Optimal. Leaf size=52 \[ -\frac{\left (a+b (c+d x)^2\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac{b (c+d x)^2}{a}+1\right )}{2 a d (p+1)} \]
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Rubi [A] time = 0.0532383, 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, Rules used = {372, 266, 65} \[ -\frac{\left (a+b (c+d x)^2\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac{b (c+d x)^2}{a}+1\right )}{2 a d (p+1)} \]
Antiderivative was successfully verified.
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Rule 372
Rule 266
Rule 65
Rubi steps
\begin{align*} \int \frac{\left (a+b (c+d x)^2\right )^p}{c+d x} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b x^2\right )^p}{x} \, dx,x,c+d x\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{(a+b x)^p}{x} \, dx,x,(c+d x)^2\right )}{2 d}\\ &=-\frac{\left (a+b (c+d x)^2\right )^{1+p} \, _2F_1\left (1,1+p;2+p;1+\frac{b (c+d x)^2}{a}\right )}{2 a d (1+p)}\\ \end{align*}
Mathematica [A] time = 0.0244537, size = 52, normalized size = 1. \[ -\frac{\left (a+b (c+d x)^2\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac{b (c+d x)^2}{a}+1\right )}{2 a d (p+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.12, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b \left ( dx+c \right ) ^{2} \right ) ^{p}}{dx+c}}\, 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 \frac{{\left ({\left (d x + c\right )}^{2} b + a\right )}^{p}}{d x + c}\,{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 (\frac{{\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right )}^{p}}{d x + c}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 \frac{{\left ({\left (d x + c\right )}^{2} b + a\right )}^{p}}{d x + c}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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