3.6.51 \(\int (g x)^m (c+d x+e x^2+f x^3) (a+b x^4)^p \, dx\) [551]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.17, antiderivative size = 269, normalized size of antiderivative = 1.00, 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 = {1847, 1350, 372, 371} \begin {gather*} \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)} \end {gather*}

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 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 1350

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 1847

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)/c^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, {
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)]

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 = 0.27, size = 174, normalized size = 0.65 \begin {gather*} x (g x)^m \left (a+b x^4\right )^p \left (1+\frac {b x^4}{a}\right )^{-p} \left (\frac {f x^3 \, _2F_1\left (1+\frac {m}{4},-p;2+\frac {m}{4};-\frac {b x^4}{a}\right )}{4+m}+\frac {c \, _2F_1\left (\frac {1+m}{4},-p;\frac {5+m}{4};-\frac {b x^4}{a}\right )}{1+m}+x \left (\frac {d \, _2F_1\left (\frac {2+m}{4},-p;\frac {6+m}{4};-\frac {b x^4}{a}\right )}{2+m}+\frac {e x \, _2F_1\left (\frac {3+m}{4},-p;\frac {7+m}{4};-\frac {b x^4}{a}\right )}{3+m}\right )\right ) \end {gather*}

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*((f*x^3*Hypergeometric2F1[1 + m/4, -p, 2 + m/4, -((b*x^4)/a)])/(4 + m) + (c*Hypergeom
etric2F1[(1 + m)/4, -p, (5 + m)/4, -((b*x^4)/a)])/(1 + m) + x*((d*Hypergeometric2F1[(2 + m)/4, -p, (6 + m)/4,
-((b*x^4)/a)])/(2 + m) + (e*x*Hypergeometric2F1[(3 + m)/4, -p, (7 + m)/4, -((b*x^4)/a)])/(3 + m))))/(1 + (b*x^
4)/a)^p

________________________________________________________________________________________

Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \left (g x \right )^{m} \left (f \,x^{3}+e \,x^{2}+d x +c \right ) \left (b \,x^{4}+a \right )^{p}\, dx\]

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)

________________________________________________________________________________________

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

________________________________________________________________________________________

Fricas [F]
time = 0.39, size = 32, normalized size = 0.12 \begin {gather*} {\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 {gather*}

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)

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

Giac [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((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 + x^2*e + d*x + c)*(b*x^4 + a)^p*(g*x)^m, x)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (g\,x\right )}^m\,{\left (b\,x^4+a\right )}^p\,\left (f\,x^3+e\,x^2+d\,x+c\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________