Optimal. Leaf size=159 \[ -\frac{9 x^{19/2}}{35 b^2 \left (a x+b x^3\right )^{5/2}}-\frac{3 x^{13/2}}{5 b^3 \left (a x+b x^3\right )^{3/2}}-\frac{3 x^{7/2}}{b^4 \sqrt{a x+b x^3}}+\frac{9 \sqrt{x} \sqrt{a x+b x^3}}{2 b^5}-\frac{9 a \tanh ^{-1}\left (\frac{\sqrt{b} x^{3/2}}{\sqrt{a x+b x^3}}\right )}{2 b^{11/2}}-\frac{x^{25/2}}{7 b \left (a x+b x^3\right )^{7/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.248376, antiderivative size = 159, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {2022, 2024, 2029, 206} \[ -\frac{9 x^{19/2}}{35 b^2 \left (a x+b x^3\right )^{5/2}}-\frac{3 x^{13/2}}{5 b^3 \left (a x+b x^3\right )^{3/2}}-\frac{3 x^{7/2}}{b^4 \sqrt{a x+b x^3}}+\frac{9 \sqrt{x} \sqrt{a x+b x^3}}{2 b^5}-\frac{9 a \tanh ^{-1}\left (\frac{\sqrt{b} x^{3/2}}{\sqrt{a x+b x^3}}\right )}{2 b^{11/2}}-\frac{x^{25/2}}{7 b \left (a x+b x^3\right )^{7/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2022
Rule 2024
Rule 2029
Rule 206
Rubi steps
\begin{align*} \int \frac{x^{29/2}}{\left (a x+b x^3\right )^{9/2}} \, dx &=-\frac{x^{25/2}}{7 b \left (a x+b x^3\right )^{7/2}}+\frac{9 \int \frac{x^{23/2}}{\left (a x+b x^3\right )^{7/2}} \, dx}{7 b}\\ &=-\frac{x^{25/2}}{7 b \left (a x+b x^3\right )^{7/2}}-\frac{9 x^{19/2}}{35 b^2 \left (a x+b x^3\right )^{5/2}}+\frac{9 \int \frac{x^{17/2}}{\left (a x+b x^3\right )^{5/2}} \, dx}{5 b^2}\\ &=-\frac{x^{25/2}}{7 b \left (a x+b x^3\right )^{7/2}}-\frac{9 x^{19/2}}{35 b^2 \left (a x+b x^3\right )^{5/2}}-\frac{3 x^{13/2}}{5 b^3 \left (a x+b x^3\right )^{3/2}}+\frac{3 \int \frac{x^{11/2}}{\left (a x+b x^3\right )^{3/2}} \, dx}{b^3}\\ &=-\frac{x^{25/2}}{7 b \left (a x+b x^3\right )^{7/2}}-\frac{9 x^{19/2}}{35 b^2 \left (a x+b x^3\right )^{5/2}}-\frac{3 x^{13/2}}{5 b^3 \left (a x+b x^3\right )^{3/2}}-\frac{3 x^{7/2}}{b^4 \sqrt{a x+b x^3}}+\frac{9 \int \frac{x^{5/2}}{\sqrt{a x+b x^3}} \, dx}{b^4}\\ &=-\frac{x^{25/2}}{7 b \left (a x+b x^3\right )^{7/2}}-\frac{9 x^{19/2}}{35 b^2 \left (a x+b x^3\right )^{5/2}}-\frac{3 x^{13/2}}{5 b^3 \left (a x+b x^3\right )^{3/2}}-\frac{3 x^{7/2}}{b^4 \sqrt{a x+b x^3}}+\frac{9 \sqrt{x} \sqrt{a x+b x^3}}{2 b^5}-\frac{(9 a) \int \frac{\sqrt{x}}{\sqrt{a x+b x^3}} \, dx}{2 b^5}\\ &=-\frac{x^{25/2}}{7 b \left (a x+b x^3\right )^{7/2}}-\frac{9 x^{19/2}}{35 b^2 \left (a x+b x^3\right )^{5/2}}-\frac{3 x^{13/2}}{5 b^3 \left (a x+b x^3\right )^{3/2}}-\frac{3 x^{7/2}}{b^4 \sqrt{a x+b x^3}}+\frac{9 \sqrt{x} \sqrt{a x+b x^3}}{2 b^5}-\frac{(9 a) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x^{3/2}}{\sqrt{a x+b x^3}}\right )}{2 b^5}\\ &=-\frac{x^{25/2}}{7 b \left (a x+b x^3\right )^{7/2}}-\frac{9 x^{19/2}}{35 b^2 \left (a x+b x^3\right )^{5/2}}-\frac{3 x^{13/2}}{5 b^3 \left (a x+b x^3\right )^{3/2}}-\frac{3 x^{7/2}}{b^4 \sqrt{a x+b x^3}}+\frac{9 \sqrt{x} \sqrt{a x+b x^3}}{2 b^5}-\frac{9 a \tanh ^{-1}\left (\frac{\sqrt{b} x^{3/2}}{\sqrt{a x+b x^3}}\right )}{2 b^{11/2}}\\ \end{align*}
