2}{x}^{2}+378\,ac{x}^{4}+189\,{b}^{2}{x}^{4}+24\,{a}^{2}{m}^{3}+916\,abm{x}^{2}+206\,{a}^{2}{m}^{2}+630\,ab{x}^{2}+744\,m{a}^{2}+945\,{a}^{2} \right ) x \left ( dx \right ) ^{m}}{ \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((d*x)^m*(c*x^4+b*x^2+a)^2,x)

[Out]

x*(c^2*m^4*x^8+16*c^2*m^3*x^8+2*b*c*m^4*x^6+86*c^2*m^2*x^8+36*b*c*m^3*x^6+176*c^2*m*x^8+2*a*c*m^4*x^4+b^2*m^4*
x^4+208*b*c*m^2*x^6+105*c^2*x^8+40*a*c*m^3*x^4+20*b^2*m^3*x^4+444*b*c*m*x^6+2*a*b*m^4*x^2+260*a*c*m^2*x^4+130*
b^2*m^2*x^4+270*b*c*x^6+44*a*b*m^3*x^2+600*a*c*m*x^4+300*b^2*m*x^4+a^2*m^4+328*a*b*m^2*x^2+378*a*c*x^4+189*b^2
*x^4+24*a^2*m^3+916*a*b*m*x^2+206*a^2*m^2+630*a*b*x^2+744*a^2*m+945*a^2)*(d*x)^m/(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((d*x)^m*(c*x^4+b*x^2+a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.47672, size = 587, normalized size = 5.81 \begin{align*} \frac{{\left ({\left (c^{2} m^{4} + 16 \, c^{2} m^{3} + 86 \, c^{2} m^{2} + 176 \, c^{2} m + 105 \, c^{2}\right )} x^{9} + 2 \,{\left (b c m^{4} + 18 \, b c m^{3} + 104 \, b c m^{2} + 222 \, b c m + 135 \, b c\right )} x^{7} +{\left ({\left (b^{2} + 2 \, a c\right )} m^{4} + 20 \,{\left (b^{2} + 2 \, a c\right )} m^{3} + 130 \,{\left (b^{2} + 2 \, a c\right )} m^{2} + 189 \, b^{2} + 378 \, a c + 300 \,{\left (b^{2} + 2 \, a c\right )} m\right )} x^{5} + 2 \,{\left (a b m^{4} + 22 \, a b m^{3} + 164 \, a b m^{2} + 458 \, a b m + 315 \, a b\right )} x^{3} +{\left (a^{2} m^{4} + 24 \, a^{2} m^{3} + 206 \, a^{2} m^{2} + 744 \, a^{2} m + 945 \, a^{2}\right )} x\right )} \left (d x\right )^{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((d*x)^m*(c*x^4+b*x^2+a)^2,x, algorithm="fricas")

[Out]

((c^2*m^4 + 16*c^2*m^3 + 86*c^2*m^2 + 176*c^2*m + 105*c^2)*x^9 + 2*(b*c*m^4 + 18*b*c*m^3 + 104*b*c*m^2 + 222*b
*c*m + 135*b*c)*x^7 + ((b^2 + 2*a*c)*m^4 + 20*(b^2 + 2*a*c)*m^3 + 130*(b^2 + 2*a*c)*m^2 + 189*b^2 + 378*a*c +
300*(b^2 + 2*a*c)*m)*x^5 + 2*(a*b*m^4 + 22*a*b*m^3 + 164*a*b*m^2 + 458*a*b*m + 315*a*b)*x^3 + (a^2*m^4 + 24*a^
2*m^3 + 206*a^2*m^2 + 744*a^2*m + 945*a^2)*x)*(d*x)^m/(m^5 + 25*m^4 + 230*m^3 + 950*m^2 + 1689*m + 945)

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Sympy [A]  time = 2.76449, size = 1486, normalized size = 14.71 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)**m*(c*x**4+b*x**2+a)**2,x)

[Out]

