Optimal. Leaf size=175 \[ \frac {2 d \left (d^2-e^2 x^2\right )^{-2+p}}{2-p}-\frac {e x \left (d^2-e^2 x^2\right )^{-2+p}}{3-2 p}-\frac {2 e (4-3 p) x \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac {1}{2},3-p;\frac {3}{2};\frac {e^2 x^2}{d^2}\right )}{d^4 (3-2 p)}+\frac {\left (d^2-e^2 x^2\right )^{-1+p} \, _2F_1\left (1,-1+p;p;1-\frac {e^2 x^2}{d^2}\right )}{2 d (1-p)} \]
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Rubi [A]
time = 0.10, antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {866, 1666, 457,
80, 67, 396, 252, 251} \begin {gather*} \frac {\left (d^2-e^2 x^2\right )^{p-1} \, _2F_1\left (1,p-1;p;1-\frac {e^2 x^2}{d^2}\right )}{2 d (1-p)}-\frac {e x \left (d^2-e^2 x^2\right )^{p-2}}{3-2 p}+\frac {2 d \left (d^2-e^2 x^2\right )^{p-2}}{2-p}-\frac {2 e (4-3 p) x \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \left (d^2-e^2 x^2\right )^p \, _2F_1\left (\frac {1}{2},3-p;\frac {3}{2};\frac {e^2 x^2}{d^2}\right )}{d^4 (3-2 p)} \end {gather*}
Antiderivative was successfully verified.
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Rule 67
Rule 80
Rule 251
Rule 252
Rule 396
Rule 457
Rule 866
Rule 1666
Rubi steps
\begin {align*} \int \frac {\left (d^2-e^2 x^2\right )^p}{x (d+e x)^3} \, dx &=\int \frac {(d-e x)^3 \left (d^2-e^2 x^2\right )^{-3+p}}{x} \, dx\\ &=\int \frac {\left (d^2-e^2 x^2\right )^{-3+p} \left (d^3+3 d e^2 x^2\right )}{x} \, dx+\int \left (d^2-e^2 x^2\right )^{-3+p} \left (-3 d^2 e-e^3 x^2\right ) \, dx\\ &=-\frac {e x \left (d^2-e^2 x^2\right )^{-2+p}}{3-2 p}+\frac {1}{2} \text {Subst}\left (\int \frac {\left (d^2-e^2 x\right )^{-3+p} \left (d^3+3 d e^2 x\right )}{x} \, dx,x,x^2\right )-\frac {\left (2 d^2 e (4-3 p)\right ) \int \left (d^2-e^2 x^2\right )^{-3+p} \, dx}{3-2 p}\\ &=\frac {2 d \left (d^2-e^2 x^2\right )^{-2+p}}{2-p}-\frac {e x \left (d^2-e^2 x^2\right )^{-2+p}}{3-2 p}+\frac {1}{2} d \text {Subst}\left (\int \frac {\left (d^2-e^2 x\right )^{-2+p}}{x} \, dx,x,x^2\right )-\frac {\left (2 e (4-3 p) \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p}\right ) \int \left (1-\frac {e^2 x^2}{d^2}\right )^{-3+p} \, dx}{d^4 (3-2 p)}\\ &=\frac {2 d \left (d^2-e^2 x^2\right )^{-2+p}}{2-p}-\frac {e x \left (d^2-e^2 x^2\right )^{-2+p}}{3-2 p}-\frac {2 e (4-3 p) x \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac {1}{2},3-p;\frac {3}{2};\frac {e^2 x^2}{d^2}\right )}{d^4 (3-2 p)}+\frac {\left (d^2-e^2 x^2\right )^{-1+p} \, _2F_1\left (1,-1+p;p;1-\frac {e^2 x^2}{d^2}\right )}{2 d (1-p)}\\ \end {align*}
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Mathematica [A]
time = 0.38, size = 328, normalized size = 1.87 \begin {gather*} \frac {2^{-3+p} \left (1-\frac {d^2}{e^2 x^2}\right )^{-p} \left (1+\frac {e x}{d}\right )^{-p} \left (d^2-e^2 x^2\right )^p \left (4 p \left (1-\frac {d^2}{e^2 x^2}\right )^p (d-e x) \, _2F_1\left (1-p,1+p;2+p;\frac {d-e x}{2 d}\right )+2 p \left (1-\frac {d^2}{e^2 x^2}\right )^p (d-e x) \, _2F_1\left (2-p,1+p;2+p;\frac {d-e x}{2 d}\right )+d p \left (1-\frac {d^2}{e^2 x^2}\right )^p \, _2F_1\left (3-p,1+p;2+p;\frac {d-e x}{2 d}\right )-e p \left (1-\frac {d^2}{e^2 x^2}\right )^p x \, _2F_1\left (3-p,1+p;2+p;\frac {d-e x}{2 d}\right )+4 d \left (\frac {1}{2}+\frac {e x}{2 d}\right )^p \, _2F_1\left (-p,-p;1-p;\frac {d^2}{e^2 x^2}\right )+4 d p \left (\frac {1}{2}+\frac {e x}{2 d}\right )^p \, _2F_1\left (-p,-p;1-p;\frac {d^2}{e^2 x^2}\right )\right )}{d^4 p (1+p)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (-e^{2} x^{2}+d^{2}\right )^{p}}{x \left (e x +d \right )^{3}}\, dx\]
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 (- \left (- d + e x\right ) \left (d + e x\right )\right )^{p}}{x \left (d + e x\right )^{3}}\, 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 = -0.01 \begin {gather*} \int \frac {{\left (d^2-e^2\,x^2\right )}^p}{x\,{\left (d+e\,x\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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