Optimal. Leaf size=192 \[ \frac{a^4 c \sqrt{c \left (a+b x^2\right )^3}}{a+b x^2}+\frac{1}{3} a^3 c \sqrt{c \left (a+b x^2\right )^3}+\frac{1}{5} a^2 c \left (a+b x^2\right ) \sqrt{c \left (a+b x^2\right )^3}-\frac{a^3 c \sqrt{c \left (a+b x^2\right )^3} \tanh ^{-1}\left (\sqrt{\frac{b x^2}{a}+1}\right )}{\left (\frac{b x^2}{a}+1\right )^{3/2}}+\frac{1}{7} a c \left (a+b x^2\right )^2 \sqrt{c \left (a+b x^2\right )^3}+\frac{1}{9} c \left (a+b x^2\right )^3 \sqrt{c \left (a+b x^2\right )^3} \]
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Rubi [A] time = 0.217076, antiderivative size = 194, normalized size of antiderivative = 1.01, number of steps used = 9, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {6720, 266, 50, 63, 208} \[ \frac{a^4 c \sqrt{c \left (a+b x^2\right )^3}}{a+b x^2}+\frac{1}{3} a^3 c \sqrt{c \left (a+b x^2\right )^3}+\frac{1}{5} a^2 c \left (a+b x^2\right ) \sqrt{c \left (a+b x^2\right )^3}-\frac{a^{9/2} c \sqrt{c \left (a+b x^2\right )^3} \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{\left (a+b x^2\right )^{3/2}}+\frac{1}{7} a c \left (a+b x^2\right )^2 \sqrt{c \left (a+b x^2\right )^3}+\frac{1}{9} c \left (a+b x^2\right )^3 \sqrt{c \left (a+b x^2\right )^3} \]
Antiderivative was successfully verified.
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Rule 6720
Rule 266
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\left (c \left (a+b x^2\right )^3\right )^{3/2}}{x} \, dx &=\frac{\left (c \sqrt{c \left (a+b x^2\right )^3}\right ) \int \frac{\left (a+b x^2\right )^{9/2}}{x} \, dx}{\left (a+b x^2\right )^{3/2}}\\ &=\frac{\left (c \sqrt{c \left (a+b x^2\right )^3}\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^{9/2}}{x} \, dx,x,x^2\right )}{2 \left (a+b x^2\right )^{3/2}}\\ &=\frac{1}{9} c \left (a+b x^2\right )^3 \sqrt{c \left (a+b x^2\right )^3}+\frac{\left (a c \sqrt{c \left (a+b x^2\right )^3}\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^{7/2}}{x} \, dx,x,x^2\right )}{2 \left (a+b x^2\right )^{3/2}}\\ &=\frac{1}{7} a c \left (a+b x^2\right )^2 \sqrt{c \left (a+b x^2\right )^3}+\frac{1}{9} c \left (a+b x^2\right )^3 \sqrt{c \left (a+b x^2\right )^3}+\frac{\left (a^2 c \sqrt{c \left (a+b x^2\right )^3}\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^{5/2}}{x} \, dx,x,x^2\right )}{2 \left (a+b x^2\right )^{3/2}}\\ &=\frac{1}{5} a^2 c \left (a+b x^2\right ) \sqrt{c \left (a+b x^2\right )^3}+\frac{1}{7} a c \left (a+b x^2\right )^2 \sqrt{c \left (a+b x^2\right )^3}+\frac{1}{9} c \left (a+b x^2\right )^3 \sqrt{c \left (a+b x^2\right )^3}+\frac{\left (a^3 c \sqrt{c \left (a+b x^2\right )^3}\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^{3/2}}{x} \, dx,x,x^2\right )}{2 \left (a+b x^2\right )^{3/2}}\\ &=\frac{1}{3} a^3 c \sqrt{c \left (a+b x^2\right )^3}+\frac{1}{5} a^2 c \left (a+b x^2\right ) \sqrt{c \left (a+b x^2\right )^3}+\frac{1}{7} a c \left (a+b x^2\right )^2 \sqrt{c \left (a+b x^2\right )^3}+\frac{1}{9} c \left (a+b x^2\right )^3 \sqrt{c \left (a+b x^2\right )^3}+\frac{\left (a^4 c \sqrt{c \left (a+b x^2\right )^3}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x} \, dx,x,x^2\right )}{2 \left (a+b x^2\right )^{3/2}}\\ &=\frac{1}{3} a^3 c \sqrt{c \left (a+b x^2\right )^3}+\frac{a^4 c \sqrt{c \left (a+b x^2\right )^3}}{a+b x^2}+\frac{1}{5} a^2 c \left (a+b x^2\right ) \sqrt{c \left (a+b x^2\right )^3}+\frac{1}{7} a c \left (a+b x^2\right )^2 \sqrt{c \left (a+b x^2\right )^3}+\frac{1}{9} c \left (a+b x^2\right )^3 \sqrt{c \left (a+b x^2\right )^3}+\frac{\left (a^5 c \sqrt{c \left (a+b x^2\right )^3}\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^2\right )}{2 \left (a+b x^2\right )^{3/2}}\\ &=\frac{1}{3} a^3 c \sqrt{c \left (a+b x^2\right )^3}+\frac{a^4 c \sqrt{c \left (a+b x^2\right )^3}}{a+b x^2}+\frac{1}{5} a^2 c \left (a+b x^2\right ) \sqrt{c \left (a+b x^2\right )^3}+\frac{1}{7} a c \left (a+b x^2\right )^2 \sqrt{c \left (a+b x^2\right )^3}+\frac{1}{9} c \left (a+b x^2\right )^3 \sqrt{c \left (a+b x^2\right )^3}+\frac{\left (a^5 c \sqrt{c \left (a+b x^2\right )^3}\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x^2}\right )}{b \left (a+b x^2\right )^{3/2}}\\ &=\frac{1}{3} a^3 c \sqrt{c \left (a+b x^2\right )^3}+\frac{a^4 c \sqrt{c \left (a+b x^2\right )^3}}{a+b x^2}+\frac{1}{5} a^2 c \left (a+b x^2\right ) \sqrt{c \left (a+b x^2\right )^3}+\frac{1}{7} a c \left (a+b x^2\right )^2 \sqrt{c \left (a+b x^2\right )^3}+\frac{1}{9} c \left (a+b x^2\right )^3 \sqrt{c \left (a+b x^2\right )^3}-\frac{a^{9/2} c \sqrt{c \left (a+b x^2\right )^3} \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{\left (a+b x^2\right )^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0749732, size = 111, normalized size = 0.58 \[ \frac{\left (c \left (a+b x^2\right )^3\right )^{3/2} \left (\sqrt{a+b x^2} \left (408 a^2 b^2 x^4+506 a^3 b x^2+563 a^4+185 a b^3 x^6+35 b^4 x^8\right )-315 a^{9/2} \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )\right )}{315 \left (a+b x^2\right )^{9/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 221, normalized size = 1.2 \begin{align*}{\frac{1}{315\,c \left ( b{x}^{2}+a \right ) ^{3}} \left ( c \left ( b{x}^{2}+a \right ) ^{3} \right ) ^{{\frac{3}{2}}} \left ( 35\,\sqrt{ac} \left ( bc{x}^{2}+ac \right ) ^{5/2}{x}^{4}{b}^{2}+115\,\sqrt{ac} \left ( bc{x}^{2}+ac \right ) ^{5/2}{x}^{2}ab-315\,\ln \left ( 2\,{\frac{\sqrt{ac}\sqrt{bc{x}^{2}+ac}+ac}{x}} \right ){a}^{5}{c}^{3}-46\,\sqrt{ac} \left ( bc{x}^{2}+ac \right ) ^{5/2}{a}^{2}+105\,\sqrt{ac} \left ( bc{x}^{2}+ac \right ) ^{3/2}{a}^{3}c+315\,\sqrt{ac}\sqrt{bc{x}^{2}+ac}{a}^{4}{c}^{2}+189\,{a}^{2} \left ( c \left ( b{x}^{2}+a \right ) \right ) ^{5/2}\sqrt{ac} \right ) \left ( c \left ( b{x}^{2}+a \right ) \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{ac}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left ({\left (b x^{2} + a\right )}^{3} c\right )^{\frac{3}{2}}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.72856, size = 864, normalized size = 4.5 \begin{align*} \left [\frac{315 \,{\left (a^{4} b c x^{2} + a^{5} c\right )} \sqrt{a c} \log \left (-\frac{b^{2} c x^{4} + 3 \, a b c x^{2} + 2 \, a^{2} c - 2 \, \sqrt{b^{3} c x^{6} + 3 \, a b^{2} c x^{4} + 3 \, a^{2} b c x^{2} + a^{3} c} \sqrt{a c}}{b x^{4} + a x^{2}}\right ) + 2 \,{\left (35 \, b^{4} c x^{8} + 185 \, a b^{3} c x^{6} + 408 \, a^{2} b^{2} c x^{4} + 506 \, a^{3} b c x^{2} + 563 \, a^{4} c\right )} \sqrt{b^{3} c x^{6} + 3 \, a b^{2} c x^{4} + 3 \, a^{2} b c x^{2} + a^{3} c}}{630 \,{\left (b x^{2} + a\right )}}, \frac{315 \,{\left (a^{4} b c x^{2} + a^{5} c\right )} \sqrt{-a c} \arctan \left (\frac{\sqrt{b^{3} c x^{6} + 3 \, a b^{2} c x^{4} + 3 \, a^{2} b c x^{2} + a^{3} c} \sqrt{-a c}}{b^{2} c x^{4} + 2 \, a b c x^{2} + a^{2} c}\right ) +{\left (35 \, b^{4} c x^{8} + 185 \, a b^{3} c x^{6} + 408 \, a^{2} b^{2} c x^{4} + 506 \, a^{3} b c x^{2} + 563 \, a^{4} c\right )} \sqrt{b^{3} c x^{6} + 3 \, a b^{2} c x^{4} + 3 \, a^{2} b c x^{2} + a^{3} c}}{315 \,{\left (b x^{2} + a\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (c \left (a + b x^{2}\right )^{3}\right )^{\frac{3}{2}}}{x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23287, size = 194, normalized size = 1.01 \begin{align*} \frac{1}{315} \,{\left (\frac{315 \, a^{5} c \arctan \left (\frac{\sqrt{b c x^{2} + a c}}{\sqrt{-a c}}\right )}{\sqrt{-a c}} + \frac{315 \, \sqrt{b c x^{2} + a c} a^{4} c^{36} + 105 \,{\left (b c x^{2} + a c\right )}^{\frac{3}{2}} a^{3} c^{35} + 63 \,{\left (b c x^{2} + a c\right )}^{\frac{5}{2}} a^{2} c^{34} + 45 \,{\left (b c x^{2} + a c\right )}^{\frac{7}{2}} a c^{33} + 35 \,{\left (b c x^{2} + a c\right )}^{\frac{9}{2}} c^{32}}{c^{36}}\right )} c \mathrm{sgn}\left (b x^{2} + a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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