Optimal. Leaf size=52 \[ -\frac{\left (a+b (c+d x)^2\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac{b (c+d x)^2}{a}+1\right )}{2 a d (p+1)} \]
[Out]
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Rubi [A] time = 0.117322, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ -\frac{\left (a+b (c+d x)^2\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac{b (c+d x)^2}{a}+1\right )}{2 a d (p+1)} \]
Antiderivative was successfully verified.
[In] Int[(a + b*(c + d*x)^2)^p/(c + d*x),x]
[Out]
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Rubi in Sympy [A] time = 10.1708, size = 39, normalized size = 0.75 \[ - \frac{\left (a + b \left (c + d x\right )^{2}\right )^{p + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, p + 1 \\ p + 2 \end{matrix}\middle |{1 + \frac{b \left (c + d x\right )^{2}}{a}} \right )}}{2 a d \left (p + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b*(d*x+c)**2)**p/(d*x+c),x)
[Out]
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Mathematica [A] time = 0.0477715, size = 66, normalized size = 1.27 \[ \frac{\left (\frac{a}{b (c+d x)^2}+1\right )^{-p} \left (a+b (c+d x)^2\right )^p \, _2F_1\left (-p,-p;1-p;-\frac{a}{b (c+d x)^2}\right )}{2 d p} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*(c + d*x)^2)^p/(c + d*x),x]
[Out]
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Maple [F] time = 0.154, size = 0, normalized size = 0. \[ \int{\frac{ \left ( a+b \left ( dx+c \right ) ^{2} \right ) ^{p}}{dx+c}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b*(d*x+c)^2)^p/(d*x+c),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left ({\left (d x + c\right )}^{2} b + a\right )}^{p}}{d x + c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((d*x + c)^2*b + a)^p/(d*x + c),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right )}^{p}}{d x + c}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((d*x + c)^2*b + a)^p/(d*x + c),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((a+b*(d*x+c)**2)**p/(d*x+c),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left ({\left (d x + c\right )}^{2} b + a\right )}^{p}}{d x + c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((d*x + c)^2*b + a)^p/(d*x + c),x, algorithm="giac")
[Out]