3.551 \(\int (g x)^m (c+d x+e x^2+f x^3) (a+b x^4)^p \, dx\)

Optimal. Leaf size=269 \[ \frac{c (g x)^{m+1} \left (a+b x^4\right )^p \left (\frac{b x^4}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+1}{4},-p;\frac{m+5}{4};-\frac{b x^4}{a}\right )}{g (m+1)}+\frac{d (g x)^{m+2} \left (a+b x^4\right )^p \left (\frac{b x^4}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+2}{4},-p;\frac{m+6}{4};-\frac{b x^4}{a}\right )}{g^2 (m+2)}+\frac{e (g x)^{m+3} \left (a+b x^4\right )^p \left (\frac{b x^4}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+3}{4},-p;\frac{m+7}{4};-\frac{b x^4}{a}\right )}{g^3 (m+3)}+\frac{f (g x)^{m+4} \left (a+b x^4\right )^p \left (\frac{b x^4}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+4}{4},-p;\frac{m+8}{4};-\frac{b x^4}{a}\right )}{g^4 (m+4)} \]

[Out]

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

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Rubi [A]  time = 0.256017, antiderivative size = 269, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 4, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {1833, 1336, 365, 364} \[ \frac{c (g x)^{m+1} \left (a+b x^4\right )^p \left (\frac{b x^4}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+1}{4},-p;\frac{m+5}{4};-\frac{b x^4}{a}\right )}{g (m+1)}+\frac{d (g x)^{m+2} \left (a+b x^4\right )^p \left (\frac{b x^4}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+2}{4},-p;\frac{m+6}{4};-\frac{b x^4}{a}\right )}{g^2 (m+2)}+\frac{e (g x)^{m+3} \left (a+b x^4\right )^p \left (\frac{b x^4}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+3}{4},-p;\frac{m+7}{4};-\frac{b x^4}{a}\right )}{g^3 (m+3)}+\frac{f (g x)^{m+4} \left (a+b x^4\right )^p \left (\frac{b x^4}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+4}{4},-p;\frac{m+8}{4};-\frac{b x^4}{a}\right )}{g^4 (m+4)} \]

Antiderivative was successfully verified.

[In]

Int[(g*x)^m*(c + d*x + e*x^2 + f*x^3)*(a + b*x^4)^p,x]

[Out]

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

Rule 1833

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Module[{q = Expon[Pq, x], j, k}, Int[
Sum[((c*x)^(m + j)*Sum[Coeff[Pq, x, j + (k*n)/2]*x^((k*n)/2), {k, 0, (2*(q - j))/n + 1}]*(a + b*x^n)^p)/c^j, {
j, 0, n/2 - 1}], x]] /; FreeQ[{a, b, c, m, p}, x] && PolyQ[Pq, x] && IGtQ[n/2, 0] &&  !PolyQ[Pq, x^(n/2)]

Rule 1336

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(f*x)^m*(d + e*x^2)^q*(a + c*x^4)^p, x], x] /; FreeQ[{a, c, d, e, f, m, p, q}, x] && (IGtQ[p, 0] || IGtQ[q,
0] || IntegersQ[m, q])

Rule 365

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 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps

\begin{align*} \int (g x)^m \left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^p \, dx &=\int \left ((g x)^m \left (c+e x^2\right ) \left (a+b x^4\right )^p+\frac{(g x)^{1+m} \left (d+f x^2\right ) \left (a+b x^4\right )^p}{g}\right ) \, dx\\ &=\frac{\int (g x)^{1+m} \left (d+f x^2\right ) \left (a+b x^4\right )^p \, dx}{g}+\int (g x)^m \left (c+e x^2\right ) \left (a+b x^4\right )^p \, dx\\ &=\frac{\int \left (d (g x)^{1+m} \left (a+b x^4\right )^p+\frac{f (g x)^{3+m} \left (a+b x^4\right )^p}{g^2}\right ) \, dx}{g}+\int \left (c (g x)^m \left (a+b x^4\right )^p+\frac{e (g x)^{2+m} \left (a+b x^4\right )^p}{g^2}\right ) \, dx\\ &=c \int (g x)^m \left (a+b x^4\right )^p \, dx+\frac{f \int (g x)^{3+m} \left (a+b x^4\right )^p \, dx}{g^3}+\frac{e \int (g x)^{2+m} \left (a+b x^4\right )^p \, dx}{g^2}+\frac{d \int (g x)^{1+m} \left (a+b x^4\right )^p \, dx}{g}\\ &=\left (c \left (a+b x^4\right )^p \left (1+\frac{b x^4}{a}\right )^{-p}\right ) \int (g x)^m \left (1+\frac{b x^4}{a}\right )^p \, dx+\frac{\left (f \left (a+b x^4\right )^p \left (1+\frac{b x^4}{a}\right )^{-p}\right ) \int (g x)^{3+m} \left (1+\frac{b x^4}{a}\right )^p \, dx}{g^3}+\frac{\left (e \left (a+b x^4\right )^p \left (1+\frac{b x^4}{a}\right )^{-p}\right ) \int (g x)^{2+m} \left (1+\frac{b x^4}{a}\right )^p \, dx}{g^2}+\frac{\left (d \left (a+b x^4\right )^p \left (1+\frac{b x^4}{a}\right )^{-p}\right ) \int (g x)^{1+m} \left (1+\frac{b x^4}{a}\right )^p \, dx}{g}\\ &=\frac{c (g x)^{1+m} \left (a+b x^4\right )^p \left (1+\frac{b x^4}{a}\right )^{-p} \, _2F_1\left (\frac{1+m}{4},-p;\frac{5+m}{4};-\frac{b x^4}{a}\right )}{g (1+m)}+\frac{d (g x)^{2+m} \left (a+b x^4\right )^p \left (1+\frac{b x^4}{a}\right )^{-p} \, _2F_1\left (\frac{2+m}{4},-p;\frac{6+m}{4};-\frac{b x^4}{a}\right )}{g^2 (2+m)}+\frac{e (g x)^{3+m} \left (a+b x^4\right )^p \left (1+\frac{b x^4}{a}\right )^{-p} \, _2F_1\left (\frac{3+m}{4},-p;\frac{7+m}{4};-\frac{b x^4}{a}\right )}{g^3 (3+m)}+\frac{f (g x)^{4+m} \left (a+b x^4\right )^p \left (1+\frac{b x^4}{a}\right )^{-p} \, _2F_1\left (\frac{4+m}{4},-p;\frac{8+m}{4};-\frac{b x^4}{a}\right )}{g^4 (4+m)}\\ \end{align*}

