Optimal. Leaf size=178 \[ -\frac{e \left (a+c x^2\right )^{p+1} \left ((2 p+3) \left (a e^2-c d^2 (2 p+5)\right )-2 c d e (p+1) (p+3) x\right )}{2 c^2 (p+2) \left (2 p^2+5 p+3\right )}-\frac{d x \left (a+c x^2\right )^p \left (\frac{c x^2}{a}+1\right )^{-p} \left (3 a e^2-c d^2 (2 p+3)\right ) \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};-\frac{c x^2}{a}\right )}{c (2 p+3)}+\frac{e (d+e x)^2 \left (a+c x^2\right )^{p+1}}{2 c (p+2)} \]
[Out]
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Rubi [A] time = 0.390373, antiderivative size = 169, normalized size of antiderivative = 0.95, number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ -\frac{e \left (a+c x^2\right )^{p+1} \left ((2 p+3) \left (a e^2-c d^2 (2 p+5)\right )-2 c d e (p+1) (p+3) x\right )}{2 c^2 (p+2) \left (2 p^2+5 p+3\right )}+d x \left (a+c x^2\right )^p \left (\frac{c x^2}{a}+1\right )^{-p} \left (d^2-\frac{3 a e^2}{2 c p+3 c}\right ) \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};-\frac{c x^2}{a}\right )+\frac{e (d+e x)^2 \left (a+c x^2\right )^{p+1}}{2 c (p+2)} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^3*(a + c*x^2)^p,x]
[Out]
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Rubi in Sympy [A] time = 47.4701, size = 160, normalized size = 0.9 \[ - \frac{d x \left (1 + \frac{c x^{2}}{a}\right )^{- p} \left (a + c x^{2}\right )^{p} \left (3 a e^{2} - 2 c d^{2} p - 3 c d^{2}\right ){{}_{2}F_{1}\left (\begin{matrix} - p, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{- \frac{c x^{2}}{a}} \right )}}{c \left (2 p + 3\right )} + \frac{e \left (a + c x^{2}\right )^{p + 1} \left (d + e x\right )^{2}}{2 c \left (p + 2\right )} - \frac{e \left (a + c x^{2}\right )^{p + 1} \left (- 4 c d e x \left (p + 1\right ) \left (p + 3\right ) + \left (4 p + 6\right ) \left (a e^{2} - 2 c d^{2} p - 5 c d^{2}\right )\right )}{4 c^{2} \left (p + 1\right ) \left (p + 2\right ) \left (2 p + 3\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**3*(c*x**2+a)**p,x)
[Out]
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Mathematica [A] time = 0.484716, size = 223, normalized size = 1.25 \[ \frac{\left (a+c x^2\right )^p \left (\frac{c x^2}{a}+1\right )^{-p} \left (e \left (-a^2 e^2 \left (\left (\frac{c x^2}{a}+1\right )^p-1\right )+c^2 x^2 \left (\frac{c x^2}{a}+1\right )^p \left (3 d^2 (p+2)+e^2 (p+1) x^2\right )+2 c^2 d e \left (p^2+3 p+2\right ) x^3 \, _2F_1\left (\frac{3}{2},-p;\frac{5}{2};-\frac{c x^2}{a}\right )+a c \left (3 d^2 (p+2) \left (\left (\frac{c x^2}{a}+1\right )^p-1\right )+e^2 p x^2 \left (\frac{c x^2}{a}+1\right )^p\right )\right )+2 c^2 d^3 \left (p^2+3 p+2\right ) x \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};-\frac{c x^2}{a}\right )\right )}{2 c^2 (p+1) (p+2)} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^3*(a + c*x^2)^p,x]
[Out]
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Maple [F] time = 0.075, size = 0, normalized size = 0. \[ \int \left ( ex+d \right ) ^{3} \left ( c{x}^{2}+a \right ) ^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^3*(c*x^2+a)^p,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (e x + d\right )}^{3}{\left (c x^{2} + a\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^3*(c*x^2 + a)^p,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}\right )}{\left (c x^{2} + a\right )}^{p}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^3*(c*x^2 + a)^p,x, algorithm="fricas")
[Out]
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Sympy [A] time = 58.5066, size = 468, normalized size = 2.63 \[ a^{p} d^{3} x{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, - p \\ \frac{3}{2} \end{matrix}\middle |{\frac{c x^{2} e^{i \pi }}{a}} \right )} + a^{p} d e^{2} x^{3}{{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, - p \\ \frac{5}{2} \end{matrix}\middle |{\frac{c x^{2} e^{i \pi }}{a}} \right )} + 3 d^{2} e \left (\begin{cases} \frac{a^{p} x^{2}}{2} & \text{for}\: c = 0 \\\frac{\begin{cases} \frac{\left (a + c x^{2}\right )^{p + 1}}{p + 1} & \text{for}\: p \neq -1 \\\log{\left (a + c x^{2} \right )} & \text{otherwise} \end{cases}}{2 c} & \text{otherwise} \end{cases}\right ) + e^{3} \left (\begin{cases} \frac{a^{p} x^{4}}{4} & \text{for}\: c = 0 \\\frac{a \log{\left (- i \sqrt{a} \sqrt{\frac{1}{c}} + x \right )}}{2 a c^{2} + 2 c^{3} x^{2}} + \frac{a \log{\left (i \sqrt{a} \sqrt{\frac{1}{c}} + x \right )}}{2 a c^{2} + 2 c^{3} x^{2}} + \frac{a}{2 a c^{2} + 2 c^{3} x^{2}} + \frac{c x^{2} \log{\left (- i \sqrt{a} \sqrt{\frac{1}{c}} + x \right )}}{2 a c^{2} + 2 c^{3} x^{2}} + \frac{c x^{2} \log{\left (i \sqrt{a} \sqrt{\frac{1}{c}} + x \right )}}{2 a c^{2} + 2 c^{3} x^{2}} & \text{for}\: p = -2 \\- \frac{a \log{\left (- i \sqrt{a} \sqrt{\frac{1}{c}} + x \right )}}{2 c^{2}} - \frac{a \log{\left (i \sqrt{a} \sqrt{\frac{1}{c}} + x \right )}}{2 c^{2}} + \frac{x^{2}}{2 c} & \text{for}\: p = -1 \\- \frac{a^{2} \left (a + c x^{2}\right )^{p}}{2 c^{2} p^{2} + 6 c^{2} p + 4 c^{2}} + \frac{a c p x^{2} \left (a + c x^{2}\right )^{p}}{2 c^{2} p^{2} + 6 c^{2} p + 4 c^{2}} + \frac{c^{2} p x^{4} \left (a + c x^{2}\right )^{p}}{2 c^{2} p^{2} + 6 c^{2} p + 4 c^{2}} + \frac{c^{2} x^{4} \left (a + c x^{2}\right )^{p}}{2 c^{2} p^{2} + 6 c^{2} p + 4 c^{2}} & \text{otherwise} \end{cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**3*(c*x**2+a)**p,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (e x + d\right )}^{3}{\left (c x^{2} + a\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^3*(c*x^2 + a)^p,x, algorithm="giac")
[Out]