Optimal. Leaf size=66 \[ -\frac{8 \sqrt{a-b x}}{3 a^2 x^{3/2}}-\frac{16 b \sqrt{a-b x}}{3 a^3 \sqrt{x}}+\frac{2}{a x^{3/2} \sqrt{a-b x}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0104674, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {45, 37} \[ -\frac{8 \sqrt{a-b x}}{3 a^2 x^{3/2}}-\frac{16 b \sqrt{a-b x}}{3 a^3 \sqrt{x}}+\frac{2}{a x^{3/2} \sqrt{a-b x}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rule 37
Rubi steps
\begin{align*} \int \frac{1}{x^{5/2} (a-b x)^{3/2}} \, dx &=\frac{2}{a x^{3/2} \sqrt{a-b x}}+\frac{4 \int \frac{1}{x^{5/2} \sqrt{a-b x}} \, dx}{a}\\ &=\frac{2}{a x^{3/2} \sqrt{a-b x}}-\frac{8 \sqrt{a-b x}}{3 a^2 x^{3/2}}+\frac{(8 b) \int \frac{1}{x^{3/2} \sqrt{a-b x}} \, dx}{3 a^2}\\ &=\frac{2}{a x^{3/2} \sqrt{a-b x}}-\frac{8 \sqrt{a-b x}}{3 a^2 x^{3/2}}-\frac{16 b \sqrt{a-b x}}{3 a^3 \sqrt{x}}\\ \end{align*}
Mathematica [A] time = 0.0112065, size = 39, normalized size = 0.59 \[ -\frac{2 \left (a^2+4 a b x-8 b^2 x^2\right )}{3 a^3 x^{3/2} \sqrt{a-b x}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.005, size = 34, normalized size = 0.5 \begin{align*} -{\frac{-16\,{b}^{2}{x}^{2}+8\,abx+2\,{a}^{2}}{3\,{a}^{3}}{x}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{-bx+a}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.03725, size = 70, normalized size = 1.06 \begin{align*} \frac{2 \, b^{2} \sqrt{x}}{\sqrt{-b x + a} a^{3}} - \frac{2 \,{\left (\frac{6 \, \sqrt{-b x + a} b}{\sqrt{x}} + \frac{{\left (-b x + a\right )}^{\frac{3}{2}}}{x^{\frac{3}{2}}}\right )}}{3 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.84799, size = 107, normalized size = 1.62 \begin{align*} -\frac{2 \,{\left (8 \, b^{2} x^{2} - 4 \, a b x - a^{2}\right )} \sqrt{-b x + a} \sqrt{x}}{3 \,{\left (a^{3} b x^{3} - a^{4} x^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 16.2018, size = 456, normalized size = 6.91 \begin{align*} \begin{cases} - \frac{2 a^{3} b^{\frac{9}{2}} \sqrt{\frac{a}{b x} - 1}}{3 a^{5} b^{4} x - 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6} x^{3}} - \frac{6 a^{2} b^{\frac{11}{2}} x \sqrt{\frac{a}{b x} - 1}}{3 a^{5} b^{4} x - 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6} x^{3}} + \frac{24 a b^{\frac{13}{2}} x^{2} \sqrt{\frac{a}{b x} - 1}}{3 a^{5} b^{4} x - 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6} x^{3}} - \frac{16 b^{\frac{15}{2}} x^{3} \sqrt{\frac{a}{b x} - 1}}{3 a^{5} b^{4} x - 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6} x^{3}} & \text{for}\: \frac{\left |{a}\right |}{\left |{b}\right | \left |{x}\right |} > 1 \\- \frac{2 i a^{3} b^{\frac{9}{2}} \sqrt{- \frac{a}{b x} + 1}}{3 a^{5} b^{4} x - 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6} x^{3}} - \frac{6 i a^{2} b^{\frac{11}{2}} x \sqrt{- \frac{a}{b x} + 1}}{3 a^{5} b^{4} x - 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6} x^{3}} + \frac{24 i a b^{\frac{13}{2}} x^{2} \sqrt{- \frac{a}{b x} + 1}}{3 a^{5} b^{4} x - 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6} x^{3}} - \frac{16 i b^{\frac{15}{2}} x^{3} \sqrt{- \frac{a}{b x} + 1}}{3 a^{5} b^{4} x - 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6} x^{3}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.09395, size = 144, normalized size = 2.18 \begin{align*} -\frac{\sqrt{-b x + a}{\left (\frac{5 \,{\left (b x - a\right )}{\left | b \right |}}{b^{2}} + \frac{6 \, a{\left | b \right |}}{b^{2}}\right )}}{24 \,{\left ({\left (b x - a\right )} b + a b\right )}^{\frac{3}{2}}} - \frac{4 \, \sqrt{-b} b^{3}}{{\left ({\left (\sqrt{-b x + a} \sqrt{-b} - \sqrt{{\left (b x - a\right )} b + a b}\right )}^{2} - a b\right )} a^{2}{\left | b \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]