3.1.88 \(\int (f x)^m (d+e x^n)^2 (a+c x^{2 n})^p \, dx\) [88]

Optimal. Leaf size=262 \[ \frac {d^2 (f x)^{1+m} \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p} \, _2F_1\left (\frac {1+m}{2 n},-p;1+\frac {1+m}{2 n};-\frac {c x^{2 n}}{a}\right )}{f (1+m)}+\frac {2 d e x^{1+n} (f x)^m \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p} \, _2F_1\left (\frac {1+m+n}{2 n},-p;\frac {1+m+3 n}{2 n};-\frac {c x^{2 n}}{a}\right )}{1+m+n}+\frac {e^2 x^{1+2 n} (f x)^m \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p} \, _2F_1\left (\frac {1+m+2 n}{2 n},-p;\frac {1+m+4 n}{2 n};-\frac {c x^{2 n}}{a}\right )}{1+m+2 n} \]

[Out]

d^2*(f*x)^(1+m)*(a+c*x^(2*n))^p*hypergeom([-p, 1/2*(1+m)/n],[1+1/2*(1+m)/n],-c*x^(2*n)/a)/f/(1+m)/((1+c*x^(2*n
)/a)^p)+2*d*e*x^(1+n)*(f*x)^m*(a+c*x^(2*n))^p*hypergeom([-p, 1/2*(1+m+n)/n],[1/2*(1+m+3*n)/n],-c*x^(2*n)/a)/(1
+m+n)/((1+c*x^(2*n)/a)^p)+e^2*x^(1+2*n)*(f*x)^m*(a+c*x^(2*n))^p*hypergeom([-p, 1/2*(1+m+2*n)/n],[1/2*(1+m+4*n)
/n],-c*x^(2*n)/a)/(1+m+2*n)/((1+c*x^(2*n)/a)^p)

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Rubi [A]
time = 0.10, antiderivative size = 262, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1575, 372, 371, 20} \begin {gather*} \frac {d^2 (f x)^{m+1} \left (a+c x^{2 n}\right )^p \left (\frac {c x^{2 n}}{a}+1\right )^{-p} \, _2F_1\left (\frac {m+1}{2 n},-p;\frac {m+1}{2 n}+1;-\frac {c x^{2 n}}{a}\right )}{f (m+1)}+\frac {2 d e x^{n+1} (f x)^m \left (a+c x^{2 n}\right )^p \left (\frac {c x^{2 n}}{a}+1\right )^{-p} \, _2F_1\left (\frac {m+n+1}{2 n},-p;\frac {m+3 n+1}{2 n};-\frac {c x^{2 n}}{a}\right )}{m+n+1}+\frac {e^2 x^{2 n+1} (f x)^m \left (a+c x^{2 n}\right )^p \left (\frac {c x^{2 n}}{a}+1\right )^{-p} \, _2F_1\left (\frac {m+2 n+1}{2 n},-p;\frac {m+4 n+1}{2 n};-\frac {c x^{2 n}}{a}\right )}{m+2 n+1} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(d^2*(f*x)^(1 + m)*(a + c*x^(2*n))^p*Hypergeometric2F1[(1 + m)/(2*n), -p, 1 + (1 + m)/(2*n), -((c*x^(2*n))/a)]
)/(f*(1 + m)*(1 + (c*x^(2*n))/a)^p) + (2*d*e*x^(1 + n)*(f*x)^m*(a + c*x^(2*n))^p*Hypergeometric2F1[(1 + m + n)
/(2*n), -p, (1 + m + 3*n)/(2*n), -((c*x^(2*n))/a)])/((1 + m + n)*(1 + (c*x^(2*n))/a)^p) + (e^2*x^(1 + 2*n)*(f*
x)^m*(a + c*x^(2*n))^p*Hypergeometric2F1[(1 + m + 2*n)/(2*n), -p, (1 + m + 4*n)/(2*n), -((c*x^(2*n))/a)])/((1
+ m + 2*n)*(1 + (c*x^(2*n))/a)^p)

Rule 20

Int[(u_.)*((a_.)*(v_))^(m_)*((b_.)*(v_))^(n_), x_Symbol] :> Dist[b^IntPart[n]*((b*v)^FracPart[n]/(a^IntPart[n]
*(a*v)^FracPart[n])), Int[u*(a*v)^(m + n), x], x] /; FreeQ[{a, b, m, n}, x] &&  !IntegerQ[m] &&  !IntegerQ[n]
&&  !IntegerQ[m + n]

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 372

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^IntPart[p]*((a + b*x^n)^FracPart[p]/
(1 + b*(x^n/a))^FracPart[p]), Int[(c*x)^m*(1 + b*(x^n/a))^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[
p, 0] &&  !(ILtQ[p, 0] || GtQ[a, 0])

