3.1106 \(\int (d x)^m \left (a+b x^2+c x^4\right )^2 \, dx\)
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 = 0.119178, 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
\[ \frac{a^2 (d x)^{m+1}}{d (m+1)}+\frac{\left (2 a c+b^2\right ) (d x)^{m+5}}{d^5 (m+5)}+\frac{2 a b (d x)^{m+3}}{d^3 (m+3)}+\frac{2 b c (d x)^{m+7}}{d^7 (m+7)}+\frac{c^2 (d x)^{m+9}}{d^9 (m+9)} \]
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))
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Rubi in Sympy [A] time = 23.2288, size = 90, normalized size = 0.89 \[ \frac{a^{2} \left (d x\right )^{m + 1}}{d \left (m + 1\right )} + \frac{2 a b \left (d x\right )^{m + 3}}{d^{3} \left (m + 3\right )} + \frac{2 b c \left (d x\right )^{m + 7}}{d^{7} \left (m + 7\right )} + \frac{c^{2} \left (d x\right )^{m + 9}}{d^{9} \left (m + 9\right )} + \frac{\left (d x\right )^{m + 5} \left (2 a c + b^{2}\right )}{d^{5} \left (m + 5\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x)**m*(c*x**4+b*x**2+a)**2,x)
[Out]
a**2*(d*x)**(m + 1)/(d*(m + 1)) + 2*a*b*(d*x)**(m + 3)/(d**3*(m + 3)) + 2*b*c*(d
*x)**(m + 7)/(d**7*(m + 7)) + c**2*(d*x)**(m + 9)/(d**9*(m + 9)) + (d*x)**(m + 5
)*(2*a*c + b**2)/(d**5*(m + 5))
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Mathematica [A] time = 0.0788425, size = 70, normalized size = 0.69 \[ (d x)^m \left (\frac{a^2 x}{m+1}+\frac{x^5 \left (2 a c+b^2\right )}{m+5}+\frac{2 a b x^3}{m+3}+\frac{2 b c x^7}{m+7}+\frac{c^2 x^9}{m+9}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(d*x)^m*(a + b*x^2 + c*x^4)^2,x]
[Out]
(d*x)^m*((a^2*x)/(1 + m) + (2*a*b*x^3)/(3 + m) + ((b^2 + 2*a*c)*x^5)/(5 + m) + (
2*b*c*x^7)/(7 + m) + (c^2*x^9)/(9 + m))
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Maple [B] time = 0.009, size = 301, normalized size = 3. \[{\frac{ \left ({c}^{2}{m}^{4}{x}^{8}+16\,{c}^{2}{m}^{3}{x}^{8}+2\,bc{m}^{4}{x}^{6}+86\,{c}^{2}{m}^{2}{x}^{8}+36\,bc{m}^{3}{x}^{6}+176\,{c}^{2}m{x}^{8}+2\,ac{m}^{4}{x}^{4}+{b}^{2}{m}^{4}{x}^{4}+208\,bc{m}^{2}{x}^{6}+105\,{c}^{2}{x}^{8}+40\,ac{m}^{3}{x}^{4}+20\,{b}^{2}{m}^{3}{x}^{4}+444\,bcm{x}^{6}+2\,ab{m}^{4}{x}^{2}+260\,ac{m}^{2}{x}^{4}+130\,{b}^{2}{m}^{2}{x}^{4}+270\,bc{x}^{6}+44\,ab{m}^{3}{x}^{2}+600\,acm{x}^{4}+300\,{b}^{2}m{x}^{4}+{a}^{2}{m}^{4}+328\,ab{m}^{2}{x}^{2}+378\,{x}^{4}ac+189\,{b}^{2}{x}^{4}+24\,{a}^{2}{m}^{3}+916\,abm{x}^{2}+206\,{a}^{2}{m}^{2}+630\,ab{x}^{2}+744\,{a}^{2}m+945\,{a}^{2} \right ) x \left ( dx \right ) ^{m}}{ \left ( 9+m \right ) \left ( 7+m \right ) \left ( 5+m \right ) \left ( 3+m \right ) \left ( 1+m \right ) }} \]
