3.601 \(\int \frac{(d x)^m}{\left (a+b x^n+c x^{2 n}\right )^3} \, dx\)

Optimal. Leaf size=615 \[ -\frac{c (d x)^{m+1} \left (-8 a^2 c^2 \left (m^2+m (2-6 n)+8 n^2-6 n+1\right )+6 a b^2 c \left (m^2+m (2-4 n)+3 n^2-4 n+1\right )+b (m-n+1) \sqrt{b^2-4 a c} \left (2 a c (2 m-7 n+2)-b^2 (m-2 n+1)\right )+b^4 \left (-\left (m^2+m (2-3 n)+2 n^2-3 n+1\right )\right )\right ) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right )}{2 a^2 d (m+1) n^2 \left (b^2-4 a c\right )^{5/2} \left (b-\sqrt{b^2-4 a c}\right )}-\frac{c (d x)^{m+1} \left (8 a^2 c^2 \left (m^2+m (2-6 n)+8 n^2-6 n+1\right )-6 a b^2 c \left (m^2+m (2-4 n)+3 n^2-4 n+1\right )+b (m-n+1) \sqrt{b^2-4 a c} \left (2 a c (2 m-7 n+2)-b^2 (m-2 n+1)\right )+b^4 \left (m^2+m (2-3 n)+2 n^2-3 n+1\right )\right ) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{2 a^2 d (m+1) n^2 \left (b^2-4 a c\right )^{5/2} \left (\sqrt{b^2-4 a c}+b\right )}-\frac{(d x)^{m+1} \left (4 a^2 c^2 (m-4 n+1)-b c x^n \left (2 a c (2 m-7 n+2)-b^2 (m-2 n+1)\right )-5 a b^2 c (m-3 n+1)+b^4 (m-2 n+1)\right )}{2 a^2 d n^2 \left (b^2-4 a c\right )^2 \left (a+b x^n+c x^{2 n}\right )}+\frac{(d x)^{m+1} \left (-2 a c+b^2+b c x^n\right )}{2 a d n \left (b^2-4 a c\right ) \left (a+b x^n+c x^{2 n}\right )^2} \]

[Out]

((d*x)^(1 + m)*(b^2 - 2*a*c + b*c*x^n))/(2*a*(b^2 - 4*a*c)*d*n*(a + b*x^n + c*x^
(2*n))^2) - ((d*x)^(1 + m)*(4*a^2*c^2*(1 + m - 4*n) - 5*a*b^2*c*(1 + m - 3*n) +
b^4*(1 + m - 2*n) - b*c*(2*a*c*(2 + 2*m - 7*n) - b^2*(1 + m - 2*n))*x^n))/(2*a^2
*(b^2 - 4*a*c)^2*d*n^2*(a + b*x^n + c*x^(2*n))) - (c*(b*Sqrt[b^2 - 4*a*c]*(2*a*c
*(2 + 2*m - 7*n) - b^2*(1 + m - 2*n))*(1 + m - n) - b^4*(1 + m^2 + m*(2 - 3*n) -
3*n + 2*n^2) + 6*a*b^2*c*(1 + m^2 + m*(2 - 4*n) - 4*n + 3*n^2) - 8*a^2*c^2*(1 +
m^2 + m*(2 - 6*n) - 6*n + 8*n^2))*(d*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/n,
(1 + m + n)/n, (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/(2*a^2*(b^2 - 4*a*c)^(5/2)*
(b - Sqrt[b^2 - 4*a*c])*d*(1 + m)*n^2) - (c*(b*Sqrt[b^2 - 4*a*c]*(2*a*c*(2 + 2*m
- 7*n) - b^2*(1 + m - 2*n))*(1 + m - n) + b^4*(1 + m^2 + m*(2 - 3*n) - 3*n + 2*
n^2) - 6*a*b^2*c*(1 + m^2 + m*(2 - 4*n) - 4*n + 3*n^2) + 8*a^2*c^2*(1 + m^2 + m*
(2 - 6*n) - 6*n + 8*n^2))*(d*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/n, (1 + m +
n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(2*a^2*(b^2 - 4*a*c)^(5/2)*(b + Sqrt
[b^2 - 4*a*c])*d*(1 + m)*n^2)

