3.153 \(\int (f x)^m \left (d+e x^n\right ) \left (a+b x^n+c x^{2 n}\right )^p \, dx\)

Optimal. Leaf size=323 \[ \frac{d (f x)^{m+1} \left (\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1\right )^{-p} \left (\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1\right )^{-p} \left (a+b x^n+c x^{2 n}\right )^p F_1\left (\frac{m+1}{n};-p,-p;\frac{m+n+1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{f (m+1)}+\frac{e x^{n+1} (f x)^m \left (\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1\right )^{-p} \left (\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1\right )^{-p} \left (a+b x^n+c x^{2 n}\right )^p F_1\left (\frac{m+n+1}{n};-p,-p;\frac{m+2 n+1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{m+n+1} \]

[Out]

(d*(f*x)^(1 + m)*(a + b*x^n + c*x^(2*n))^p*AppellF1[(1 + m)/n, -p, -p, (1 + m +
n)/n, (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c]), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(
f*(1 + m)*(1 + (2*c*x^n)/(b - Sqrt[b^2 - 4*a*c]))^p*(1 + (2*c*x^n)/(b + Sqrt[b^2
- 4*a*c]))^p) + (e*x^(1 + n)*(f*x)^m*(a + b*x^n + c*x^(2*n))^p*AppellF1[(1 + m
+ n)/n, -p, -p, (1 + m + 2*n)/n, (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c]), (-2*c*x^n)/
(b + Sqrt[b^2 - 4*a*c])])/((1 + m + n)*(1 + (2*c*x^n)/(b - Sqrt[b^2 - 4*a*c]))^p
*(1 + (2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]))^p)

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Rubi [A]  time = 0.824632, antiderivative size = 323, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138 \[ \frac{d (f x)^{m+1} \left (\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1\right )^{-p} \left (\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1\right )^{-p} \left (a+b x^n+c x^{2 n}\right )^p F_1\left (\frac{m+1}{n};-p,-p;\frac{m+n+1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{f (m+1)}+\frac{e x^{n+1} (f x)^m \left (\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1\right )^{-p} \left (\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1\right )^{-p} \left (a+b x^n+c x^{2 n}\right )^p F_1\left (\frac{m+n+1}{n};-p,-p;\frac{m+2 n+1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{m+n+1} \]

Antiderivative was successfully verified.

[In]  Int[(f*x)^m*(d + e*x^n)*(a + b*x^n + c*x^(2*n))^p,x]

[Out]

(d*(f*x)^(1 + m)*(a + b*x^n + c*x^(2*n))^p*AppellF1[(1 + m)/n, -p, -p, (1 + m +
n)/n, (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c]), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(
f*(1 + m)*(1 + (2*c*x^n)/(b - Sqrt[b^2 - 4*a*c]))^p*(1 + (2*c*x^n)/(b + Sqrt[b^2
- 4*a*c]))^p) + (e*x^(1 + n)*(f*x)^m*(a + b*x^n + c*x^(2*n))^p*AppellF1[(1 + m
+ n)/n, -p, -p, (1 + m + 2*n)/n, (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c]), (-2*c*x^n)/
(b + Sqrt[b^2 - 4*a*c])])/((1 + m + n)*(1 + (2*c*x^n)/(b - Sqrt[b^2 - 4*a*c]))^p
*(1 + (2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]))^p)

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Rubi in Sympy [A]  time = 86.8058, size = 279, normalized size = 0.86 \[ \frac{d \left (f x\right )^{m + 1} \left (\frac{2 c x^{n}}{b - \sqrt{- 4 a c + b^{2}}} + 1\right )^{- p} \left (\frac{2 c x^{n}}{b + \sqrt{- 4 a c + b^{2}}} + 1\right )^{- p} \left (a + b x^{n} + c x^{2 n}\right )^{p} \operatorname{appellf_{1}}{\left (\frac{m + 1}{n},- p,- p,\frac{m + n + 1}{n},- \frac{2 c x^{n}}{b - \sqrt{- 4 a c + b^{2}}},- \frac{2 c x^{n}}{b + \sqrt{- 4 a c + b^{2}}} \right )}}{f \left (m + 1\right )} + \frac{e x^{n} \left (f x\right )^{- n} \left (f x\right )^{m + n + 1} \left (\frac{2 c x^{n}}{b - \sqrt{- 4 a c + b^{2}}} + 1\right )^{- p} \left (\frac{2 c x^{n}}{b + \sqrt{- 4 a c + b^{2}}} + 1\right )^{- p} \left (a + b x^{n} + c x^{2 n}\right )^{p} \operatorname{appellf_{1}}{\left (\frac{m + n + 1}{n},- p,- p,\frac{m + 2 n + 1}{n},- \frac{2 c x^{n}}{b - \sqrt{- 4 a c + b^{2}}},- \frac{2 c x^{n}}{b + \sqrt{- 4 a c + b^{2}}} \right )}}{f \left (m + n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((f*x)**m*(d+e*x**n)*(a+b*x**n+c*x**(2*n))**p,x)

[Out]

d*(f*x)**(m + 1)*(2*c*x**n/(b - sqrt(-4*a*c + b**2)) + 1)**(-p)*(2*c*x**n/(b + s
qrt(-4*a*c + b**2)) + 1)**(-p)*(a + b*x**n + c*x**(2*n))**p*appellf1((m + 1)/n,
-p, -p, (m + n + 1)/n, -2*c*x**n/(b - sqrt(-4*a*c + b**2)), -2*c*x**n/(b + sqrt(
-4*a*c + b**2)))/(f*(m + 1)) + e*x**n*(f*x)**(-n)*(f*x)**(m + n + 1)*(2*c*x**n/(
b - sqrt(-4*a*c + b**2)) + 1)**(-p)*(2*c*x**n/(b + sqrt(-4*a*c + b**2)) + 1)**(-
p)*(a + b*x**n + c*x**(2*n))**p*appellf1((m + n + 1)/n, -p, -p, (m + 2*n + 1)/n,
-2*c*x**n/(b - sqrt(-4*a*c + b**2)), -2*c*x**n/(b + sqrt(-4*a*c + b**2)))/(f*(m
+ n + 1))

