3.600 \(\int \frac{(d x)^m}{\left (a+b x^n+c x^{2 n}\right )^2} \, dx\)
Optimal. Leaf size=328 \[ \frac{c (d x)^{m+1} \left (\frac{4 a c (m-2 n+1)-b^2 (m-n+1)}{\sqrt{b^2-4 a c}}-b (m-n+1)\right ) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right )}{a d (m+1) n \left (b^2-4 a c\right ) \left (b-\sqrt{b^2-4 a c}\right )}-\frac{c (d x)^{m+1} \left (b (m-n+1) \sqrt{b^2-4 a c}+4 a c (m-2 n+1)+b^2 (-(m-n+1))\right ) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{a d (m+1) n \left (b^2-4 a c\right )^{3/2} \left (\sqrt{b^2-4 a c}+b\right )}+\frac{(d x)^{m+1} \left (-2 a c+b^2+b c x^n\right )}{a d n \left (b^2-4 a c\right ) \left (a+b x^n+c x^{2 n}\right )} \]
[Out]
((d*x)^(1 + m)*(b^2 - 2*a*c + b*c*x^n))/(a*(b^2 - 4*a*c)*d*n*(a + b*x^n + c*x^(2
*n))) + (c*((4*a*c*(1 + m - 2*n) - b^2*(1 + m - n))/Sqrt[b^2 - 4*a*c] - b*(1 + m
- n))*(d*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, (-2*c*x^n)/(
b - Sqrt[b^2 - 4*a*c])])/(a*(b^2 - 4*a*c)*(b - Sqrt[b^2 - 4*a*c])*d*(1 + m)*n) -
(c*(4*a*c*(1 + m - 2*n) - b^2*(1 + m - n) + b*Sqrt[b^2 - 4*a*c]*(1 + m - n))*(d
*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, (-2*c*x^n)/(b + Sqrt[
b^2 - 4*a*c])])/(a*(b^2 - 4*a*c)^(3/2)*(b + Sqrt[b^2 - 4*a*c])*d*(1 + m)*n)
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Rubi [A] time = 1.88159, antiderivative size = 328, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136
\[ \frac{c (d x)^{m+1} \left (\frac{4 a c (m-2 n+1)-b^2 (m-n+1)}{\sqrt{b^2-4 a c}}-b (m-n+1)\right ) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right )}{a d (m+1) n \left (b^2-4 a c\right ) \left (b-\sqrt{b^2-4 a c}\right )}-\frac{c (d x)^{m+1} \left (b (m-n+1) \sqrt{b^2-4 a c}+4 a c (m-2 n+1)+b^2 (-(m-n+1))\right ) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{a d (m+1) n \left (b^2-4 a c\right )^{3/2} \left (\sqrt{b^2-4 a c}+b\right )}+\frac{(d x)^{m+1} \left (-2 a c+b^2+b c x^n\right )}{a d n \left (b^2-4 a c\right ) \left (a+b x^n+c x^{2 n}\right )} \]
Antiderivative was successfully verified.
[In] Int[(d*x)^m/(a + b*x^n + c*x^(2*n))^2,x]
[Out]
((d*x)^(1 + m)*(b^2 - 2*a*c + b*c*x^n))/(a*(b^2 - 4*a*c)*d*n*(a + b*x^n + c*x^(2
*n))) + (c*((4*a*c*(1 + m - 2*n) - b^2*(1 + m - n))/Sqrt[b^2 - 4*a*c] - b*(1 + m
- n))*(d*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, (-2*c*x^n)/(
b - Sqrt[b^2 - 4*a*c])])/(a*(b^2 - 4*a*c)*(b - Sqrt[b^2 - 4*a*c])*d*(1 + m)*n) -
(c*(4*a*c*(1 + m - 2*n) - b^2*(1 + m - n) + b*Sqrt[b^2 - 4*a*c]*(1 + m - n))*(d
*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, (-2*c*x^n)/(b + Sqrt[
b^2 - 4*a*c])])/(a*(b^2 - 4*a*c)^(3/2)*(b + Sqrt[b^2 - 4*a*c])*d*(1 + m)*n)
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Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x)**m/(a+b*x**n+c*x**(2*n))**2,x)
[Out]
Timed out
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Mathematica [B] time = 6.45158, size = 3515, normalized size = 10.72 \[ \text{Result too large to show} \]
Antiderivative was successfully verified.
