Optimal. Leaf size=135 \[ -\frac{2 a (c+d x)^{5/2}}{5 b^2}-\frac{2 a (c+d x)^{3/2} (b c-a d)}{3 b^3}-\frac{2 a \sqrt{c+d x} (b c-a d)^2}{b^4}+\frac{2 a (b c-a d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{b^{9/2}}+\frac{2 (c+d x)^{7/2}}{7 b d} \]
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Rubi [A] time = 0.0746933, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {80, 50, 63, 208} \[ -\frac{2 a (c+d x)^{5/2}}{5 b^2}-\frac{2 a (c+d x)^{3/2} (b c-a d)}{3 b^3}-\frac{2 a \sqrt{c+d x} (b c-a d)^2}{b^4}+\frac{2 a (b c-a d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{b^{9/2}}+\frac{2 (c+d x)^{7/2}}{7 b d} \]
Antiderivative was successfully verified.
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Rule 80
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{x (c+d x)^{5/2}}{a+b x} \, dx &=\frac{2 (c+d x)^{7/2}}{7 b d}-\frac{a \int \frac{(c+d x)^{5/2}}{a+b x} \, dx}{b}\\ &=-\frac{2 a (c+d x)^{5/2}}{5 b^2}+\frac{2 (c+d x)^{7/2}}{7 b d}-\frac{(a (b c-a d)) \int \frac{(c+d x)^{3/2}}{a+b x} \, dx}{b^2}\\ &=-\frac{2 a (b c-a d) (c+d x)^{3/2}}{3 b^3}-\frac{2 a (c+d x)^{5/2}}{5 b^2}+\frac{2 (c+d x)^{7/2}}{7 b d}-\frac{\left (a (b c-a d)^2\right ) \int \frac{\sqrt{c+d x}}{a+b x} \, dx}{b^3}\\ &=-\frac{2 a (b c-a d)^2 \sqrt{c+d x}}{b^4}-\frac{2 a (b c-a d) (c+d x)^{3/2}}{3 b^3}-\frac{2 a (c+d x)^{5/2}}{5 b^2}+\frac{2 (c+d x)^{7/2}}{7 b d}-\frac{\left (a (b c-a d)^3\right ) \int \frac{1}{(a+b x) \sqrt{c+d x}} \, dx}{b^4}\\ &=-\frac{2 a (b c-a d)^2 \sqrt{c+d x}}{b^4}-\frac{2 a (b c-a d) (c+d x)^{3/2}}{3 b^3}-\frac{2 a (c+d x)^{5/2}}{5 b^2}+\frac{2 (c+d x)^{7/2}}{7 b d}-\frac{\left (2 a (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^4 d}\\ &=-\frac{2 a (b c-a d)^2 \sqrt{c+d x}}{b^4}-\frac{2 a (b c-a d) (c+d x)^{3/2}}{3 b^3}-\frac{2 a (c+d x)^{5/2}}{5 b^2}+\frac{2 (c+d x)^{7/2}}{7 b d}+\frac{2 a (b c-a d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{b^{9/2}}\\ \end{align*}
Mathematica [A] time = 0.23325, size = 126, normalized size = 0.93 \[ -\frac{2 a (c+d x)^{5/2}}{5 b^2}-\frac{2 a (b c-a d) \left (\sqrt{b} \sqrt{c+d x} (-3 a d+4 b c+b d x)-3 (b c-a d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )\right )}{3 b^{9/2}}+\frac{2 (c+d x)^{7/2}}{7 b d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.009, size = 291, normalized size = 2.2 \begin{align*}{\frac{2}{7\,bd} \left ( dx+c \right ) ^{{\frac{7}{2}}}}-{\frac{2\,a}{5\,{b}^{2}} \left ( dx+c \right ) ^{{\frac{5}{2}}}}+{\frac{2\,d{a}^{2}}{3\,{b}^{3}} \left ( dx+c \right ) ^{{\frac{3}{2}}}}-{\frac{2\,ac}{3\,{b}^{2}} \left ( dx+c \right ) ^{{\frac{3}{2}}}}-2\,{\frac{{d}^{2}{a}^{3}\sqrt{dx+c}}{{b}^{4}}}+4\,{\frac{d{a}^{2}c\sqrt{dx+c}}{{b}^{3}}}-2\,{\frac{a{c}^{2}\sqrt{dx+c}}{{b}^{2}}}+2\,{\frac{{d}^{3}{a}^{4}}{{b}^{4}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{b\sqrt{dx+c}}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }-6\,{\frac{{d}^{2}{a}^{3}c}{{b}^{3}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{b\sqrt{dx+c}}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }+6\,{\frac{d{a}^{2}{c}^{2}}{{b}^{2}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{b\sqrt{dx+c}}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }-2\,{\frac{a{c}^{3}}{b\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 = 2.59758, size = 907, normalized size = 6.72 \begin{align*} \left [\frac{105 \,{\left (a b^{2} c^{2} d - 2 \, a^{2} b c d^{2} + a^{3} d^{3}\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 (15 \, b^{3} d^{3} x^{3} + 15 \, b^{3} c^{3} - 161 \, a b^{2} c^{2} d + 245 \, a^{2} b c d^{2} - 105 \, a^{3} d^{3} + 3 \,{\left (15 \, b^{3} c d^{2} - 7 \, a b^{2} d^{3}\right )} x^{2} +{\left (45 \, b^{3} c^{2} d - 77 \, a b^{2} c d^{2} + 35 \, a^{2} b d^{3}\right )} x\right )} \sqrt{d x + c}}{105 \, b^{4} d}, \frac{2 \,{\left (105 \,{\left (a b^{2} c^{2} d - 2 \, a^{2} b c d^{2} + a^{3} d^{3}\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 (15 \, b^{3} d^{3} x^{3} + 15 \, b^{3} c^{3} - 161 \, a b^{2} c^{2} d + 245 \, a^{2} b c d^{2} - 105 \, a^{3} d^{3} + 3 \,{\left (15 \, b^{3} c d^{2} - 7 \, a b^{2} d^{3}\right )} x^{2} +{\left (45 \, b^{3} c^{2} d - 77 \, a b^{2} c d^{2} + 35 \, a^{2} b d^{3}\right )} x\right )} \sqrt{d x + c}\right )}}{105 \, b^{4} d}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 41.9367, size = 148, normalized size = 1.1 \begin{align*} - \frac{2 a \left (c + d x\right )^{\frac{5}{2}}}{5 b^{2}} + \frac{2 a \left (a d - b c\right )^{3} \operatorname{atan}{\left (\frac{\sqrt{c + d x}}{\sqrt{\frac{a d - b c}{b}}} \right )}}{b^{5} \sqrt{\frac{a d - b c}{b}}} + \frac{2 \left (c + d x\right )^{\frac{7}{2}}}{7 b d} + \frac{\left (c + d x\right )^{\frac{3}{2}} \left (2 a^{2} d - 2 a b c\right )}{3 b^{3}} + \frac{\sqrt{c + d x} \left (- 2 a^{3} d^{2} + 4 a^{2} b c d - 2 a b^{2} c^{2}\right )}{b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25262, size = 286, normalized size = 2.12 \begin{align*} -\frac{2 \,{\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} 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^{4}} + \frac{2 \,{\left (15 \,{\left (d x + c\right )}^{\frac{7}{2}} b^{6} d^{6} - 21 \,{\left (d x + c\right )}^{\frac{5}{2}} a b^{5} d^{7} - 35 \,{\left (d x + c\right )}^{\frac{3}{2}} a b^{5} c d^{7} - 105 \, \sqrt{d x + c} a b^{5} c^{2} d^{7} + 35 \,{\left (d x + c\right )}^{\frac{3}{2}} a^{2} b^{4} d^{8} + 210 \, \sqrt{d x + c} a^{2} b^{4} c d^{8} - 105 \, \sqrt{d x + c} a^{3} b^{3} d^{9}\right )}}{105 \, b^{7} d^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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