Optimal. Leaf size=128 \[ \frac{\left (b x^2+c x^4\right )^{3/2} (4 A c+b B)}{4 b x^2}+\frac{3}{8} \sqrt{b x^2+c x^4} (4 A c+b B)+\frac{3 b (4 A c+b B) \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{8 \sqrt{c}}-\frac{A \left (b x^2+c x^4\right )^{5/2}}{b x^6} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.277322, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {2034, 792, 664, 620, 206} \[ \frac{\left (b x^2+c x^4\right )^{3/2} (4 A c+b B)}{4 b x^2}+\frac{3}{8} \sqrt{b x^2+c x^4} (4 A c+b B)+\frac{3 b (4 A c+b B) \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{8 \sqrt{c}}-\frac{A \left (b x^2+c x^4\right )^{5/2}}{b x^6} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2034
Rule 792
Rule 664
Rule 620
Rule 206
Rubi steps
\begin{align*} \int \frac{\left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2}}{x^5} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(A+B x) \left (b x+c x^2\right )^{3/2}}{x^3} \, dx,x,x^2\right )\\ &=-\frac{A \left (b x^2+c x^4\right )^{5/2}}{b x^6}+\frac{\left (-3 (-b B+A c)+\frac{5}{2} (-b B+2 A c)\right ) \operatorname{Subst}\left (\int \frac{\left (b x+c x^2\right )^{3/2}}{x^2} \, dx,x,x^2\right )}{b}\\ &=\frac{(b B+4 A c) \left (b x^2+c x^4\right )^{3/2}}{4 b x^2}-\frac{A \left (b x^2+c x^4\right )^{5/2}}{b x^6}+\frac{1}{8} (3 (b B+4 A c)) \operatorname{Subst}\left (\int \frac{\sqrt{b x+c x^2}}{x} \, dx,x,x^2\right )\\ &=\frac{3}{8} (b B+4 A c) \sqrt{b x^2+c x^4}+\frac{(b B+4 A c) \left (b x^2+c x^4\right )^{3/2}}{4 b x^2}-\frac{A \left (b x^2+c x^4\right )^{5/2}}{b x^6}+\frac{1}{16} (3 b (b B+4 A c)) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b x+c x^2}} \, dx,x,x^2\right )\\ &=\frac{3}{8} (b B+4 A c) \sqrt{b x^2+c x^4}+\frac{(b B+4 A c) \left (b x^2+c x^4\right )^{3/2}}{4 b x^2}-\frac{A \left (b x^2+c x^4\right )^{5/2}}{b x^6}+\frac{1}{8} (3 b (b B+4 A c)) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x^2}{\sqrt{b x^2+c x^4}}\right )\\ &=\frac{3}{8} (b B+4 A c) \sqrt{b x^2+c x^4}+\frac{(b B+4 A c) \left (b x^2+c x^4\right )^{3/2}}{4 b x^2}-\frac{A \left (b x^2+c x^4\right )^{5/2}}{b x^6}+\frac{3 b (b B+4 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{8 \sqrt{c}}\\ \end{align*}
Mathematica [A] time = 0.174219, size = 96, normalized size = 0.75 \[ \frac{\sqrt{x^2 \left (b+c x^2\right )} \left (\frac{3 \sqrt{b} x (4 A c+b B) \sinh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{\sqrt{c} \sqrt{\frac{c x^2}{b}+1}}-8 A b+4 A c x^2+5 b B x^2+2 B c x^4\right )}{8 x^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.009, size = 174, normalized size = 1.4 \begin{align*}{\frac{1}{8\,b{x}^{4}} \left ( c{x}^{4}+b{x}^{2} \right ) ^{{\frac{3}{2}}} \left ( 8\,A{c}^{3/2} \left ( c{x}^{2}+b \right ) ^{3/2}{x}^{2}+12\,A{c}^{3/2}\sqrt{c{x}^{2}+b}{x}^{2}b+2\,B\sqrt{c} \left ( c{x}^{2}+b \right ) ^{3/2}{x}^{2}b-8\,A\sqrt{c} \left ( c{x}^{2}+b \right ) ^{5/2}+3\,B\sqrt{c}\sqrt{c{x}^{2}+b}{x}^{2}{b}^{2}+12\,A\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+b} \right ) x{b}^{2}c+3\,B\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+b} \right ) x{b}^{3} \right ) \left ( c{x}^{2}+b \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{c}}}} \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 = 1.11389, size = 467, normalized size = 3.65 \begin{align*} \left [\frac{3 \,{\left (B b^{2} + 4 \, A b c\right )} \sqrt{c} x^{2} \log \left (-2 \, c x^{2} - b - 2 \, \sqrt{c x^{4} + b x^{2}} \sqrt{c}\right ) + 2 \,{\left (2 \, B c^{2} x^{4} - 8 \, A b c +{\left (5 \, B b c + 4 \, A c^{2}\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2}}}{16 \, c x^{2}}, -\frac{3 \,{\left (B b^{2} + 4 \, A b c\right )} \sqrt{-c} x^{2} \arctan \left (\frac{\sqrt{c x^{4} + b x^{2}} \sqrt{-c}}{c x^{2} + b}\right ) -{\left (2 \, B c^{2} x^{4} - 8 \, A b c +{\left (5 \, B b c + 4 \, A c^{2}\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2}}}{8 \, c x^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac{3}{2}} \left (A + B x^{2}\right )}{x^{5}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.23806, size = 170, normalized size = 1.33 \begin{align*} \frac{2 \, A b^{2} \sqrt{c} \mathrm{sgn}\left (x\right )}{{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{2} - b} + \frac{1}{8} \,{\left (2 \, B c x^{2} \mathrm{sgn}\left (x\right ) + \frac{5 \, B b c^{2} \mathrm{sgn}\left (x\right ) + 4 \, A c^{3} \mathrm{sgn}\left (x\right )}{c^{2}}\right )} \sqrt{c x^{2} + b} x - \frac{3 \,{\left (B b^{2} \sqrt{c} \mathrm{sgn}\left (x\right ) + 4 \, A b c^{\frac{3}{2}} \mathrm{sgn}\left (x\right )\right )} \log \left ({\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{2}\right )}{16 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]