[In] Integrate[(d*x)^m/(a + b*x^n + c*x^(2*n))^2,x]
[Out]
(x*(d*x)^m*(-b^2 + 2*a*c - b*c*x^n))/(a*(-b^2 + 4*a*c)*n*(a + b*x^n + c*x^(2*n))
) - (b*c*x^(1 + n)*(d*x)^m*(x^n)^((1 + m)/n - (1 + m + n)/n)*(-(((x^n/(-(-b - Sq
rt[b^2 - 4*a*c])/(2*c) + x^n))^(-n^(-1) - m/n)*Hypergeometric2F1[-((1 + m)/n), -
((1 + m)/n), 1 - (1 + m)/n, -(-b - Sqrt[b^2 - 4*a*c])/(2*c*(-(-b - Sqrt[b^2 - 4*
a*c])/(2*c) + x^n))])/Sqrt[b^2 - 4*a*c]) + ((x^n/(-(-b + Sqrt[b^2 - 4*a*c])/(2*c
) + x^n))^(-n^(-1) - m/n)*Hypergeometric2F1[-((1 + m)/n), -((1 + m)/n), 1 - (1 +
m)/n, -(-b + Sqrt[b^2 - 4*a*c])/(2*c*(-(-b + Sqrt[b^2 - 4*a*c])/(2*c) + x^n))])
/Sqrt[b^2 - 4*a*c]))/(a*(-b^2 + 4*a*c)*(1 + m)) + (b*c*x^(1 + n)*(d*x)^m*(x^n)^(
(1 + m)/n - (1 + m + n)/n)*(-(((x^n/(-(-b - Sqrt[b^2 - 4*a*c])/(2*c) + x^n))^(-n
^(-1) - m/n)*Hypergeometric2F1[-((1 + m)/n), -((1 + m)/n), 1 - (1 + m)/n, -(-b -
Sqrt[b^2 - 4*a*c])/(2*c*(-(-b - Sqrt[b^2 - 4*a*c])/(2*c) + x^n))])/Sqrt[b^2 - 4
*a*c]) + ((x^n/(-(-b + Sqrt[b^2 - 4*a*c])/(2*c) + x^n))^(-n^(-1) - m/n)*Hypergeo
metric2F1[-((1 + m)/n), -((1 + m)/n), 1 - (1 + m)/n, -(-b + Sqrt[b^2 - 4*a*c])/(
2*c*(-(-b + Sqrt[b^2 - 4*a*c])/(2*c) + x^n))])/Sqrt[b^2 - 4*a*c]))/(a*(-b^2 + 4*
a*c)*(1 + m)*n) + (b*c*m*x^(1 + n)*(d*x)^m*(x^n)^((1 + m)/n - (1 + m + n)/n)*(-(
((x^n/(-(-b - Sqrt[b^2 - 4*a*c])/(2*c) + x^n))^(-n^(-1) - m/n)*Hypergeometric2F1
[-((1 + m)/n), -((1 + m)/n), 1 - (1 + m)/n, -(-b - Sqrt[b^2 - 4*a*c])/(2*c*(-(-b
- Sqrt[b^2 - 4*a*c])/(2*c) + x^n))])/Sqrt[b^2 - 4*a*c]) + ((x^n/(-(-b + Sqrt[b^
2 - 4*a*c])/(2*c) + x^n))^(-n^(-1) - m/n)*Hypergeometric2F1[-((1 + m)/n), -((1 +
m)/n), 1 - (1 + m)/n, -(-b + Sqrt[b^2 - 4*a*c])/(2*c*(-(-b + Sqrt[b^2 - 4*a*c])
/(2*c) + x^n))])/Sqrt[b^2 - 4*a*c]))/(a*(-b^2 + 4*a*c)*(1 + m)*n) + (b^2*x*(d*x)
^m*((1 - (x^n/(-(-b - Sqrt[b^2 - 4*a*c])/(2*c) + x^n))^(-n^(-1) - m/n)*Hypergeom
etric2F1[-((1 + m)/n), -((1 + m)/n), 1 - (1 + m)/n, -(-b - Sqrt[b^2 - 4*a*c])/(2
*c*(-(-b - Sqrt[b^2 - 4*a*c])/(2*c) + x^n))])/((b*(-b - Sqrt[b^2 - 4*a*c]))/(2*c
) + (-b - Sqrt[b^2 - 4*a*c])^2/(2*c)) + (1 - (x^n/(-(-b + Sqrt[b^2 - 4*a*c])/(2*
c) + x^n))^(-n^(-1) - m/n)*Hypergeometric2F1[-((1 + m)/n), -((1 + m)/n), 1 - (1
+ m)/n, -(-b + Sqrt[b^2 - 4*a*c])/(2*c*(-(-b + Sqrt[b^2 - 4*a*c])/(2*c) + x^n))]
)/((b*(-b + Sqrt[b^2 - 4*a*c]))/(2*c) + (-b + Sqrt[b^2 - 4*a*c])^2/(2*c))))/(a*(
-b^2 + 4*a*c)*(1 + m)) - (4*c*x*(d*x)^m*((1 - (x^n/(-(-b - Sqrt[b^2 - 4*a*c])/(2
*c) + x^n))^(-n^(-1) - m/n)*Hypergeometric2F1[-((1 + m)/n), -((1 + m)/n), 1 - (1
+ m)/n, -(-b - Sqrt[b^2 - 4*a*c])/(2*c*(-(-b - Sqrt[b^2 - 4*a*c])/(2*c) + x^n))
