Optimal. Leaf size=79 \[ -\frac {e^2 (a+b x) (d+e x)^{1+m} \, _2F_1\left (3,1+m;2+m;\frac {b (d+e x)}{b d-a e}\right )}{(b d-a e)^3 (1+m) \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A]
time = 0.03, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {660, 70}
\begin {gather*} -\frac {e^2 (a+b x) (d+e x)^{m+1} \, _2F_1\left (3,m+1;m+2;\frac {b (d+e x)}{b d-a e}\right )}{(m+1) \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 70
Rule 660
Rubi steps
\begin {align*} \int \frac {(d+e x)^m}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx &=\frac {\left (b^2 \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^m}{\left (a b+b^2 x\right )^3} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {e^2 (a+b x) (d+e x)^{1+m} \, _2F_1\left (3,1+m;2+m;\frac {b (d+e x)}{b d-a e}\right )}{(b d-a e)^3 (1+m) \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 72, normalized size = 0.91 \begin {gather*} \frac {e^2 (a+b x)^3 (d+e x)^{1+m} \, _2F_1\left (3,1+m;2+m;-\frac {b (d+e x)}{-b d+a e}\right )}{(-b d+a e)^3 (1+m) \left ((a+b x)^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.30, size = 0, normalized size = 0.00 \[\int \frac {\left (e x +d \right )^{m}}{\left (b^{2} x^{2}+2 a b x +a^{2}\right )^{\frac {3}{2}}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: HeuristicGCDFailed} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\left (d+e\,x\right )}^m}{{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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