Optimal. Leaf size=159 \[ 2 d e^2 (1-p) x \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};\frac {e^2 x^2}{d^2}\right )-\frac {3 e \left (d^2-e^2 x^2\right )^{p+1} \, _2F_1\left (1,p+1;p+2;1-\frac {e^2 x^2}{d^2}\right )}{2 (p+1)}-\frac {e \left (d^2-e^2 x^2\right )^{p+1}}{2 (p+1)}-\frac {d \left (d^2-e^2 x^2\right )^{p+1}}{x} \]
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Rubi [A] time = 0.18, antiderivative size = 159, 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 = {1807, 1652, 446, 80, 65, 12, 246, 245} \[ 2 d e^2 (1-p) x \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};\frac {e^2 x^2}{d^2}\right )-\frac {3 e \left (d^2-e^2 x^2\right )^{p+1} \, _2F_1\left (1,p+1;p+2;1-\frac {e^2 x^2}{d^2}\right )}{2 (p+1)}-\frac {e \left (d^2-e^2 x^2\right )^{p+1}}{2 (p+1)}-\frac {d \left (d^2-e^2 x^2\right )^{p+1}}{x} \]
Antiderivative was successfully verified.
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Rule 12
Rule 65
Rule 80
Rule 245
Rule 246
Rule 446
Rule 1652
Rule 1807
Rubi steps
\begin {align*} \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^p}{x^2} \, dx &=-\frac {d \left (d^2-e^2 x^2\right )^{1+p}}{x}-\frac {\int \frac {\left (d^2-e^2 x^2\right )^p \left (-3 d^4 e-2 d^3 e^2 (1-p) x-d^2 e^3 x^2\right )}{x} \, dx}{d^2}\\ &=-\frac {d \left (d^2-e^2 x^2\right )^{1+p}}{x}-\frac {\int -2 d^3 e^2 (1-p) \left (d^2-e^2 x^2\right )^p \, dx}{d^2}-\frac {\int \frac {\left (d^2-e^2 x^2\right )^p \left (-3 d^4 e-d^2 e^3 x^2\right )}{x} \, dx}{d^2}\\ &=-\frac {d \left (d^2-e^2 x^2\right )^{1+p}}{x}-\frac {\operatorname {Subst}\left (\int \frac {\left (d^2-e^2 x\right )^p \left (-3 d^4 e-d^2 e^3 x\right )}{x} \, dx,x,x^2\right )}{2 d^2}+\left (2 d e^2 (1-p)\right ) \int \left (d^2-e^2 x^2\right )^p \, dx\\ &=-\frac {e \left (d^2-e^2 x^2\right )^{1+p}}{2 (1+p)}-\frac {d \left (d^2-e^2 x^2\right )^{1+p}}{x}+\frac {1}{2} \left (3 d^2 e\right ) \operatorname {Subst}\left (\int \frac {\left (d^2-e^2 x\right )^p}{x} \, dx,x,x^2\right )+\left (2 d e^2 (1-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 )^p \, dx\\ &=-\frac {e \left (d^2-e^2 x^2\right )^{1+p}}{2 (1+p)}-\frac {d \left (d^2-e^2 x^2\right )^{1+p}}{x}+2 d e^2 (1-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},-p;\frac {3}{2};\frac {e^2 x^2}{d^2}\right )-\frac {3 e \left (d^2-e^2 x^2\right )^{1+p} \, _2F_1\left (1,1+p;2+p;1-\frac {e^2 x^2}{d^2}\right )}{2 (1+p)}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 158, normalized size = 0.99 \[ \frac {\left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \left (e x \left (6 d e (p+1) x \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};\frac {e^2 x^2}{d^2}\right )-\left (d^2-e^2 x^2\right ) \left (1-\frac {e^2 x^2}{d^2}\right )^p \left (3 \, _2F_1\left (1,p+1;p+2;1-\frac {e^2 x^2}{d^2}\right )+1\right )\right )-2 d^3 (p+1) \, _2F_1\left (-\frac {1}{2},-p;\frac {1}{2};\frac {e^2 x^2}{d^2}\right )\right )}{2 (p+1) x} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.86, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}\right )} {\left (-e^{2} x^{2} + d^{2}\right )}^{p}}{x^{2}}, x\right ) \]
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 \[ \int \frac {{\left (e x + d\right )}^{3} {\left (-e^{2} x^{2} + d^{2}\right )}^{p}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.04, size = 0, normalized size = 0.00 \[ \int \frac {\left (e x +d \right )^{3} \left (-e^{2} x^{2}+d^{2}\right )^{p}}{x^{2}}\, dx \]
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 \[ \int \frac {{\left (e x + d\right )}^{3} {\left (-e^{2} x^{2} + d^{2}\right )}^{p}}{x^{2}}\,{d x} \]
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 \[ \int \frac {{\left (d^2-e^2\,x^2\right )}^p\,{\left (d+e\,x\right )}^3}{x^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 7.45, size = 177, normalized size = 1.11 \[ - \frac {d^{3} d^{2 p} {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - p \\ \frac {1}{2} \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{x} - \frac {3 d^{2} e e^{2 p} x^{2 p} e^{i \pi p} \Gamma \left (- p\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, - p \\ 1 - p \end {matrix}\middle | {\frac {d^{2}}{e^{2} x^{2}}} \right )}}{2 \Gamma \left (1 - p\right )} + 3 d d^{2 p} e^{2} x {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, - p \\ \frac {3}{2} \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )} + e^{3} \left (\begin {cases} \frac {x^{2} \left (d^{2}\right )^{p}}{2} & \text {for}\: e^{2} = 0 \\- \frac {\begin {cases} \frac {\left (d^{2} - e^{2} x^{2}\right )^{p + 1}}{p + 1} & \text {for}\: p \neq -1 \\\log {\left (d^{2} - e^{2} x^{2} \right )} & \text {otherwise} \end {cases}}{2 e^{2}} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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