3.123 \(\int \frac{\left (c+d x^4\right )^q}{\left (a+b x^4\right )^2} \, dx\)

Optimal. Leaf size=57 \[ \frac{x \left (c+d x^4\right )^q \left (\frac{d x^4}{c}+1\right )^{-q} F_1\left (\frac{1}{4};2,-q;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{a^2} \]

[Out]

(x*(c + d*x^4)^q*AppellF1[1/4, 2, -q, 5/4, -((b*x^4)/a), -((d*x^4)/c)])/(a^2*(1
+ (d*x^4)/c)^q)

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Rubi [A]  time = 0.0831709, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{x \left (c+d x^4\right )^q \left (\frac{d x^4}{c}+1\right )^{-q} F_1\left (\frac{1}{4};2,-q;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{a^2} \]

Antiderivative was successfully verified.

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

[Out]

(x*(c + d*x^4)^q*AppellF1[1/4, 2, -q, 5/4, -((b*x^4)/a), -((d*x^4)/c)])/(a^2*(1
+ (d*x^4)/c)^q)

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Rubi in Sympy [A]  time = 20.2223, size = 44, normalized size = 0.77 \[ \frac{x \left (1 + \frac{d x^{4}}{c}\right )^{- q} \left (c + d x^{4}\right )^{q} \operatorname{appellf_{1}}{\left (\frac{1}{4},2,- q,\frac{5}{4},- \frac{b x^{4}}{a},- \frac{d x^{4}}{c} \right )}}{a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(1 + d*x**4/c)**(-q)*(c + d*x**4)**q*appellf1(1/4, 2, -q, 5/4, -b*x**4/a, -d*x
**4/c)/a**2

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Mathematica [B]  time = 0.307696, size = 162, normalized size = 2.84 \[ \frac{5 a c x \left (c+d x^4\right )^q F_1\left (\frac{1}{4};2,-q;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{\left (a+b x^4\right )^2 \left (4 x^4 \left (a d q F_1\left (\frac{5}{4};2,1-q;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )-2 b c F_1\left (\frac{5}{4};3,-q;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )+5 a c F_1\left (\frac{1}{4};2,-q;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(5*a*c*x*(c + d*x^4)^q*AppellF1[1/4, 2, -q, 5/4, -((b*x^4)/a), -((d*x^4)/c)])/((
a + b*x^4)^2*(5*a*c*AppellF1[1/4, 2, -q, 5/4, -((b*x^4)/a), -((d*x^4)/c)] + 4*x^
4*(a*d*q*AppellF1[5/4, 2, 1 - q, 9/4, -((b*x^4)/a), -((d*x^4)/c)] - 2*b*c*Appell
F1[5/4, 3, -q, 9/4, -((b*x^4)/a), -((d*x^4)/c)])))

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Maple [F]  time = 0.064, size = 0, normalized size = 0. \[ \int{\frac{ \left ( d{x}^{4}+c \right ) ^{q}}{ \left ( b{x}^{4}+a \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x^4+c)^q/(b*x^4+a)^2,x)

[Out]

int((d*x^4+c)^q/(b*x^4+a)^2,x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((d*x^4 + c)^q/(b*x^4 + a)^2, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (d x^{4} + c\right )}^{q}}{b^{2} x^{8} + 2 \, a b x^{4} + a^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((d*x^4 + c)^q/(b^2*x^8 + 2*a*b*x^4 + a^2), 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((d*x**4+c)**q/(b*x**4+a)**2,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((d*x^4 + c)^q/(b*x^4 + a)^2, x)