3.184 \(\int \frac{\left (a+b x^4\right )^p}{c+e x^2} \, dx\)

Optimal. Leaf size=123 \[ \frac{x \left (a+b x^4\right )^p \left (\frac{b x^4}{a}+1\right )^{-p} F_1\left (\frac{1}{4};-p,1;\frac{5}{4};-\frac{b x^4}{a},\frac{e^2 x^4}{c^2}\right )}{c}-\frac{e x^3 \left (a+b x^4\right )^p \left (\frac{b x^4}{a}+1\right )^{-p} F_1\left (\frac{3}{4};-p,1;\frac{7}{4};-\frac{b x^4}{a},\frac{e^2 x^4}{c^2}\right )}{3 c^2} \]

[Out]

(x*(a + b*x^4)^p*AppellF1[1/4, -p, 1, 5/4, -((b*x^4)/a), (e^2*x^4)/c^2])/(c*(1 +
(b*x^4)/a)^p) - (e*x^3*(a + b*x^4)^p*AppellF1[3/4, -p, 1, 7/4, -((b*x^4)/a), (e
^2*x^4)/c^2])/(3*c^2*(1 + (b*x^4)/a)^p)

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Rubi [A]  time = 0.280909, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ \frac{x \left (a+b x^4\right )^p \left (\frac{b x^4}{a}+1\right )^{-p} F_1\left (\frac{1}{4};-p,1;\frac{5}{4};-\frac{b x^4}{a},\frac{e^2 x^4}{c^2}\right )}{c}-\frac{e x^3 \left (a+b x^4\right )^p \left (\frac{b x^4}{a}+1\right )^{-p} F_1\left (\frac{3}{4};-p,1;\frac{7}{4};-\frac{b x^4}{a},\frac{e^2 x^4}{c^2}\right )}{3 c^2} \]

Antiderivative was successfully verified.

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

[Out]

(x*(a + b*x^4)^p*AppellF1[1/4, -p, 1, 5/4, -((b*x^4)/a), (e^2*x^4)/c^2])/(c*(1 +
(b*x^4)/a)^p) - (e*x^3*(a + b*x^4)^p*AppellF1[3/4, -p, 1, 7/4, -((b*x^4)/a), (e
^2*x^4)/c^2])/(3*c^2*(1 + (b*x^4)/a)^p)

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Rubi in Sympy [A]  time = 57.1198, size = 97, normalized size = 0.79 \[ \frac{x \left (1 + \frac{b x^{4}}{a}\right )^{- p} \left (a + b x^{4}\right )^{p} \operatorname{appellf_{1}}{\left (\frac{1}{4},1,- p,\frac{5}{4},\frac{e^{2} x^{4}}{c^{2}},- \frac{b x^{4}}{a} \right )}}{c} - \frac{e x^{3} \left (1 + \frac{b x^{4}}{a}\right )^{- p} \left (a + b x^{4}\right )^{p} \operatorname{appellf_{1}}{\left (\frac{3}{4},1,- p,\frac{7}{4},\frac{e^{2} x^{4}}{c^{2}},- \frac{b x^{4}}{a} \right )}}{3 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(1 + b*x**4/a)**(-p)*(a + b*x**4)**p*appellf1(1/4, 1, -p, 5/4, e**2*x**4/c**2,
-b*x**4/a)/c - e*x**3*(1 + b*x**4/a)**(-p)*(a + b*x**4)**p*appellf1(3/4, 1, -p,
7/4, e**2*x**4/c**2, -b*x**4/a)/(3*c**2)

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Mathematica [A]  time = 0.0454001, size = 0, normalized size = 0. \[ \int \frac{\left (a+b x^4\right )^p}{c+e x^2} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.066, size = 0, normalized size = 0. \[ \int{\frac{ \left ( b{x}^{4}+a \right ) ^{p}}{e{x}^{2}+c}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^4+a)^p/(e*x^2+c),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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