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 = 0.05, antiderivative size = 52, normalized size of antiderivative = 1.00, 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 65
Rule 266
Rule 372
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*}
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Mathematica [A] time = 0.04, size = 52, normalized size = 1.00 \[ -\frac {\left (a+b (c+d x)^2\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac {b (c+d x)^2}{a}+1\right )}{2 a d (p+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.78, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right )}^{p}}{d x + c}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left ({\left (d x + c\right )}^{2} b + a\right )}^{p}}{d x + c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.27, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +\left (d x +c \right )^{2} b \right )^{p}}{d x +c}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left ({\left (d x + c\right )}^{2} b + a\right )}^{p}}{d x + c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (a+b\,{\left (c+d\,x\right )}^2\right )}^p}{c+d\,x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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