Optimal. Leaf size=249 \[ -\frac{\left (-24 a^2 c^2+60 a b c+5 b^2\right ) \left (c+d x^2\right ) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{48 a d^3}-\frac{b \left (-24 a^2 c^2+12 a b c+b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{\sqrt{a}}\right )}{16 a^{3/2} d^3}-\frac{b c^2 \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{d^3}+\frac{\left (c+d x^2\right )^3 \left (\frac{a c+a d x^2+b}{c+d x^2}\right )^{5/2}}{6 a d^3}-\frac{(12 a c+b) \left (c+d x^2\right )^2 \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{24 d^3} \]
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Rubi [A] time = 0.730147, antiderivative size = 311, normalized size of antiderivative = 1.25, number of steps used = 10, number of rules used = 9, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {6722, 1975, 446, 89, 80, 50, 63, 217, 206} \[ -\frac{\left (-24 a^2 c^2+12 a b c+b^2\right ) \left (c+d x^2\right ) \sqrt{a+\frac{b}{c+d x^2}} \left (a \left (c+d x^2\right )+b\right )}{24 a b d^3}-\frac{\left (-24 a^2 c^2+12 a b c+b^2\right ) \left (c+d x^2\right ) \sqrt{a+\frac{b}{c+d x^2}}}{16 a d^3}-\frac{b \left (-24 a^2 c^2+12 a b c+b^2\right ) \sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{c+d x^2}}{\sqrt{a \left (c+d x^2\right )+b}}\right )}{16 a^{3/2} d^3 \sqrt{a \left (c+d x^2\right )+b}}-\frac{c^2 \sqrt{a+\frac{b}{c+d x^2}} \left (a \left (c+d x^2\right )+b\right )^2}{b d^3}+\frac{\left (c+d x^2\right ) \sqrt{a+\frac{b}{c+d x^2}} \left (a \left (c+d x^2\right )+b\right )^2}{6 a d^3} \]
Antiderivative was successfully verified.
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Rule 6722
Rule 1975
Rule 446
Rule 89
Rule 80
Rule 50
Rule 63
Rule 217
Rule 206
Rubi steps
\begin{align*} \int x^5 \left (a+\frac{b}{c+d x^2}\right )^{3/2} \, dx &=\frac{\left (\sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}\right ) \int \frac{x^5 \left (b+a \left (c+d x^2\right )\right )^{3/2}}{\left (c+d x^2\right )^{3/2}} \, dx}{\sqrt{b+a \left (c+d x^2\right )}}\\ &=\frac{\left (\sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}\right ) \int \frac{x^5 \left (b+a c+a d x^2\right )^{3/2}}{\left (c+d x^2\right )^{3/2}} \, dx}{\sqrt{b+a \left (c+d x^2\right )}}\\ &=\frac{\left (\sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}\right ) \operatorname{Subst}\left (\int \frac{x^2 (b+a c+a d x)^{3/2}}{(c+d x)^{3/2}} \, dx,x,x^2\right )}{2 \sqrt{b+a \left (c+d x^2\right )}}\\ &=-\frac{c^2 \sqrt{a+\frac{b}{c+d x^2}} \left (b+a \left (c+d x^2\right )\right )^2}{b d^3}+\frac{\left (\sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}\right ) \operatorname{Subst}\left (\int \frac{(b+a c+a d x)^{3/2} \left (-\frac{1}{2} c (b-4 a c) d+\frac{1}{2} b d^2 x\right )}{\sqrt{c+d x}} \, dx,x,x^2\right )}{b d^3 \sqrt{b+a \left (c+d x^2\right )}}\\ &=-\frac{c^2 \sqrt{a+\frac{b}{c+d x^2}} \left (b+a \left (c+d x^2\right )\right )^2}{b d^3}+\frac{\left (c+d x^2\right ) \sqrt{a+\frac{b}{c+d x^2}} \left (b+a \left (c+d x^2\right )\right )^2}{6 a d^3}--\frac{\left (\left (-\frac{3}{2} a c (b-4 a c) d^3-\frac{1}{2} b d^2 \left (\frac{5 a c d}{2}+\frac{1}{2} (b+a c) d\right )\right ) \sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}\right ) \operatorname{Subst}\left (\int \frac{(b+a c+a d x)^{3/2}}{\sqrt{c+d x}} \, dx,x,x^2\right )}{3 a b d^5 \sqrt{b+a \left (c+d x^2\right )}}\\ &=-\frac{\left (b^2+12 a b c-24 a^2 c^2\right ) \left (c+d x^2\right ) \sqrt{a+\frac{b}{c+d x^2}} \left (b+a \left (c+d x^2\right )\right )}{24 a b d^3}-\frac{c^2 \sqrt{a+\frac{b}{c+d x^2}} \left (b+a \left (c+d x^2\right )\right )^2}{b d^3}+\frac{\left (c+d x^2\right ) \sqrt{a+\frac{b}{c+d x^2}} \left (b+a \left (c+d x^2\right )\right )^2}{6 a d^3}--\frac{\left (\left (-\frac{3}{2} a c (b-4 a c) d^3-\frac{1}{2} b d^2 \left (\frac{5 a c d}{2}+\frac{1}{2} (b+a c) d\right )\right ) \sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{b+a c+a d x}}{\sqrt{c+d x}} \, dx,x,x^2\right )}{4 a d^5 \sqrt{b+a \left (c+d x^2\right )}}\\ &=-\frac{\left (b^2+12 a b c-24 a^2 c^2\right ) \left (c+d x^2\right ) \sqrt{a+\frac{b}{c+d x^2}}}{16 a d^3}-\frac{\left (b^2+12 a b c-24 a^2 c^2\right ) \left (c+d x^2\right ) \sqrt{a+\frac{b}{c+d x^2}} \left (b+a \left (c+d x^2\right )\right )}{24 a b d^3}-\frac{c^2 \sqrt{a+\frac{b}{c+d x^2}} \left (b+a \left (c+d x^2\right )\right )^2}{b d^3}+\frac{\left (c+d x^2\right ) \sqrt{a+\frac{b}{c+d x^2}} \left (b+a \left (c+d x^2\right )\right )^2}{6 a d^3}--\frac{\left (b \left (-\frac{3}{2} a c (b-4 a c) d^3-\frac{1}{2} b d^2 \left (\frac{5 a c d}{2}+\frac{1}{2} (b+a c) d\right )\right ) \sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c+d x} \sqrt{b+a c+a d x}} \, dx,x,x^2\right )}{8 a d^5 \sqrt{b+a \left (c+d x^2\right )}}\\ &=-\frac{\left (b^2+12 a b c-24 a^2 c^2\right ) \left (c+d x^2\right ) \sqrt{a+\frac{b}{c+d x^2}}}{16 a d^3}-\frac{\left (b^2+12 a b c-24 a^2 c^2\right ) \left (c+d x^2\right ) \sqrt{a+\frac{b}{c+d x^2}} \left (b+a \left (c+d x^2\right )\right )}{24 a b d^3}-\frac{c^2 \sqrt{a+\frac{b}{c+d x^2}} \left (b+a \left (c+d x^2\right )\right )^2}{b d^3}+\frac{\left (c+d x^2\right ) \sqrt{a+\frac{b}{c+d x^2}} \left (b+a \left (c+d x^2\right )\right )^2}{6 a d^3}--\frac{\left (b \left (-\frac{3}{2} a c (b-4 a c) d^3-\frac{1}{2} b d^2 \left (\frac{5 a c d}{2}+\frac{1}{2} (b+a c) d\right )\right ) \sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b+a x^2}} \, dx,x,\sqrt{c+d x^2}\right )}{4 a d^6 \sqrt{b+a \left (c+d x^2\right )}}\\ &=-\frac{\left (b^2+12 a b c-24 a^2 c^2\right ) \left (c+d x^2\right ) \sqrt{a+\frac{b}{c+d x^2}}}{16 a d^3}-\frac{\left (b^2+12 a b c-24 a^2 c^2\right ) \left (c+d x^2\right ) \sqrt{a+\frac{b}{c+d x^2}} \left (b+a \left (c+d x^2\right )\right )}{24 a b d^3}-\frac{c^2 \sqrt{a+\frac{b}{c+d x^2}} \left (b+a \left (c+d x^2\right )\right )^2}{b d^3}+\frac{\left (c+d x^2\right ) \sqrt{a+\frac{b}{c+d x^2}} \left (b+a \left (c+d x^2\right )\right )^2}{6 a d^3}--\frac{\left (b \left (-\frac{3}{2} a c (b-4 a c) d^3-\frac{1}{2} b d^2 \left (\frac{5 a c d}{2}+\frac{1}{2} (b+a c) d\right )\right ) \sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}\right ) \operatorname{Subst}\left (\int \frac{1}{1-a x^2} \, dx,x,\frac{\sqrt{c+d x^2}}{\sqrt{b+a \left (c+d x^2\right )}}\right )}{4 a d^6 \sqrt{b+a \left (c+d x^2\right )}}\\ &=-\frac{\left (b^2+12 a b c-24 a^2 c^2\right ) \left (c+d x^2\right ) \sqrt{a+\frac{b}{c+d x^2}}}{16 a d^3}-\frac{\left (b^2+12 a b c-24 a^2 c^2\right ) \left (c+d x^2\right ) \sqrt{a+\frac{b}{c+d x^2}} \left (b+a \left (c+d x^2\right )\right )}{24 a b d^3}-\frac{c^2 \sqrt{a+\frac{b}{c+d x^2}} \left (b+a \left (c+d x^2\right )\right )^2}{b d^3}+\frac{\left (c+d x^2\right ) \sqrt{a+\frac{b}{c+d x^2}} \left (b+a \left (c+d x^2\right )\right )^2}{6 a d^3}-\frac{b \left (b^2+12 a b c-24 a^2 c^2\right ) \sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{c+d x^2}}{\sqrt{b+a \left (c+d x^2\right )}}\right )}{16 a^{3/2} d^3 \sqrt{b+a \left (c+d x^2\right )}}\\ \end{align*}
Mathematica [A] time = 0.342408, size = 142, normalized size = 0.57 \[ \frac{\sqrt{a} \sqrt{\frac{a c+a d x^2+b}{c+d x^2}} \left (8 a^2 \left (c^3+d^3 x^6\right )-2 a b \left (47 c^2+16 c d x^2-7 d^2 x^4\right )+3 b^2 \left (c+d x^2\right )\right )-3 b \left (-24 a^2 c^2+12 a b c+b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{c+d x^2}}}{\sqrt{a}}\right )}{48 a^{3/2} d^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.031, size = 1018, normalized size = 4.1 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [A] time = 3.1079, size = 954, normalized size = 3.83 \begin{align*} \left [\frac{3 \,{\left (24 \, a^{2} b c^{2} - 12 \, a b^{2} c - b^{3}\right )} \sqrt{a} \log \left (8 \, a^{2} d^{2} x^{4} + 8 \, a^{2} c^{2} + 8 \,{\left (2 \, a^{2} c + a b\right )} d x^{2} + 8 \, a b c + b^{2} + 4 \,{\left (2 \, a d^{2} x^{4} +{\left (4 \, a c + b\right )} d x^{2} + 2 \, a c^{2} + b c\right )} \sqrt{a} \sqrt{\frac{a d x^{2} + a c + b}{d x^{2} + c}}\right ) + 4 \,{\left (8 \, a^{3} d^{3} x^{6} + 14 \, a^{2} b d^{2} x^{4} + 8 \, a^{3} c^{3} - 94 \, a^{2} b c^{2} + 3 \, a b^{2} c -{\left (32 \, a^{2} b c - 3 \, a b^{2}\right )} d x^{2}\right )} \sqrt{\frac{a d x^{2} + a c + b}{d x^{2} + c}}}{192 \, a^{2} d^{3}}, -\frac{3 \,{\left (24 \, a^{2} b c^{2} - 12 \, a b^{2} c - b^{3}\right )} \sqrt{-a} \arctan \left (\frac{{\left (2 \, a d x^{2} + 2 \, a c + b\right )} \sqrt{-a} \sqrt{\frac{a d x^{2} + a c + b}{d x^{2} + c}}}{2 \,{\left (a^{2} d x^{2} + a^{2} c + a b\right )}}\right ) - 2 \,{\left (8 \, a^{3} d^{3} x^{6} + 14 \, a^{2} b d^{2} x^{4} + 8 \, a^{3} c^{3} - 94 \, a^{2} b c^{2} + 3 \, a b^{2} c -{\left (32 \, a^{2} b c - 3 \, a b^{2}\right )} d x^{2}\right )} \sqrt{\frac{a d x^{2} + a c + b}{d x^{2} + c}}}{96 \, a^{2} d^{3}}\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 x^{5} \left (\frac{a c + a d x^{2} + b}{c + d x^{2}}\right )^{\frac{3}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a + \frac{b}{d x^{2} + c}\right )}^{\frac{3}{2}} x^{5}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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