Mathematica [A]  time = 0.214322, size = 174, normalized size = 0.65 \[ x (g x)^m \left (a+b x^4\right )^p \left (\frac{b x^4}{a}+1\right )^{-p} \left (\frac{c \, _2F_1\left (\frac{m+1}{4},-p;\frac{m+5}{4};-\frac{b x^4}{a}\right )}{m+1}+x \left (\frac{d \, _2F_1\left (\frac{m+2}{4},-p;\frac{m+6}{4};-\frac{b x^4}{a}\right )}{m+2}+x \left (\frac{e \, _2F_1\left (\frac{m+3}{4},-p;\frac{m+7}{4};-\frac{b x^4}{a}\right )}{m+3}+\frac{f x \, _2F_1\left (\frac{m+4}{4},-p;\frac{m+8}{4};-\frac{b x^4}{a}\right )}{m+4}\right )\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(g*x)^m*(c + d*x + e*x^2 + f*x^3)*(a + b*x^4)^p,x]

[Out]

(x*(g*x)^m*(a + b*x^4)^p*((c*Hypergeometric2F1[(1 + m)/4, -p, (5 + m)/4, -((b*x^4)/a)])/(1 + m) + x*((d*Hyperg
eometric2F1[(2 + m)/4, -p, (6 + m)/4, -((b*x^4)/a)])/(2 + m) + x*((e*Hypergeometric2F1[(3 + m)/4, -p, (7 + m)/
4, -((b*x^4)/a)])/(3 + m) + (f*x*Hypergeometric2F1[(4 + m)/4, -p, (8 + m)/4, -((b*x^4)/a)])/(4 + m)))))/(1 + (
b*x^4)/a)^p

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Maple [F]  time = 0.314, size = 0, normalized size = 0. \begin{align*} \int \left ( gx \right ) ^{m} \left ( f{x}^{3}+e{x}^{2}+dx+c \right ) \left ( b{x}^{4}+a \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*x)^m*(f*x^3+e*x^2+d*x+c)*(b*x^4+a)^p,x)

[Out]

int((g*x)^m*(f*x^3+e*x^2+d*x+c)*(b*x^4+a)^p,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (f x^{3} + e x^{2} + d x + c\right )}{\left (b x^{4} + a\right )}^{p} \left (g x\right )^{m}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x)^m*(f*x^3+e*x^2+d*x+c)*(b*x^4+a)^p,x, algorithm="maxima")

[Out]

integrate((f*x^3 + e*x^2 + d*x + c)*(b*x^4 + a)^p*(g*x)^m, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (f x^{3} + e x^{2} + d x + c\right )}{\left (b x^{4} + a\right )}^{p} \left (g x\right )^{m}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x)^m*(f*x^3+e*x^2+d*x+c)*(b*x^4+a)^p,x, algorithm="fricas")

[Out]

integral((f*x^3 + e*x^2 + d*x + c)*(b*x^4 + a)^p*(g*x)^m, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x)**m*(f*x**3+e*x**2+d*x+c)*(b*x**4+a)**p,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (f x^{3} + e x^{2} + d x + c\right )}{\left (b x^{4} + a\right )}^{p} \left (g x\right )^{m}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x)^m*(f*x^3+e*x^2+d*x+c)*(b*x^4+a)^p,x, algorithm="giac")

[Out]

integrate((f*x^3 + e*x^2 + d*x + c)*(b*x^4 + a)^p*(g*x)^m, x)