Optimal. Leaf size=181 \[ -\frac{\sqrt{d+e x} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{b^2 c \left (b x+c x^2\right )}-\frac{\sqrt{c d-b e} \left (-b c (A e+2 B d)+4 A c^2 d+b^2 (-B) e\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^3 c^{3/2}}-\frac{\sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) (3 A b e-4 A c d+2 b B d)}{b^3} \]
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Rubi [A] time = 0.337044, antiderivative size = 181, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {818, 826, 1166, 208} \[ -\frac{\sqrt{d+e x} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{b^2 c \left (b x+c x^2\right )}-\frac{\sqrt{c d-b e} \left (-b c (A e+2 B d)+4 A c^2 d+b^2 (-B) e\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^3 c^{3/2}}-\frac{\sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) (3 A b e-4 A c d+2 b B d)}{b^3} \]
Antiderivative was successfully verified.
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Rule 818
Rule 826
Rule 1166
Rule 208
Rubi steps
\begin{align*} \int \frac{(A+B x) (d+e x)^{3/2}}{\left (b x+c x^2\right )^2} \, dx &=-\frac{\sqrt{d+e x} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{b^2 c \left (b x+c x^2\right )}+\frac{\int \frac{\frac{1}{2} c d (2 b B d-4 A c d+3 A b e)-\frac{1}{2} e \left (2 A c^2 d-b^2 B e-b c (B d+A e)\right ) x}{\sqrt{d+e x} \left (b x+c x^2\right )} \, dx}{b^2 c}\\ &=-\frac{\sqrt{d+e x} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{b^2 c \left (b x+c x^2\right )}+\frac{2 \operatorname{Subst}\left (\int \frac{\frac{1}{2} c d e (2 b B d-4 A c d+3 A b e)+\frac{1}{2} d e \left (2 A c^2 d-b^2 B e-b c (B d+A e)\right )-\frac{1}{2} e \left (2 A c^2 d-b^2 B e-b c (B d+A e)\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^2 c}\\ &=-\frac{\sqrt{d+e x} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{b^2 c \left (b x+c x^2\right )}+\frac{(c d (2 b B d-4 A c d+3 A b e)) \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 )}{b^3}+\frac{\left ((c d-b e) \left (4 A c^2 d-b^2 B e-b c (2 B d+A e)\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 )}{b^3 c}\\ &=-\frac{\sqrt{d+e x} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{b^2 c \left (b x+c x^2\right )}-\frac{\sqrt{d} (2 b B d-4 A c d+3 A b e) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b^3}-\frac{\sqrt{c d-b e} \left (4 A c^2 d-b^2 B e-b c (2 B d+A e)\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^3 c^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.365193, size = 171, normalized size = 0.94 \[ \frac{-\frac{\sqrt{c d-b e} \left (-b c (A e+2 B d)+4 A c^2 d+b^2 (-B) e\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{c^{3/2}}+\frac{b \sqrt{d+e x} (A c (-b d+b e x-2 c d x)+b B x (c d-b e))}{c x (b+c x)}+\sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) (-3 A b e+4 A c d-2 b B d)}{b^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.019, size = 443, normalized size = 2.5 \begin{align*} -{\frac{Ad}{{b}^{2}x}\sqrt{ex+d}}-3\,{\frac{e\sqrt{d}A}{{b}^{2}}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) }+4\,{\frac{{d}^{3/2}Ac}{{b}^{3}}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) }-2\,{\frac{{d}^{3/2}B}{{b}^{2}}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) }+{\frac{{e}^{2}A}{b \left ( cex+be \right ) }\sqrt{ex+d}}-{\frac{Acde}{{b}^{2} \left ( cex+be \right ) }\sqrt{ex+d}}-{\frac{{e}^{2}B}{c \left ( cex+be \right ) }\sqrt{ex+d}}+{\frac{Bed}{b \left ( cex+be \right ) }\sqrt{ex+d}}+{\frac{{e}^{2}A}{b}\arctan \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( be-cd \right ) c}}}} \right ){\frac{1}{\sqrt{ \left ( be-cd \right ) c}}}}-5\,{\frac{Acde}{{b}^{2}\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{\sqrt{ex+d}c}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }+4\,{\frac{A{c}^{2}{d}^{2}}{{b}^{3}\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{\sqrt{ex+d}c}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }+{\frac{{e}^{2}B}{c}\arctan \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( be-cd \right ) c}}}} \right ){\frac{1}{\sqrt{ \left ( be-cd \right ) c}}}}+{\frac{Bed}{b}\arctan \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( be-cd \right ) c}}}} \right ){\frac{1}{\sqrt{ \left ( be-cd \right ) c}}}}-2\,{\frac{Bc{d}^{2}}{{b}^{2}\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{\sqrt{ex+d}c}{\sqrt{ \left ( be-cd \right ) c}}} \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 = 7.25157, size = 2452, normalized size = 13.55 \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 = 1.25983, size = 451, normalized size = 2.49 \begin{align*} \frac{{\left (2 \, B b d^{2} - 4 \, A c d^{2} + 3 \, A b d e\right )} \arctan \left (\frac{\sqrt{x e + d}}{\sqrt{-d}}\right )}{b^{3} \sqrt{-d}} - \frac{{\left (2 \, B b c^{2} d^{2} - 4 \, A c^{3} d^{2} - B b^{2} c d e + 5 \, A b c^{2} d e - B b^{3} e^{2} - A b^{2} c e^{2}\right )} \arctan \left (\frac{\sqrt{x e + d} c}{\sqrt{-c^{2} d + b c e}}\right )}{\sqrt{-c^{2} d + b c e} b^{3} c} + \frac{{\left (x e + d\right )}^{\frac{3}{2}} B b c d e - 2 \,{\left (x e + d\right )}^{\frac{3}{2}} A c^{2} d e - \sqrt{x e + d} B b c d^{2} e + 2 \, \sqrt{x e + d} A c^{2} d^{2} e -{\left (x e + d\right )}^{\frac{3}{2}} B b^{2} e^{2} +{\left (x e + d\right )}^{\frac{3}{2}} A b c e^{2} + \sqrt{x e + d} B b^{2} d e^{2} - 2 \, \sqrt{x e + d} A b c d e^{2}}{{\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 )} b^{2} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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