Optimal. Leaf size=59 \[ -\frac{2 a^2 \left (a+\frac{b}{x}\right )^{5/2}}{5 b^3}-\frac{2 \left (a+\frac{b}{x}\right )^{9/2}}{9 b^3}+\frac{4 a \left (a+\frac{b}{x}\right )^{7/2}}{7 b^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0256244, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ -\frac{2 a^2 \left (a+\frac{b}{x}\right )^{5/2}}{5 b^3}-\frac{2 \left (a+\frac{b}{x}\right )^{9/2}}{9 b^3}+\frac{4 a \left (a+\frac{b}{x}\right )^{7/2}}{7 b^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (a+\frac{b}{x}\right )^{3/2}}{x^4} \, dx &=-\operatorname{Subst}\left (\int x^2 (a+b x)^{3/2} \, dx,x,\frac{1}{x}\right )\\ &=-\operatorname{Subst}\left (\int \left (\frac{a^2 (a+b x)^{3/2}}{b^2}-\frac{2 a (a+b x)^{5/2}}{b^2}+\frac{(a+b x)^{7/2}}{b^2}\right ) \, dx,x,\frac{1}{x}\right )\\ &=-\frac{2 a^2 \left (a+\frac{b}{x}\right )^{5/2}}{5 b^3}+\frac{4 a \left (a+\frac{b}{x}\right )^{7/2}}{7 b^3}-\frac{2 \left (a+\frac{b}{x}\right )^{9/2}}{9 b^3}\\ \end{align*}
Mathematica [A] time = 0.0203657, size = 47, normalized size = 0.8 \[ -\frac{2 \sqrt{a+\frac{b}{x}} (a x+b)^2 \left (8 a^2 x^2-20 a b x+35 b^2\right )}{315 b^3 x^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.006, size = 44, normalized size = 0.8 \begin{align*} -{\frac{ \left ( 2\,ax+2\,b \right ) \left ( 8\,{a}^{2}{x}^{2}-20\,xab+35\,{b}^{2} \right ) }{315\,{b}^{3}{x}^{3}} \left ({\frac{ax+b}{x}} \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.33718, size = 63, normalized size = 1.07 \begin{align*} -\frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{9}{2}}}{9 \, b^{3}} + \frac{4 \,{\left (a + \frac{b}{x}\right )}^{\frac{7}{2}} a}{7 \, b^{3}} - \frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{5}{2}} a^{2}}{5 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.70602, size = 135, normalized size = 2.29 \begin{align*} -\frac{2 \,{\left (8 \, a^{4} x^{4} - 4 \, a^{3} b x^{3} + 3 \, a^{2} b^{2} x^{2} + 50 \, a b^{3} x + 35 \, b^{4}\right )} \sqrt{\frac{a x + b}{x}}}{315 \, b^{3} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 1.96814, size = 986, normalized size = 16.71 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.18513, size = 281, normalized size = 4.76 \begin{align*} \frac{2 \,{\left (420 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{6} a^{3} \mathrm{sgn}\left (x\right ) + 1575 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{5} a^{\frac{5}{2}} b \mathrm{sgn}\left (x\right ) + 2583 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{4} a^{2} b^{2} \mathrm{sgn}\left (x\right ) + 2310 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{3} a^{\frac{3}{2}} b^{3} \mathrm{sgn}\left (x\right ) + 1170 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{2} a b^{4} \mathrm{sgn}\left (x\right ) + 315 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )} \sqrt{a} b^{5} \mathrm{sgn}\left (x\right ) + 35 \, b^{6} \mathrm{sgn}\left (x\right )\right )}}{315 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]