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

Optimal. Leaf size=93 \[ \frac{b x \left (c+d x^4\right )^{q+1}}{d (4 q+5)}-\frac{x \left (c+d x^4\right )^q \left (\frac{d x^4}{c}+1\right )^{-q} (b c-a d (4 q+5)) \, _2F_1\left (\frac{1}{4},-q;\frac{5}{4};-\frac{d x^4}{c}\right )}{d (4 q+5)} \]

[Out]

(b*x*(c + d*x^4)^(1 + q))/(d*(5 + 4*q)) - ((b*c - a*d*(5 + 4*q))*x*(c + d*x^4)^q
*Hypergeometric2F1[1/4, -q, 5/4, -((d*x^4)/c)])/(d*(5 + 4*q)*(1 + (d*x^4)/c)^q)

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Rubi [A]  time = 0.111122, antiderivative size = 85, normalized size of antiderivative = 0.91, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ x \left (c+d x^4\right )^q \left (\frac{d x^4}{c}+1\right )^{-q} \left (a-\frac{b c}{4 d q+5 d}\right ) \, _2F_1\left (\frac{1}{4},-q;\frac{5}{4};-\frac{d x^4}{c}\right )+\frac{b x \left (c+d x^4\right )^{q+1}}{d (4 q+5)} \]

Antiderivative was successfully verified.

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

[Out]

(b*x*(c + d*x^4)^(1 + q))/(d*(5 + 4*q)) + ((a - (b*c)/(5*d + 4*d*q))*x*(c + d*x^
4)^q*Hypergeometric2F1[1/4, -q, 5/4, -((d*x^4)/c)])/(1 + (d*x^4)/c)^q

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Rubi in Sympy [A]  time = 11.3714, size = 73, normalized size = 0.78 \[ \frac{b x \left (c + d x^{4}\right )^{q + 1}}{d \left (4 q + 5\right )} - \frac{x \left (1 + \frac{d x^{4}}{c}\right )^{- q} \left (c + d x^{4}\right )^{q} \left (- a d \left (4 q + 5\right ) + b c\right ){{}_{2}F_{1}\left (\begin{matrix} - q, \frac{1}{4} \\ \frac{5}{4} \end{matrix}\middle |{- \frac{d x^{4}}{c}} \right )}}{d \left (4 q + 5\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0347268, size = 75, normalized size = 0.81 \[ \frac{1}{5} x \left (c+d x^4\right )^q \left (\frac{d x^4}{c}+1\right )^{-q} \left (5 a \, _2F_1\left (\frac{1}{4},-q;\frac{5}{4};-\frac{d x^4}{c}\right )+b x^4 \, _2F_1\left (\frac{5}{4},-q;\frac{9}{4};-\frac{d x^4}{c}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(x*(c + d*x^4)^q*(5*a*Hypergeometric2F1[1/4, -q, 5/4, -((d*x^4)/c)] + b*x^4*Hype
rgeometric2F1[5/4, -q, 9/4, -((d*x^4)/c)]))/(5*(1 + (d*x^4)/c)^q)

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Maple [F]  time = 0.049, size = 0, normalized size = 0. \[ \int \left ( b{x}^{4}+a \right ) \left ( d{x}^{4}+c \right ) ^{q}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^4+a)*(d*x^4+c)^q,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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