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

Optimal. Leaf size=204 \[ c^3 x \left (a+b x^4\right )^p \left (\frac{b x^4}{a}+1\right )^{-p} \, _2F_1\left (\frac{1}{4},-p;\frac{5}{4};-\frac{b x^4}{a}\right )-\frac{e x^3 \left (a+b x^4\right )^p \left (\frac{b x^4}{a}+1\right )^{-p} \left (a e^2-b c^2 (4 p+7)\right ) \, _2F_1\left (\frac{3}{4},-p;\frac{7}{4};-\frac{b x^4}{a}\right )}{b (4 p+7)}+\frac{3}{5} c e^2 x^5 \left (a+b x^4\right )^p \left (\frac{b x^4}{a}+1\right )^{-p} \, _2F_1\left (\frac{5}{4},-p;\frac{9}{4};-\frac{b x^4}{a}\right )+\frac{e^3 x^3 \left (a+b x^4\right )^{p+1}}{b (4 p+7)} \]

[Out]

(e^3*x^3*(a + b*x^4)^(1 + p))/(b*(7 + 4*p)) + (c^3*x*(a + b*x^4)^p*Hypergeometri
c2F1[1/4, -p, 5/4, -((b*x^4)/a)])/(1 + (b*x^4)/a)^p - (e*(a*e^2 - b*c^2*(7 + 4*p
))*x^3*(a + b*x^4)^p*Hypergeometric2F1[3/4, -p, 7/4, -((b*x^4)/a)])/(b*(7 + 4*p)
*(1 + (b*x^4)/a)^p) + (3*c*e^2*x^5*(a + b*x^4)^p*Hypergeometric2F1[5/4, -p, 9/4,
-((b*x^4)/a)])/(5*(1 + (b*x^4)/a)^p)

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Rubi [A]  time = 0.406175, antiderivative size = 196, normalized size of antiderivative = 0.96, number of steps used = 9, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316 \[ c^3 x \left (a+b x^4\right )^p \left (\frac{b x^4}{a}+1\right )^{-p} \, _2F_1\left (\frac{1}{4},-p;\frac{5}{4};-\frac{b x^4}{a}\right )+e x^3 \left (a+b x^4\right )^p \left (\frac{b x^4}{a}+1\right )^{-p} \left (c^2-\frac{a e^2}{4 b p+7 b}\right ) \, _2F_1\left (\frac{3}{4},-p;\frac{7}{4};-\frac{b x^4}{a}\right )+\frac{3}{5} c e^2 x^5 \left (a+b x^4\right )^p \left (\frac{b x^4}{a}+1\right )^{-p} \, _2F_1\left (\frac{5}{4},-p;\frac{9}{4};-\frac{b x^4}{a}\right )+\frac{e^3 x^3 \left (a+b x^4\right )^{p+1}}{b (4 p+7)} \]

Antiderivative was successfully verified.

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

[Out]

(e^3*x^3*(a + b*x^4)^(1 + p))/(b*(7 + 4*p)) + (c^3*x*(a + b*x^4)^p*Hypergeometri
c2F1[1/4, -p, 5/4, -((b*x^4)/a)])/(1 + (b*x^4)/a)^p + (e*(c^2 - (a*e^2)/(7*b + 4
*b*p))*x^3*(a + b*x^4)^p*Hypergeometric2F1[3/4, -p, 7/4, -((b*x^4)/a)])/(1 + (b*
x^4)/a)^p + (3*c*e^2*x^5*(a + b*x^4)^p*Hypergeometric2F1[5/4, -p, 9/4, -((b*x^4)
/a)])/(5*(1 + (b*x^4)/a)^p)

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Rubi in Sympy [A]  time = 33.9907, size = 168, normalized size = 0.82 \[ c^{3} x \left (1 + \frac{b x^{4}}{a}\right )^{- p} \left (a + b x^{4}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{1}{4} \\ \frac{5}{4} \end{matrix}\middle |{- \frac{b x^{4}}{a}} \right )} + c^{2} e x^{3} \left (1 + \frac{b x^{4}}{a}\right )^{- p} \left (a + b x^{4}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{- \frac{b x^{4}}{a}} \right )} + \frac{3 c e^{2} x^{5} \left (1 + \frac{b x^{4}}{a}\right )^{- p} \left (a + b x^{4}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{- \frac{b x^{4}}{a}} \right )}}{5} + \frac{e^{3} x^{7} \left (1 + \frac{b x^{4}}{a}\right )^{- p} \left (a + b x^{4}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{- \frac{b x^{4}}{a}} \right )}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0934814, size = 136, normalized size = 0.67 \[ \frac{1}{35} x \left (a+b x^4\right )^p \left (\frac{b x^4}{a}+1\right )^{-p} \left (35 c^3 \, _2F_1\left (\frac{1}{4},-p;\frac{5}{4};-\frac{b x^4}{a}\right )+e x^2 \left (35 c^2 \, _2F_1\left (\frac{3}{4},-p;\frac{7}{4};-\frac{b x^4}{a}\right )+e x^2 \left (21 c \, _2F_1\left (\frac{5}{4},-p;\frac{9}{4};-\frac{b x^4}{a}\right )+5 e x^2 \, _2F_1\left (\frac{7}{4},-p;\frac{11}{4};-\frac{b x^4}{a}\right )\right )\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(x*(a + b*x^4)^p*(35*c^3*Hypergeometric2F1[1/4, -p, 5/4, -((b*x^4)/a)] + e*x^2*(
35*c^2*Hypergeometric2F1[3/4, -p, 7/4, -((b*x^4)/a)] + e*x^2*(21*c*Hypergeometri
c2F1[5/4, -p, 9/4, -((b*x^4)/a)] + 5*e*x^2*Hypergeometric2F1[7/4, -p, 11/4, -((b
*x^4)/a)]))))/(35*(1 + (b*x^4)/a)^p)

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Maple [F]  time = 0.073, size = 0, normalized size = 0. \[ \int \left ( e{x}^{2}+c \right ) ^{3} \left ( b{x}^{4}+a \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x^2+c)^3*(b*x^4+a)^p,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x^2 + c)^3*(b*x^4 + a)^p,x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (e^{3} x^{6} + 3 \, c e^{2} x^{4} + 3 \, c^{2} e x^{2} + c^{3}\right )}{\left (b x^{4} + a\right )}^{p}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x^2 + c)^3*(b*x^4 + a)^p,x, algorithm="fricas")

[Out]

integral((e^3*x^6 + 3*c*e^2*x^4 + 3*c^2*e*x^2 + c^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((e*x**2+c)**3*(b*x**4+a)**p,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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