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Mathematica [B]  time = 2.48676, size = 922, normalized size = 2.85 \[ \frac{2^{-p-1} \left (b+\sqrt{b^2-4 a c}\right ) x (f x)^m \left (x^n+\frac{b-\sqrt{b^2-4 a c}}{2 c}\right )^{-p} \left (\frac{2 c x^n+b-\sqrt{b^2-4 a c}}{c}\right )^p \left (\left (\sqrt{b^2-4 a c}-b\right ) x^n-2 a\right )^2 \left (\left (c x^n+b\right ) x^n+a\right )^{p-1} \left (\frac{e (m+2 n+1) \left (2 c x^n+b-\sqrt{b^2-4 a c}\right ) F_1\left (\frac{m+n+1}{n};-p,-p;\frac{m+2 n+1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right ) x^n}{n p x^n \left (\left (\sqrt{b^2-4 a c}-b\right ) F_1\left (\frac{m+2 n+1}{n};1-p,-p;\frac{m+3 n+1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right )-\left (b+\sqrt{b^2-4 a c}\right ) F_1\left (\frac{m+2 n+1}{n};-p,1-p;\frac{m+3 n+1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right )\right )-2 a (m+2 n+1) F_1\left (\frac{m+n+1}{n};-p,-p;\frac{m+2 n+1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right )}+\frac{d (m+n+1)^2 \left (-2 c x^n-b+\sqrt{b^2-4 a c}\right ) F_1\left (\frac{m+1}{n};-p,-p;\frac{m+n+1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right )}{(m+1) \left (2 a (m+n+1) F_1\left (\frac{m+1}{n};-p,-p;\frac{m+n+1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right )-n p x^n \left (\left (\sqrt{b^2-4 a c}-b\right ) F_1\left (\frac{m+n+1}{n};1-p,-p;\frac{m+2 n+1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right )-\left (b+\sqrt{b^2-4 a c}\right ) F_1\left (\frac{m+n+1}{n};-p,1-p;\frac{m+2 n+1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right )\right )\right )}\right )}{\left (\sqrt{b^2-4 a c}-b\right ) (m+n+1) \left (2 c x^n+b+\sqrt{b^2-4 a c}\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(f*x)^m*(d + e*x^n)*(a + b*x^n + c*x^(2*n))^p,x]

[Out]

(2^(-1 - p)*(b + Sqrt[b^2 - 4*a*c])*x*(f*x)^m*((b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)
/c)^p*(-2*a + (-b + Sqrt[b^2 - 4*a*c])*x^n)^2*(a + x^n*(b + c*x^n))^(-1 + p)*((d
*(1 + m + n)^2*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^n)*AppellF1[(1 + m)/n, -p, -p, (1
+ m + n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*
c])])/((1 + m)*(2*a*(1 + m + n)*AppellF1[(1 + m)/n, -p, -p, (1 + m + n)/n, (-2*c
*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])] - n*p*x^n*((-
b + Sqrt[b^2 - 4*a*c])*AppellF1[(1 + m + n)/n, 1 - p, -p, (1 + m + 2*n)/n, (-2*c
*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])] - (b + Sqrt[b
^2 - 4*a*c])*AppellF1[(1 + m + n)/n, -p, 1 - p, (1 + m + 2*n)/n, (-2*c*x^n)/(b +
Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])]))) + (e*(1 + m + 2*n)*x
^n*(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)*AppellF1[(1 + m + n)/n, -p, -p, (1 + m + 2*
n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])])/(
-2*a*(1 + m + 2*n)*AppellF1[(1 + m + n)/n, -p, -p, (1 + m + 2*n)/n, (-2*c*x^n)/(
b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])] + n*p*x^n*((-b + Sqr
t[b^2 - 4*a*c])*AppellF1[(1 + m + 2*n)/n, 1 - p, -p, (1 + m + 3*n)/n, (-2*c*x^n)
/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])] - (b + Sqrt[b^2 -
4*a*c])*AppellF1[(1 + m + 2*n)/n, -p, 1 - p, (1 + m + 3*n)/n, (-2*c*x^n)/(b + Sq
rt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])]))))/((-b + Sqrt[b^2 - 4*a*
c])*(1 + m + n)*((b - Sqrt[b^2 - 4*a*c])/(2*c) + x^n)^p*(b + Sqrt[b^2 - 4*a*c] +
2*c*x^n))

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Maple [F]  time = 0.079, size = 0, normalized size = 0. \[ \int \left ( fx \right ) ^{m} \left ( d+e{x}^{n} \right ) \left ( a+b{x}^{n}+c{x}^{2\,n} \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((f*x)^m*(d+e*x^n)*(a+b*x^n+c*x^(2*n))^p,x)

[Out]

int((f*x)^m*(d+e*x^n)*(a+b*x^n+c*x^(2*n))^p,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (e x^{n} + d\right )}{\left (c x^{2 \, n} + b x^{n} + a\right )}^{p} \left (f x\right )^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((e*x^n + d)*(c*x^(2*n) + b*x^n + a)^p*(f*x)^m, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((e*x^n + d)*(c*x^(2*n) + b*x^n + a)^p*(f*x)^m, 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((f*x)**m*(d+e*x**n)*(a+b*x**n+c*x**(2*n))**p,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((e*x^n + d)*(c*x^(2*n) + b*x^n + a)^p*(f*x)^m, x)