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Rubi [A]  time = 19.0273, antiderivative size = 637, normalized size of antiderivative = 1.04, number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{c (d x)^{m+1} \left (-8 a^2 c^2 \left (m^2+m (2-6 n)+8 n^2-6 n+1\right )+6 a b^2 c \left (m^2+m (2-4 n)+3 n^2-4 n+1\right )+b (m-n+1) \sqrt{b^2-4 a c} \left (2 a c (2 m-7 n+2)-b^2 (m-2 n+1)\right )+b^4 \left (-\left (m^2+m (2-3 n)+2 n^2-3 n+1\right )\right )\right ) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right )}{2 a^2 d (m+1) n^2 \left (b^2-4 a c\right )^{5/2} \left (b-\sqrt{b^2-4 a c}\right )}-\frac{c (d x)^{m+1} \left (8 a^2 c^2 \left (m^2+m (2-6 n)+8 n^2-6 n+1\right )-6 a b^2 c \left (m^2+m (2-4 n)+3 n^2-4 n+1\right )+b (m-n+1) \sqrt{b^2-4 a c} \left (2 a c (2 m-7 n+2)-b^2 (m-2 n+1)\right )+b^4 \left (m^2+m (2-3 n)+2 n^2-3 n+1\right )\right ) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{2 a^2 d (m+1) n^2 \left (b^2-4 a c\right )^{5/2} \left (\sqrt{b^2-4 a c}+b\right )}-\frac{(d x)^{m+1} \left (4 a^2 c^2 (m-4 n+1)-b c x^n \left (2 a c (2 m-7 n+2)-b^2 (m-2 n+1)\right )-5 a b^2 c (m-3 n+1)+b^4 (m-2 n+1)\right )}{2 a^2 d n^2 \left (b^2-4 a c\right )^2 \left (a+b x^n+c x^{2 n}\right )}+\frac{(d x)^{m+1} \left (-2 a c+b^2+b c x^n\right )}{2 a d n \left (b^2-4 a c\right ) \left (a+b x^n+c x^{2 n}\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[(d*x)^m/(a + b*x^n + c*x^(2*n))^3,x]

[Out]

((d*x)^(1 + m)*(b^2 - 2*a*c + b*c*x^n))/(2*a*(b^2 - 4*a*c)*d*n*(a + b*x^n + c*x^
(2*n))^2) - ((d*x)^(1 + m)*(4*a^2*c^2*(1 + m - 4*n) - 5*a*b^2*c*(1 + m - 3*n) +
b^4*(1 + m - 2*n) - b*c*(2*a*c*(2 + 2*m - 7*n) - b^2*(1 + m - 2*n))*x^n))/(2*a^2
*(b^2 - 4*a*c)^2*d*n^2*(a + b*x^n + c*x^(2*n))) - (c*(b*Sqrt[b^2 - 4*a*c]*(2*a*c
*(2 + 2*m - 7*n) - b^2*(1 + m - 2*n))*(1 + m - n) - b^4*(1 + m^2 + m*(2 - 3*n) -
3*n + 2*n^2) + 6*a*b^2*c*(1 + m^2 + m*(2 - 4*n) - 4*n + 3*n^2) - 8*a^2*c^2*(1 +
m^2 + m*(2 - 6*n) - 6*n + 8*n^2))*(d*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/n,
(1 + m + n)/n, (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/(2*a^2*(b^2 - 4*a*c)^(5/2)*
(b - Sqrt[b^2 - 4*a*c])*d*(1 + m)*n^2) - (c*(b*Sqrt[b^2 - 4*a*c]*(2*a*c*(2 + 2*m
- 7*n) - b^2*(1 + m - 2*n))*(1 + m - n) + b^4*(1 + m^2 + m*(2 - 3*n) - 3*n + 2*
n^2) - 6*a*b^2*c*(1 + m^2 + m*(2 - 4*n) - 4*n + 3*n^2) + 8*a^2*c^2*(1 + m^2 + m*
(2 - 6*n) - 6*n + 8*n^2))*(d*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/n, (1 + m +
n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(2*a^2*(b^2 - 4*a*c)^(5/2)*(b + Sqrt
[b^2 - 4*a*c])*d*(1 + m)*n^2)

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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))**3,x)

[Out]

Timed out

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Mathematica [B]  time = 7.10444, size = 12289, normalized size = 19.98 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

[In]  Integrate[(d*x)^m/(a + b*x^n + c*x^(2*n))^3,x]

[Out]

