Optimal. Leaf size=176 \[ -\frac{e \left (a+b x^2\right )^{p+1} \left (a e^2-3 b d^2 (p+2)\right )}{2 b^2 (p+1) (p+2)}-\frac{d x \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \left (3 a e^2-b d^2 (2 p+3)\right ) \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right )}{b (2 p+3)}+\frac{3 d e^2 x \left (a+b x^2\right )^{p+1}}{b (2 p+3)}+\frac{e^3 x^2 \left (a+b x^2\right )^{p+1}}{2 b (p+2)} \]
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Rubi [A] time = 0.149465, antiderivative size = 169, normalized size of antiderivative = 0.96, 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 = {743, 780, 246, 245} \[ -\frac{e \left (a+b x^2\right )^{p+1} \left ((2 p+3) \left (a e^2-b d^2 (2 p+5)\right )-2 b d e (p+1) (p+3) x\right )}{2 b^2 (p+2) \left (2 p^2+5 p+3\right )}+d x \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \left (d^2-\frac{3 a e^2}{2 b p+3 b}\right ) \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right )+\frac{e (d+e x)^2 \left (a+b x^2\right )^{p+1}}{2 b (p+2)} \]
Antiderivative was successfully verified.
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Rule 743
Rule 780
Rule 246
Rule 245
Rubi steps
\begin{align*} \int (d+e x)^3 \left (a+b x^2\right )^p \, dx &=\frac{e (d+e x)^2 \left (a+b x^2\right )^{1+p}}{2 b (2+p)}+\frac{\int (d+e x) \left (-2 \left (a e^2-b d^2 (2+p)\right )+2 b d e (3+p) x\right ) \left (a+b x^2\right )^p \, dx}{2 b (2+p)}\\ &=\frac{e (d+e x)^2 \left (a+b x^2\right )^{1+p}}{2 b (2+p)}-\frac{e \left ((3+2 p) \left (a e^2-b d^2 (5+2 p)\right )-2 b d e (1+p) (3+p) x\right ) \left (a+b x^2\right )^{1+p}}{2 b^2 (2+p) \left (3+5 p+2 p^2\right )}+\left (d \left (d^2-\frac{3 a e^2}{3 b+2 b p}\right )\right ) \int \left (a+b x^2\right )^p \, dx\\ &=\frac{e (d+e x)^2 \left (a+b x^2\right )^{1+p}}{2 b (2+p)}-\frac{e \left ((3+2 p) \left (a e^2-b d^2 (5+2 p)\right )-2 b d e (1+p) (3+p) x\right ) \left (a+b x^2\right )^{1+p}}{2 b^2 (2+p) \left (3+5 p+2 p^2\right )}+\left (d \left (d^2-\frac{3 a e^2}{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{e (d+e x)^2 \left (a+b x^2\right )^{1+p}}{2 b (2+p)}-\frac{e \left ((3+2 p) \left (a e^2-b d^2 (5+2 p)\right )-2 b d e (1+p) (3+p) x\right ) \left (a+b x^2\right )^{1+p}}{2 b^2 (2+p) \left (3+5 p+2 p^2\right )}+d \left (d^2-\frac{3 a e^2}{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*}
Mathematica [A] time = 0.218668, size = 223, normalized size = 1.27 \[ \frac{\left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \left (e \left (-a^2 e^2 \left (\left (\frac{b x^2}{a}+1\right )^p-1\right )+b^2 x^2 \left (\frac{b x^2}{a}+1\right )^p \left (3 d^2 (p+2)+e^2 (p+1) x^2\right )+2 b^2 d e \left (p^2+3 p+2\right ) x^3 \, _2F_1\left (\frac{3}{2},-p;\frac{5}{2};-\frac{b x^2}{a}\right )+a b \left (3 d^2 (p+2) \left (\left (\frac{b x^2}{a}+1\right )^p-1\right )+e^2 p x^2 \left (\frac{b x^2}{a}+1\right )^p\right )\right )+2 b^2 d^3 \left (p^2+3 p+2\right ) x \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right )\right )}{2 b^2 (p+1) (p+2)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.535, size = 0, normalized size = 0. \begin{align*} \int \left ( ex+d \right ) ^{3} \left ( b{x}^{2}+a \right ) ^{p}\, 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 (e x + d\right )}^{3}{\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 (e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}\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 = 21.1549, size = 468, normalized size = 2.66 \begin{align*} a^{p} d^{3} 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 )} + a^{p} d e^{2} 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 d^{2} e \left (\begin{cases} \frac{a^{p} x^{2}}{2} & \text{for}\: b = 0 \\\frac{\begin{cases} \frac{\left (a + b x^{2}\right )^{p + 1}}{p + 1} & \text{for}\: p \neq -1 \\\log{\left (a + b x^{2} \right )} & \text{otherwise} \end{cases}}{2 b} & \text{otherwise} \end{cases}\right ) + e^{3} \left (\begin{cases} \frac{a^{p} x^{4}}{4} & \text{for}\: b = 0 \\\frac{a \log{\left (- i \sqrt{a} \sqrt{\frac{1}{b}} + x \right )}}{2 a b^{2} + 2 b^{3} x^{2}} + \frac{a \log{\left (i \sqrt{a} \sqrt{\frac{1}{b}} + x \right )}}{2 a b^{2} + 2 b^{3} x^{2}} + \frac{a}{2 a b^{2} + 2 b^{3} x^{2}} + \frac{b x^{2} \log{\left (- i \sqrt{a} \sqrt{\frac{1}{b}} + x \right )}}{2 a b^{2} + 2 b^{3} x^{2}} + \frac{b x^{2} \log{\left (i \sqrt{a} \sqrt{\frac{1}{b}} + x \right )}}{2 a b^{2} + 2 b^{3} x^{2}} & \text{for}\: p = -2 \\- \frac{a \log{\left (- i \sqrt{a} \sqrt{\frac{1}{b}} + x \right )}}{2 b^{2}} - \frac{a \log{\left (i \sqrt{a} \sqrt{\frac{1}{b}} + x \right )}}{2 b^{2}} + \frac{x^{2}}{2 b} & \text{for}\: p = -1 \\- \frac{a^{2} \left (a + b x^{2}\right )^{p}}{2 b^{2} p^{2} + 6 b^{2} p + 4 b^{2}} + \frac{a b p x^{2} \left (a + b x^{2}\right )^{p}}{2 b^{2} p^{2} + 6 b^{2} p + 4 b^{2}} + \frac{b^{2} p x^{4} \left (a + b x^{2}\right )^{p}}{2 b^{2} p^{2} + 6 b^{2} p + 4 b^{2}} + \frac{b^{2} x^{4} \left (a + b x^{2}\right )^{p}}{2 b^{2} p^{2} + 6 b^{2} p + 4 b^{2}} & \text{otherwise} \end{cases}\right ) \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 (e x + d\right )}^{3}{\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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