Optimal. Leaf size=55 \[ \frac{(c+d x)^3 \left (a+b (c+d x)^2\right )^{p+1} \, _2F_1\left (1,p+\frac{5}{2};\frac{5}{2};-\frac{b (c+d x)^2}{a}\right )}{3 a d} \]
[Out]
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Rubi [A] time = 0.118178, antiderivative size = 68, normalized size of antiderivative = 1.24, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{(c+d x)^3 \left (a+b (c+d x)^2\right )^p \left (\frac{b (c+d x)^2}{a}+1\right )^{-p} \, _2F_1\left (\frac{3}{2},-p;\frac{5}{2};-\frac{b (c+d x)^2}{a}\right )}{3 d} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x)^2*(a + b*(c + d*x)^2)^p,x]
[Out]
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Rubi in Sympy [A] time = 13.4556, size = 53, normalized size = 0.96 \[ \frac{\left (1 + \frac{b \left (c + d x\right )^{2}}{a}\right )^{- p} \left (a + b \left (c + d x\right )^{2}\right )^{p} \left (c + d x\right )^{3}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{- \frac{b \left (c + d x\right )^{2}}{a}} \right )}}{3 d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x+c)**2*(a+b*(d*x+c)**2)**p,x)
[Out]
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Mathematica [A] time = 0.0348957, size = 68, normalized size = 1.24 \[ \frac{(c+d x)^3 \left (a+b (c+d x)^2\right )^p \left (\frac{b (c+d x)^2}{a}+1\right )^{-p} \, _2F_1\left (\frac{3}{2},-p;\frac{5}{2};-\frac{b (c+d x)^2}{a}\right )}{3 d} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x)^2*(a + b*(c + d*x)^2)^p,x]
[Out]
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Maple [F] time = 0.152, size = 0, normalized size = 0. \[ \int \left ( dx+c \right ) ^{2} \left ( a+b \left ( dx+c \right ) ^{2} \right ) ^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x+c)^2*(a+b*(d*x+c)^2)^p,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (d x + c\right )}^{2}{\left ({\left (d x + c\right )}^{2} b + a\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^2*((d*x + c)^2*b + a)^p,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )}{\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right )}^{p}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^2*((d*x + c)^2*b + a)^p,x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x+c)**2*(a+b*(d*x+c)**2)**p,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (d x + c\right )}^{2}{\left ({\left (d x + c\right )}^{2} b + a\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^2*((d*x + c)^2*b + a)^p,x, algorithm="giac")
[Out]