Optimal. Leaf size=67 \[ \frac{4 b (b d-a e)}{e^3 \sqrt{d+e x}}-\frac{2 (b d-a e)^2}{3 e^3 (d+e x)^{3/2}}+\frac{2 b^2 \sqrt{d+e x}}{e^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0222164, antiderivative size = 67, 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} \[ \frac{4 b (b d-a e)}{e^3 \sqrt{d+e x}}-\frac{2 (b d-a e)^2}{3 e^3 (d+e x)^{3/2}}+\frac{2 b^2 \sqrt{d+e x}}{e^3} \]
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)^{5/2}} \, dx &=\int \frac{(a+b x)^2}{(d+e x)^{5/2}} \, dx\\ &=\int \left (\frac{(-b d+a e)^2}{e^2 (d+e x)^{5/2}}-\frac{2 b (b d-a e)}{e^2 (d+e x)^{3/2}}+\frac{b^2}{e^2 \sqrt{d+e x}}\right ) \, dx\\ &=-\frac{2 (b d-a e)^2}{3 e^3 (d+e x)^{3/2}}+\frac{4 b (b d-a e)}{e^3 \sqrt{d+e x}}+\frac{2 b^2 \sqrt{d+e x}}{e^3}\\ \end{align*}
Mathematica [A] time = 0.0360681, size = 62, normalized size = 0.93 \[ \frac{-2 a^2 e^2-4 a b e (2 d+3 e x)+2 b^2 \left (8 d^2+12 d e x+3 e^2 x^2\right )}{3 e^3 (d+e x)^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.046, size = 62, normalized size = 0.9 \begin{align*} -{\frac{-6\,{b}^{2}{x}^{2}{e}^{2}+12\,xab{e}^{2}-24\,x{b}^{2}de+2\,{a}^{2}{e}^{2}+8\,abde-16\,{b}^{2}{d}^{2}}{3\,{e}^{3}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.0363, size = 97, normalized size = 1.45 \begin{align*} \frac{2 \,{\left (\frac{3 \, \sqrt{e x + d} b^{2}}{e^{2}} - \frac{b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2} - 6 \,{\left (b^{2} d - a b e\right )}{\left (e x + d\right )}}{{\left (e x + d\right )}^{\frac{3}{2}} e^{2}}\right )}}{3 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.44072, size = 174, normalized size = 2.6 \begin{align*} \frac{2 \,{\left (3 \, b^{2} e^{2} x^{2} + 8 \, b^{2} d^{2} - 4 \, a b d e - a^{2} e^{2} + 6 \,{\left (2 \, b^{2} d e - a b e^{2}\right )} x\right )} \sqrt{e x + d}}{3 \,{\left (e^{5} x^{2} + 2 \, d e^{4} x + d^{2} e^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 1.45511, size = 265, normalized size = 3.96 \begin{align*} \begin{cases} - \frac{2 a^{2} e^{2}}{3 d e^{3} \sqrt{d + e x} + 3 e^{4} x \sqrt{d + e x}} - \frac{8 a b d e}{3 d e^{3} \sqrt{d + e x} + 3 e^{4} x \sqrt{d + e x}} - \frac{12 a b e^{2} x}{3 d e^{3} \sqrt{d + e x} + 3 e^{4} x \sqrt{d + e x}} + \frac{16 b^{2} d^{2}}{3 d e^{3} \sqrt{d + e x} + 3 e^{4} x \sqrt{d + e x}} + \frac{24 b^{2} d e x}{3 d e^{3} \sqrt{d + e x} + 3 e^{4} x \sqrt{d + e x}} + \frac{6 b^{2} e^{2} x^{2}}{3 d e^{3} \sqrt{d + e x} + 3 e^{4} x \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{5}{2}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.24103, size = 101, normalized size = 1.51 \begin{align*} 2 \, \sqrt{x e + d} b^{2} e^{\left (-3\right )} + \frac{2 \,{\left (6 \,{\left (x e + d\right )} b^{2} d - b^{2} d^{2} - 6 \,{\left (x e + d\right )} a b e + 2 \, a b d e - a^{2} e^{2}\right )} e^{\left (-3\right )}}{3 \,{\left (x e + d\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]