Optimal. Leaf size=86 \[ 5 b^2 \sqrt{x} \sqrt{a+b x}+5 a b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )-\frac{2 (a+b x)^{5/2}}{3 x^{3/2}}-\frac{10 b (a+b x)^{3/2}}{3 \sqrt{x}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0262959, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {47, 50, 63, 217, 206} \[ 5 b^2 \sqrt{x} \sqrt{a+b x}+5 a b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )-\frac{2 (a+b x)^{5/2}}{3 x^{3/2}}-\frac{10 b (a+b x)^{3/2}}{3 \sqrt{x}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 47
Rule 50
Rule 63
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{(a+b x)^{5/2}}{x^{5/2}} \, dx &=-\frac{2 (a+b x)^{5/2}}{3 x^{3/2}}+\frac{1}{3} (5 b) \int \frac{(a+b x)^{3/2}}{x^{3/2}} \, dx\\ &=-\frac{10 b (a+b x)^{3/2}}{3 \sqrt{x}}-\frac{2 (a+b x)^{5/2}}{3 x^{3/2}}+\left (5 b^2\right ) \int \frac{\sqrt{a+b x}}{\sqrt{x}} \, dx\\ &=5 b^2 \sqrt{x} \sqrt{a+b x}-\frac{10 b (a+b x)^{3/2}}{3 \sqrt{x}}-\frac{2 (a+b x)^{5/2}}{3 x^{3/2}}+\frac{1}{2} \left (5 a b^2\right ) \int \frac{1}{\sqrt{x} \sqrt{a+b x}} \, dx\\ &=5 b^2 \sqrt{x} \sqrt{a+b x}-\frac{10 b (a+b x)^{3/2}}{3 \sqrt{x}}-\frac{2 (a+b x)^{5/2}}{3 x^{3/2}}+\left (5 a b^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,\sqrt{x}\right )\\ &=5 b^2 \sqrt{x} \sqrt{a+b x}-\frac{10 b (a+b x)^{3/2}}{3 \sqrt{x}}-\frac{2 (a+b x)^{5/2}}{3 x^{3/2}}+\left (5 a b^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{\sqrt{x}}{\sqrt{a+b x}}\right )\\ &=5 b^2 \sqrt{x} \sqrt{a+b x}-\frac{10 b (a+b x)^{3/2}}{3 \sqrt{x}}-\frac{2 (a+b x)^{5/2}}{3 x^{3/2}}+5 a b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )\\ \end{align*}
Mathematica [C] time = 0.0102402, size = 50, normalized size = 0.58 \[ -\frac{2 a^2 \sqrt{a+b x} \, _2F_1\left (-\frac{5}{2},-\frac{3}{2};-\frac{1}{2};-\frac{b x}{a}\right )}{3 x^{3/2} \sqrt{\frac{b x}{a}+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.017, size = 82, normalized size = 1. \begin{align*} -{\frac{-3\,{b}^{2}{x}^{2}+14\,abx+2\,{a}^{2}}{3}\sqrt{bx+a}{x}^{-{\frac{3}{2}}}}+{\frac{5\,a}{2}{b}^{{\frac{3}{2}}}\ln \left ({ \left ({\frac{a}{2}}+bx \right ){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+ax} \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 = 1.83555, size = 362, normalized size = 4.21 \begin{align*} \left [\frac{15 \, a b^{\frac{3}{2}} x^{2} \log \left (2 \, b x + 2 \, \sqrt{b x + a} \sqrt{b} \sqrt{x} + a\right ) + 2 \,{\left (3 \, b^{2} x^{2} - 14 \, a b x - 2 \, a^{2}\right )} \sqrt{b x + a} \sqrt{x}}{6 \, x^{2}}, -\frac{15 \, a \sqrt{-b} b x^{2} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right ) -{\left (3 \, b^{2} x^{2} - 14 \, a b x - 2 \, a^{2}\right )} \sqrt{b x + a} \sqrt{x}}{3 \, x^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 11.8364, size = 99, normalized size = 1.15 \begin{align*} - \frac{2 a^{2} \sqrt{b} \sqrt{\frac{a}{b x} + 1}}{3 x} - \frac{14 a b^{\frac{3}{2}} \sqrt{\frac{a}{b x} + 1}}{3} - \frac{5 a b^{\frac{3}{2}} \log{\left (\frac{a}{b x} \right )}}{2} + 5 a b^{\frac{3}{2}} \log{\left (\sqrt{\frac{a}{b x} + 1} + 1 \right )} + b^{\frac{5}{2}} x \sqrt{\frac{a}{b x} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]