Optimal. Leaf size=195 \[ \frac {(d+e x)^{1-2 p} \left (a+b x+c x^2\right )^p \left (1-\frac {2 c (d+e x)}{2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}\right )^{-p} \left (1-\frac {2 c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}\right )^{-p} F_1\left (1-2 p;-p,-p;2-2 p;\frac {2 c (d+e x)}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e},\frac {2 c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{e (1-2 p)} \]
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Rubi [A] time = 0.11, antiderivative size = 195, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {759, 133} \[ \frac {(d+e x)^{1-2 p} \left (a+b x+c x^2\right )^p \left (1-\frac {2 c (d+e x)}{2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}\right )^{-p} \left (1-\frac {2 c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}\right )^{-p} F_1\left (1-2 p;-p,-p;2-2 p;\frac {2 c (d+e x)}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e},\frac {2 c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{e (1-2 p)} \]
Antiderivative was successfully verified.
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Rule 133
Rule 759
Rubi steps
\begin {align*} \int (d+e x)^{-2 p} \left (a+b x+c x^2\right )^p \, dx &=\frac {\left (\left (a+b x+c x^2\right )^p \left (1-\frac {d+e x}{d-\frac {\left (b-\sqrt {b^2-4 a c}\right ) e}{2 c}}\right )^{-p} \left (1-\frac {d+e x}{d-\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{2 c}}\right )^{-p}\right ) \operatorname {Subst}\left (\int x^{-2 p} \left (1-\frac {2 c x}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}\right )^p \left (1-\frac {2 c x}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )^p \, dx,x,d+e x\right )}{e}\\ &=\frac {(d+e x)^{1-2 p} \left (a+b x+c x^2\right )^p \left (1-\frac {2 c (d+e x)}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}\right )^{-p} \left (1-\frac {2 c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )^{-p} F_1\left (1-2 p;-p,-p;2-2 p;\frac {2 c (d+e x)}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e},\frac {2 c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{e (1-2 p)}\\ \end {align*}
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Mathematica [A] time = 0.29, size = 214, normalized size = 1.10 \[ -\frac {(d+e x)^{1-2 p} (a+x (b+c x))^p \left (\frac {e \left (\sqrt {b^2-4 a c}-b-2 c x\right )}{e \left (\sqrt {b^2-4 a c}-b\right )+2 c d}\right )^{-p} \left (\frac {e \left (\sqrt {b^2-4 a c}+b+2 c x\right )}{e \left (\sqrt {b^2-4 a c}+b\right )-2 c d}\right )^{-p} F_1\left (1-2 p;-p,-p;2-2 p;\frac {2 c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e},\frac {2 c (d+e x)}{2 c d+\left (\sqrt {b^2-4 a c}-b\right ) e}\right )}{e (2 p-1)} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.10, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (c x^{2} + b x + a\right )}^{p}}{{\left (e x + d\right )}^{2 \, p}}, x\right ) \]
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 \[ \int \frac {{\left (c x^{2} + b x + a\right )}^{p}}{{\left (e x + d\right )}^{2 \, p}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.54, size = 0, normalized size = 0.00 \[ \int \left (e x +d \right )^{-2 p} \left (c \,x^{2}+b x +a \right )^{p}\, dx \]
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 \[ \int \frac {{\left (c x^{2} + b x + a\right )}^{p}}{{\left (e x + d\right )}^{2 \, p}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (c\,x^2+b\,x+a\right )}^p}{{\left (d+e\,x\right )}^{2\,p}} \,d x \]
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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