Rule 1575

Int[((f_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[Expan
dIntegrand[(f*x)^m*(d + e*x^n)^q*(a + c*x^(2*n))^p, x], x] /; FreeQ[{a, c, d, e, f, m, n, p, q}, x] && EqQ[n2,
2*n] && (IGtQ[p, 0] || IGtQ[q, 0])

Rubi steps

\begin {align*} \int (f x)^m \left (d+e x^n\right )^2 \left (a+c x^{2 n}\right )^p \, dx &=\int \left (d^2 (f x)^m \left (a+c x^{2 n}\right )^p+2 d e x^n (f x)^m \left (a+c x^{2 n}\right )^p+e^2 x^{2 n} (f x)^m \left (a+c x^{2 n}\right )^p\right ) \, dx\\ &=d^2 \int (f x)^m \left (a+c x^{2 n}\right )^p \, dx+(2 d e) \int x^n (f x)^m \left (a+c x^{2 n}\right )^p \, dx+e^2 \int x^{2 n} (f x)^m \left (a+c x^{2 n}\right )^p \, dx\\ &=\left (2 d e x^{-m} (f x)^m\right ) \int x^{m+n} \left (a+c x^{2 n}\right )^p \, dx+\left (e^2 x^{-m} (f x)^m\right ) \int x^{m+2 n} \left (a+c x^{2 n}\right )^p \, dx+\left (d^2 \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p}\right ) \int (f x)^m \left (1+\frac {c x^{2 n}}{a}\right )^p \, dx\\ &=\frac {d^2 (f x)^{1+m} \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p} \, _2F_1\left (\frac {1+m}{2 n},-p;1+\frac {1+m}{2 n};-\frac {c x^{2 n}}{a}\right )}{f (1+m)}+\left (2 d e x^{-m} (f x)^m \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p}\right ) \int x^{m+n} \left (1+\frac {c x^{2 n}}{a}\right )^p \, dx+\left (e^2 x^{-m} (f x)^m \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p}\right ) \int x^{m+2 n} \left (1+\frac {c x^{2 n}}{a}\right )^p \, dx\\ &=\frac {d^2 (f x)^{1+m} \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p} \, _2F_1\left (\frac {1+m}{2 n},-p;1+\frac {1+m}{2 n};-\frac {c x^{2 n}}{a}\right )}{f (1+m)}+\frac {2 d e x^{1+n} (f x)^m \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p} \, _2F_1\left (\frac {1+m+n}{2 n},-p;\frac {1+m+3 n}{2 n};-\frac {c x^{2 n}}{a}\right )}{1+m+n}+\frac {e^2 x^{1+2 n} (f x)^m \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p} \, _2F_1\left (\frac {1+m+2 n}{2 n},-p;\frac {1+m+4 n}{2 n};-\frac {c x^{2 n}}{a}\right )}{1+m+2 n}\\ \end {align*}

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Mathematica [A]
time = 0.21, size = 189, normalized size = 0.72 \begin {gather*} x (f x)^m \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p} \left (\frac {d^2 \, _2F_1\left (\frac {1+m}{2 n},-p;1+\frac {1+m}{2 n};-\frac {c x^{2 n}}{a}\right )}{1+m}+e x^n \left (\frac {2 d \, _2F_1\left (\frac {1+m+n}{2 n},-p;\frac {1+m+3 n}{2 n};-\frac {c x^{2 n}}{a}\right )}{1+m+n}+\frac {e x^n \, _2F_1\left (\frac {1+m+2 n}{2 n},-p;\frac {1+m+4 n}{2 n};-\frac {c x^{2 n}}{a}\right )}{1+m+2 n}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(f*x)^m*(a + c*x^(2*n))^p*((d^2*Hypergeometric2F1[(1 + m)/(2*n), -p, 1 + (1 + m)/(2*n), -((c*x^(2*n))/a)])/
(1 + m) + e*x^n*((2*d*Hypergeometric2F1[(1 + m + n)/(2*n), -p, (1 + m + 3*n)/(2*n), -((c*x^(2*n))/a)])/(1 + m
+ n) + (e*x^n*Hypergeometric2F1[(1 + m + 2*n)/(2*n), -p, (1 + m + 4*n)/(2*n), -((c*x^(2*n))/a)])/(1 + m + 2*n)
)))/(1 + (c*x^(2*n))/a)^p

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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \left (f x \right )^{m} \left (d +e \,x^{n}\right )^{2} \left (a +c \,x^{2 n}\right )^{p}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Simplification assuming sageVARx near 0Simplification assuming sageVARf near 0Simplification assuming sageV
ARx near 0S

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (a+c\,x^{2\,n}\right )}^p\,{\left (f\,x\right )}^m\,{\left (d+e\,x^n\right )}^2 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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