Optimal. Leaf size=218 \[ \frac{\left (15 a^2 b c d^2+5 a^3 d^3-5 a b^2 c^2 d+b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 \sqrt{b} d^{5/2}}-2 a^{5/2} \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )-\frac{\sqrt{a+b x} \sqrt{c+d x} (b c-5 a d) (a d+b c)}{8 d^2}+\frac{1}{3} (a+b x)^{5/2} \sqrt{c+d x}+\frac{(a+b x)^{3/2} \sqrt{c+d x} (5 a d+b c)}{12 d} \]
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Rubi [A] time = 0.250781, antiderivative size = 218, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {101, 154, 157, 63, 217, 206, 93, 208} \[ \frac{\left (15 a^2 b c d^2+5 a^3 d^3-5 a b^2 c^2 d+b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 \sqrt{b} d^{5/2}}-2 a^{5/2} \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )-\frac{\sqrt{a+b x} \sqrt{c+d x} (b c-5 a d) (a d+b c)}{8 d^2}+\frac{1}{3} (a+b x)^{5/2} \sqrt{c+d x}+\frac{(a+b x)^{3/2} \sqrt{c+d x} (5 a d+b c)}{12 d} \]
Antiderivative was successfully verified.
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Rule 101
Rule 154
Rule 157
Rule 63
Rule 217
Rule 206
Rule 93
Rule 208
Rubi steps
\begin{align*} \int \frac{(a+b x)^{5/2} \sqrt{c+d x}}{x} \, dx &=\frac{1}{3} (a+b x)^{5/2} \sqrt{c+d x}-\frac{1}{3} \int \frac{(a+b x)^{3/2} \left (-3 a c+\frac{1}{2} (-b c-5 a d) x\right )}{x \sqrt{c+d x}} \, dx\\ &=\frac{(b c+5 a d) (a+b x)^{3/2} \sqrt{c+d x}}{12 d}+\frac{1}{3} (a+b x)^{5/2} \sqrt{c+d x}-\frac{\int \frac{\sqrt{a+b x} \left (-6 a^2 c d+\frac{3}{4} (b c-5 a d) (b c+a d) x\right )}{x \sqrt{c+d x}} \, dx}{6 d}\\ &=-\frac{(b c-5 a d) (b c+a d) \sqrt{a+b x} \sqrt{c+d x}}{8 d^2}+\frac{(b c+5 a d) (a+b x)^{3/2} \sqrt{c+d x}}{12 d}+\frac{1}{3} (a+b x)^{5/2} \sqrt{c+d x}-\frac{\int \frac{-6 a^3 c d^2-\frac{3}{8} \left (16 a^2 b c d^2+(b c-5 a d) (b c-a d) (b c+a d)\right ) x}{x \sqrt{a+b x} \sqrt{c+d x}} \, dx}{6 d^2}\\ &=-\frac{(b c-5 a d) (b c+a d) \sqrt{a+b x} \sqrt{c+d x}}{8 d^2}+\frac{(b c+5 a d) (a+b x)^{3/2} \sqrt{c+d x}}{12 d}+\frac{1}{3} (a+b x)^{5/2} \sqrt{c+d x}+\left (a^3 c\right ) \int \frac{1}{x \sqrt{a+b x} \sqrt{c+d x}} \, dx+\frac{\left (b^3 c^3-5 a b^2 c^2 d+15 a^2 b c d^2+5 a^3 d^3\right ) \int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx}{16 d^2}\\ &=-\frac{(b c-5 a d) (b c+a d) \sqrt{a+b x} \sqrt{c+d x}}{8 d^2}+\frac{(b c+5 a d) (a+b x)^{3/2} \sqrt{c+d x}}{12 d}+\frac{1}{3} (a+b x)^{5/2} \sqrt{c+d x}+\left (2 a^3 c\right ) \operatorname{Subst}\left (\int \frac{1}{-a+c x^2} \, dx,x,\frac{\sqrt{a+b x}}{\sqrt{c+d x}}\right )+\frac{\left (b^3 c^3-5 a b^2 c^2 d+15 a^2 b c d^2+5 a^3 d^3\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 )}{8 b d^2}\\ &=-\frac{(b c-5 a d) (b c+a d) \sqrt{a+b x} \sqrt{c+d x}}{8 d^2}+\frac{(b c+5 a d) (a+b x)^{3/2} \sqrt{c+d x}}{12 d}+\frac{1}{3} (a+b x)^{5/2} \sqrt{c+d x}-2 a^{5/2} \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )+\frac{\left (b^3 c^3-5 a b^2 c^2 d+15 a^2 b c d^2+5 a^3 d^3\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 )}{8 b d^2}\\ &=-\frac{(b c-5 a d) (b c+a d) \sqrt{a+b x} \sqrt{c+d x}}{8 d^2}+\frac{(b c+5 a d) (a+b x)^{3/2} \sqrt{c+d x}}{12 d}+\frac{1}{3} (a+b x)^{5/2} \sqrt{c+d x}-2 