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 = 0.13, antiderivative size = 176, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, 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 245
Rule 246
Rule 388
Rule 416
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*}
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Mathematica [A] time = 0.06, size = 106, normalized size = 0.60 \[ \frac {1}{45} x \left (c+d x^4\right )^q \left (\frac {d x^4}{c}+1\right )^{-q} \left (45 a^2 \, _2F_1\left (\frac {1}{4},-q;\frac {5}{4};-\frac {d x^4}{c}\right )+b x^4 \left (18 a \, _2F_1\left (\frac {5}{4},-q;\frac {9}{4};-\frac {d x^4}{c}\right )+5 b x^4 \, _2F_1\left (\frac {9}{4},-q;\frac {13}{4};-\frac {d x^4}{c}\right )\right )\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 1.77, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b^{2} x^{8} + 2 \, a b x^{4} + a^{2}\right )} {\left (d x^{4} + c\right )}^{q}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b x^{4} + a\right )}^{2} {\left (d x^{4} + c\right )}^{q}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.53, size = 0, normalized size = 0.00 \[ \int \left (b \,x^{4}+a \right )^{2} \left (d \,x^{4}+c \right )^{q}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b x^{4} + a\right )}^{2} {\left (d x^{4} + c\right )}^{q}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (b\,x^4+a\right )}^2\,{\left (d\,x^4+c\right )}^q \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 166.03, size = 119, normalized size = 0.68 \[ \frac {a^{2} c^{q} x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, - q \\ \frac {5}{4} \end {matrix}\middle | {\frac {d x^{4} e^{i \pi }}{c}} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} + \frac {a b c^{q} x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{4}, - q \\ \frac {9}{4} \end {matrix}\middle | {\frac {d x^{4} e^{i \pi }}{c}} \right )}}{2 \Gamma \left (\frac {9}{4}\right )} + \frac {b^{2} c^{q} x^{9} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {9}{4}, - q \\ \frac {13}{4} \end {matrix}\middle | {\frac {d x^{4} e^{i \pi }}{c}} \right )}}{4 \Gamma \left (\frac {13}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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