Optimal. Leaf size=93 \[ \frac {d x \left (a+b x^2\right )^{1+p}}{b (3+2 p)}-\frac {(a d-b c (3+2 p)) x \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};-\frac {b x^2}{a}\right )}{b (3+2 p)} \]
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Rubi [A]
time = 0.03, antiderivative size = 85, normalized size of antiderivative = 0.91, number of steps
used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {396, 252, 251}
\begin {gather*} x \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (c-\frac {a d}{2 b p+3 b}\right ) \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};-\frac {b x^2}{a}\right )+\frac {d x \left (a+b x^2\right )^{p+1}}{b (2 p+3)} \end {gather*}
Antiderivative was successfully verified.
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Rule 251
Rule 252
Rule 396
Rubi steps
\begin {align*} \int \left (a+b x^2\right )^p \left (c+d x^2\right ) \, dx &=\frac {d x \left (a+b x^2\right )^{1+p}}{b (3+2 p)}-\left (-c+\frac {a d}{3 b+2 b p}\right ) \int \left (a+b x^2\right )^p \, dx\\ &=\frac {d x \left (a+b x^2\right )^{1+p}}{b (3+2 p)}-\left (\left (-c+\frac {a d}{3 b+2 b p}\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\\ &=\frac {d x \left (a+b x^2\right )^{1+p}}{b (3+2 p)}+\left (c-\frac {a d}{3 b+2 b p}\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 )\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 90, normalized size = 0.97 \begin {gather*} \frac {x \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \left (d \left (a+b x^2\right ) \left (1+\frac {b x^2}{a}\right )^p+(-a d+b c (3+2 p)) \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};-\frac {b x^2}{a}\right )\right )}{b (3+2 p)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \left (b \,x^{2}+a \right )^{p} \left (d \,x^{2}+c \right )\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 4.68, size = 53, normalized size = 0.57 \begin {gather*} a^{p} c 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 {a^{p} 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} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int {\left (b\,x^2+a\right )}^p\,\left (d\,x^2+c\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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