Optimal. Leaf size=176 \[ \frac{x \left (c+d x^4\right )^q \left (\frac{d x^4}{c}+1\right )^{-q} \left (a^2 d^2 \left (16 q^2+56 q+45\right )-2 a b c d (4 q+9)+5 b^2 c^2\right ) \, _2F_1\left (\frac{1}{4},-q;\frac{5}{4};-\frac{d x^4}{c}\right )}{d^2 (4 q+5) (4 q+9)}-\frac{b x \left (c+d x^4\right )^{q+1} (5 b c-a d (4 q+13))}{d^2 (4 q+5) (4 q+9)}+\frac{b x \left (a+b x^4\right ) \left (c+d x^4\right )^{q+1}}{d (4 q+9)} \]
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Rubi [A] time = 0.133064, 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, Rules used = {416, 388, 246, 245} \[ \frac{x \left (c+d x^4\right )^q \left (\frac{d x^4}{c}+1\right )^{-q} \left (a^2 d^2 \left (16 q^2+56 q+45\right )-2 a b c d (4 q+9)+5 b^2 c^2\right ) \, _2F_1\left (\frac{1}{4},-q;\frac{5}{4};-\frac{d x^4}{c}\right )}{d^2 (4 q+5) (4 q+9)}-\frac{b x \left (c+d x^4\right )^{q+1} (5 b c-a d (4 q+13))}{d^2 (4 q+5) (4 q+9)}+\frac{b x \left (a+b x^4\right ) \left (c+d x^4\right )^{q+1}}{d (4 q+9)} \]
Antiderivative was successfully verified.
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Rule 416
Rule 388
Rule 246
Rule 245
Rubi steps
\begin{align*} \int \left (a+b x^4\right )^2 \left (c+d x^4\right )^q \, dx &=\frac{b x \left (a+b x^4\right ) \left (c+d x^4\right )^{1+q}}{d (9+4 q)}+\frac{\int \left (c+d x^4\right )^q \left (-a (b c-a d (9+4 q))-b (5 b c-a d (13+4 q)) x^4\right ) \, dx}{d (9+4 q)}\\ &=-\frac{b (5 b c-a d (13+4 q)) x \left (c+d x^4\right )^{1+q}}{d^2 (5+4 q) (9+4 q)}+\frac{b x \left (a+b x^4\right ) \left (c+d x^4\right )^{1+q}}{d (9+4 q)}+\frac{\left (5 b^2 c^2-2 a b c d (9+4 q)+a^2 d^2 \left (45+56 q+16 q^2\right )\right ) \int \left (c+d x^4\right )^q \, dx}{d^2 (5+4 q) (9+4 q)}\\ &=-\frac{b (5 b c-a d (13+4 q)) x \left (c+d x^4\right )^{1+q}}{d^2 (5+4 q) (9+4 q)}+\frac{b x \left (a+b x^4\right ) \left (c+d x^4\right )^{1+q}}{d (9+4 q)}+\frac{\left (\left (5 b^2 c^2-2 a b c d (9+4 q)+a^2 d^2 \left (45+56 q+16 q^2\right )\right ) \left (c+d x^4\right )^q \left (1+\frac{d x^4}{c}\right )^{-q}\right ) \int \left (1+\frac{d x^4}{c}\right )^q \, dx}{d^2 (5+4 q) (9+4 q)}\\ &=-\frac{b (5 b c-a d (13+4 q)) x \left (c+d x^4\right )^{1+q}}{d^2 (5+4 q) (9+4 q)}+\frac{b x \left (a+b x^4\right ) \left (c+d x^4\right )^{1+q}}{d (9+4 q)}+\frac{\left (5 b^2 c^2-2 a b c d (9+4 q)+a^2 d^2 \left (45+56 q+16 q^2\right )\right ) x \left (c+d x^4\right )^q \left (1+\frac{d x^4}{c}\right )^{-q} \, _2F_1\left (\frac{1}{4},-q;\frac{5}{4};-\frac{d x^4}{c}\right )}{d^2 (5+4 q) (9+4 q)}\\ \end{align*}
Mathematica [A] time = 0.052535, 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.
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Maple [F] time = 0.361, size = 0, normalized size = 0. \begin{align*} \int \left ( b{x}^{4}+a \right ) ^{2} \left ( d{x}^{4}+c \right ) ^{q}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x^{4} + a\right )}^{2}{\left (d x^{4} + c\right )}^{q}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x^{4} + a\right )}^{2}{\left (d x^{4} + c\right )}^{q}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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