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

Optimal. Leaf size=175 \[ \frac{c \left (a+b x^4\right )^{p+1}}{4 b (p+1)}+\frac{1}{5} d x^5 \left (a+b x^4\right )^p \left (\frac{b x^4}{a}+1\right )^{-p} \, _2F_1\left (\frac{5}{4},-p;\frac{9}{4};-\frac{b x^4}{a}\right )+\frac{1}{6} e x^6 \left (a+b x^4\right )^p \left (\frac{b x^4}{a}+1\right )^{-p} \, _2F_1\left (\frac{3}{2},-p;\frac{5}{2};-\frac{b x^4}{a}\right )+\frac{1}{7} f x^7 \left (a+b x^4\right )^p \left (\frac{b x^4}{a}+1\right )^{-p} \, _2F_1\left (\frac{7}{4},-p;\frac{11}{4};-\frac{b x^4}{a}\right ) \]

[Out]

(c*(a + b*x^4)^(1 + p))/(4*b*(1 + p)) + (d*x^5*(a + b*x^4)^p*Hypergeometric2F1[5
/4, -p, 9/4, -((b*x^4)/a)])/(5*(1 + (b*x^4)/a)^p) + (e*x^6*(a + b*x^4)^p*Hyperge
ometric2F1[3/2, -p, 5/2, -((b*x^4)/a)])/(6*(1 + (b*x^4)/a)^p) + (f*x^7*(a + b*x^
4)^p*Hypergeometric2F1[7/4, -p, 11/4, -((b*x^4)/a)])/(7*(1 + (b*x^4)/a)^p)

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Rubi [A]  time = 0.403432, antiderivative size = 175, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{c \left (a+b x^4\right )^{p+1}}{4 b (p+1)}+\frac{1}{5} d x^5 \left (a+b x^4\right )^p \left (\frac{b x^4}{a}+1\right )^{-p} \, _2F_1\left (\frac{5}{4},-p;\frac{9}{4};-\frac{b x^4}{a}\right )+\frac{1}{6} e x^6 \left (a+b x^4\right )^p \left (\frac{b x^4}{a}+1\right )^{-p} \, _2F_1\left (\frac{3}{2},-p;\frac{5}{2};-\frac{b x^4}{a}\right )+\frac{1}{7} f x^7 \left (a+b x^4\right )^p \left (\frac{b x^4}{a}+1\right )^{-p} \, _2F_1\left (\frac{7}{4},-p;\frac{11}{4};-\frac{b x^4}{a}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(c*(a + b*x^4)^(1 + p))/(4*b*(1 + p)) + (d*x^5*(a + b*x^4)^p*Hypergeometric2F1[5
/4, -p, 9/4, -((b*x^4)/a)])/(5*(1 + (b*x^4)/a)^p) + (e*x^6*(a + b*x^4)^p*Hyperge
ometric2F1[3/2, -p, 5/2, -((b*x^4)/a)])/(6*(1 + (b*x^4)/a)^p) + (f*x^7*(a + b*x^
4)^p*Hypergeometric2F1[7/4, -p, 11/4, -((b*x^4)/a)])/(7*(1 + (b*x^4)/a)^p)

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Rubi in Sympy [A]  time = 46.9727, size = 139, normalized size = 0.79 \[ \frac{d x^{5} \left (1 + \frac{b x^{4}}{a}\right )^{- p} \left (a + b x^{4}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{- \frac{b x^{4}}{a}} \right )}}{5} + \frac{e x^{6} \left (1 + \frac{b x^{4}}{a}\right )^{- p} \left (a + b x^{4}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{- \frac{b x^{4}}{a}} \right )}}{6} + \frac{f x^{7} \left (1 + \frac{b x^{4}}{a}\right )^{- p} \left (a + b x^{4}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{- \frac{b x^{4}}{a}} \right )}}{7} + \frac{c \left (a + b x^{4}\right )^{p + 1}}{4 b \left (p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d*x**5*(1 + b*x**4/a)**(-p)*(a + b*x**4)**p*hyper((-p, 5/4), (9/4,), -b*x**4/a)/
5 + e*x**6*(1 + b*x**4/a)**(-p)*(a + b*x**4)**p*hyper((-p, 3/2), (5/2,), -b*x**4
/a)/6 + f*x**7*(1 + b*x**4/a)**(-p)*(a + b*x**4)**p*hyper((-p, 7/4), (11/4,), -b
*x**4/a)/7 + c*(a + b*x**4)**(p + 1)/(4*b*(p + 1))

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Mathematica [A]  time = 0.239194, size = 160, normalized size = 0.91 \[ \frac{\left (a+b x^4\right )^p \left (\frac{b x^4}{a}+1\right )^{-p} \left (105 c \left (b x^4 \left (\frac{b x^4}{a}+1\right )^p+a \left (\left (\frac{b x^4}{a}+1\right )^p-1\right )\right )+84 b d (p+1) x^5 \, _2F_1\left (\frac{5}{4},-p;\frac{9}{4};-\frac{b x^4}{a}\right )+70 b e (p+1) x^6 \, _2F_1\left (\frac{3}{2},-p;\frac{5}{2};-\frac{b x^4}{a}\right )+60 b f (p+1) x^7 \, _2F_1\left (\frac{7}{4},-p;\frac{11}{4};-\frac{b x^4}{a}\right )\right )}{420 b (p+1)} \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x^4)^p*(105*c*(b*x^4*(1 + (b*x^4)/a)^p + a*(-1 + (1 + (b*x^4)/a)^p)) + 8
4*b*d*(1 + p)*x^5*Hypergeometric2F1[5/4, -p, 9/4, -((b*x^4)/a)] + 70*b*e*(1 + p)
*x^6*Hypergeometric2F1[3/2, -p, 5/2, -((b*x^4)/a)] + 60*b*f*(1 + p)*x^7*Hypergeo
metric2F1[7/4, -p, 11/4, -((b*x^4)/a)]))/(420*b*(1 + p)*(1 + (b*x^4)/a)^p)

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Maple [F]  time = 0.077, size = 0, normalized size = 0. \[ \int{x}^{3} \left ( f{x}^{3}+e{x}^{2}+dx+c \right ) \left ( b{x}^{4}+a \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(f*x^3+e*x^2+d*x+c)*(b*x^4+a)^p,x)

[Out]

int(x^3*(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. \[ \text{Exception raised: ValueError} \]

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*x^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (f x^{6} + e x^{5} + d x^{4} + c x^{3}\right )}{\left (b x^{4} + a\right )}^{p}, x\right ) \]

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*x^3,x, algorithm="fricas")

[Out]

integral((f*x^6 + e*x^5 + d*x^4 + c*x^3)*(b*x^4 + a)^p, 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(x**3*(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. \[ \int{\left (f x^{3} + e x^{2} + d x + c\right )}{\left (b x^{4} + a\right )}^{p} x^{3}\,{d x} \]

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*x^3,x, algorithm="giac")

[Out]

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