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] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^4 + b*x^2 + a)^2*(d*x)^m,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 0.298807, size = 325, normalized size = 3.22 \[ \frac{{\left ({\left (c^{2} m^{4} + 16 \, c^{2} m^{3} + 86 \, c^{2} m^{2} + 176 \, c^{2} m + 105 \, c^{2}\right )} x^{9} + 2 \,{\left (b c m^{4} + 18 \, b c m^{3} + 104 \, b c m^{2} + 222 \, b c m + 135 \, b c\right )} x^{7} +{\left ({\left (b^{2} + 2 \, a c\right )} m^{4} + 20 \,{\left (b^{2} + 2 \, a c\right )} m^{3} + 130 \,{\left (b^{2} + 2 \, a c\right )} m^{2} + 189 \, b^{2} + 378 \, a c + 300 \,{\left (b^{2} + 2 \, a c\right )} m\right )} x^{5} + 2 \,{\left (a b m^{4} + 22 \, a b m^{3} + 164 \, a b m^{2} + 458 \, a b m + 315 \, a b\right )} x^{3} +{\left (a^{2} m^{4} + 24 \, a^{2} m^{3} + 206 \, a^{2} m^{2} + 744 \, a^{2} m + 945 \, a^{2}\right )} x\right )} \left (d x\right )^{m}}{m^{5} + 25 \, m^{4} + 230 \, m^{3} + 950 \, m^{2} + 1689 \, m + 945} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^4 + b*x^2 + a)^2*(d*x)^m,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 = 8.39635, size = 1486, normalized size = 14.71 \[ \text{result too large to display} \]
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 + 1689*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 + 945) + 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 + 94
5) + 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/XCAS [A] time = 0.273654, size = 687, normalized size = 6.8 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^4 + b*x^2 + a)^2*(d*x)^m,x, algorithm="giac")
[Out]
(c^2*m^4*x^9*e^(m*ln(d*x)) + 16*c^2*m^3*x^9*e^(m*ln(d*x)) + 2*b*c*m^4*x^7*e^(m*l
n(d*x)) + 86*c^2*m^2*x^9*e^(m*ln(d*x)) + 36*b*c*m^3*x^7*e^(m*ln(d*x)) + 176*c^2*
m*x^9*e^(m*ln(d*x)) + b^2*m^4*x^5*e^(m*ln(d*x)) + 2*a*c*m^4*x^5*e^(m*ln(d*x)) +
208*b*c*m^2*x^7*e^(m*ln(d*x)) + 105*c^2*x^9*e^(m*ln(d*x)) + 20*b^2*m^3*x^5*e^(m*
ln(d*x)) + 40*a*c*m^3*x^5*e^(m*ln(d*x)) + 444*b*c*m*x^7*e^(m*ln(d*x)) + 2*a*b*m^
4*x^3*e^(m*ln(d*x)) + 130*b^2*m^2*x^5*e^(m*ln(d*x)) + 260*a*c*m^2*x^5*e^(m*ln(d*
x)) + 270*b*c*x^7*e^(m*ln(d*x)) + 44*a*b*m^3*x^3*e^(m*ln(d*x)) + 300*b^2*m*x^5*e
^(m*ln(d*x)) + 600*a*c*m*x^5*e^(m*ln(d*x)) + a^2*m^4*x*e^(m*ln(d*x)) + 328*a*b*m
^2*x^3*e^(m*ln(d*x)) + 189*b^2*x^5*e^(m*ln(d*x)) + 378*a*c*x^5*e^(m*ln(d*x)) + 2
4*a^2*m^3*x*e^(m*ln(d*x)) + 916*a*b*m*x^3*e^(m*ln(d*x)) + 206*a^2*m^2*x*e^(m*ln(
d*x)) + 630*a*b*x^3*e^(m*ln(d*x)) + 744*a^2*m*x*e^(m*ln(d*x)) + 945*a^2*x*e^(m*l
n(d*x)))/(m^5 + 25*m^4 + 230*m^3 + 950*m^2 + 1689*m + 945)