3.182 \(\int \left (c+e x^2\right ) \left (a+b x^4\right )^p \, dx\)

Optimal. Leaf size=96 \[ c x \left (a+b x^4\right )^p \left (\frac{b x^4}{a}+1\right )^{-p} \, _2F_1\left (\frac{1}{4},-p;\frac{5}{4};-\frac{b x^4}{a}\right )+\frac{1}{3} e x^3 \left (a+b x^4\right )^p \left (\frac{b x^4}{a}+1\right )^{-p} \, _2F_1\left (\frac{3}{4},-p;\frac{7}{4};-\frac{b x^4}{a}\right ) \]

[Out]

(c*x*(a + b*x^4)^p*Hypergeometric2F1[1/4, -p, 5/4, -((b*x^4)/a)])/(1 + (b*x^4)/a
)^p + (e*x^3*(a + b*x^4)^p*Hypergeometric2F1[3/4, -p, 7/4, -((b*x^4)/a)])/(3*(1
+ (b*x^4)/a)^p)

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Rubi [A]  time = 0.112348, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294 \[ c x \left (a+b x^4\right )^p \left (\frac{b x^4}{a}+1\right )^{-p} \, _2F_1\left (\frac{1}{4},-p;\frac{5}{4};-\frac{b x^4}{a}\right )+\frac{1}{3} e x^3 \left (a+b x^4\right )^p \left (\frac{b x^4}{a}+1\right )^{-p} \, _2F_1\left (\frac{3}{4},-p;\frac{7}{4};-\frac{b x^4}{a}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(c*x*(a + b*x^4)^p*Hypergeometric2F1[1/4, -p, 5/4, -((b*x^4)/a)])/(1 + (b*x^4)/a
)^p + (e*x^3*(a + b*x^4)^p*Hypergeometric2F1[3/4, -p, 7/4, -((b*x^4)/a)])/(3*(1
+ (b*x^4)/a)^p)

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Rubi in Sympy [A]  time = 14.8423, size = 76, normalized size = 0.79 \[ c x \left (1 + \frac{b x^{4}}{a}\right )^{- p} \left (a + b x^{4}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{1}{4} \\ \frac{5}{4} \end{matrix}\middle |{- \frac{b x^{4}}{a}} \right )} + \frac{e x^{3} \left (1 + \frac{b x^{4}}{a}\right )^{- p} \left (a + b x^{4}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{- \frac{b x^{4}}{a}} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c*x*(1 + b*x**4/a)**(-p)*(a + b*x**4)**p*hyper((-p, 1/4), (5/4,), -b*x**4/a) + e
*x**3*(1 + b*x**4/a)**(-p)*(a + b*x**4)**p*hyper((-p, 3/4), (7/4,), -b*x**4/a)/3

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Mathematica [A]  time = 0.0298074, size = 75, normalized size = 0.78 \[ \frac{1}{3} x \left (a+b x^4\right )^p \left (\frac{b x^4}{a}+1\right )^{-p} \left (3 c \, _2F_1\left (\frac{1}{4},-p;\frac{5}{4};-\frac{b x^4}{a}\right )+e x^2 \, _2F_1\left (\frac{3}{4},-p;\frac{7}{4};-\frac{b x^4}{a}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(x*(a + b*x^4)^p*(3*c*Hypergeometric2F1[1/4, -p, 5/4, -((b*x^4)/a)] + e*x^2*Hype
rgeometric2F1[3/4, -p, 7/4, -((b*x^4)/a)]))/(3*(1 + (b*x^4)/a)^p)

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Maple [F]  time = 0.044, size = 0, normalized size = 0. \[ \int \left ( e{x}^{2}+c \right ) \left ( b{x}^{4}+a \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x^2+c)*(b*x^4+a)^p,x)

[Out]

int((e*x^2+c)*(b*x^4+a)^p,x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x^2 + c)*(b*x^4 + a)^p,x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (e x^{2} + c\right )}{\left (b x^{4} + a\right )}^{p}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x^2 + c)*(b*x^4 + a)^p,x, algorithm="fricas")

[Out]

integral((e*x^2 + c)*(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((e*x**2+c)*(b*x**4+a)**p,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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