Optimal. Leaf size=133 \[ \frac {d e (2+p) \left (a+c x^2\right )^{1+p}}{c (1+p) (3+2 p)}+\frac {e (d+e x) \left (a+c x^2\right )^{1+p}}{c (3+2 p)}-\frac {\left (a e^2-c d^2 (3+2 p)\right ) x \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p} \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};-\frac {c x^2}{a}\right )}{c (3+2 p)} \]
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Rubi [A]
time = 0.05, antiderivative size = 125, normalized size of antiderivative = 0.94, number of steps
used = 4, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {757, 655, 252,
251} \begin {gather*} x \left (a+c x^2\right )^p \left (\frac {c x^2}{a}+1\right )^{-p} \left (d^2-\frac {a e^2}{2 c p+3 c}\right ) \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};-\frac {c x^2}{a}\right )+\frac {e (d+e x) \left (a+c x^2\right )^{p+1}}{c (2 p+3)}+\frac {d e (p+2) \left (a+c x^2\right )^{p+1}}{c (p+1) (2 p+3)} \end {gather*}
Antiderivative was successfully verified.
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Rule 251
Rule 252
Rule 655
Rule 757
Rubi steps
\begin {align*} \int (d+e x)^2 \left (a+c x^2\right )^p \, dx &=\frac {e (d+e x) \left (a+c x^2\right )^{1+p}}{c (3+2 p)}+\frac {\int \left (-a e^2+c d^2 (3+2 p)+2 c d e (2+p) x\right ) \left (a+c x^2\right )^p \, dx}{c (3+2 p)}\\ &=\frac {d e (2+p) \left (a+c x^2\right )^{1+p}}{c (1+p) (3+2 p)}+\frac {e (d+e x) \left (a+c x^2\right )^{1+p}}{c (3+2 p)}+\left (d^2-\frac {a e^2}{3 c+2 c p}\right ) \int \left (a+c x^2\right )^p \, dx\\ &=\frac {d e (2+p) \left (a+c x^2\right )^{1+p}}{c (1+p) (3+2 p)}+\frac {e (d+e x) \left (a+c x^2\right )^{1+p}}{c (3+2 p)}+\left (\left (d^2-\frac {a e^2}{3 c+2 c p}\right ) \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p}\right ) \int \left (1+\frac {c x^2}{a}\right )^p \, dx\\ &=\frac {d e (2+p) \left (a+c x^2\right )^{1+p}}{c (1+p) (3+2 p)}+\frac {e (d+e x) \left (a+c x^2\right )^{1+p}}{c (3+2 p)}+\left (d^2-\frac {a e^2}{3 c+2 c p}\right ) x \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p} \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};-\frac {c x^2}{a}\right )\\ \end {align*}
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Mathematica [A]
time = 0.19, size = 133, normalized size = 1.00 \begin {gather*} \frac {\left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p} \left (3 c d^2 (1+p) x \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};-\frac {c x^2}{a}\right )+e \left (3 d \left (c x^2 \left (1+\frac {c x^2}{a}\right )^p+a \left (-1+\left (1+\frac {c x^2}{a}\right )^p\right )\right )+c e (1+p) x^3 \, _2F_1\left (\frac {3}{2},-p;\frac {5}{2};-\frac {c x^2}{a}\right )\right )\right )}{3 c (1+p)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.12, size = 0, normalized size = 0.00 \[\int \left (e x +d \right )^{2} \left (c \,x^{2}+a \right )^{p}\, dx\]
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 \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 [A]
time = 5.75, size = 97, normalized size = 0.73 \begin {gather*} a^{p} d^{2} x {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, - p \\ \frac {3}{2} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )} + \frac {a^{p} e^{2} x^{3} {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, - p \\ \frac {5}{2} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{3} + 2 d e \left (\begin {cases} \frac {a^{p} x^{2}}{2} & \text {for}\: c = 0 \\\frac {\begin {cases} \frac {\left (a + c x^{2}\right )^{p + 1}}{p + 1} & \text {for}\: p \neq -1 \\\log {\left (a + c x^{2} \right )} & \text {otherwise} \end {cases}}{2 c} & \text {otherwise} \end {cases}\right ) \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 (c\,x^2+a\right )}^p\,{\left (d+e\,x\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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