Optimal. Leaf size=187 \[ -\frac{a^{3/2} (5 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x^2}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 c^{3/2}}-\frac{b^{3/2} (b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b} \sqrt{c+d x^2}}\right )}{2 d^{3/2}}-\frac{a \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{2 c x^2}+\frac{b \sqrt{a+b x^2} \sqrt{c+d x^2} (a d+b c)}{2 c d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.229505, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.346, Rules used = {446, 98, 154, 157, 63, 217, 206, 93, 208} \[ -\frac{a^{3/2} (5 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x^2}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 c^{3/2}}-\frac{b^{3/2} (b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b} \sqrt{c+d x^2}}\right )}{2 d^{3/2}}-\frac{a \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{2 c x^2}+\frac{b \sqrt{a+b x^2} \sqrt{c+d x^2} (a d+b c)}{2 c d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 446
Rule 98
Rule 154
Rule 157
Rule 63
Rule 217
Rule 206
Rule 93
Rule 208
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^{5/2}}{x^3 \sqrt{c+d x^2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^{5/2}}{x^2 \sqrt{c+d x}} \, dx,x,x^2\right )\\ &=-\frac{a \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{2 c x^2}-\frac{\operatorname{Subst}\left (\int \frac{\sqrt{a+b x} \left (-\frac{1}{2} a (5 b c-a d)-b (b c+a d) x\right )}{x \sqrt{c+d x}} \, dx,x,x^2\right )}{2 c}\\ &=\frac{b (b c+a d) \sqrt{a+b x^2} \sqrt{c+d x^2}}{2 c d}-\frac{a \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{2 c x^2}-\frac{\operatorname{Subst}\left (\int \frac{-\frac{1}{2} a^2 d (5 b c-a d)+\frac{1}{2} b^2 c (b c-5 a d) x}{x \sqrt{a+b x} \sqrt{c+d x}} \, dx,x,x^2\right )}{2 c d}\\ &=\frac{b (b c+a d) \sqrt{a+b x^2} \sqrt{c+d x^2}}{2 c d}-\frac{a \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{2 c x^2}-\frac{\left (b^2 (b c-5 a d)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx,x,x^2\right )}{4 d}+\frac{\left (a^2 (5 b c-a d)\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x} \sqrt{c+d x}} \, dx,x,x^2\right )}{4 c}\\ &=\frac{b (b c+a d) \sqrt{a+b x^2} \sqrt{c+d x^2}}{2 c d}-\frac{a \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{2 c x^2}-\frac{(b (b c-5 a d)) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c-\frac{a d}{b}+\frac{d x^2}{b}}} \, dx,x,\sqrt{a+b x^2}\right )}{2 d}+\frac{\left (a^2 (5 b c-a d)\right ) \operatorname{Subst}\left (\int \frac{1}{-a+c x^2} \, dx,x,\frac{\sqrt{a+b x^2}}{\sqrt{c+d x^2}}\right )}{2 c}\\ &=\frac{b (b c+a d) \sqrt{a+b x^2} \sqrt{c+d x^2}}{2 c d}-\frac{a \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{2 c x^2}-\frac{a^{3/2} (5 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x^2}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 c^{3/2}}-\frac{(b (b c-5 a d)) \operatorname{Subst}\left (\int \frac{1}{1-\frac{d x^2}{b}} \, dx,x,\frac{\sqrt{a+b x^2}}{\sqrt{c+d x^2}}\right )}{2 d}\\ &=\frac{b (b c+a d) \sqrt{a+b x^2} \sqrt{c+d x^2}}{2 c d}-\frac{a \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{2 c x^2}-\frac{a^{3/2} (5 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x^2}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 c^{3/2}}-\frac{b^{3/2} (b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b} \sqrt{c+d x^2}}\right )}{2 d^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.965392, size = 196, normalized size = 1.05 \[ \frac{1}{2} \left (\frac{\sqrt{a+b x^2} \sqrt{c+d x^2} \left (b^2 c x^2-a^2 d\right )}{c d x^2}+\frac{a^{3/2} (a d-5 b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x^2}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{c^{3/2}}-\frac{(b c-5 a d) (b c-a d)^{3/2} \left (\frac{b \left (c+d x^2\right )}{b c-a d}\right )^{3/2} \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b c-a d}}\right )}{d^{3/2} \left (c+d x^2\right )^{3/2}}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.017, size = 423, normalized size = 2.3 \begin{align*}{\frac{1}{4\,cd{x}^{2}}\sqrt{b{x}^{2}+a}\sqrt{d{x}^{2}+c} \left ( 5\,\ln \left ( 1/2\,{\frac{2\,d{x}^{2}b+2\,\sqrt{bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){x}^{2}a{b}^{2}cd\sqrt{ac}-\ln \left ({\frac{1}{2} \left ( 2\,d{x}^{2}b+2\,\sqrt{bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac}\sqrt{bd}+ad+bc \right ){\frac{1}{\sqrt{bd}}}} \right ){x}^{2}{b}^{3}{c}^{2}\sqrt{ac}+\ln \left ({\frac{1}{{x}^{2}} \left ( ad{x}^{2}+bc{x}^{2}+2\,\sqrt{ac}\sqrt{bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac}+2\,ac \right ) } \right ){x}^{2}{a}^{3}{d}^{2}\sqrt{bd}-5\,\ln \left ({\frac{ad{x}^{2}+bc{x}^{2}+2\,\sqrt{ac}\sqrt{bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac}+2\,ac}{{x}^{2}}} \right ){x}^{2}{a}^{2}bcd\sqrt{bd}+2\,{x}^{2}{b}^{2}c\sqrt{bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac}\sqrt{bd}\sqrt{ac}-2\,{a}^{2}d\sqrt{bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac}\sqrt{bd}\sqrt{ac} \right ){\frac{1}{\sqrt{bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac}}}{\frac{1}{\sqrt{bd}}}{\frac{1}{\sqrt{ac}}}} \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 = 14.4561, size = 2340, normalized size = 12.51 \begin{align*} \text{result too large to display} \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 (a + b x^{2}\right )^{\frac{5}{2}}}{x^{3} \sqrt{c + d x^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 2.02664, size = 752, normalized size = 4.02 \begin{align*} \frac{b^{3}{\left (\frac{2 \, \sqrt{b^{2} c +{\left (b x^{2} + a\right )} b d - a b d} \sqrt{b x^{2} + a}}{b d} + \frac{{\left (\sqrt{b d} b c - 5 \, \sqrt{b d} a d\right )} \log \left ({\left (\sqrt{b x^{2} + a} \sqrt{b d} - \sqrt{b^{2} c +{\left (b x^{2} + a\right )} b d - a b d}\right )}^{2}\right )}{b d^{2}} - \frac{2 \,{\left (5 \, \sqrt{b d} a^{2} b c - \sqrt{b d} a^{3} d\right )} \arctan \left (-\frac{b^{2} c + a b d -{\left (\sqrt{b x^{2} + a} \sqrt{b d} - \sqrt{b^{2} c +{\left (b x^{2} + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt{-a b c d} b}\right )}{\sqrt{-a b c d} b^{2} c} - \frac{4 \,{\left (\sqrt{b d} a^{2} b^{3} c^{2} - 2 \, \sqrt{b d} a^{3} b^{2} c d + \sqrt{b d} a^{4} b d^{2} - \sqrt{b d}{\left (\sqrt{b x^{2} + a} \sqrt{b d} - \sqrt{b^{2} c +{\left (b x^{2} + a\right )} b d - a b d}\right )}^{2} a^{2} b c - \sqrt{b d}{\left (\sqrt{b x^{2} + a} \sqrt{b d} - \sqrt{b^{2} c +{\left (b x^{2} + a\right )} b d - a b d}\right )}^{2} a^{3} d\right )}}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2} - 2 \,{\left (\sqrt{b x^{2} + a} \sqrt{b d} - \sqrt{b^{2} c +{\left (b x^{2} + a\right )} b d - a b d}\right )}^{2} b^{2} c - 2 \,{\left (\sqrt{b x^{2} + a} \sqrt{b d} - \sqrt{b^{2} c +{\left (b x^{2} + a\right )} b d - a b d}\right )}^{2} a b d +{\left (\sqrt{b x^{2} + a} \sqrt{b d} - \sqrt{b^{2} c +{\left (b x^{2} + a\right )} b d - a b d}\right )}^{4}\right )} b c}\right )}}{4 \,{\left | b \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]