Optimal. Leaf size=246 \[ -\frac{3 \left (b^2 e^2-12 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{4 b^5 \sqrt{d}}+\frac{3 \sqrt{c} \left (5 b^2 e^2-20 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{4 b^5 \sqrt{c d-b e}}-\frac{\sqrt{d+e x} (x (2 c d-b e)+b d)}{2 b^2 \left (b x+c x^2\right )^2}+\frac{\sqrt{d+e x} (12 c x (2 c d-b e) (c d-b e)+b (12 c d-7 b e) (c d-b e))}{4 b^4 \left (b x+c x^2\right ) (c d-b e)} \]
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Rubi [A] time = 0.464637, antiderivative size = 246, 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, 822, 826, 1166, 208} \[ -\frac{3 \left (b^2 e^2-12 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{4 b^5 \sqrt{d}}+\frac{3 \sqrt{c} \left (5 b^2 e^2-20 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{4 b^5 \sqrt{c d-b e}}-\frac{\sqrt{d+e x} (x (2 c d-b e)+b d)}{2 b^2 \left (b x+c x^2\right )^2}+\frac{\sqrt{d+e x} (12 c x (2 c d-b e) (c d-b e)+b (12 c d-7 b e) (c d-b e))}{4 b^4 \left (b x+c x^2\right ) (c d-b e)} \]
Antiderivative was successfully verified.
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Rule 738
Rule 822
Rule 826
Rule 1166
Rule 208
Rubi steps
\begin{align*} \int \frac{(d+e x)^{3/2}}{\left (b x+c x^2\right )^3} \, dx &=-\frac{\sqrt{d+e x} (b d+(2 c d-b e) x)}{2 b^2 \left (b x+c x^2\right )^2}-\frac{\int \frac{\frac{1}{2} d (12 c d-7 b e)+\frac{5}{2} e (2 c d-b e) x}{\sqrt{d+e x} \left (b x+c x^2\right )^2} \, dx}{2 b^2}\\ &=-\frac{\sqrt{d+e x} (b d+(2 c d-b e) x)}{2 b^2 \left (b x+c x^2\right )^2}+\frac{\sqrt{d+e x} (b (12 c d-7 b e) (c d-b e)+12 c (c d-b e) (2 c d-b e) x)}{4 b^4 (c d-b e) \left (b x+c x^2\right )}+\frac{\int \frac{\frac{3}{4} d (c d-b e) \left (16 c^2 d^2-12 b c d e+b^2 e^2\right )+3 c d e (c d-b e) (2 c d-b e) x}{\sqrt{d+e x} \left (b x+c x^2\right )} \, dx}{2 b^4 d (c d-b e)}\\ &=-\frac{\sqrt{d+e x} (b d+(2 c d-b e) x)}{2 b^2 \left (b x+c x^2\right )^2}+\frac{\sqrt{d+e x} (b (12 c d-7 b e) (c d-b e)+12 c (c d-b e) (2 c d-b e) x)}{4 b^4 (c d-b e) \left (b x+c x^2\right )}+\frac{\operatorname{Subst}\left (\int \frac{-3 c d^2 e (c d-b e) (2 c d-b e)+\frac{3}{4} d e (c d-b e) \left (16 c^2 d^2-12 b c d e+b^2 e^2\right )+3 c d e (c d-b e) (2 c d-b e) 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 d (c d-b e)}\\ &=-\frac{\sqrt{d+e x} (b d+(2 c d-b e) x)}{2 b^2 \left (b x+c x^2\right )^2}+\frac{\sqrt{d+e x} (b (12 c d-7 b e) (c d-b e)+12 c (c d-b e) (2 c d-b e) x)}{4 b^4 (c d-b e) \left (b x+c x^2\right )}+\frac{\left (3 c \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 \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{\sqrt{d+e x} (b d+(2 c d-b e) x)}{2 b^2 \left (b x+c x^2\right )^2}+\frac{\sqrt{d+e x} (b (12 c d-7 b e) (c d-b e)+12 c (c d-b e) (2 c d-b e) x)}{4 b^4 (c d-b e) \left (b x+c x^2\right )}-\frac{3 \left (16 c^2 d^2-12 b c d e+b^2 e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{4 b^5 \sqrt{d}}+\frac{3 \sqrt{c} \left (16 c^2 d^2-20 b c d e+5 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 d-b e}}\\ \end{align*}
