Optimal. Leaf size=157 \[ \frac{2 \sqrt{a+b x} \left (\frac{a^2}{c^2}-\frac{b^2}{d^2}\right )}{\sqrt{c+d x}}-\frac{2 a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{c^{5/2}}+\frac{2 b^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{d^{5/2}}-\frac{2 (a+b x)^{3/2} (b c-a d)}{3 c d (c+d x)^{3/2}} \]
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Rubi [A] time = 0.10863, antiderivative size = 157, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {98, 150, 157, 63, 217, 206, 93, 208} \[ \frac{2 \sqrt{a+b x} \left (\frac{a^2}{c^2}-\frac{b^2}{d^2}\right )}{\sqrt{c+d x}}-\frac{2 a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{c^{5/2}}+\frac{2 b^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{d^{5/2}}-\frac{2 (a+b x)^{3/2} (b c-a d)}{3 c d (c+d x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 98
Rule 150
Rule 157
Rule 63
Rule 217
Rule 206
Rule 93
Rule 208
Rubi steps
\begin{align*} \int \frac{(a+b x)^{5/2}}{x (c+d x)^{5/2}} \, dx &=-\frac{2 (b c-a d) (a+b x)^{3/2}}{3 c d (c+d x)^{3/2}}+\frac{2 \int \frac{\sqrt{a+b x} \left (\frac{3 a^2 d}{2}+\frac{3}{2} b^2 c x\right )}{x (c+d x)^{3/2}} \, dx}{3 c d}\\ &=-\frac{2 (b c-a d) (a+b x)^{3/2}}{3 c d (c+d x)^{3/2}}+\frac{2 \left (\frac{a^2}{c^2}-\frac{b^2}{d^2}\right ) \sqrt{a+b x}}{\sqrt{c+d x}}-\frac{4 \int \frac{-\frac{3}{4} a^3 d^2-\frac{3}{4} b^3 c^2 x}{x \sqrt{a+b x} \sqrt{c+d x}} \, dx}{3 c^2 d^2}\\ &=-\frac{2 (b c-a d) (a+b x)^{3/2}}{3 c d (c+d x)^{3/2}}+\frac{2 \left (\frac{a^2}{c^2}-\frac{b^2}{d^2}\right ) \sqrt{a+b x}}{\sqrt{c+d x}}+\frac{a^3 \int \frac{1}{x \sqrt{a+b x} \sqrt{c+d x}} \, dx}{c^2}+\frac{b^3 \int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx}{d^2}\\ &=-\frac{2 (b c-a d) (a+b x)^{3/2}}{3 c d (c+d x)^{3/2}}+\frac{2 \left (\frac{a^2}{c^2}-\frac{b^2}{d^2}\right ) \sqrt{a+b x}}{\sqrt{c+d x}}+\frac{\left (2 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{-a+c x^2} \, dx,x,\frac{\sqrt{a+b x}}{\sqrt{c+d x}}\right )}{c^2}+\frac{\left (2 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c-\frac{a d}{b}+\frac{d x^2}{b}}} \, dx,x,\sqrt{a+b x}\right )}{d^2}\\ &=-\frac{2 (b c-a d) (a+b x)^{3/2}}{3 c d (c+d x)^{3/2}}+\frac{2 \left (\frac{a^2}{c^2}-\frac{b^2}{d^2}\right ) \sqrt{a+b x}}{\sqrt{c+d x}}-\frac{2 a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{c^{5/2}}+\frac{\left (2 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-\frac{d x^2}{b}} \, dx,x,\frac{\sqrt{a+b x}}{\sqrt{c+d x}}\right )}{d^2}\\ &=-\frac{2 (b c-a d) (a+b x)^{3/2}}{3 c d (c+d x)^{3/2}}+\frac{2 \left (\frac{a^2}{c^2}-\frac{b^2}{d^2}\right ) \sqrt{a+b x}}{\sqrt{c+d x}}-\frac{2 a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{c^{5/2}}+\frac{2 b^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{d^{5/2}}\\ \end{align*}
Mathematica [A] time = 1.00806, size = 206, normalized size = 1.31 \[ -\frac{2 (b c-a d) \left (a^2 d (4 c+3 d x)+a b \left (3 c^2+8 c d x+3 d^2 x^2\right )+b^2 c x (3 c+4 d x)\right )}{3 c^2 d^2 \sqrt{a+b x} (c+d x)^{3/2}}-\frac{2 a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{c^{5/2}}+\frac{2 (b c-a d)^{5/2} \left (\frac{b (c+d x)}{b c-a d}\right )^{5/2} \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b c-a d}}\right )}{d^{5/2} (c+d x)^{5/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.023, size = 566, normalized size = 3.6 \begin{align*}{\frac{1}{3\,{c}^{2}{d}^{2}}\sqrt{bx+a} \left ( 3\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){x}^{2}{b}^{3}{c}^{2}{d}^{2}\sqrt{ac}-3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{2}{a}^{3}{d}^{4}\sqrt{bd}+6\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) x{b}^{3}{c}^{3}d\sqrt{ac}-6\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ) x{a}^{3}c{d}^{3}\sqrt{bd}+3\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){b}^{3}{c}^{4}\sqrt{ac}-3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){a}^{3}{c}^{2}{d}^{2}\sqrt{bd}+6\,x{a}^{2}{d}^{3}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}\sqrt{ac}+2\,xabc{d}^{2}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}\sqrt{ac}-8\,x{b}^{2}{c}^{2}d\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}\sqrt{ac}+8\,{a}^{2}c{d}^{2}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}\sqrt{ac}-2\,ab{c}^{2}d\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}\sqrt{ac}-6\,{b}^{2}{c}^{3}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}\sqrt{ac} \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}{\frac{1}{\sqrt{bd}}}{\frac{1}{\sqrt{ac}}} \left ( dx+c \right ) ^{-{\frac{3}{2}}}} \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 = 16.6077, size = 2917, normalized size = 18.58 \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 [B] time = 3.03395, size = 494, normalized size = 3.15 \begin{align*} \frac{\sqrt{b d} c^{4} \log \left ({\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{32 \,{\left (b^{2} c d^{4} - a b d^{5}\right )}} + \frac{2 \, \sqrt{b d} a^{3} b \arctan \left (\frac{b^{2} c + a b d -{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt{-a b c d} b}\right )}{\sqrt{-a b c d} c^{2}{\left | b \right |}} + \frac{\sqrt{b x + a}{\left (\frac{{\left (4 \, b^{8} c^{5} d^{2} - 5 \, a b^{7} c^{4} d^{3} - 2 \, a^{2} b^{6} c^{3} d^{4} + 3 \, a^{3} b^{5} c^{2} d^{5}\right )}{\left (b x + a\right )}}{b^{8} c^{2} d^{4} - 2 \, a b^{7} c d^{5} + a^{2} b^{6} d^{6}} + \frac{3 \,{\left (b^{9} c^{6} d - 2 \, a b^{8} c^{5} d^{2} + 2 \, a^{3} b^{6} c^{3} d^{4} - a^{4} b^{5} c^{2} d^{5}\right )}}{b^{8} c^{2} d^{4} - 2 \, a b^{7} c d^{5} + a^{2} b^{6} d^{6}}\right )}}{48 \,{\left (b^{2} c +{\left (b x + a\right )} b d - a b d\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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