Optimal. Leaf size=130 \[ \frac{2 (d+e x)^{3/2} (A b-a B)}{3 b^2}+\frac{2 \sqrt{d+e x} (A b-a B) (b d-a e)}{b^3}-\frac{2 (A b-a B) (b d-a e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{7/2}}+\frac{2 B (d+e x)^{5/2}}{5 b e} \]
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Rubi [A] time = 0.0764276, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {80, 50, 63, 208} \[ \frac{2 (d+e x)^{3/2} (A b-a B)}{3 b^2}+\frac{2 \sqrt{d+e x} (A b-a B) (b d-a e)}{b^3}-\frac{2 (A b-a B) (b d-a e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{7/2}}+\frac{2 B (d+e x)^{5/2}}{5 b e} \]
Antiderivative was successfully verified.
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Rule 80
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{(A+B x) (d+e x)^{3/2}}{a+b x} \, dx &=\frac{2 B (d+e x)^{5/2}}{5 b e}+\frac{\left (2 \left (\frac{5 A b e}{2}-\frac{5 a B e}{2}\right )\right ) \int \frac{(d+e x)^{3/2}}{a+b x} \, dx}{5 b e}\\ &=\frac{2 (A b-a B) (d+e x)^{3/2}}{3 b^2}+\frac{2 B (d+e x)^{5/2}}{5 b e}+\frac{((A b-a B) (b d-a e)) \int \frac{\sqrt{d+e x}}{a+b x} \, dx}{b^2}\\ &=\frac{2 (A b-a B) (b d-a e) \sqrt{d+e x}}{b^3}+\frac{2 (A b-a B) (d+e x)^{3/2}}{3 b^2}+\frac{2 B (d+e x)^{5/2}}{5 b e}+\frac{\left ((A b-a B) (b d-a e)^2\right ) \int \frac{1}{(a+b x) \sqrt{d+e x}} \, dx}{b^3}\\ &=\frac{2 (A b-a B) (b d-a e) \sqrt{d+e x}}{b^3}+\frac{2 (A b-a B) (d+e x)^{3/2}}{3 b^2}+\frac{2 B (d+e x)^{5/2}}{5 b e}+\frac{\left (2 (A b-a B) (b d-a e)^2\right ) \operatorname{Subst}\left (\int \frac{1}{a-\frac{b d}{e}+\frac{b x^2}{e}} \, dx,x,\sqrt{d+e x}\right )}{b^3 e}\\ &=\frac{2 (A b-a B) (b d-a e) \sqrt{d+e x}}{b^3}+\frac{2 (A b-a B) (d+e x)^{3/2}}{3 b^2}+\frac{2 B (d+e x)^{5/2}}{5 b e}-\frac{2 (A b-a B) (b d-a e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.183098, size = 109, normalized size = 0.84 \[ \frac{2 (A b-a B) \left (\sqrt{b} \sqrt{d+e x} (-3 a e+4 b d+b e x)-3 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )\right )}{3 b^{7/2}}+\frac{2 B (d+e x)^{5/2}}{5 b e} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.009, size = 370, normalized size = 2.9 \begin{align*}{\frac{2\,B}{5\,be} \left ( ex+d \right ) ^{{\frac{5}{2}}}}+{\frac{2\,A}{3\,b} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{2\,Ba}{3\,{b}^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-2\,{\frac{Aae\sqrt{ex+d}}{{b}^{2}}}+2\,{\frac{Ad\sqrt{ex+d}}{b}}+2\,{\frac{{a}^{2}eB\sqrt{ex+d}}{{b}^{3}}}-2\,{\frac{Bad\sqrt{ex+d}}{{b}^{2}}}+2\,{\frac{{e}^{2}A{a}^{2}}{{b}^{2}\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{b\sqrt{ex+d}}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) }-4\,{\frac{aAde}{b\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{b\sqrt{ex+d}}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) }+2\,{\frac{A{d}^{2}}{\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{b\sqrt{ex+d}}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) }-2\,{\frac{B{a}^{3}{e}^{2}}{{b}^{3}\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{b\sqrt{ex+d}}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) }+4\,{\frac{{a}^{2}eBd}{{b}^{2}\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{b\sqrt{ex+d}}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) }-2\,{\frac{Ba{d}^{2}}{b\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{b\sqrt{ex+d}}{\sqrt{ \left ( ae-bd \right ) b}}} \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 = 1.55856, size = 795, normalized size = 6.12 \begin{align*} \left [-\frac{15 \,{\left ({\left (B a b - A b^{2}\right )} d e -{\left (B a^{2} - A a b\right )} e^{2}\right )} \sqrt{\frac{b d - a e}{b}} \log \left (\frac{b e x + 2 \, b d - a e - 2 \, \sqrt{e x + d} b \sqrt{\frac{b d - a e}{b}}}{b x + a}\right ) - 2 \,{\left (3 \, B b^{2} e^{2} x^{2} + 3 \, B b^{2} d^{2} - 20 \,{\left (B a b - A b^{2}\right )} d e + 15 \,{\left (B a^{2} - A a b\right )} e^{2} +{\left (6 \, B b^{2} d e - 5 \,{\left (B a b - A b^{2}\right )} e^{2}\right )} x\right )} \sqrt{e x + d}}{15 \, b^{3} e}, \frac{2 \,{\left (15 \,{\left ({\left (B a b - A b^{2}\right )} d e -{\left (B a^{2} - A a b\right )} e^{2}\right )} \sqrt{-\frac{b d - a e}{b}} \arctan \left (-\frac{\sqrt{e x + d} b \sqrt{-\frac{b d - a e}{b}}}{b d - a e}\right ) +{\left (3 \, B b^{2} e^{2} x^{2} + 3 \, B b^{2} d^{2} - 20 \,{\left (B a b - A b^{2}\right )} d e + 15 \,{\left (B a^{2} - A a b\right )} e^{2} +{\left (6 \, B b^{2} d e - 5 \,{\left (B a b - A b^{2}\right )} e^{2}\right )} x\right )} \sqrt{e x + d}\right )}}{15 \, b^{3} e}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 31.0056, size = 139, normalized size = 1.07 \begin{align*} \frac{2 B \left (d + e x\right )^{\frac{5}{2}}}{5 b e} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (2 A b - 2 B a\right )}{3 b^{2}} + \frac{\sqrt{d + e x} \left (- 2 A a b e + 2 A b^{2} d + 2 B a^{2} e - 2 B a b d\right )}{b^{3}} - \frac{2 \left (- A b + B a\right ) \left (a e - b d\right )^{2} \operatorname{atan}{\left (\frac{\sqrt{d + e x}}{\sqrt{\frac{a e - b d}{b}}} \right )}}{b^{4} \sqrt{\frac{a e - b d}{b}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 2.2389, size = 308, normalized size = 2.37 \begin{align*} -\frac{2 \,{\left (B a b^{2} d^{2} - A b^{3} d^{2} - 2 \, B a^{2} b d e + 2 \, A a b^{2} d e + B a^{3} e^{2} - A a^{2} b e^{2}\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{\sqrt{-b^{2} d + a b e} b^{3}} + \frac{2 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} B b^{4} e^{4} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} B a b^{3} e^{5} + 5 \,{\left (x e + d\right )}^{\frac{3}{2}} A b^{4} e^{5} - 15 \, \sqrt{x e + d} B a b^{3} d e^{5} + 15 \, \sqrt{x e + d} A b^{4} d e^{5} + 15 \, \sqrt{x e + d} B a^{2} b^{2} e^{6} - 15 \, \sqrt{x e + d} A a b^{3} e^{6}\right )} e^{\left (-5\right )}}{15 \, b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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