Optimal. Leaf size=60 \[ -\frac {(d+e x)^m \, _2F_1\left (1,-3+m;-\frac {1}{2}+m;\frac {d+e x}{2 d}\right )}{d e (3-2 m) \left (d^2-e^2 x^2\right )^{3/2}} \]
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Rubi [A]
time = 0.04, antiderivative size = 83, normalized size of antiderivative = 1.38, number of steps
used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {694, 692, 71}
\begin {gather*} \frac {2^{m-\frac {3}{2}} (d+e x)^m \left (\frac {e x}{d}+1\right )^{\frac {3}{2}-m} \, _2F_1\left (-\frac {3}{2},\frac {5}{2}-m;-\frac {1}{2};\frac {d-e x}{2 d}\right )}{3 d e \left (d^2-e^2 x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 71
Rule 692
Rule 694
Rubi steps
\begin {align*} \int \frac {(d+e x)^m}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx &=\left ((d+e x)^m \left (1+\frac {e x}{d}\right )^{-m}\right ) \int \frac {\left (1+\frac {e x}{d}\right )^m}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx\\ &=\frac {\left ((d+e x)^m \left (1+\frac {e x}{d}\right )^{\frac {3}{2}-m} \left (d^2-d e x\right )^{3/2}\right ) \int \frac {\left (1+\frac {e x}{d}\right )^{-\frac {5}{2}+m}}{\left (d^2-d e x\right )^{5/2}} \, dx}{\left (d^2-e^2 x^2\right )^{3/2}}\\ &=\frac {2^{-\frac {3}{2}+m} (d+e x)^m \left (1+\frac {e x}{d}\right )^{\frac {3}{2}-m} \, _2F_1\left (-\frac {3}{2},\frac {5}{2}-m;-\frac {1}{2};\frac {d-e x}{2 d}\right )}{3 d e \left (d^2-e^2 x^2\right )^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.53, size = 92, normalized size = 1.53 \begin {gather*} \frac {2^{-\frac {3}{2}+m} (d+e x)^m \left (1+\frac {e x}{d}\right )^{\frac {1}{2}-m} \, _2F_1\left (-\frac {3}{2},\frac {5}{2}-m;-\frac {1}{2};\frac {d-e x}{2 d}\right )}{\left (3 d^3 e-3 d^2 e^2 x\right ) \sqrt {d^2-e^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.12, size = 0, normalized size = 0.00 \[\int \frac {\left (e x +d \right )^{m}}{\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}\, dx\]
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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (d + e x\right )^{m}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {5}{2}}}\, dx \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 = -0.02 \begin {gather*} \int \frac {{\left (d+e\,x\right )}^m}{{\left (d^2-e^2\,x^2\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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