Optimal. Leaf size=173 \[ \frac{e^2 (6-p) \left (d^2-e^2 x^2\right )^{p-2} \, _2F_1\left (1,p-2;p-1;1-\frac{e^2 x^2}{d^2}\right )}{2 d (2-p)}-\frac{2 e^3 (8-3 p) x \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \left (d^2-e^2 x^2\right )^p \, _2F_1\left (\frac{1}{2},3-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right )}{d^6}+\frac{3 e \left (d^2-e^2 x^2\right )^{p-2}}{x}-\frac{d \left (d^2-e^2 x^2\right )^{p-2}}{2 x^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.265752, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ \frac{e^2 (6-p) \left (d^2-e^2 x^2\right )^{p-2} \, _2F_1\left (1,p-2;p-1;1-\frac{e^2 x^2}{d^2}\right )}{2 d (2-p)}-\frac{2 e^3 (8-3 p) x \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \left (d^2-e^2 x^2\right )^p \, _2F_1\left (\frac{1}{2},3-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right )}{d^6}+\frac{3 e \left (d^2-e^2 x^2\right )^{p-2}}{x}-\frac{d \left (d^2-e^2 x^2\right )^{p-2}}{2 x^2} \]
Antiderivative was successfully verified.
[In]
[Out]
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^3 (d+e x)^3} \, dx &=\int \frac{(d-e x)^3 \left (d^2-e^2 x^2\right )^{-3+p}}{x^3} \, dx\\ &=-\frac{d \left (d^2-e^2 x^2\right )^{-2+p}}{2 x^2}-\frac{\int \frac{\left (d^2-e^2 x^2\right )^{-3+p} \left (6 d^4 e-2 d^3 e^2 (6-p) x+2 d^2 e^3 x^2\right )}{x^2} \, dx}{2 d^2}\\ &=-\frac{d \left (d^2-e^2 x^2\right )^{-2+p}}{2 x^2}+\frac{3 e \left (d^2-e^2 x^2\right )^{-2+p}}{x}+\frac{\int \frac{\left (2 d^5 e^2 (6-p)-4 d^4 e^3 (8-3 p) x\right ) \left (d^2-e^2 x^2\right )^{-3+p}}{x} \, dx}{2 d^4}\\ &=-\frac{d \left (d^2-e^2 x^2\right )^{-2+p}}{2 x^2}+\frac{3 e \left (d^2-e^2 x^2\right )^{-2+p}}{x}-\left (2 e^3 (8-3 p)\right ) \int \left (d^2-e^2 x^2\right )^{-3+p} \, dx+\left (d e^2 (6-p)\right ) \int \frac{\left (d^2-e^2 x^2\right )^{-3+p}}{x} \, dx\\ &=-\frac{d \left (d^2-e^2 x^2\right )^{-2+p}}{2 x^2}+\frac{3 e \left (d^2-e^2 x^2\right )^{-2+p}}{x}+\frac{1}{2} \left (d e^2 (6-p)\right ) \operatorname{Subst}\left (\int \frac{\left (d^2-e^2 x\right )^{-3+p}}{x} \, dx,x,x^2\right )-\frac{\left (2 e^3 (8-3 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^6}\\ &=-\frac{d \left (d^2-e^2 x^2\right )^{-2+p}}{2 x^2}+\frac{3 e \left (d^2-e^2 x^2\right )^{-2+p}}{x}-\frac{2 e^3 (8-3 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^6}+\frac{e^2 (6-p) \left (d^2-e^2 x^2\right )^{-2+p} \, _2F_1\left (1,-2+p;-1+p;1-\frac{e^2 x^2}{d^2}\right )}{2 d (2-p)}\\ \end{align*}
Mathematica [A] time = 0.680982, size = 341, normalized size = 1.97 \[ \frac{\left (d^2-e^2 x^2\right )^p \left (\frac{4 d^3 \left (1-\frac{d^2}{e^2 x^2}\right )^{-p} \, _2F_1\left (1-p,-p;2-p;\frac{d^2}{e^2 x^2}\right )}{(p-1) x^2}+\frac{24 d e^2 \left (1-\frac{d^2}{e^2 x^2}\right )^{-p} \, _2F_1\left (-p,-p;1-p;\frac{d^2}{e^2 x^2}\right )}{p}+\frac{24 d^2 e \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (-\frac{1}{2},-p;\frac{1}{2};\frac{e^2 x^2}{d^2}\right )}{x}+\frac{3 e^2 2^{p+3} (d-e x) \left (\frac{e x}{d}+1\right )^{-p} \, _2F_1\left (1-p,p+1;p+2;\frac{d-e x}{2 d}\right )}{p+1}+\frac{3 e^2 2^{p+1} (d-e x) \left (\frac{e x}{d}+1\right )^{-p} \, _2F_1\left (2-p,p+1;p+2;\frac{d-e x}{2 d}\right )}{p+1}+\frac{e^2 2^p (d-e x) \left (\frac{e x}{d}+1\right )^{-p} \, _2F_1\left (3-p,p+1;p+2;\frac{d-e x}{2 d}\right )}{p+1}\right )}{8 d^6} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.68, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{p}}{{x}^{3} \left ( ex+d \right ) ^{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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 )}^{3} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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^{3} x^{6} + 3 \, d e^{2} x^{5} + 3 \, d^{2} e x^{4} + d^{3} x^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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^{3} \left (d + e x\right )^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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 )}^{3} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]