Result too large to show

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Maple [F]  time = 0.074, size = 0, normalized size = 0. \[ \int{\frac{ \left ( dx \right ) ^{m}}{ \left ( a+b{x}^{n}+c{x}^{2\,n} \right ) ^{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x)^m/(a+b*x^n+c*x^(2*n))^3,x)

[Out]

int((d*x)^m/(a+b*x^n+c*x^(2*n))^3,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x)^m/(c*x^(2*n) + b*x^n + a)^3,x, algorithm="maxima")

[Out]

1/2*((a^2*b^2*c*d^m*(5*m - 21*n + 5) - a*b^4*d^m*(m - 3*n + 1) - 4*a^3*c^2*d^m*(
m - 6*n + 1))*x*x^m + (2*a*b*c^3*d^m*(2*m - 7*n + 2) - b^3*c^2*d^m*(m - 2*n + 1)
)*x*e^(m*log(x) + 3*n*log(x)) + (a*b^2*c^2*d^m*(9*m - 29*n + 9) - 2*b^4*c*d^m*(m
- 2*n + 1) - 4*a^2*c^3*d^m*(m - 4*n + 1))*x*e^(m*log(x) + 2*n*log(x)) - (b^5*d^
m*(m - 2*n + 1) - 4*a*b^3*c*d^m*(m - 3*n + 1) + 2*a^2*b*c^2*d^m*n)*x*e^(m*log(x)
+ n*log(x)))/(a^4*b^4*n^2 - 8*a^5*b^2*c*n^2 + 16*a^6*c^2*n^2 + (a^2*b^4*c^2*n^2
- 8*a^3*b^2*c^3*n^2 + 16*a^4*c^4*n^2)*x^(4*n) + 2*(a^2*b^5*c*n^2 - 8*a^3*b^3*c^
2*n^2 + 16*a^4*b*c^3*n^2)*x^(3*n) + (a^2*b^6*n^2 - 6*a^3*b^4*c*n^2 + 32*a^5*c^3*
n^2)*x^(2*n) + 2*(a^3*b^5*n^2 - 8*a^4*b^3*c*n^2 + 16*a^5*b*c^2*n^2)*x^n) - integ
rate(-1/2*(((m^2 - m*(3*n - 2) + 2*n^2 - 3*n + 1)*b^4*d^m - (5*m^2 - m*(21*n - 1
0) + 16*n^2 - 21*n + 5)*a*b^2*c*d^m + 4*(m^2 - 2*m*(3*n - 1) + 8*n^2 - 6*n + 1)*
a^2*c^2*d^m)*x^m + ((m^2 - m*(3*n - 2) + 2*n^2 - 3*n + 1)*b^3*c*d^m - 2*(2*m^2 -
m*(9*n - 4) + 7*n^2 - 9*n + 2)*a*b*c^2*d^m)*e^(m*log(x) + n*log(x)))/(a^3*b^4*n
^2 - 8*a^4*b^2*c*n^2 + 16*a^5*c^2*n^2 + (a^2*b^4*c*n^2 - 8*a^3*b^2*c^2*n^2 + 16*
a^4*c^3*n^2)*x^(2*n) + (a^2*b^5*n^2 - 8*a^3*b^3*c*n^2 + 16*a^4*b*c^2*n^2)*x^n),
x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\left (d x\right )^{m}}{c^{3} x^{6 \, n} + b^{3} x^{3 \, n} + 3 \, a^{2} b x^{n} + a^{3} + 3 \,{\left (b c^{2} x^{n} + b^{2} c + a c^{2}\right )} x^{4 \, n} + 3 \,{\left (2 \, a b c x^{n} + a b^{2} + a^{2} c\right )} x^{2 \, n}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x)^m/(c*x^(2*n) + b*x^n + a)^3,x, algorithm="fricas")

[Out]

integral((d*x)^m/(c^3*x^(6*n) + b^3*x^(3*n) + 3*a^2*b*x^n + a^3 + 3*(b*c^2*x^n +
b^2*c + a*c^2)*x^(4*n) + 3*(2*a*b*c*x^n + a*b^2 + a^2*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))**3,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d x\right )^{m}}{{\left (c x^{2 \, n} + b x^{n} + a\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x)^m/(c*x^(2*n) + b*x^n + a)^3,x, algorithm="giac")

[Out]

integrate((d*x)^m/(c*x^(2*n) + b*x^n + a)^3, x)