Optimal. Leaf size=232 \[ \frac{3 \sqrt{d+e x} \left (x \left (b^2 e^2-8 b c d e+8 c^2 d^2\right )+b d (4 c d-3 b e)\right )}{4 b^4 \left (b x+c x^2\right )}-\frac{3 \sqrt{d} \left (5 b^2 e^2-20 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{4 b^5}+\frac{3 \sqrt{c d-b e} \left (b^2 e^2-12 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{4 b^5 \sqrt{c}}-\frac{(d+e x)^{3/2} (x (2 c d-b e)+b d)}{2 b^2 \left (b x+c x^2\right )^2} \]
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Rubi [A] time = 0.307329, antiderivative size = 232, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {738, 820, 826, 1166, 208} \[ \frac{3 \sqrt{d+e x} \left (x \left (b^2 e^2-8 b c d e+8 c^2 d^2\right )+b d (4 c d-3 b e)\right )}{4 b^4 \left (b x+c x^2\right )}-\frac{3 \sqrt{d} \left (5 b^2 e^2-20 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{4 b^5}+\frac{3 \sqrt{c d-b e} \left (b^2 e^2-12 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{4 b^5 \sqrt{c}}-\frac{(d+e x)^{3/2} (x (2 c d-b e)+b d)}{2 b^2 \left (b x+c x^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 738
Rule 820
Rule 826
Rule 1166
Rule 208
Rubi steps
\begin{align*} \int \frac{(d+e x)^{5/2}}{\left (b x+c x^2\right )^3} \, dx &=-\frac{(d+e x)^{3/2} (b d+(2 c d-b e) x)}{2 b^2 \left (b x+c x^2\right )^2}-\frac{\int \frac{\sqrt{d+e x} \left (\frac{3}{2} d (4 c d-3 b e)+\frac{3}{2} e (2 c d-b e) x\right )}{\left (b x+c x^2\right )^2} \, dx}{2 b^2}\\ &=-\frac{(d+e x)^{3/2} (b d+(2 c d-b e) x)}{2 b^2 \left (b x+c x^2\right )^2}+\frac{3 \sqrt{d+e x} \left (b d (4 c d-3 b e)+\left (8 c^2 d^2-8 b c d e+b^2 e^2\right ) x\right )}{4 b^4 \left (b x+c x^2\right )}+\frac{\int \frac{\frac{3}{4} d \left (16 c^2 d^2-20 b c d e+5 b^2 e^2\right )+\frac{3}{4} e \left (8 c^2 d^2-8 b c d e+b^2 e^2\right ) x}{\sqrt{d+e x} \left (b x+c x^2\right )} \, dx}{2 b^4}\\ &=-\frac{(d+e x)^{3/2} (b d+(2 c d-b e) x)}{2 b^2 \left (b x+c x^2\right )^2}+\frac{3 \sqrt{d+e x} \left (b d (4 c d-3 b e)+\left (8 c^2 d^2-8 b c d e+b^2 e^2\right ) x\right )}{4 b^4 \left (b x+c x^2\right )}+\frac{\operatorname{Subst}\left (\int \frac{-\frac{3}{4} d e \left (8 c^2 d^2-8 b c d e+b^2 e^2\right )+\frac{3}{4} d e \left (16 c^2 d^2-20 b c d e+5 b^2 e^2\right )+\frac{3}{4} e \left (8 c^2 d^2-8 b c d e+b^2 e^2\right ) x^2}{c d^2-b d e+(-2 c d+b e) x^2+c x^4} \, dx,x,\sqrt{d+e x}\right )}{b^4}\\ &=-\frac{(d+e x)^{3/2} (b d+(2 c d-b e) x)}{2 b^2 \left (b x+c x^2\right )^2}+\frac{3 \sqrt{d+e x} \left (b d (4 c d-3 b e)+\left (8 c^2 d^2-8 b c d e+b^2 e^2\right ) x\right )}{4 b^4 \left (b x+c x^2\right )}-\frac{\left (3 (c d-b e) \left (16 c^2 d^2-12 b c d e+b^2 e^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{b e}{2}+\frac{1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt{d+e x}\right )}{4 b^5}+\frac{\left (3 c d \left (16 c^2 d^2-20 b c d e+5 b^2 e^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{b e}{2}+\frac{1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt{d+e x}\right )}{4 b^5}\\ &=-\frac{(d+e x)^{3/2} (b d+(2 c d-b e) x)}{2 b^2 \left (b x+c x^2\right )^2}+\frac{3 \sqrt{d+e x} \left (b d (4 c d-3 b e)+\left (8 c^2 d^2-8 b c d e+b^2 e^2\right ) x\right )}{4 b^4 \left (b x+c x^2\right )}-\frac{3 \sqrt{d} \left (16 c^2 d^2-20 b c d e+5 b^2 e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{4 b^5}+\frac{3 \sqrt{c d-b e} \left (16 c^2 d^2-12 b c d e+b^2 e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{4 b^5 \sqrt{c}}\\ \end{align*}
