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

Optimal. Leaf size=158 \[ \frac{(d x)^{m+1} \left (\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1\right )^{-p} \left (\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1\right )^{-p} \left (a+b x^n+c x^{2 n}\right )^p F_1\left (\frac{m+1}{n};-p,-p;\frac{m+n+1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{d (m+1)} \]

[Out]

((d*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])])/(d*
(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)

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Rubi [A]  time = 0.320689, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{(d x)^{m+1} \left (\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1\right )^{-p} \left (\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1\right )^{-p} \left (a+b x^n+c x^{2 n}\right )^p F_1\left (\frac{m+1}{n};-p,-p;\frac{m+n+1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{d (m+1)} \]

Antiderivative was successfully verified.

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

[Out]

((d*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])])/(d*
(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)

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Rubi in Sympy [A]  time = 35.0149, size = 129, normalized size = 0.82 \[ \frac{\left (d x\right )^{m + 1} \left (\frac{2 c x^{n}}{b - \sqrt{- 4 a c + b^{2}}} + 1\right )^{- p} \left (\frac{2 c x^{n}}{b + \sqrt{- 4 a c + b^{2}}} + 1\right )^{- p} \left (a + b x^{n} + c x^{2 n}\right )^{p} \operatorname{appellf_{1}}{\left (\frac{m + 1}{n},- p,- p,\frac{m + n + 1}{n},- \frac{2 c x^{n}}{b - \sqrt{- 4 a c + b^{2}}},- \frac{2 c x^{n}}{b + \sqrt{- 4 a c + b^{2}}} \right )}}{d \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(d*x)**(m + 1)*(2*c*x**n/(b - sqrt(-4*a*c + b**2)) + 1)**(-p)*(2*c*x**n/(b + sqr
t(-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)))/(d*(m + 1))

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Mathematica [B]  time = 5.6303, size = 534, normalized size = 3.38 \[ -\frac{2^{-p-1} x (m+n+1) \left (\sqrt{b^2-4 a c}+b\right ) (d x)^m \left (\sqrt{b^2-4 a c}-b-2 c x^n\right ) \left (x^n \left (\sqrt{b^2-4 a c}-b\right )-2 a\right )^2 \left (\frac{b-\sqrt{b^2-4 a c}}{2 c}+x^n\right )^{-p} \left (\frac{-\sqrt{b^2-4 a c}+b+2 c x^n}{c}\right )^p \left (a+x^n \left (b+c x^n\right )\right )^{p-1} F_1\left (\frac{m+1}{n};-p,-p;\frac{m+n+1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right )}{(m+1) \left (\sqrt{b^2-4 a c}-b\right ) \left (\sqrt{b^2-4 a c}+b+2 c x^n\right ) \left (n p x^n \left (\left (\sqrt{b^2-4 a c}-b\right ) F_1\left (\frac{m+n+1}{n};1-p,-p;\frac{m+2 n+1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right )-\left (\sqrt{b^2-4 a c}+b\right ) F_1\left (\frac{m+n+1}{n};-p,1-p;\frac{m+2 n+1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right )\right )-2 a (m+n+1) F_1\left (\frac{m+1}{n};-p,-p;\frac{m+n+1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [F]  time = 0.081, size = 0, normalized size = 0. \[ \int \left ( dx \right ) ^{m} \left ( a+b{x}^{n}+c{x}^{2\,n} \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((c*x^(2*n) + b*x^n + a)^p*(d*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((d*x)**m*(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. \[ \int{\left (c x^{2 \, n} + b x^{n} + a\right )}^{p} \left (d x\right )^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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