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 = 0.120852, 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} \[ \frac{x \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \left (3 a^2 d^2-2 a b c d (2 p+5)+b^2 c^2 \left (4 p^2+16 p+15\right )\right ) \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right )}{b^2 (2 p+3) (2 p+5)}-\frac{d x \left (a+b x^2\right )^{p+1} (3 a d-b c (2 p+7))}{b^2 (2 p+3) (2 p+5)}+\frac{d x \left (c+d x^2\right ) \left (a+b x^2\right )^{p+1}}{b (2 p+5)} \]
Antiderivative was successfully verified.
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Rule 416
Rule 388
Rule 246
Rule 245
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*}
Mathematica [A] time = 5.04245, size = 106, normalized size = 0.6 \[ \frac{1}{15} x \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \left (15 c^2 \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right )+d x^2 \left (10 c \, _2F_1\left (\frac{3}{2},-p;\frac{5}{2};-\frac{b x^2}{a}\right )+3 d x^2 \, _2F_1\left (\frac{5}{2},-p;\frac{7}{2};-\frac{b x^2}{a}\right )\right )\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.043, size = 0, normalized size = 0. \begin{align*} \int \left ( b{x}^{2}+a \right ) ^{p} \left ( d{x}^{2}+c \right ) ^{2}\, 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 (d x^{2} + c\right )}^{2}{\left (b x^{2} + a\right )}^{p}\,{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 (d^{2} x^{4} + 2 \, c d x^{2} + c^{2}\right )}{\left (b x^{2} + a\right )}^{p}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 22.8613, size = 88, normalized size = 0.5 \begin{align*} 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} \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 (d x^{2} + c\right )}^{2}{\left (b x^{2} + a\right )}^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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