Mathematica [A] time = 0.426926, size = 222, normalized size = 0.96 \[ \frac{\frac{b \sqrt{d+e x} \left (b^2 c x \left (8 d^2-37 d e x+3 e^2 x^2\right )+b^3 \left (-2 d^2-9 d e x+5 e^2 x^2\right )+12 b c^2 d x^2 (3 d-2 e x)+24 c^3 d^2 x^3\right )}{x^2 (b+c x)^2}-3 \sqrt{d} \left (5 b^2 e^2-20 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )+\frac{3 \sqrt{c d-b e} \left (b^2 e^2-12 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{\sqrt{c}}}{4 b^5} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.244, size = 521, normalized size = 2.3 \begin{align*}{\frac{3\,{e}^{3}c}{4\,{b}^{2} \left ( cex+be \right ) ^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{15\,{e}^{2}{c}^{2}d}{4\,{b}^{3} \left ( cex+be \right ) ^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+3\,{\frac{e \left ( ex+d \right ) ^{3/2}{d}^{2}{c}^{3}}{{b}^{4} \left ( cex+be \right ) ^{2}}}+{\frac{5\,{e}^{4}}{4\,b \left ( cex+be \right ) ^{2}}\sqrt{ex+d}}-{\frac{11\,{e}^{3}cd}{2\,{b}^{2} \left ( cex+be \right ) ^{2}}\sqrt{ex+d}}+{\frac{29\,{e}^{2}{c}^{2}{d}^{2}}{4\,{b}^{3} \left ( cex+be \right ) ^{2}}\sqrt{ex+d}}-3\,{\frac{e\sqrt{ex+d}{c}^{3}{d}^{3}}{{b}^{4} \left ( cex+be \right ) ^{2}}}+{\frac{3\,{e}^{3}}{4\,{b}^{2}}\arctan \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( be-cd \right ) c}}}} \right ){\frac{1}{\sqrt{ \left ( be-cd \right ) c}}}}-{\frac{39\,{e}^{2}cd}{4\,{b}^{3}}\arctan \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( be-cd \right ) c}}}} \right ){\frac{1}{\sqrt{ \left ( be-cd \right ) c}}}}+21\,{\frac{{c}^{2}e{d}^{2}}{{b}^{4}\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{\sqrt{ex+d}c}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-12\,{\frac{{c}^{3}{d}^{3}}{{b}^{5}\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{\sqrt{ex+d}c}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-{\frac{9\,d}{4\,{b}^{3}{x}^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+3\,{\frac{{d}^{2} \left ( ex+d \right ) ^{3/2}c}{e{b}^{4}{x}^{2}}}+{\frac{7\,{d}^{2}}{4\,{b}^{3}{x}^{2}}\sqrt{ex+d}}-3\,{\frac{{d}^{3}\sqrt{ex+d}c}{e{b}^{4}{x}^{2}}}-{\frac{15\,{e}^{2}}{4\,{b}^{3}}\sqrt{d}{\it Artanh} \left ({\sqrt{ex+d}{\frac{1}{\sqrt{d}}}} \right ) }+15\,{\frac{e{d}^{3/2}c}{{b}^{4}}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) }-12\,{\frac{{d}^{5/2}{c}^{2}}{{b}^{5}}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) } \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 [B] time = 2.91168, size = 3510, normalized size = 15.13 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Giac [B] time = 1.44165, size = 605, normalized size = 2.61 \begin{align*} -\frac{3 \,{\left (16 \, c^{3} d^{3} - 28 \, b c^{2} d^{2} e + 13 \, b^{2} c d e^{2} - b^{3} e^{3}\right )} \arctan \left (\frac{\sqrt{x e + d} c}{\sqrt{-c^{2} d + b c e}}\right )}{4 \, \sqrt{-c^{2} d + b c e} b^{5}} + \frac{3 \,{\left (16 \, c^{2} d^{3} - 20 \, b c d^{2} e + 5 \, b^{2} d e^{2}\right )} \arctan \left (\frac{\sqrt{x e + d}}{\sqrt{-d}}\right )}{4 \, b^{5} \sqrt{-d}} + \frac{24 \,{\left (x e + d\right )}^{\frac{7}{2}} c^{3} d^{2} e - 72 \,{\left (x e + d\right )}^{\frac{5}{2}} c^{3} d^{3} e + 72 \,{\left (x e + d\right )}^{\frac{3}{2}} c^{3} d^{4} e - 24 \, \sqrt{x e + d} c^{3} d^{5} e - 24 \,{\left (x e + d\right )}^{\frac{7}{2}} b c^{2} d e^{2} + 108 \,{\left (x e + d\right )}^{\frac{5}{2}} b c^{2} d^{2} e^{2} - 144 \,{\left (x e + d\right )}^{\frac{3}{2}} b c^{2} d^{3} e^{2} + 60 \, \sqrt{x e + d} b c^{2} d^{4} e^{2} + 3 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{2} c e^{3} - 46 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{2} c d e^{3} + 91 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{2} c d^{2} e^{3} - 48 \, \sqrt{x e + d} b^{2} c d^{3} e^{3} + 5 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{3} e^{4} - 19 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{3} d e^{4} + 12 \, \sqrt{x e + d} b^{3} d^{2} e^{4}}{4 \,{\left ({\left (x e + d\right )}^{2} c - 2 \,{\left (x e + d\right )} c d + c d^{2} +{\left (x e + d\right )} b e - b d e\right )}^{2} b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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