Optimal. Leaf size=131 \[ \frac{2 A b^2 \sqrt{b x+c x^2}}{\sqrt{x}}-2 A b^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )+\frac{2 A \left (b x+c x^2\right )^{5/2}}{5 x^{5/2}}+\frac{2 A b \left (b x+c x^2\right )^{3/2}}{3 x^{3/2}}+\frac{2 B \left (b x+c x^2\right )^{7/2}}{7 c x^{7/2}} \]
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Rubi [A] time = 0.115803, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {794, 664, 660, 207} \[ \frac{2 A b^2 \sqrt{b x+c x^2}}{\sqrt{x}}-2 A b^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )+\frac{2 A \left (b x+c x^2\right )^{5/2}}{5 x^{5/2}}+\frac{2 A b \left (b x+c x^2\right )^{3/2}}{3 x^{3/2}}+\frac{2 B \left (b x+c x^2\right )^{7/2}}{7 c x^{7/2}} \]
Antiderivative was successfully verified.
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Rule 794
Rule 664
Rule 660
Rule 207
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (b x+c x^2\right )^{5/2}}{x^{7/2}} \, dx &=\frac{2 B \left (b x+c x^2\right )^{7/2}}{7 c x^{7/2}}+A \int \frac{\left (b x+c x^2\right )^{5/2}}{x^{7/2}} \, dx\\ &=\frac{2 A \left (b x+c x^2\right )^{5/2}}{5 x^{5/2}}+\frac{2 B \left (b x+c x^2\right )^{7/2}}{7 c x^{7/2}}+(A b) \int \frac{\left (b x+c x^2\right )^{3/2}}{x^{5/2}} \, dx\\ &=\frac{2 A b \left (b x+c x^2\right )^{3/2}}{3 x^{3/2}}+\frac{2 A \left (b x+c x^2\right )^{5/2}}{5 x^{5/2}}+\frac{2 B \left (b x+c x^2\right )^{7/2}}{7 c x^{7/2}}+\left (A b^2\right ) \int \frac{\sqrt{b x+c x^2}}{x^{3/2}} \, dx\\ &=\frac{2 A b^2 \sqrt{b x+c x^2}}{\sqrt{x}}+\frac{2 A b \left (b x+c x^2\right )^{3/2}}{3 x^{3/2}}+\frac{2 A \left (b x+c x^2\right )^{5/2}}{5 x^{5/2}}+\frac{2 B \left (b x+c x^2\right )^{7/2}}{7 c x^{7/2}}+\left (A b^3\right ) \int \frac{1}{\sqrt{x} \sqrt{b x+c x^2}} \, dx\\ &=\frac{2 A b^2 \sqrt{b x+c x^2}}{\sqrt{x}}+\frac{2 A b \left (b x+c x^2\right )^{3/2}}{3 x^{3/2}}+\frac{2 A \left (b x+c x^2\right )^{5/2}}{5 x^{5/2}}+\frac{2 B \left (b x+c x^2\right )^{7/2}}{7 c x^{7/2}}+\left (2 A b^3\right ) \operatorname{Subst}\left (\int \frac{1}{-b+x^2} \, dx,x,\frac{\sqrt{b x+c x^2}}{\sqrt{x}}\right )\\ &=\frac{2 A b^2 \sqrt{b x+c x^2}}{\sqrt{x}}+\frac{2 A b \left (b x+c x^2\right )^{3/2}}{3 x^{3/2}}+\frac{2 A \left (b x+c x^2\right )^{5/2}}{5 x^{5/2}}+\frac{2 B \left (b x+c x^2\right )^{7/2}}{7 c x^{7/2}}-2 A b^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )\\ \end{align*}
