Optimal. Leaf size=81 \[ -\frac {2 \left (a^2+2 a b x+b^2 x^2\right )^p \left (-\frac {e (a+b x)}{b d-a e}\right )^{-2 p} \, _2F_1\left (-\frac {1}{2},-2 p;\frac {1}{2};\frac {b (d+e x)}{b d-a e}\right )}{e \sqrt {d+e x}} \]
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Rubi [A] time = 0.04, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {646, 70, 69} \[ -\frac {2 \left (a^2+2 a b x+b^2 x^2\right )^p \left (-\frac {e (a+b x)}{b d-a e}\right )^{-2 p} \, _2F_1\left (-\frac {1}{2},-2 p;\frac {1}{2};\frac {b (d+e x)}{b d-a e}\right )}{e \sqrt {d+e x}} \]
Antiderivative was successfully verified.
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Rule 69
Rule 70
Rule 646
Rubi steps
\begin {align*} \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^p}{(d+e x)^{3/2}} \, dx &=\left (\left (a b+b^2 x\right )^{-2 p} \left (a^2+2 a b x+b^2 x^2\right )^p\right ) \int \frac {\left (a b+b^2 x\right )^{2 p}}{(d+e x)^{3/2}} \, dx\\ &=\left (\left (\frac {e \left (a b+b^2 x\right )}{-b^2 d+a b e}\right )^{-2 p} \left (a^2+2 a b x+b^2 x^2\right )^p\right ) \int \frac {\left (-\frac {a e}{b d-a e}-\frac {b e x}{b d-a e}\right )^{2 p}}{(d+e x)^{3/2}} \, dx\\ &=-\frac {2 \left (-\frac {e (a+b x)}{b d-a e}\right )^{-2 p} \left (a^2+2 a b x+b^2 x^2\right )^p \, _2F_1\left (-\frac {1}{2},-2 p;\frac {1}{2};\frac {b (d+e x)}{b d-a e}\right )}{e \sqrt {d+e x}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 71, normalized size = 0.88 \[ -\frac {2 \left ((a+b x)^2\right )^p \left (\frac {e (a+b x)}{a e-b d}\right )^{-2 p} \, _2F_1\left (-\frac {1}{2},-2 p;\frac {1}{2};\frac {b (d+e x)}{b d-a e}\right )}{e \sqrt {d+e x}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.05, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e x + d} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p}}{e^{2} x^{2} + 2 \, d e x + d^{2}}, x\right ) \]
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 \[ \int \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p}}{{\left (e x + d\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.80, size = 0, normalized size = 0.00 \[ \int \frac {\left (b^{2} x^{2}+2 a b x +a^{2}\right )^{p}}{\left (e x +d \right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p}}{{\left (e x + d\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^p}{{\left (d+e\,x\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\left (a + b x\right )^{2}\right )^{p}}{\left (d + e x\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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