Optimal. Leaf size=210 \[ \frac{4 e^4 \left (p^2-17 p+48\right ) x \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \left (d^2-e^2 x^2\right )^p \, _2F_1\left (\frac{1}{2},4-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right )}{3 d^8}-\frac{2 e^3 (5-p) \left (d^2-e^2 x^2\right )^{p-3} \, _2F_1\left (1,p-3;p-2;1-\frac{e^2 x^2}{d^2}\right )}{d (3-p)}-\frac{e^2 (27-2 p) \left (d^2-e^2 x^2\right )^{p-3}}{3 x}+\frac{2 d e \left (d^2-e^2 x^2\right )^{p-3}}{x^2}-\frac{d^2 \left (d^2-e^2 x^2\right )^{p-3}}{3 x^3} \]
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Rubi [A] time = 0.390951, antiderivative size = 210, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28, Rules used = {852, 1807, 764, 266, 65, 246, 245} \[ \frac{4 e^4 \left (p^2-17 p+48\right ) x \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \left (d^2-e^2 x^2\right )^p \, _2F_1\left (\frac{1}{2},4-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right )}{3 d^8}-\frac{2 e^3 (5-p) \left (d^2-e^2 x^2\right )^{p-3} \, _2F_1\left (1,p-3;p-2;1-\frac{e^2 x^2}{d^2}\right )}{d (3-p)}-\frac{e^2 (27-2 p) \left (d^2-e^2 x^2\right )^{p-3}}{3 x}+\frac{2 d e \left (d^2-e^2 x^2\right )^{p-3}}{x^2}-\frac{d^2 \left (d^2-e^2 x^2\right )^{p-3}}{3 x^3} \]
Antiderivative was successfully verified.
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Rule 852
Rule 1807
Rule 764
Rule 266
Rule 65
Rule 246
Rule 245
Rubi steps
\begin{align*} \int \frac{\left (d^2-e^2 x^2\right )^p}{x^4 (d+e x)^4} \, dx &=\int \frac{(d-e x)^4 \left (d^2-e^2 x^2\right )^{-4+p}}{x^4} \, dx\\ &=-\frac{d^2 \left (d^2-e^2 x^2\right )^{-3+p}}{3 x^3}-\frac{\int \frac{\left (d^2-e^2 x^2\right )^{-4+p} \left (12 d^5 e-d^4 e^2 (27-2 p) x+12 d^3 e^3 x^2-3 d^2 e^4 x^3\right )}{x^3} \, dx}{3 d^2}\\ &=-\frac{d^2 \left (d^2-e^2 x^2\right )^{-3+p}}{3 x^3}+\frac{2 d e \left (d^2-e^2 x^2\right )^{-3+p}}{x^2}+\frac{\int \frac{\left (d^2-e^2 x^2\right )^{-4+p} \left (2 d^6 e^2 (27-2 p)-24 d^5 e^3 (5-p) x+6 d^4 e^4 x^2\right )}{x^2} \, dx}{6 d^4}\\ &=-\frac{d^2 \left (d^2-e^2 x^2\right )^{-3+p}}{3 x^3}+\frac{2 d e \left (d^2-e^2 x^2\right )^{-3+p}}{x^2}-\frac{e^2 (27-2 p) \left (d^2-e^2 x^2\right )^{-3+p}}{3 x}-\frac{\int \frac{\left (24 d^7 e^3 (5-p)-8 d^6 e^4 \left (48-17 p+p^2\right ) x\right ) \left (d^2-e^2 x^2\right )^{-4+p}}{x} \, dx}{6 d^6}\\ &=-\frac{d^2 \left (d^2-e^2 x^2\right )^{-3+p}}{3 x^3}+\frac{2 d e \left (d^2-e^2 x^2\right )^{-3+p}}{x^2}-\frac{e^2 (27-2 p) \left (d^2-e^2 x^2\right )^{-3+p}}{3 x}-\left (4 d e^3 (5-p)\right ) \int \frac{\left (d^2-e^2 x^2\right )^{-4+p}}{x} \, dx+\frac{1}{3} \left (4 e^4 \left (48-17 p+p^2\right )\right ) \int \left (d^2-e^2 x^2\right )^{-4+p} \, dx\\ &=-\frac{d^2 \left (d^2-e^2 x^2\right )^{-3+p}}{3 x^3}+\frac{2 d e \left (d^2-e^2 x^2\right )^{-3+p}}{x^2}-\frac{e^2 (27-2 p) \left (d^2-e^2 x^2\right )^{-3+p}}{3 x}-\left (2 d e^3 (5-p)\right ) \operatorname{Subst}\left (\int \frac{\left (d^2-e^2 x\right )^{-4+p}}{x} \, dx,x,x^2\right )+\frac{\left (4 e^4 \left (48-17 p+p^2\right ) \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 )^{-4+p} \, dx}{3 d^8}\\ &=-\frac{d^2 \left (d^2-e^2 x^2\right )^{-3+p}}{3 x^3}+\frac{2 d e \left (d^2-e^2 x^2\right )^{-3+p}}{x^2}-\frac{e^2 (27-2 p) \left (d^2-e^2 x^2\right )^{-3+p}}{3 x}+\frac{4 e^4 \left (48-17 p+p^2\right ) 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},4-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right )}{3 d^8}-\frac{2 e^3 (5-p) \left (d^2-e^2 x^2\right )^{-3+p} \, _2F_1\left (1,-3+p;-2+p;1-\frac{e^2 x^2}{d^2}\right )}{d (3-p)}\\ \end{align*}
Mathematica [B] time = 0.674403, size = 452, normalized size = 2.15 \[ \frac{\left (d^2-e^2 x^2\right )^p \left (-\frac{96 d^3 e \left (1-\frac{d^2}{e^2 x^2}\right )^{-p} \, _2F_1\left (1-p,-p;2-p;\frac{d^2}{e^2 x^2}\right )}{(p-1) x^2}-\frac{480 d e^3 \left (1-\frac{d^2}{e^2 x^2}\right )^{-p} \, _2F_1\left (-p,-p;1-p;\frac{d^2}{e^2 x^2}\right )}{p}-\frac{16 d^4 \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (-\frac{3}{2},-p;-\frac{1}{2};\frac{e^2 x^2}{d^2}\right )}{x^3}-\frac{480 d^2 e^2 \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (-\frac{1}{2},-p;\frac{1}{2};\frac{e^2 x^2}{d^2}\right )}{x}+\frac{15 e^3 2^{p+5} (e x-d) \left (\frac{e x}{d}+1\right )^{-p} \, _2F_1\left (1-p,p+1;p+2;\frac{d-e x}{2 d}\right )}{p+1}+\frac{15 e^3 2^{p+3} (e x-d) \left (\frac{e x}{d}+1\right )^{-p} \, _2F_1\left (2-p,p+1;p+2;\frac{d-e x}{2 d}\right )}{p+1}+\frac{3 e^3 2^{p+3} (e x-d) \left (\frac{e x}{d}+1\right )^{-p} \, _2F_1\left (3-p,p+1;p+2;\frac{d-e x}{2 d}\right )}{p+1}+\frac{3 e^3 2^p (e x-d) \left (\frac{e x}{d}+1\right )^{-p} \, _2F_1\left (4-p,p+1;p+2;\frac{d-e x}{2 d}\right )}{p+1}\right )}{48 d^8} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.677, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{p}}{{x}^{4} \left ( ex+d \right ) ^{4}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{p}}{{\left (e x + d\right )}^{4} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{p}}{e^{4} x^{8} + 4 \, d e^{3} x^{7} + 6 \, d^{2} e^{2} x^{6} + 4 \, d^{3} e x^{5} + d^{4} x^{4}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{p}}{x^{4} \left (d + e x\right )^{4}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{p}}{{\left (e x + d\right )}^{4} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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