])/((b*(-b - Sqrt[b^2 - 4*a*c]))/(2*c) + (-b - Sqrt[b^2 - 4*a*c])^2/(2*c)) + (1
- (x^n/(-(-b + Sqrt[b^2 - 4*a*c])/(2*c) + x^n))^(-n^(-1) - m/n)*Hypergeometric2F
1[-((1 + m)/n), -((1 + m)/n), 1 - (1 + m)/n, -(-b + Sqrt[b^2 - 4*a*c])/(2*c*(-(-
b + Sqrt[b^2 - 4*a*c])/(2*c) + x^n))])/((b*(-b + Sqrt[b^2 - 4*a*c]))/(2*c) + (-b
+ Sqrt[b^2 - 4*a*c])^2/(2*c))))/((-b^2 + 4*a*c)*(1 + m)) - (b^2*x*(d*x)^m*((1 -
(x^n/(-(-b - Sqrt[b^2 - 4*a*c])/(2*c) + x^n))^(-n^(-1) - m/n)*Hypergeometric2F1
[-((1 + m)/n), -((1 + m)/n), 1 - (1 + m)/n, -(-b - Sqrt[b^2 - 4*a*c])/(2*c*(-(-b
- Sqrt[b^2 - 4*a*c])/(2*c) + x^n))])/((b*(-b - Sqrt[b^2 - 4*a*c]))/(2*c) + (-b
- Sqrt[b^2 - 4*a*c])^2/(2*c)) + (1 - (x^n/(-(-b + Sqrt[b^2 - 4*a*c])/(2*c) + x^n
))^(-n^(-1) - m/n)*Hypergeometric2F1[-((1 + m)/n), -((1 + m)/n), 1 - (1 + m)/n,
-(-b + Sqrt[b^2 - 4*a*c])/(2*c*(-(-b + Sqrt[b^2 - 4*a*c])/(2*c) + x^n))])/((b*(-
b + Sqrt[b^2 - 4*a*c]))/(2*c) + (-b + Sqrt[b^2 - 4*a*c])^2/(2*c))))/(a*(-b^2 + 4
*a*c)*(1 + m)*n) + (2*c*x*(d*x)^m*((1 - (x^n/(-(-b - Sqrt[b^2 - 4*a*c])/(2*c) +
x^n))^(-n^(-1) - m/n)*Hypergeometric2F1[-((1 + m)/n), -((1 + m)/n), 1 - (1 + m)/
n, -(-b - Sqrt[b^2 - 4*a*c])/(2*c*(-(-b - Sqrt[b^2 - 4*a*c])/(2*c) + x^n))])/((b
*(-b - Sqrt[b^2 - 4*a*c]))/(2*c) + (-b - Sqrt[b^2 - 4*a*c])^2/(2*c)) + (1 - (x^n
/(-(-b + Sqrt[b^2 - 4*a*c])/(2*c) + x^n))^(-n^(-1) - m/n)*Hypergeometric2F1[-((1
+ m)/n), -((1 + m)/n), 1 - (1 + m)/n, -(-b + Sqrt[b^2 - 4*a*c])/(2*c*(-(-b + Sq
rt[b^2 - 4*a*c])/(2*c) + x^n))])/((b*(-b + Sqrt[b^2 - 4*a*c]))/(2*c) + (-b + Sqr
t[b^2 - 4*a*c])^2/(2*c))))/((-b^2 + 4*a*c)*(1 + m)*n) - (b^2*m*x*(d*x)^m*((1 - (
x^n/(-(-b - Sqrt[b^2 - 4*a*c])/(2*c) + x^n))^(-n^(-1) - m/n)*Hypergeometric2F1[-
((1 + m)/n), -((1 + m)/n), 1 - (1 + m)/n, -(-b - Sqrt[b^2 - 4*a*c])/(2*c*(-(-b -
Sqrt[b^2 - 4*a*c])/(2*c) + x^n))])/((b*(-b - Sqrt[b^2 - 4*a*c]))/(2*c) + (-b -
Sqrt[b^2 - 4*a*c])^2/(2*c)) + (1 - (x^n/(-(-b + Sqrt[b^2 - 4*a*c])/(2*c) + x^n))
^(-n^(-1) - m/n)*Hypergeometric2F1[-((1 + m)/n), -((1 + m)/n), 1 - (1 + m)/n, -(
-b + Sqrt[b^2 - 4*a*c])/(2*c*(-(-b + Sqrt[b^2 - 4*a*c])/(2*c) + x^n))])/((b*(-b
+ Sqrt[b^2 - 4*a*c]))/(2*c) + (-b + Sqrt[b^2 - 4*a*c])^2/(2*c))))/(a*(-b^2 + 4*a
*c)*(1 + m)*n) + (2*c*m*x*(d*x)^m*((1 - (x^n/(-(-b - Sqrt[b^2 - 4*a*c])/(2*c) +
x^n))^(-n^(-1) - m/n)*Hypergeometric2F1[-((1 + m)/n), -((1 + m)/n), 1 - (1 + m)/
n, -(-b - Sqrt[b^2 - 4*a*c])/(2*c*(-(-b - Sqrt[b^2 - 4*a*c])/(2*c) + x^n))])/((b
*(-b - Sqrt[b^2 - 4*a*c]))/(2*c) + (-b - Sqrt[b^2 - 4*a*c])^2/(2*c)) + (1 - (x^n
