Optimal. Leaf size=69 \[ \frac{4 b (b d-a e)}{3 e^3 (d+e x)^{3/2}}-\frac{2 (b d-a e)^2}{5 e^3 (d+e x)^{5/2}}-\frac{2 b^2}{e^3 \sqrt{d+e x}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0260152, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {27, 43} \[ \frac{4 b (b d-a e)}{3 e^3 (d+e x)^{3/2}}-\frac{2 (b d-a e)^2}{5 e^3 (d+e x)^{5/2}}-\frac{2 b^2}{e^3 \sqrt{d+e x}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 27
Rule 43
Rubi steps
\begin{align*} \int \frac{a^2+2 a b x+b^2 x^2}{(d+e x)^{7/2}} \, dx &=\int \frac{(a+b x)^2}{(d+e x)^{7/2}} \, dx\\ &=\int \left (\frac{(-b d+a e)^2}{e^2 (d+e x)^{7/2}}-\frac{2 b (b d-a e)}{e^2 (d+e x)^{5/2}}+\frac{b^2}{e^2 (d+e x)^{3/2}}\right ) \, dx\\ &=-\frac{2 (b d-a e)^2}{5 e^3 (d+e x)^{5/2}}+\frac{4 b (b d-a e)}{3 e^3 (d+e x)^{3/2}}-\frac{2 b^2}{e^3 \sqrt{d+e x}}\\ \end{align*}
Mathematica [A] time = 0.0326638, size = 61, normalized size = 0.88 \[ -\frac{2 \left (3 a^2 e^2+2 a b e (2 d+5 e x)+b^2 \left (8 d^2+20 d e x+15 e^2 x^2\right )\right )}{15 e^3 (d+e x)^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.044, size = 63, normalized size = 0.9 \begin{align*} -{\frac{30\,{b}^{2}{x}^{2}{e}^{2}+20\,xab{e}^{2}+40\,x{b}^{2}de+6\,{a}^{2}{e}^{2}+8\,abde+16\,{b}^{2}{d}^{2}}{15\,{e}^{3}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.03004, size = 88, normalized size = 1.28 \begin{align*} -\frac{2 \,{\left (15 \,{\left (e x + d\right )}^{2} b^{2} + 3 \, b^{2} d^{2} - 6 \, a b d e + 3 \, a^{2} e^{2} - 10 \,{\left (b^{2} d - a b e\right )}{\left (e x + d\right )}\right )}}{15 \,{\left (e x + d\right )}^{\frac{5}{2}} e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.47153, size = 204, normalized size = 2.96 \begin{align*} -\frac{2 \,{\left (15 \, b^{2} e^{2} x^{2} + 8 \, b^{2} d^{2} + 4 \, a b d e + 3 \, a^{2} e^{2} + 10 \,{\left (2 \, b^{2} d e + a b e^{2}\right )} x\right )} \sqrt{e x + d}}{15 \,{\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 3.13579, size = 389, normalized size = 5.64 \begin{align*} \begin{cases} - \frac{6 a^{2} e^{2}}{15 d^{2} e^{3} \sqrt{d + e x} + 30 d e^{4} x \sqrt{d + e x} + 15 e^{5} x^{2} \sqrt{d + e x}} - \frac{8 a b d e}{15 d^{2} e^{3} \sqrt{d + e x} + 30 d e^{4} x \sqrt{d + e x} + 15 e^{5} x^{2} \sqrt{d + e x}} - \frac{20 a b e^{2} x}{15 d^{2} e^{3} \sqrt{d + e x} + 30 d e^{4} x \sqrt{d + e x} + 15 e^{5} x^{2} \sqrt{d + e x}} - \frac{16 b^{2} d^{2}}{15 d^{2} e^{3} \sqrt{d + e x} + 30 d e^{4} x \sqrt{d + e x} + 15 e^{5} x^{2} \sqrt{d + e x}} - \frac{40 b^{2} d e x}{15 d^{2} e^{3} \sqrt{d + e x} + 30 d e^{4} x \sqrt{d + e x} + 15 e^{5} x^{2} \sqrt{d + e x}} - \frac{30 b^{2} e^{2} x^{2}}{15 d^{2} e^{3} \sqrt{d + e x} + 30 d e^{4} x \sqrt{d + e x} + 15 e^{5} x^{2} \sqrt{d + e x}} & \text{for}\: e \neq 0 \\\frac{a^{2} x + a b x^{2} + \frac{b^{2} x^{3}}{3}}{d^{\frac{7}{2}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.19207, size = 97, normalized size = 1.41 \begin{align*} -\frac{2 \,{\left (15 \,{\left (x e + d\right )}^{2} b^{2} - 10 \,{\left (x e + d\right )} b^{2} d + 3 \, b^{2} d^{2} + 10 \,{\left (x e + d\right )} a b e - 6 \, a b d e + 3 \, a^{2} e^{2}\right )} e^{\left (-3\right )}}{15 \,{\left (x e + d\right )}^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]