Mathematica [A] time = 0.18079, size = 130, normalized size = 0.82 \[ \frac{\sqrt{x} \left (\sqrt{b} x \left (1218 a^2 b^2 x^4+1050 a^3 b x^2+315 a^4+528 a b^3 x^6+35 b^4 x^8\right )-\frac{315 \sqrt{a} \left (a+b x^2\right )^4 \sinh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{\frac{b x^2}{a}+1}}\right )}{70 b^{11/2} \left (a+b x^2\right )^3 \sqrt{x \left (a+b x^2\right )}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.04, size = 212, normalized size = 1.3 \begin{align*} -{\frac{1}{70\, \left ( b{x}^{2}+a \right ) ^{4}}\sqrt{x \left ( b{x}^{2}+a \right ) } \left ( -35\,{x}^{9}{b}^{9/2}+315\,\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){x}^{6}a{b}^{3}\sqrt{b{x}^{2}+a}-528\,{b}^{7/2}{x}^{7}a+945\,\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){x}^{4}{a}^{2}{b}^{2}\sqrt{b{x}^{2}+a}-1218\,{b}^{5/2}{x}^{5}{a}^{2}+945\,\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){x}^{2}{a}^{3}b\sqrt{b{x}^{2}+a}-1050\,{b}^{3/2}{x}^{3}{a}^{3}+315\,\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){a}^{4}\sqrt{b{x}^{2}+a}-315\,\sqrt{b}x{a}^{4} \right ){b}^{-{\frac{11}{2}}}{\frac{1}{\sqrt{x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{\frac{29}{2}}}{{\left (b x^{3} + a x\right )}^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.58916, size = 845, normalized size = 5.31 \begin{align*} \left [\frac{315 \,{\left (a b^{4} x^{8} + 4 \, a^{2} b^{3} x^{6} + 6 \, a^{3} b^{2} x^{4} + 4 \, a^{4} b x^{2} + a^{5}\right )} \sqrt{b} \log \left (2 \, b x^{2} - 2 \, \sqrt{b x^{3} + a x} \sqrt{b} \sqrt{x} + a\right ) + 2 \,{\left (35 \, b^{5} x^{8} + 528 \, a b^{4} x^{6} + 1218 \, a^{2} b^{3} x^{4} + 1050 \, a^{3} b^{2} x^{2} + 315 \, a^{4} b\right )} \sqrt{b x^{3} + a x} \sqrt{x}}{140 \,{\left (b^{10} x^{8} + 4 \, a b^{9} x^{6} + 6 \, a^{2} b^{8} x^{4} + 4 \, a^{3} b^{7} x^{2} + a^{4} b^{6}\right )}}, \frac{315 \,{\left (a b^{4} x^{8} + 4 \, a^{2} b^{3} x^{6} + 6 \, a^{3} b^{2} x^{4} + 4 \, a^{4} b x^{2} + a^{5}\right )} \sqrt{-b} \arctan \left (\frac{\sqrt{b x^{3} + a x} \sqrt{-b}}{b x^{\frac{3}{2}}}\right ) +{\left (35 \, b^{5} x^{8} + 528 \, a b^{4} x^{6} + 1218 \, a^{2} b^{3} x^{4} + 1050 \, a^{3} b^{2} x^{2} + 315 \, a^{4} b\right )} \sqrt{b x^{3} + a x} \sqrt{x}}{70 \,{\left (b^{10} x^{8} + 4 \, a b^{9} x^{6} + 6 \, a^{2} b^{8} x^{4} + 4 \, a^{3} b^{7} x^{2} + a^{4} b^{6}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [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]
________________________________________________________________________________________
Giac [A] time = 1.32596, size = 123, normalized size = 0.77 \begin{align*} \frac{{\left ({\left ({\left (x^{2}{\left (\frac{35 \, x^{2}}{b} + \frac{528 \, a}{b^{2}}\right )} + \frac{1218 \, a^{2}}{b^{3}}\right )} x^{2} + \frac{1050 \, a^{3}}{b^{4}}\right )} x^{2} + \frac{315 \, a^{4}}{b^{5}}\right )} x}{70 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}}} + \frac{9 \, a \log \left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{2 \, b^{\frac{11}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]