/(-(-b + Sqrt[b^2 - 4*a*c])/(2*c) + x^n))^(-n^(-1) - m/n)*Hypergeometric2F1[-((1
+ m)/n), -((1 + m)/n), 1 - (1 + m)/n, -(-b + Sqrt[b^2 - 4*a*c])/(2*c*(-(-b + Sq
rt[b^2 - 4*a*c])/(2*c) + x^n))])/((b*(-b + Sqrt[b^2 - 4*a*c]))/(2*c) + (-b + Sqr
t[b^2 - 4*a*c])^2/(2*c))))/((-b^2 + 4*a*c)*(1 + m)*n)
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Maple [F] time = 0.058, size = 0, normalized size = 0. \[ \int{\frac{ \left ( dx \right ) ^{m}}{ \left ( a+b{x}^{n}+c{x}^{2\,n} \right ) ^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x)^m/(a+b*x^n+c*x^(2*n))^2,x)
[Out]
int((d*x)^m/(a+b*x^n+c*x^(2*n))^2,x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \frac{b c d^{m} x e^{\left (m \log \left (x\right ) + n \log \left (x\right )\right )} +{\left (b^{2} d^{m} - 2 \, a c d^{m}\right )} x x^{m}}{a^{2} b^{2} n - 4 \, a^{3} c n +{\left (a b^{2} c n - 4 \, a^{2} c^{2} n\right )} x^{2 \, n} +{\left (a b^{3} n - 4 \, a^{2} b c n\right )} x^{n}} + \int -\frac{b c d^{m}{\left (m - n + 1\right )} e^{\left (m \log \left (x\right ) + n \log \left (x\right )\right )} +{\left (b^{2} d^{m}{\left (m - n + 1\right )} - 2 \, a c d^{m}{\left (m - 2 \, n + 1\right )}\right )} x^{m}}{a^{2} b^{2} n - 4 \, a^{3} c n +{\left (a b^{2} c n - 4 \, a^{2} c^{2} n\right )} x^{2 \, n} +{\left (a b^{3} n - 4 \, a^{2} b c n\right )} x^{n}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x)^m/(c*x^(2*n) + b*x^n + a)^2,x, algorithm="maxima")
[Out]
(b*c*d^m*x*e^(m*log(x) + n*log(x)) + (b^2*d^m - 2*a*c*d^m)*x*x^m)/(a^2*b^2*n - 4
*a^3*c*n + (a*b^2*c*n - 4*a^2*c^2*n)*x^(2*n) + (a*b^3*n - 4*a^2*b*c*n)*x^n) + in
tegrate(-(b*c*d^m*(m - n + 1)*e^(m*log(x) + n*log(x)) + (b^2*d^m*(m - n + 1) - 2
*a*c*d^m*(m - 2*n + 1))*x^m)/(a^2*b^2*n - 4*a^3*c*n + (a*b^2*c*n - 4*a^2*c^2*n)*
x^(2*n) + (a*b^3*n - 4*a^2*b*c*n)*x^n), x)
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\left (d x\right )^{m}}{c^{2} x^{4 \, n} + 2 \, a b x^{n} + a^{2} +{\left (2 \, b c x^{n} + b^{2} + 2 \, a c\right )} x^{2 \, n}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x)^m/(c*x^(2*n) + b*x^n + a)^2,x, algorithm="fricas")
[Out]
integral((d*x)^m/(c^2*x^(4*n) + 2*a*b*x^n + a^2 + (2*b*c*x^n + b^2 + 2*a*c)*x^(2
*n)), x)
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x)**m/(a+b*x**n+c*x**(2*n))**2,x)
[Out]
Timed out
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d x\right )^{m}}{{\left (c x^{2 \, n} + b x^{n} + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x)^m/(c*x^(2*n) + b*x^n + a)^2,x, algorithm="giac")
[Out]
integrate((d*x)^m/(c*x^(2*n) + b*x^n + a)^2, x)