Optimal. Leaf size=51 \[ \frac{2 d x}{3 a^2 \sqrt{a+c x^2}}-\frac{a e-c d x}{3 a c \left (a+c x^2\right )^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0114975, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {639, 191} \[ \frac{2 d x}{3 a^2 \sqrt{a+c x^2}}-\frac{a e-c d x}{3 a c \left (a+c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 639
Rule 191
Rubi steps
\begin{align*} \int \frac{d+e x}{\left (a+c x^2\right )^{5/2}} \, dx &=-\frac{a e-c d x}{3 a c \left (a+c x^2\right )^{3/2}}+\frac{(2 d) \int \frac{1}{\left (a+c x^2\right )^{3/2}} \, dx}{3 a}\\ &=-\frac{a e-c d x}{3 a c \left (a+c x^2\right )^{3/2}}+\frac{2 d x}{3 a^2 \sqrt{a+c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0182051, size = 43, normalized size = 0.84 \[ \frac{-a^2 e+3 a c d x+2 c^2 d x^3}{3 a^2 c \left (a+c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.043, size = 39, normalized size = 0.8 \begin{align*} -{\frac{-2\,{c}^{2}d{x}^{3}-3\,dxac+{a}^{2}e}{3\,{a}^{2}c} \left ( c{x}^{2}+a \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.17246, size = 65, normalized size = 1.27 \begin{align*} \frac{2 \, d x}{3 \, \sqrt{c x^{2} + a} a^{2}} + \frac{d x}{3 \,{\left (c x^{2} + a\right )}^{\frac{3}{2}} a} - \frac{e}{3 \,{\left (c x^{2} + a\right )}^{\frac{3}{2}} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.18817, size = 126, normalized size = 2.47 \begin{align*} \frac{{\left (2 \, c^{2} d x^{3} + 3 \, a c d x - a^{2} e\right )} \sqrt{c x^{2} + a}}{3 \,{\left (a^{2} c^{3} x^{4} + 2 \, a^{3} c^{2} x^{2} + a^{4} c\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 13.3911, size = 146, normalized size = 2.86 \begin{align*} d \left (\frac{3 a x}{3 a^{\frac{7}{2}} \sqrt{1 + \frac{c x^{2}}{a}} + 3 a^{\frac{5}{2}} c x^{2} \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{2 c x^{3}}{3 a^{\frac{7}{2}} \sqrt{1 + \frac{c x^{2}}{a}} + 3 a^{\frac{5}{2}} c x^{2} \sqrt{1 + \frac{c x^{2}}{a}}}\right ) + e \left (\begin{cases} - \frac{1}{3 a c \sqrt{a + c x^{2}} + 3 c^{2} x^{2} \sqrt{a + c x^{2}}} & \text{for}\: c \neq 0 \\\frac{x^{2}}{2 a^{\frac{5}{2}}} & \text{otherwise} \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.6705, size = 51, normalized size = 1. \begin{align*} \frac{{\left (\frac{2 \, c d x^{2}}{a^{2}} + \frac{3 \, d}{a}\right )} x - \frac{e}{c}}{3 \,{\left (c x^{2} + a\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]