Optimal. Leaf size=112 \[ \frac{2 \sqrt{c+d x} (b c-a d)^2}{b^3}+\frac{2 (c+d x)^{3/2} (b c-a d)}{3 b^2}-\frac{2 (b c-a d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{b^{7/2}}+\frac{2 (c+d x)^{5/2}}{5 b} \]
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Rubi [A] time = 0.0579813, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {50, 63, 208} \[ \frac{2 \sqrt{c+d x} (b c-a d)^2}{b^3}+\frac{2 (c+d x)^{3/2} (b c-a d)}{3 b^2}-\frac{2 (b c-a d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{b^{7/2}}+\frac{2 (c+d x)^{5/2}}{5 b} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{(c+d x)^{5/2}}{a+b x} \, dx &=\frac{2 (c+d x)^{5/2}}{5 b}+\frac{(b c-a d) \int \frac{(c+d x)^{3/2}}{a+b x} \, dx}{b}\\ &=\frac{2 (b c-a d) (c+d x)^{3/2}}{3 b^2}+\frac{2 (c+d x)^{5/2}}{5 b}+\frac{(b c-a d)^2 \int \frac{\sqrt{c+d x}}{a+b x} \, dx}{b^2}\\ &=\frac{2 (b c-a d)^2 \sqrt{c+d x}}{b^3}+\frac{2 (b c-a d) (c+d x)^{3/2}}{3 b^2}+\frac{2 (c+d x)^{5/2}}{5 b}+\frac{(b c-a d)^3 \int \frac{1}{(a+b x) \sqrt{c+d x}} \, dx}{b^3}\\ &=\frac{2 (b c-a d)^2 \sqrt{c+d x}}{b^3}+\frac{2 (b c-a d) (c+d x)^{3/2}}{3 b^2}+\frac{2 (c+d x)^{5/2}}{5 b}+\frac{\left (2 (b c-a d)^3\right ) \operatorname{Subst}\left (\int \frac{1}{a-\frac{b c}{d}+\frac{b x^2}{d}} \, dx,x,\sqrt{c+d x}\right )}{b^3 d}\\ &=\frac{2 (b c-a d)^2 \sqrt{c+d x}}{b^3}+\frac{2 (b c-a d) (c+d x)^{3/2}}{3 b^2}+\frac{2 (c+d x)^{5/2}}{5 b}-\frac{2 (b c-a d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{b^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.150215, size = 105, normalized size = 0.94 \[ \frac{2 (b c-a d) \left (\sqrt{b} \sqrt{c+d x} (-3 a d+4 b c+b d x)-3 (b c-a d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )\right )}{3 b^{7/2}}+\frac{2 (c+d x)^{5/2}}{5 b} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.007, size = 263, normalized size = 2.4 \begin{align*}{\frac{2}{5\,b} \left ( dx+c \right ) ^{{\frac{5}{2}}}}-{\frac{2\,ad}{3\,{b}^{2}} \left ( dx+c \right ) ^{{\frac{3}{2}}}}+{\frac{2\,c}{3\,b} \left ( dx+c \right ) ^{{\frac{3}{2}}}}+2\,{\frac{{a}^{2}{d}^{2}\sqrt{dx+c}}{{b}^{3}}}-4\,{\frac{acd\sqrt{dx+c}}{{b}^{2}}}+2\,{\frac{{c}^{2}\sqrt{dx+c}}{b}}-2\,{\frac{{a}^{3}{d}^{3}}{{b}^{3}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{b\sqrt{dx+c}}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }+6\,{\frac{{a}^{2}c{d}^{2}}{{b}^{2}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{b\sqrt{dx+c}}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }-6\,{\frac{a{c}^{2}d}{b\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{b\sqrt{dx+c}}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }+2\,{\frac{{c}^{3}}{\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{b\sqrt{dx+c}}{\sqrt{ \left ( ad-bc \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 = 1.8973, size = 644, normalized size = 5.75 \begin{align*} \left [\frac{15 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{\frac{b c - a d}{b}} \log \left (\frac{b d x + 2 \, b c - a d - 2 \, \sqrt{d x + c} b \sqrt{\frac{b c - a d}{b}}}{b x + a}\right ) + 2 \,{\left (3 \, b^{2} d^{2} x^{2} + 23 \, b^{2} c^{2} - 35 \, a b c d + 15 \, a^{2} d^{2} +{\left (11 \, b^{2} c d - 5 \, a b d^{2}\right )} x\right )} \sqrt{d x + c}}{15 \, b^{3}}, -\frac{2 \,{\left (15 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{-\frac{b c - a d}{b}} \arctan \left (-\frac{\sqrt{d x + c} b \sqrt{-\frac{b c - a d}{b}}}{b c - a d}\right ) -{\left (3 \, b^{2} d^{2} x^{2} + 23 \, b^{2} c^{2} - 35 \, a b c d + 15 \, a^{2} d^{2} +{\left (11 \, b^{2} c d - 5 \, a b d^{2}\right )} x\right )} \sqrt{d x + c}\right )}}{15 \, b^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 19.9037, size = 121, normalized size = 1.08 \begin{align*} \frac{2 \left (c + d x\right )^{\frac{5}{2}}}{5 b} + \frac{\left (c + d x\right )^{\frac{3}{2}} \left (- 2 a d + 2 b c\right )}{3 b^{2}} + \frac{\sqrt{c + d x} \left (2 a^{2} d^{2} - 4 a b c d + 2 b^{2} c^{2}\right )}{b^{3}} - \frac{2 \left (a d - b c\right )^{3} \operatorname{atan}{\left (\frac{\sqrt{c + d x}}{\sqrt{\frac{a d - b c}{b}}} \right )}}{b^{4} \sqrt{\frac{a d - b c}{b}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07464, size = 231, normalized size = 2.06 \begin{align*} \frac{2 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} b^{3}} + \frac{2 \,{\left (3 \,{\left (d x + c\right )}^{\frac{5}{2}} b^{4} + 5 \,{\left (d x + c\right )}^{\frac{3}{2}} b^{4} c + 15 \, \sqrt{d x + c} b^{4} c^{2} - 5 \,{\left (d x + c\right )}^{\frac{3}{2}} a b^{3} d - 30 \, \sqrt{d x + c} a b^{3} c d + 15 \, \sqrt{d x + c} a^{2} b^{2} d^{2}\right )}}{15 \, b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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