Optimal. Leaf size=100 \[ -\frac{15 a^2 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )}{4 b^{7/2}}+\frac{5 x^{3/2} \sqrt{a-b x}}{2 b^2}+\frac{15 a \sqrt{x} \sqrt{a-b x}}{4 b^3}+\frac{2 x^{5/2}}{b \sqrt{a-b x}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0300671, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {47, 50, 63, 217, 203} \[ -\frac{15 a^2 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )}{4 b^{7/2}}+\frac{5 x^{3/2} \sqrt{a-b x}}{2 b^2}+\frac{15 a \sqrt{x} \sqrt{a-b x}}{4 b^3}+\frac{2 x^{5/2}}{b \sqrt{a-b x}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 47
Rule 50
Rule 63
Rule 217
Rule 203
Rubi steps
\begin{align*} \int \frac{x^{5/2}}{(a-b x)^{3/2}} \, dx &=\frac{2 x^{5/2}}{b \sqrt{a-b x}}-\frac{5 \int \frac{x^{3/2}}{\sqrt{a-b x}} \, dx}{b}\\ &=\frac{2 x^{5/2}}{b \sqrt{a-b x}}+\frac{5 x^{3/2} \sqrt{a-b x}}{2 b^2}-\frac{(15 a) \int \frac{\sqrt{x}}{\sqrt{a-b x}} \, dx}{4 b^2}\\ &=\frac{2 x^{5/2}}{b \sqrt{a-b x}}+\frac{15 a \sqrt{x} \sqrt{a-b x}}{4 b^3}+\frac{5 x^{3/2} \sqrt{a-b x}}{2 b^2}-\frac{\left (15 a^2\right ) \int \frac{1}{\sqrt{x} \sqrt{a-b x}} \, dx}{8 b^3}\\ &=\frac{2 x^{5/2}}{b \sqrt{a-b x}}+\frac{15 a \sqrt{x} \sqrt{a-b x}}{4 b^3}+\frac{5 x^{3/2} \sqrt{a-b x}}{2 b^2}-\frac{\left (15 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a-b x^2}} \, dx,x,\sqrt{x}\right )}{4 b^3}\\ &=\frac{2 x^{5/2}}{b \sqrt{a-b x}}+\frac{15 a \sqrt{x} \sqrt{a-b x}}{4 b^3}+\frac{5 x^{3/2} \sqrt{a-b x}}{2 b^2}-\frac{\left (15 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{1+b x^2} \, dx,x,\frac{\sqrt{x}}{\sqrt{a-b x}}\right )}{4 b^3}\\ &=\frac{2 x^{5/2}}{b \sqrt{a-b x}}+\frac{15 a \sqrt{x} \sqrt{a-b x}}{4 b^3}+\frac{5 x^{3/2} \sqrt{a-b x}}{2 b^2}-\frac{15 a^2 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )}{4 b^{7/2}}\\ \end{align*}
Mathematica [C] time = 0.0125512, size = 51, normalized size = 0.51 \[ \frac{2 x^{7/2} \sqrt{1-\frac{b x}{a}} \, _2F_1\left (\frac{3}{2},\frac{7}{2};\frac{9}{2};\frac{b x}{a}\right )}{7 a \sqrt{a-b x}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.023, size = 127, normalized size = 1.3 \begin{align*}{\frac{2\,bx+7\,a}{4\,{b}^{3}}\sqrt{x}\sqrt{-bx+a}}+{ \left ( -{\frac{15\,{a}^{2}}{8}\arctan \left ({\sqrt{b} \left ( x-{\frac{a}{2\,b}} \right ){\frac{1}{\sqrt{-b{x}^{2}+ax}}}} \right ){b}^{-{\frac{7}{2}}}}-2\,{\frac{{a}^{2}}{{b}^{4}}\sqrt{-b \left ( x-{\frac{a}{b}} \right ) ^{2}-a \left ( x-{\frac{a}{b}} \right ) } \left ( x-{\frac{a}{b}} \right ) ^{-1}} \right ) \sqrt{x \left ( -bx+a \right ) }{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{-bx+a}}}} \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 [A] time = 1.94584, size = 433, normalized size = 4.33 \begin{align*} \left [-\frac{15 \,{\left (a^{2} b x - a^{3}\right )} \sqrt{-b} \log \left (-2 \, b x - 2 \, \sqrt{-b x + a} \sqrt{-b} \sqrt{x} + a\right ) - 2 \,{\left (2 \, b^{3} x^{2} + 5 \, a b^{2} x - 15 \, a^{2} b\right )} \sqrt{-b x + a} \sqrt{x}}{8 \,{\left (b^{5} x - a b^{4}\right )}}, \frac{15 \,{\left (a^{2} b x - a^{3}\right )} \sqrt{b} \arctan \left (\frac{\sqrt{-b x + a}}{\sqrt{b} \sqrt{x}}\right ) +{\left (2 \, b^{3} x^{2} + 5 \, a b^{2} x - 15 \, a^{2} b\right )} \sqrt{-b x + a} \sqrt{x}}{4 \,{\left (b^{5} x - a b^{4}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 15.6244, size = 226, normalized size = 2.26 \begin{align*} \begin{cases} - \frac{15 i a^{\frac{3}{2}} \sqrt{x}}{4 b^{3} \sqrt{-1 + \frac{b x}{a}}} + \frac{5 i \sqrt{a} x^{\frac{3}{2}}}{4 b^{2} \sqrt{-1 + \frac{b x}{a}}} + \frac{15 i a^{2} \operatorname{acosh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{4 b^{\frac{7}{2}}} + \frac{i x^{\frac{5}{2}}}{2 \sqrt{a} b \sqrt{-1 + \frac{b x}{a}}} & \text{for}\: \frac{\left |{b x}\right |}{\left |{a}\right |} > 1 \\\frac{15 a^{\frac{3}{2}} \sqrt{x}}{4 b^{3} \sqrt{1 - \frac{b x}{a}}} - \frac{5 \sqrt{a} x^{\frac{3}{2}}}{4 b^{2} \sqrt{1 - \frac{b x}{a}}} - \frac{15 a^{2} \operatorname{asin}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{4 b^{\frac{7}{2}}} - \frac{x^{\frac{5}{2}}}{2 \sqrt{a} b \sqrt{1 - \frac{b x}{a}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 59.2308, size = 208, normalized size = 2.08 \begin{align*} \frac{{\left (2 \, \sqrt{{\left (b x - a\right )} b + a b} \sqrt{-b x + a}{\left (\frac{2 \,{\left (b x - a\right )}}{b^{3}} + \frac{9 \, a}{b^{3}}\right )} - \frac{32 \, a^{3} \sqrt{-b}}{{\left ({\left (\sqrt{-b x + a} \sqrt{-b} - \sqrt{{\left (b x - a\right )} b + a b}\right )}^{2} - a b\right )} b^{2}} + \frac{15 \, a^{2} \sqrt{-b} \log \left ({\left (\sqrt{-b x + a} \sqrt{-b} - \sqrt{{\left (b x - a\right )} b + a b}\right )}^{2}\right )}{b^{3}}\right )}{\left | b \right |}}{8 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]