Optimal. Leaf size=59 \[ \frac {(d+e x)^m \left (d^2-e^2 x^2\right )^{5/2} \, _2F_1\left (1,5+m;\frac {7}{2}+m;\frac {d+e x}{2 d}\right )}{d e (5+2 m)} \]
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Rubi [A]
time = 0.03, antiderivative size = 83, normalized size of antiderivative = 1.41, 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 {5}{2}} \left (d^2-e^2 x^2\right )^{5/2} (d+e x)^m \left (\frac {e x}{d}+1\right )^{-m-\frac {5}{2}} \, _2F_1\left (\frac {5}{2},-m-\frac {3}{2};\frac {7}{2};\frac {d-e x}{2 d}\right )}{5 d e} \end {gather*}
Antiderivative was successfully verified.
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Rule 71
Rule 692
Rule 694
Rubi steps
\begin {align*} \int (d+e x)^m \left (d^2-e^2 x^2\right )^{3/2} \, dx &=\left ((d+e x)^m \left (1+\frac {e x}{d}\right )^{-m}\right ) \int \left (1+\frac {e x}{d}\right )^m \left (d^2-e^2 x^2\right )^{3/2} \, dx\\ &=\frac {\left ((d+e x)^m \left (1+\frac {e x}{d}\right )^{-\frac {5}{2}-m} \left (d^2-e^2 x^2\right )^{5/2}\right ) \int \left (1+\frac {e x}{d}\right )^{\frac {3}{2}+m} \left (d^2-d e x\right )^{3/2} \, dx}{\left (d^2-d e x\right )^{5/2}}\\ &=-\frac {2^{\frac {5}{2}+m} (d+e x)^m \left (1+\frac {e x}{d}\right )^{-\frac {5}{2}-m} \left (d^2-e^2 x^2\right )^{5/2} \, _2F_1\left (\frac {5}{2},-\frac {3}{2}-m;\frac {7}{2};\frac {d-e x}{2 d}\right )}{5 d e}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 5 in
optimal.
time = 0.48, size = 191, normalized size = 3.24 \begin {gather*} -\frac {2^m (d+e x)^m \left (1+\frac {e x}{d}\right )^{-\frac {1}{2}-2 m} \left (e^3 x^3 \sqrt {d-e x} \sqrt {d+e x} \left (\frac {1}{2}+\frac {e x}{2 d}\right )^m F_1\left (3;-\frac {1}{2},-\frac {1}{2}-m;4;\frac {e x}{d},-\frac {e x}{d}\right )+2 d^2 (d-e x) \sqrt {2-\frac {2 e x}{d}} \left (1+\frac {e x}{d}\right )^m \sqrt {d^2-e^2 x^2} \, _2F_1\left (\frac {3}{2},-\frac {1}{2}-m;\frac {5}{2};\frac {d-e x}{2 d}\right )\right )}{3 e \sqrt {1-\frac {e x}{d}}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.12, size = 0, normalized size = 0.00 \[\int \left (e x +d \right )^{m} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}\, dx\]
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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {3}{2}} \left (d + e x\right )^{m}\, 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 {\left (d^2-e^2\,x^2\right )}^{3/2}\,{\left (d+e\,x\right )}^m \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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