Mathematica [A] time = 0.53347, size = 289, normalized size = 1.17 \[ \frac{3 x^2 (b+c x)^2 \left (13 b^2 c d e^2-b^3 e^3-28 b c^2 d^2 e+16 c^3 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )-\sqrt{d} \left (b \sqrt{d+e x} \left (b^2 c^2 x \left (8 d^2-55 d e x+12 e^2 x^2\right )+b^3 c \left (-2 d^2-13 d e x+19 e^2 x^2\right )+b^4 e (2 d+5 e x)+36 b c^3 d x^2 (d-e x)+24 c^4 d^2 x^3\right )+3 \sqrt{c} x^2 (b+c x)^2 \sqrt{c d-b e} \left (5 b^2 e^2-20 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )\right )}{4 b^5 \sqrt{d} x^2 (b+c x)^2 (b e-c d)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.232, size = 414, normalized size = 1.7 \begin{align*} -{\frac{7\,{e}^{2}{c}^{2}}{4\,{b}^{3} \left ( cex+be \right ) ^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+3\,{\frac{e{c}^{3} \left ( ex+d \right ) ^{3/2}d}{{b}^{4} \left ( cex+be \right ) ^{2}}}-{\frac{9\,{e}^{3}c}{4\,{b}^{2} \left ( cex+be \right ) ^{2}}\sqrt{ex+d}}+{\frac{21\,{e}^{2}{c}^{2}d}{4\,{b}^{3} \left ( cex+be \right ) ^{2}}\sqrt{ex+d}}-3\,{\frac{e{c}^{3}\sqrt{ex+d}{d}^{2}}{{b}^{4} \left ( cex+be \right ) ^{2}}}-{\frac{15\,{e}^{2}c}{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}}}}+15\,{\frac{{c}^{2}ed}{{b}^{4}\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{\sqrt{ex+d}c}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-12\,{\frac{{d}^{2}{c}^{3}}{{b}^{5}\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{\sqrt{ex+d}c}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-{\frac{5}{4\,{b}^{3}{x}^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+3\,{\frac{ \left ( ex+d \right ) ^{3/2}cd}{e{b}^{4}{x}^{2}}}-3\,{\frac{\sqrt{ex+d}c{d}^{2}}{e{b}^{4}{x}^{2}}}+{\frac{3\,d}{4\,{b}^{3}{x}^{2}}\sqrt{ex+d}}-{\frac{3\,{e}^{2}}{4\,{b}^{3}}{\it Artanh} \left ({\sqrt{ex+d}{\frac{1}{\sqrt{d}}}} \right ){\frac{1}{\sqrt{d}}}}+9\,{\frac{e\sqrt{d}c}{{b}^{4}}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) }-12\,{\frac{{d}^{3/2}{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 [A] time = 3.01614, size = 3532, normalized size = 14.36 \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 [A] time = 1.37057, size = 529, normalized size = 2.15 \begin{align*} -\frac{3 \,{\left (16 \, c^{3} d^{2} - 20 \, b c^{2} d e + 5 \, b^{2} c e^{2}\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^{2} - 12 \, b c d e + b^{2} 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 e - 72 \,{\left (x e + d\right )}^{\frac{5}{2}} c^{3} d^{2} e + 72 \,{\left (x e + d\right )}^{\frac{3}{2}} c^{3} d^{3} e - 24 \, \sqrt{x e + d} c^{3} d^{4} e - 12 \,{\left (x e + d\right )}^{\frac{7}{2}} b c^{2} e^{2} + 72 \,{\left (x e + d\right )}^{\frac{5}{2}} b c^{2} d e^{2} - 108 \,{\left (x e + d\right )}^{\frac{3}{2}} b c^{2} d^{2} e^{2} + 48 \, \sqrt{x e + d} b c^{2} d^{3} e^{2} - 19 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{2} c e^{3} + 46 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{2} c d e^{3} - 27 \, \sqrt{x e + d} b^{2} c d^{2} e^{3} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{3} e^{4} + 3 \, \sqrt{x e + d} b^{3} d 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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