Optimal. Leaf size=176 \[ \frac {x \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (3 a^2 d^2-2 a b c d (2 p+5)+b^2 c^2 \left (4 p^2+16 p+15\right )\right ) \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};-\frac {b x^2}{a}\right )}{b^2 (2 p+3) (2 p+5)}-\frac {d x \left (a+b x^2\right )^{p+1} (3 a d-b c (2 p+7))}{b^2 (2 p+3) (2 p+5)}+\frac {d x \left (c+d x^2\right ) \left (a+b x^2\right )^{p+1}}{b (2 p+5)} \]
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Rubi [A] time = 0.12, 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} \[ \frac {x \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (3 a^2 d^2-2 a b c d (2 p+5)+b^2 c^2 \left (4 p^2+16 p+15\right )\right ) \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};-\frac {b x^2}{a}\right )}{b^2 (2 p+3) (2 p+5)}-\frac {d x \left (a+b x^2\right )^{p+1} (3 a d-b c (2 p+7))}{b^2 (2 p+3) (2 p+5)}+\frac {d x \left (c+d x^2\right ) \left (a+b x^2\right )^{p+1}}{b (2 p+5)} \]
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^2\right )^p \left (c+d x^2\right )^2 \, dx &=\frac {d x \left (a+b x^2\right )^{1+p} \left (c+d x^2\right )}{b (5+2 p)}+\frac {\int \left (a+b x^2\right )^p \left (-c (a d-b c (5+2 p))-d (3 a d-b c (7+2 p)) x^2\right ) \, dx}{b (5+2 p)}\\ &=-\frac {d (3 a d-b c (7+2 p)) x \left (a+b x^2\right )^{1+p}}{b^2 (3+2 p) (5+2 p)}+\frac {d x \left (a+b x^2\right )^{1+p} \left (c+d x^2\right )}{b (5+2 p)}+\frac {\left (3 a^2 d^2-2 a b c d (5+2 p)+b^2 c^2 \left (15+16 p+4 p^2\right )\right ) \int \left (a+b x^2\right )^p \, dx}{b^2 (3+2 p) (5+2 p)}\\ &=-\frac {d (3 a d-b c (7+2 p)) x \left (a+b x^2\right )^{1+p}}{b^2 (3+2 p) (5+2 p)}+\frac {d x \left (a+b x^2\right )^{1+p} \left (c+d x^2\right )}{b (5+2 p)}+\frac {\left (\left (3 a^2 d^2-2 a b c d (5+2 p)+b^2 c^2 \left (15+16 p+4 p^2\right )\right ) \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p}\right ) \int \left (1+\frac {b x^2}{a}\right )^p \, dx}{b^2 (3+2 p) (5+2 p)}\\ &=-\frac {d (3 a d-b c (7+2 p)) x \left (a+b x^2\right )^{1+p}}{b^2 (3+2 p) (5+2 p)}+\frac {d x \left (a+b x^2\right )^{1+p} \left (c+d x^2\right )}{b (5+2 p)}+\frac {\left (3 a^2 d^2-2 a b c d (5+2 p)+b^2 c^2 \left (15+16 p+4 p^2\right )\right ) x \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};-\frac {b x^2}{a}\right )}{b^2 (3+2 p) (5+2 p)}\\ \end {align*}
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Mathematica [A] time = 5.05, size = 106, normalized size = 0.60 \[ \frac {1}{15} x \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (15 c^2 \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};-\frac {b x^2}{a}\right )+d x^2 \left (10 c \, _2F_1\left (\frac {3}{2},-p;\frac {5}{2};-\frac {b x^2}{a}\right )+3 d x^2 \, _2F_1\left (\frac {5}{2},-p;\frac {7}{2};-\frac {b x^2}{a}\right )\right )\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 1.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (d^{2} x^{4} + 2 \, c d x^{2} + c^{2}\right )} {\left (b x^{2} + a\right )}^{p}, x\right ) \]
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 \[ \int {\left (d x^{2} + c\right )}^{2} {\left (b x^{2} + a\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.31, size = 0, normalized size = 0.00 \[ \int \left (d \,x^{2}+c \right )^{2} \left (b \,x^{2}+a \right )^{p}\, dx \]
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 \[ \int {\left (d x^{2} + c\right )}^{2} {\left (b x^{2} + a\right )}^{p}\,{d x} \]
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 \[ \int {\left (b\,x^2+a\right )}^p\,{\left (d\,x^2+c\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 20.40, size = 88, normalized size = 0.50 \[ a^{p} c^{2} x {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, - p \\ \frac {3}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )} + \frac {2 a^{p} c d x^{3} {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, - p \\ \frac {5}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{3} + \frac {a^{p} d^{2} x^{5} {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{2}, - p \\ \frac {7}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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