a^{5/2} \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )+\frac{\left (b^3 c^3-5 a b^2 c^2 d+15 a^2 b c d^2+5 a^3 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 \sqrt{b} d^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.793212, size = 242, normalized size = 1.11 \[ \frac{\sqrt{d} \left (\sqrt{a+b x} (c+d x) \left (33 a^2 d^2+2 a b d (7 c+13 d x)+b^2 \left (-3 c^2+2 c d x+8 d^2 x^2\right )\right )-48 a^{5/2} \sqrt{c} d^2 \sqrt{c+d x} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )\right )+\frac{3 \sqrt{b c-a d} \left (15 a^2 b c d^2+5 a^3 d^3-5 a b^2 c^2 d+b^3 c^3\right ) \sqrt{\frac{b (c+d x)}{b c-a d}} \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b c-a d}}\right )}{b}}{24 d^{5/2} \sqrt{c+d x}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.014, size = 583, normalized size = 2.7 \begin{align*}{\frac{1}{48\,{d}^{2}}\sqrt{bx+a}\sqrt{dx+c} \left ( 16\,{x}^{2}{b}^{2}{d}^{2}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}\sqrt{ac}+15\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) \sqrt{ac}{a}^{3}{d}^{3}+45\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) \sqrt{ac}{a}^{2}bc{d}^{2}-15\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) \sqrt{ac}a{b}^{2}{c}^{2}d+3\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) \sqrt{ac}{b}^{3}{c}^{3}-48\,\sqrt{bd}\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){a}^{3}c{d}^{2}+52\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}\sqrt{ac}xab{d}^{2}+4\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}\sqrt{ac}x{b}^{2}cd+66\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}\sqrt{ac}{a}^{2}{d}^{2}+28\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}\sqrt{ac}abcd-6\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}\sqrt{ac}{b}^{2}{c}^{2} \right ){\frac{1}{\sqrt{d{x}^{2}b+adx+bcx+ac}}}{\frac{1}{\sqrt{bd}}}{\frac{1}{\sqrt{ac}}}} \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 = 36.1846, size = 2732, normalized size = 12.53 \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b x\right )^{\frac{5}{2}} \sqrt{c + d x}}{x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.84542, size = 443, normalized size = 2.03 \begin{align*} -\frac{2 \, \sqrt{b d} a^{3} c{\left | b \right |} \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} b} + \frac{1}{24} \, \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \sqrt{b x + a}{\left (2 \,{\left (b x + a\right )}{\left (\frac{4 \,{\left (b x + a\right )}{\left | b \right |}}{b^{2}} + \frac{b^{3} c d^{3}{\left | b \right |} + 5 \, a b^{2} d^{4}{\left | b \right |}}{b^{4} d^{4}}\right )} - \frac{3 \,{\left (b^{4} c^{2} d^{2}{\left | b \right |} - 4 \, a b^{3} c d^{3}{\left | b \right |} - 5 \, a^{2} b^{2} d^{4}{\left | b \right |}\right )}}{b^{4} d^{4}}\right )} - \frac{{\left (\sqrt{b d} b^{3} c^{3}{\left | b \right |} - 5 \, \sqrt{b d} a b^{2} c^{2} d{\left | b \right |} + 15 \, \sqrt{b d} a^{2} b c d^{2}{\left | b \right |} + 5 \, \sqrt{b d} a^{3} d^{3}{\left | b \right |}\right )} \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 )}{16 \, b^{2} d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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