3.119 \(\int \left (a+b x^4\right )^p \left (c+d x^4\right )^q \, dx\)

Optimal. Leaf size=79 \[ x \left (a+b x^4\right )^p \left (\frac{b x^4}{a}+1\right )^{-p} \left (c+d x^4\right )^q \left (\frac{d x^4}{c}+1\right )^{-q} F_1\left (\frac{1}{4};-p,-q;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right ) \]

[Out]

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

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Rubi [A]  time = 0.12334, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ x \left (a+b x^4\right )^p \left (\frac{b x^4}{a}+1\right )^{-p} \left (c+d x^4\right )^q \left (\frac{d x^4}{c}+1\right )^{-q} F_1\left (\frac{1}{4};-p,-q;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 21.9447, size = 61, normalized size = 0.77 \[ x \left (1 + \frac{b x^{4}}{a}\right )^{- p} \left (1 + \frac{d x^{4}}{c}\right )^{- q} \left (a + b x^{4}\right )^{p} \left (c + d x^{4}\right )^{q} \operatorname{appellf_{1}}{\left (\frac{1}{4},- p,- q,\frac{5}{4},- \frac{b x^{4}}{a},- \frac{d x^{4}}{c} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(1 + b*x**4/a)**(-p)*(1 + d*x**4/c)**(-q)*(a + b*x**4)**p*(c + d*x**4)**q*appe
llf1(1/4, -p, -q, 5/4, -b*x**4/a, -d*x**4/c)

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Mathematica [B]  time = 0.358687, size = 172, normalized size = 2.18 \[ \frac{5 a c x \left (a+b x^4\right )^p \left (c+d x^4\right )^q F_1\left (\frac{1}{4};-p,-q;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{4 x^4 \left (b c p F_1\left (\frac{5}{4};1-p,-q;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )+a d q F_1\left (\frac{5}{4};-p,1-q;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )+5 a c F_1\left (\frac{1}{4};-p,-q;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(5*a*c*x*(a + b*x^4)^p*(c + d*x^4)^q*AppellF1[1/4, -p, -q, 5/4, -((b*x^4)/a), -(
(d*x^4)/c)])/(5*a*c*AppellF1[1/4, -p, -q, 5/4, -((b*x^4)/a), -((d*x^4)/c)] + 4*x
^4*(b*c*p*AppellF1[5/4, 1 - p, -q, 9/4, -((b*x^4)/a), -((d*x^4)/c)] + a*d*q*Appe
llF1[5/4, -p, 1 - q, 9/4, -((b*x^4)/a), -((d*x^4)/c)]))

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Maple [F]  time = 0.115, size = 0, normalized size = 0. \[ \int \left ( b{x}^{4}+a \right ) ^{p} \left ( d{x}^{4}+c \right ) ^{q}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^4+a)^p*(d*x^4+c)^q,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b*x^4 + a)^p*(d*x^4 + c)^q, 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*(d*x**4+c)**q,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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