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

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*Hyperge
ometric2F1[(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 = 0.520783, 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 \[ \frac{c (g x)^{m+1} \left (a+b x^4\right )^p \left (\frac{b x^4}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+1}{4},-p;\frac{m+5}{4};-\frac{b x^4}{a}\right )}{g (m+1)}+\frac{d (g x)^{m+2} \left (a+b x^4\right )^p \left (\frac{b x^4}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+2}{4},-p;\frac{m+6}{4};-\frac{b x^4}{a}\right )}{g^2 (m+2)}+\frac{e (g x)^{m+3} \left (a+b x^4\right )^p \left (\frac{b x^4}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+3}{4},-p;\frac{m+7}{4};-\frac{b x^4}{a}\right )}{g^3 (m+3)}+\frac{f (g x)^{m+4} \left (a+b x^4\right )^p \left (\frac{b x^4}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+4}{4},-p;\frac{m+8}{4};-\frac{b x^4}{a}\right )}{g^4 (m+4)} \]

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*Hyperge
ometric2F1[(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 in Sympy [A]  time = 58.1998, size = 218, normalized size = 0.81 \[ \frac{c \left (g x\right )^{m + 1} \left (1 + \frac{b x^{4}}{a}\right )^{- p} \left (a + b x^{4}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{m}{4} + \frac{1}{4} \\ \frac{m}{4} + \frac{5}{4} \end{matrix}\middle |{- \frac{b x^{4}}{a}} \right )}}{g \left (m + 1\right )} + \frac{d \left (g x\right )^{m + 2} \left (1 + \frac{b x^{4}}{a}\right )^{- p} \left (a + b x^{4}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{m}{4} + \frac{1}{2} \\ \frac{m}{4} + \frac{3}{2} \end{matrix}\middle |{- \frac{b x^{4}}{a}} \right )}}{g^{2} \left (m + 2\right )} + \frac{e \left (g x\right )^{m + 3} \left (1 + \frac{b x^{4}}{a}\right )^{- p} \left (a + b x^{4}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{m}{4} + \frac{3}{4} \\ \frac{m}{4} + \frac{7}{4} \end{matrix}\middle |{- \frac{b x^{4}}{a}} \right )}}{g^{3} \left (m + 3\right )} + \frac{f \left (g x\right )^{m + 4} \left (1 + \frac{b x^{4}}{a}\right )^{- p} \left (a + b x^{4}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{m}{4} + 1 \\ \frac{m}{4} + 2 \end{matrix}\middle |{- \frac{b x^{4}}{a}} \right )}}{g^{4} \left (m + 4\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c*(g*x)**(m + 1)*(1 + b*x**4/a)**(-p)*(a + b*x**4)**p*hyper((-p, m/4 + 1/4), (m/
4 + 5/4,), -b*x**4/a)/(g*(m + 1)) + d*(g*x)**(m + 2)*(1 + b*x**4/a)**(-p)*(a + b
*x**4)**p*hyper((-p, m/4 + 1/2), (m/4 + 3/2,), -b*x**4/a)/(g**2*(m + 2)) + e*(g*
x)**(m + 3)*(1 + b*x**4/a)**(-p)*(a + b*x**4)**p*hyper((-p, m/4 + 3/4), (m/4 + 7
/4,), -b*x**4/a)/(g**3*(m + 3)) + f*(g*x)**(m + 4)*(1 + b*x**4/a)**(-p)*(a + b*x
**4)**p*hyper((-p, m/4 + 1), (m/4 + 2,), -b*x**4/a)/(g**4*(m + 4))

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Mathematica [A]  time = 0.345973, size = 174, normalized size = 0.65 \[ x (g x)^m \left (a+b x^4\right )^p \left (\frac{b x^4}{a}+1\right )^{-p} \left (\frac{c \, _2F_1\left (\frac{m+1}{4},-p;\frac{m+5}{4};-\frac{b x^4}{a}\right )}{m+1}+x \left (\frac{d \, _2F_1\left (\frac{m+2}{4},-p;\frac{m+6}{4};-\frac{b x^4}{a}\right )}{m+2}+\frac{e x \, _2F_1\left (\frac{m+3}{4},-p;\frac{m+7}{4};-\frac{b x^4}{a}\right )}{m+3}\right )+\frac{f x^3 \, _2F_1\left (\frac{m}{4}+1,-p;\frac{m}{4}+2;-\frac{b x^4}{a}\right )}{m+4}\right ) \]

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

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Maple [F]  time = 0.099, size = 0, normalized size = 0. \[ \int \left ( gx \right ) ^{m} \left ( f{x}^{3}+e{x}^{2}+dx+c \right ) \left ( b{x}^{4}+a \right ) ^{p}\, dx \]

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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^3 + e*x^2 + d*x + c)*(b*x^4 + a)^p*(g*x)^m,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. \[{\rm integral}\left ({\left (f x^{3} + e x^{2} + d x + c\right )}{\left (b x^{4} + a\right )}^{p} \left (g x\right )^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^3 + e*x^2 + d*x + c)*(b*x^4 + a)^p*(g*x)^m,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. \[ \text{Timed out} \]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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