Optimal. Leaf size=157 \[ \frac{e (-a B e-3 A b e+4 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 b^{3/2} (b d-a e)^{5/2}}-\frac{\sqrt{d+e x} (A b-a B)}{2 b (a+b x)^2 (b d-a e)}-\frac{\sqrt{d+e x} (-a B e-3 A b e+4 b B d)}{4 b (a+b x) (b d-a e)^2} \]
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Rubi [A] time = 0.124942, antiderivative size = 157, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {78, 51, 63, 208} \[ \frac{e (-a B e-3 A b e+4 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 b^{3/2} (b d-a e)^{5/2}}-\frac{\sqrt{d+e x} (A b-a B)}{2 b (a+b x)^2 (b d-a e)}-\frac{\sqrt{d+e x} (-a B e-3 A b e+4 b B d)}{4 b (a+b x) (b d-a e)^2} \]
Antiderivative was successfully verified.
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Rule 78
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{A+B x}{(a+b x)^3 \sqrt{d+e x}} \, dx &=-\frac{(A b-a B) \sqrt{d+e x}}{2 b (b d-a e) (a+b x)^2}+\frac{(4 b B d-3 A b e-a B e) \int \frac{1}{(a+b x)^2 \sqrt{d+e x}} \, dx}{4 b (b d-a e)}\\ &=-\frac{(A b-a B) \sqrt{d+e x}}{2 b (b d-a e) (a+b x)^2}-\frac{(4 b B d-3 A b e-a B e) \sqrt{d+e x}}{4 b (b d-a e)^2 (a+b x)}-\frac{(e (4 b B d-3 A b e-a B e)) \int \frac{1}{(a+b x) \sqrt{d+e x}} \, dx}{8 b (b d-a e)^2}\\ &=-\frac{(A b-a B) \sqrt{d+e x}}{2 b (b d-a e) (a+b x)^2}-\frac{(4 b B d-3 A b e-a B e) \sqrt{d+e x}}{4 b (b d-a e)^2 (a+b x)}-\frac{(4 b B d-3 A b e-a B e) \operatorname{Subst}\left (\int \frac{1}{a-\frac{b d}{e}+\frac{b x^2}{e}} \, dx,x,\sqrt{d+e x}\right )}{4 b (b d-a e)^2}\\ &=-\frac{(A b-a B) \sqrt{d+e x}}{2 b (b d-a e) (a+b x)^2}-\frac{(4 b B d-3 A b e-a B e) \sqrt{d+e x}}{4 b (b d-a e)^2 (a+b x)}+\frac{e (4 b B d-3 A b e-a B e) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 b^{3/2} (b d-a e)^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.289756, size = 149, normalized size = 0.95 \[ \frac{\sqrt{d+e x} \left (\frac{e (-a B e-3 A b e+4 b B d) \left (\frac{a e-b d}{e (a+b x)}+\frac{\sqrt{a e-b d} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{a e-b d}}\right )}{\sqrt{b} \sqrt{d+e x}}\right )}{2 (b d-a e)^2}+\frac{a B-A b}{(a+b x)^2}\right )}{2 b (b d-a e)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.016, size = 436, normalized size = 2.8 \begin{align*}{\frac{3\,Ab{e}^{2}}{4\, \left ( bxe+ae \right ) ^{2} \left ({a}^{2}{e}^{2}-2\,abde+{b}^{2}{d}^{2} \right ) } \left ( ex+d \right ) ^{{\frac{3}{2}}}}+{\frac{Ba{e}^{2}}{4\, \left ( bxe+ae \right ) ^{2} \left ({a}^{2}{e}^{2}-2\,abde+{b}^{2}{d}^{2} \right ) } \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{bBde}{ \left ( bxe+ae \right ) ^{2} \left ({a}^{2}{e}^{2}-2\,abde+{b}^{2}{d}^{2} \right ) } \left ( ex+d \right ) ^{{\frac{3}{2}}}}+{\frac{5\,A{e}^{2}}{4\, \left ( bxe+ae \right ) ^{2} \left ( ae-bd \right ) }\sqrt{ex+d}}-{\frac{Ba{e}^{2}}{4\, \left ( bxe+ae \right ) ^{2} \left ( ae-bd \right ) b}\sqrt{ex+d}}-{\frac{eBd}{ \left ( bxe+ae \right ) ^{2} \left ( ae-bd \right ) }\sqrt{ex+d}}+{\frac{3\,A{e}^{2}}{4\,{a}^{2}{e}^{2}-8\,abde+4\,{b}^{2}{d}^{2}}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}}+{\frac{Ba{e}^{2}}{ \left ( 4\,{a}^{2}{e}^{2}-8\,abde+4\,{b}^{2}{d}^{2} \right ) b}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}}-{\frac{eBd}{{a}^{2}{e}^{2}-2\,abde+{b}^{2}{d}^{2}}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}} \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 = 1.4934, size = 1646, normalized size = 10.48 \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 = 2.17817, size = 359, normalized size = 2.29 \begin{align*} -\frac{{\left (4 \, B b d e - B a e^{2} - 3 \, A b e^{2}\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{4 \,{\left (b^{3} d^{2} - 2 \, a b^{2} d e + a^{2} b e^{2}\right )} \sqrt{-b^{2} d + a b e}} - \frac{4 \,{\left (x e + d\right )}^{\frac{3}{2}} B b^{2} d e - 4 \, \sqrt{x e + d} B b^{2} d^{2} e -{\left (x e + d\right )}^{\frac{3}{2}} B a b e^{2} - 3 \,{\left (x e + d\right )}^{\frac{3}{2}} A b^{2} e^{2} + 3 \, \sqrt{x e + d} B a b d e^{2} + 5 \, \sqrt{x e + d} A b^{2} d e^{2} + \sqrt{x e + d} B a^{2} e^{3} - 5 \, \sqrt{x e + d} A a b e^{3}}{4 \,{\left (b^{3} d^{2} - 2 \, a b^{2} d e + a^{2} b e^{2}\right )}{\left ({\left (x e + d\right )} b - b d + a e\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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