Piecewise(((-a**2/(8*x**8) - a*b/(3*x**6) - a*c/(2*x**4) - b**2/(4*x**4) - b*c/x**2 + c**2*log(x))/d**9, Eq(m,
-9)), ((-a**2/(6*x**6) - a*b/(2*x**4) - a*c/x**2 - b**2/(2*x**2) + 2*b*c*log(x) + c**2*x**2/2)/d**7, Eq(m, -7
)), ((-a**2/(4*x**4) - a*b/x**2 + 2*a*c*log(x) + b**2*log(x) + b*c*x**2 + c**2*x**4/4)/d**5, Eq(m, -5)), ((-a*
*2/(2*x**2) + 2*a*b*log(x) + a*c*x**2 + b**2*x**2/2 + b*c*x**4/2 + c**2*x**6/6)/d**3, Eq(m, -3)), ((a**2*log(x
) + a*b*x**2 + a*c*x**4/2 + b**2*x**4/4 + b*c*x**6/3 + c**2*x**8/8)/d, Eq(m, -1)), (a**2*d**m*m**4*x*x**m/(m**
5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 24*a**2*d**m*m**3*x*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*
m**2 + 1689*m + 945) + 206*a**2*d**m*m**2*x*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 744*a
**2*d**m*m*x*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 945*a**2*d**m*x*x**m/(m**5 + 25*m**4
+ 230*m**3 + 950*m**2 + 1689*m + 945) + 2*a*b*d**m*m**4*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 168
9*m + 945) + 44*a*b*d**m*m**3*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 328*a*b*d**m*m
**2*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 916*a*b*d**m*m*x**3*x**m/(m**5 + 25*m**4
+ 230*m**3 + 950*m**2 + 1689*m + 945) + 630*a*b*d**m*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m
+ 945) + 2*a*c*d**m*m**4*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 40*a*c*d**m*m**3*x
**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 260*a*c*d**m*m**2*x**5*x**m/(m**5 + 25*m**4 +
230*m**3 + 950*m**2 + 1689*m + 945) + 600*a*c*d**m*m*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m
+ 945) + 378*a*c*d**m*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + b**2*d**m*m**4*x**5*x
**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 20*b**2*d**m*m**3*x**5*x**m/(m**5 + 25*m**4 + 230*
m**3 + 950*m**2 + 1689*m + 945) + 130*b**2*d**m*m**2*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m
+ 945) + 300*b**2*d**m*m*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 189*b**2*d**m*x**5*
x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 2*b*c*d**m*m**4*x**7*x**m/(m**5 + 25*m**4 + 230*m
**3 + 950*m**2 + 1689*m + 945) + 36*b*c*d**m*m**3*x**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 9
45) + 208*b*c*d**m*m**2*x**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 444*b*c*d**m*m*x**7*
x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 270*b*c*d**m*x**7*x**m/(m**5 + 25*m**4 + 230*m**3
+ 950*m**2 + 1689*m + 945) + c**2*d**m*m**4*x**9*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) +
16*c**2*d**m*m**3*x**9*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 86*c**2*d**m*m**2*x**9*x*
*m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 176*c**2*d**m*m*x**9*x**m/(m**5 + 25*m**4 + 230*m**
3 + 950*m**2 + 1689*m + 945) + 105*c**2*d**m*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 = 1.1323, size = 606, normalized size = 6. \begin{align*} \frac{\left (d x\right )^{m} c^{2} m^{4} x^{9} + 16 \, \left (d x\right )^{m} c^{2} m^{3} x^{9} + 2 \, \left (d x\right )^{m} b c m^{4} x^{7} + 86 \, \left (d x\right )^{m} c^{2} m^{2} x^{9} + 36 \, \left (d x\right )^{m} b c m^{3} x^{7} + 176 \, \left (d x\right )^{m} c^{2} m x^{9} + \left (d x\right )^{m} b^{2} m^{4} x^{5} + 2 \, \left (d x\right )^{m} a c m^{4} x^{5} + 208 \, \left (d x\right )^{m} b c m^{2} x^{7} + 105 \, \left (d x\right )^{m} c^{2} x^{9} + 20 \, \left (d x\right )^{m} b^{2} m^{3} x^{5} + 40 \, \left (d x\right )^{m} a c m^{3} x^{5} + 444 \, \left (d x\right )^{m} b c m x^{7} + 2 \, \left (d x\right )^{m} a b m^{4} x^{3} + 130 \, \left (d x\right )^{m} b^{2} m^{2} x^{5} + 260 \, \left (d x\right )^{m} a c m^{2} x^{5} + 270 \, \left (d x\right )^{m} b c x^{7} + 44 \, \left (d x\right )^{m} a b m^{3} x^{3} + 300 \, \left (d x\right )^{m} b^{2} m x^{5} + 600 \, \left (d x\right )^{m} a c m x^{5} + \left (d x\right )^{m} a^{2} m^{4} x + 328 \, \left (d x\right )^{m} a b m^{2} x^{3} + 189 \, \left (d x\right )^{m} b^{2} x^{5} + 378 \, \left (d x\right )^{m} a c x^{5} + 24 \, \left (d x\right )^{m} a^{2} m^{3} x + 916 \, \left (d x\right )^{m} a b m x^{3} + 206 \, \left (d x\right )^{m} a^{2} m^{2} x + 630 \, \left (d x\right )^{m} a b x^{3} + 744 \, \left (d x\right )^{m} a^{2} m x + 945 \, \left (d x\right )^{m} a^{2} x}{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((d*x)^m*(c*x^4+b*x^2+a)^2,x, algorithm="giac")

[Out]

((d*x)^m*c^2*m^4*x^9 + 16*(d*x)^m*c^2*m^3*x^9 + 2*(d*x)^m*b*c*m^4*x^7 + 86*(d*x)^m*c^2*m^2*x^9 + 36*(d*x)^m*b*
c*m^3*x^7 + 176*(d*x)^m*c^2*m*x^9 + (d*x)^m*b^2*m^4*x^5 + 2*(d*x)^m*a*c*m^4*x^5 + 208*(d*x)^m*b*c*m^2*x^7 + 10
5*(d*x)^m*c^2*x^9 + 20*(d*x)^m*b^2*m^3*x^5 + 40*(d*x)^m*a*c*m^3*x^5 + 444*(d*x)^m*b*c*m*x^7 + 2*(d*x)^m*a*b*m^
4*x^3 + 130*(d*x)^m*b^2*m^2*x^5 + 260*(d*x)^m*a*c*m^2*x^5 + 270*(d*x)^m*b*c*x^7 + 44*(d*x)^m*a*b*m^3*x^3 + 300
*(d*x)^m*b^2*m*x^5 + 600*(d*x)^m*a*c*m*x^5 + (d*x)^m*a^2*m^4*x + 328*(d*x)^m*a*b*m^2*x^3 + 189*(d*x)^m*b^2*x^5
+ 378*(d*x)^m*a*c*x^5 + 24*(d*x)^m*a^2*m^3*x + 916*(d*x)^m*a*b*m*x^3 + 206*(d*x)^m*a^2*m^2*x + 630*(d*x)^m*a*
b*x^3 + 744*(d*x)^m*a^2*m*x + 945*(d*x)^m*a^2*x)/(m^5 + 25*m^4 + 230*m^3 + 950*m^2 + 1689*m + 945)