3.120 \(\int \left (a+b x^4\right )^2 \left (c+d x^4\right )^q \, dx\)

Optimal. Leaf size=176 \[ \frac{x \left (c+d x^4\right )^q \left (\frac{d x^4}{c}+1\right )^{-q} \left (a^2 d^2 \left (16 q^2+56 q+45\right )-2 a b c d (4 q+9)+5 b^2 c^2\right ) \, _2F_1\left (\frac{1}{4},-q;\frac{5}{4};-\frac{d x^4}{c}\right )}{d^2 (4 q+5) (4 q+9)}-\frac{b x \left (c+d x^4\right )^{q+1} (5 b c-a d (4 q+13))}{d^2 (4 q+5) (4 q+9)}+\frac{b x \left (a+b x^4\right ) \left (c+d x^4\right )^{q+1}}{d (4 q+9)} \]

[Out]

-((b*(5*b*c - a*d*(13 + 4*q))*x*(c + d*x^4)^(1 + q))/(d^2*(5 + 4*q)*(9 + 4*q)))
+ (b*x*(a + b*x^4)*(c + d*x^4)^(1 + q))/(d*(9 + 4*q)) + ((5*b^2*c^2 - 2*a*b*c*d*
(9 + 4*q) + a^2*d^2*(45 + 56*q + 16*q^2))*x*(c + d*x^4)^q*Hypergeometric2F1[1/4,
-q, 5/4, -((d*x^4)/c)])/(d^2*(5 + 4*q)*(9 + 4*q)*(1 + (d*x^4)/c)^q)

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Rubi [A]  time = 0.316393, antiderivative size = 176, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{x \left (c+d x^4\right )^q \left (\frac{d x^4}{c}+1\right )^{-q} \left (a^2 d^2 \left (16 q^2+56 q+45\right )-2 a b c d (4 q+9)+5 b^2 c^2\right ) \, _2F_1\left (\frac{1}{4},-q;\frac{5}{4};-\frac{d x^4}{c}\right )}{d^2 (4 q+5) (4 q+9)}-\frac{b x \left (c+d x^4\right )^{q+1} (5 b c-a d (4 q+13))}{d^2 (4 q+5) (4 q+9)}+\frac{b x \left (a+b x^4\right ) \left (c+d x^4\right )^{q+1}}{d (4 q+9)} \]

Antiderivative was successfully verified.

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

[Out]

-((b*(5*b*c - a*d*(13 + 4*q))*x*(c + d*x^4)^(1 + q))/(d^2*(5 + 4*q)*(9 + 4*q)))
+ (b*x*(a + b*x^4)*(c + d*x^4)^(1 + q))/(d*(9 + 4*q)) + ((5*b^2*c^2 - 2*a*b*c*d*
(9 + 4*q) + a^2*d^2*(45 + 56*q + 16*q^2))*x*(c + d*x^4)^q*Hypergeometric2F1[1/4,
-q, 5/4, -((d*x^4)/c)])/(d^2*(5 + 4*q)*(9 + 4*q)*(1 + (d*x^4)/c)^q)

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Rubi in Sympy [A]  time = 32.8228, size = 153, normalized size = 0.87 \[ \frac{b x \left (a + b x^{4}\right ) \left (c + d x^{4}\right )^{q + 1}}{d \left (4 q + 9\right )} - \frac{b x \left (c + d x^{4}\right )^{q + 1} \left (- a d \left (4 q + 13\right ) + 5 b c\right )}{d^{2} \left (4 q + 5\right ) \left (4 q + 9\right )} + \frac{x \left (1 + \frac{d x^{4}}{c}\right )^{- q} \left (c + d x^{4}\right )^{q} \left (- a d \left (4 q + 5\right ) \left (- a d \left (4 q + 9\right ) + b c\right ) + b c \left (- a d \left (4 q + 13\right ) + 5 b c\right )\right ){{}_{2}F_{1}\left (\begin{matrix} - q, \frac{1}{4} \\ \frac{5}{4} \end{matrix}\middle |{- \frac{d x^{4}}{c}} \right )}}{d^{2} \left (4 q + 5\right ) \left (4 q + 9\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b*x*(a + b*x**4)*(c + d*x**4)**(q + 1)/(d*(4*q + 9)) - b*x*(c + d*x**4)**(q + 1)
*(-a*d*(4*q + 13) + 5*b*c)/(d**2*(4*q + 5)*(4*q + 9)) + x*(1 + d*x**4/c)**(-q)*(
c + d*x**4)**q*(-a*d*(4*q + 5)*(-a*d*(4*q + 9) + b*c) + b*c*(-a*d*(4*q + 13) + 5
*b*c))*hyper((-q, 1/4), (5/4,), -d*x**4/c)/(d**2*(4*q + 5)*(4*q + 9))

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Mathematica [A]  time = 0.0788624, size = 106, normalized size = 0.6 \[ \frac{1}{45} x \left (c+d x^4\right )^q \left (\frac{d x^4}{c}+1\right )^{-q} \left (45 a^2 \, _2F_1\left (\frac{1}{4},-q;\frac{5}{4};-\frac{d x^4}{c}\right )+b x^4 \left (18 a \, _2F_1\left (\frac{5}{4},-q;\frac{9}{4};-\frac{d x^4}{c}\right )+5 b x^4 \, _2F_1\left (\frac{9}{4},-q;\frac{13}{4};-\frac{d x^4}{c}\right )\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(x*(c + d*x^4)^q*(45*a^2*Hypergeometric2F1[1/4, -q, 5/4, -((d*x^4)/c)] + b*x^4*(
18*a*Hypergeometric2F1[5/4, -q, 9/4, -((d*x^4)/c)] + 5*b*x^4*Hypergeometric2F1[9
/4, -q, 13/4, -((d*x^4)/c)])))/(45*(1 + (d*x^4)/c)^q)

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Maple [F]  time = 0.057, size = 0, normalized size = 0. \[ \int \left ( b{x}^{4}+a \right ) ^{2} \left ( d{x}^{4}+c \right ) ^{q}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^4+a)^2*(d*x^4+c)^q,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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