Optimal. Leaf size=33 \[ -\frac{\left (a^2-b^2 x^2\right )^{5/2}}{5 a b (a+b x)^5} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0090839, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {651} \[ -\frac{\left (a^2-b^2 x^2\right )^{5/2}}{5 a b (a+b x)^5} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 651
Rubi steps
\begin{align*} \int \frac{\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^5} \, dx &=-\frac{\left (a^2-b^2 x^2\right )^{5/2}}{5 a b (a+b x)^5}\\ \end{align*}
Mathematica [A] time = 0.0455292, size = 41, normalized size = 1.24 \[ -\frac{(a-b x)^2 \sqrt{a^2-b^2 x^2}}{5 a b (a+b x)^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.043, size = 36, normalized size = 1.1 \begin{align*} -{\frac{-bx+a}{5\, \left ( bx+a \right ) ^{4}ba} \left ( -{b}^{2}{x}^{2}+{a}^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.13154, size = 198, normalized size = 6. \begin{align*} -\frac{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3} +{\left (b^{2} x^{2} - 2 \, a b x + a^{2}\right )} \sqrt{-b^{2} x^{2} + a^{2}}}{5 \,{\left (a b^{4} x^{3} + 3 \, a^{2} b^{3} x^{2} + 3 \, a^{3} b^{2} x + a^{4} b\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 (- a + b x\right ) \left (a + b x\right )\right )^{\frac{3}{2}}}{\left (a + b x\right )^{5}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [C] time = 1.29834, size = 139, normalized size = 4.21 \begin{align*} -\frac{1}{15} \,{\left (-\frac{3 i \, \mathrm{sgn}\left (\frac{1}{b x + a}\right ) \mathrm{sgn}\left (b\right )}{a b^{2}} - \frac{5 \,{\left (\frac{2 \, a}{b x + a} - 1\right )}^{\frac{3}{2}} \mathrm{sgn}\left (\frac{1}{b x + a}\right ) \mathrm{sgn}\left (b\right ) -{\left (3 \,{\left (\frac{2 \, a}{b x + a} - 1\right )}^{\frac{5}{2}} + 5 \,{\left (\frac{2 \, a}{b x + a} - 1\right )}^{\frac{3}{2}}\right )} \mathrm{sgn}\left (\frac{1}{b x + a}\right ) \mathrm{sgn}\left (b\right )}{a b^{2}}\right )}{\left | b \right |} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]