Optimal. Leaf size=30 \[ -\frac{1}{2 b c (a+b x) \sqrt{c (a+b x)^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0112982, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {247, 15, 30} \[ -\frac{1}{2 b c (a+b x) \sqrt{c (a+b x)^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 247
Rule 15
Rule 30
Rubi steps
\begin{align*} \int \frac{1}{\left (c (a+b x)^2\right )^{3/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\left (c x^2\right )^{3/2}} \, dx,x,a+b x\right )}{b}\\ &=\frac{(a+b x) \operatorname{Subst}\left (\int \frac{1}{x^3} \, dx,x,a+b x\right )}{b c \sqrt{c (a+b x)^2}}\\ &=-\frac{1}{2 b c (a+b x) \sqrt{c (a+b x)^2}}\\ \end{align*}
Mathematica [A] time = 0.0085386, size = 25, normalized size = 0.83 \[ -\frac{a+b x}{2 b \left (c (a+b x)^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.002, size = 22, normalized size = 0.7 \begin{align*} -{\frac{bx+a}{2\,b} \left ( c \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.955827, size = 24, normalized size = 0.8 \begin{align*} -\frac{1}{2 \, \left (b^{2} c\right )^{\frac{3}{2}}{\left (x + \frac{a}{b}\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.34497, size = 140, normalized size = 4.67 \begin{align*} -\frac{\sqrt{b^{2} c x^{2} + 2 \, a b c x + a^{2} c}}{2 \,{\left (b^{4} c^{2} x^{3} + 3 \, a b^{3} c^{2} x^{2} + 3 \, a^{2} b^{2} c^{2} x + a^{3} b c^{2}\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{1}{\left (c \left (a + b x\right )^{2}\right )^{\frac{3}{2}}}\, 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*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]