Mathematica [A] time = 0.225577, size = 95, normalized size = 0.73 \[ \frac{(x (b+c x))^{5/2} \left (A \left (\frac{14 b^2}{(b+c x)^2}+\frac{14 b}{3 (b+c x)}+\frac{14}{5}\right )-\frac{14 A b^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b+c x}}{\sqrt{b}}\right )}{(b+c x)^{5/2}}+\frac{2 B (b+c x)}{c}\right )}{7 x^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 151, normalized size = 1.2 \begin{align*} -{\frac{2}{105\,c}\sqrt{x \left ( cx+b \right ) } \left ( -15\,B{x}^{3}{c}^{3}\sqrt{cx+b}-21\,A{x}^{2}{c}^{3}\sqrt{cx+b}-45\,B{x}^{2}b{c}^{2}\sqrt{cx+b}+105\,A{b}^{5/2}c{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ) -77\,Axb{c}^{2}\sqrt{cx+b}-45\,Bx{b}^{2}c\sqrt{cx+b}-161\,A{b}^{2}c\sqrt{cx+b}-15\,B{b}^{3}\sqrt{cx+b} \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{cx+b}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} A b^{2} \int \frac{\sqrt{c x + b}}{x}\,{d x} + \frac{2 \,{\left (35 \,{\left (B b^{2} c + 2 \, A b c^{2}\right )} x^{3} +{\left (15 \, B c^{3} x^{3} + 3 \, B b c^{2} x^{2} - 4 \, B b^{2} c x + 8 \, B b^{3}\right )} x^{2} + 35 \,{\left (B b^{3} + 2 \, A b^{2} c\right )} x^{2} + 7 \,{\left (3 \,{\left (2 \, B b c^{2} + A c^{3}\right )} x^{3} +{\left (2 \, B b^{2} c + A b c^{2}\right )} x^{2} - 2 \,{\left (2 \, B b^{3} + A b^{2} c\right )} x\right )} x\right )} \sqrt{c x + b}}{105 \, c x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.65635, size = 597, normalized size = 4.56 \begin{align*} \left [\frac{105 \, A b^{\frac{5}{2}} c x \log \left (-\frac{c x^{2} + 2 \, b x - 2 \, \sqrt{c x^{2} + b x} \sqrt{b} \sqrt{x}}{x^{2}}\right ) + 2 \,{\left (15 \, B c^{3} x^{3} + 15 \, B b^{3} + 161 \, A b^{2} c + 3 \,{\left (15 \, B b c^{2} + 7 \, A c^{3}\right )} x^{2} +{\left (45 \, B b^{2} c + 77 \, A b c^{2}\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{x}}{105 \, c x}, \frac{2 \,{\left (105 \, A \sqrt{-b} b^{2} c x \arctan \left (\frac{\sqrt{-b} \sqrt{x}}{\sqrt{c x^{2} + b x}}\right ) +{\left (15 \, B c^{3} x^{3} + 15 \, B b^{3} + 161 \, A b^{2} c + 3 \,{\left (15 \, B b c^{2} + 7 \, A c^{3}\right )} x^{2} +{\left (45 \, B b^{2} c + 77 \, A b c^{2}\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{x}\right )}}{105 \, c x}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (x \left (b + c x\right )\right )^{\frac{5}{2}} \left (A + B x\right )}{x^{\frac{7}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20922, size = 188, normalized size = 1.44 \begin{align*} \frac{2 \, A b^{3} \arctan \left (\frac{\sqrt{c x + b}}{\sqrt{-b}}\right )}{\sqrt{-b}} - \frac{2 \,{\left (105 \, A b^{3} c \arctan \left (\frac{\sqrt{b}}{\sqrt{-b}}\right ) + 15 \, B \sqrt{-b} b^{\frac{7}{2}} + 161 \, A \sqrt{-b} b^{\frac{5}{2}} c\right )}}{105 \, \sqrt{-b} c} + \frac{2 \,{\left (15 \,{\left (c x + b\right )}^{\frac{7}{2}} B c^{6} + 21 \,{\left (c x + b\right )}^{\frac{5}{2}} A c^{7} + 35 \,{\left (c x + b\right )}^{\frac{3}{2}} A b c^{7} + 105 \, \sqrt{c x + b} A b^{2} c^{7}\right )}}{105 \, c^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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