Optimal. Leaf size=239 \[ \frac{3 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^4}{128 a^2 c^3 x}-\frac{3 (b c-a d)^5 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{128 a^{5/2} c^{7/2}}-\frac{\sqrt{a+b x} (c+d x)^{3/2} (b c-a d)^3}{64 a c^3 x^2}-\frac{\sqrt{a+b x} (c+d x)^{5/2} (b c-a d)^2}{16 c^3 x^3}-\frac{(a+b x)^{3/2} (c+d x)^{5/2} (b c-a d)}{8 c^2 x^4}-\frac{(a+b x)^{5/2} (c+d x)^{5/2}}{5 c x^5} \]
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Rubi [A] time = 0.13113, antiderivative size = 239, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {94, 93, 208} \[ \frac{3 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^4}{128 a^2 c^3 x}-\frac{3 (b c-a d)^5 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{128 a^{5/2} c^{7/2}}-\frac{\sqrt{a+b x} (c+d x)^{3/2} (b c-a d)^3}{64 a c^3 x^2}-\frac{\sqrt{a+b x} (c+d x)^{5/2} (b c-a d)^2}{16 c^3 x^3}-\frac{(a+b x)^{3/2} (c+d x)^{5/2} (b c-a d)}{8 c^2 x^4}-\frac{(a+b x)^{5/2} (c+d x)^{5/2}}{5 c x^5} \]
Antiderivative was successfully verified.
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Rule 94
Rule 93
Rule 208
Rubi steps
\begin{align*} \int \frac{(a+b x)^{5/2} (c+d x)^{3/2}}{x^6} \, dx &=-\frac{(a+b x)^{5/2} (c+d x)^{5/2}}{5 c x^5}+\frac{(b c-a d) \int \frac{(a+b x)^{3/2} (c+d x)^{3/2}}{x^5} \, dx}{2 c}\\ &=-\frac{(b c-a d) (a+b x)^{3/2} (c+d x)^{5/2}}{8 c^2 x^4}-\frac{(a+b x)^{5/2} (c+d x)^{5/2}}{5 c x^5}+\frac{\left (3 (b c-a d)^2\right ) \int \frac{\sqrt{a+b x} (c+d x)^{3/2}}{x^4} \, dx}{16 c^2}\\ &=-\frac{(b c-a d)^2 \sqrt{a+b x} (c+d x)^{5/2}}{16 c^3 x^3}-\frac{(b c-a d) (a+b x)^{3/2} (c+d x)^{5/2}}{8 c^2 x^4}-\frac{(a+b x)^{5/2} (c+d x)^{5/2}}{5 c x^5}+\frac{(b c-a d)^3 \int \frac{(c+d x)^{3/2}}{x^3 \sqrt{a+b x}} \, dx}{32 c^3}\\ &=-\frac{(b c-a d)^3 \sqrt{a+b x} (c+d x)^{3/2}}{64 a c^3 x^2}-\frac{(b c-a d)^2 \sqrt{a+b x} (c+d x)^{5/2}}{16 c^3 x^3}-\frac{(b c-a d) (a+b x)^{3/2} (c+d x)^{5/2}}{8 c^2 x^4}-\frac{(a+b x)^{5/2} (c+d x)^{5/2}}{5 c x^5}-\frac{\left (3 (b c-a d)^4\right ) \int \frac{\sqrt{c+d x}}{x^2 \sqrt{a+b x}} \, dx}{128 a c^3}\\ &=\frac{3 (b c-a d)^4 \sqrt{a+b x} \sqrt{c+d x}}{128 a^2 c^3 x}-\frac{(b c-a d)^3 \sqrt{a+b x} (c+d x)^{3/2}}{64 a c^3 x^2}-\frac{(b c-a d)^2 \sqrt{a+b x} (c+d x)^{5/2}}{16 c^3 x^3}-\frac{(b c-a d) (a+b x)^{3/2} (c+d x)^{5/2}}{8 c^2 x^4}-\frac{(a+b x)^{5/2} (c+d x)^{5/2}}{5 c x^5}+\frac{\left (3 (b c-a d)^5\right ) \int \frac{1}{x \sqrt{a+b x} \sqrt{c+d x}} \, dx}{256 a^2 c^3}\\ &=\frac{3 (b c-a d)^4 \sqrt{a+b x} \sqrt{c+d x}}{128 a^2 c^3 x}-\frac{(b c-a d)^3 \sqrt{a+b x} (c+d x)^{3/2}}{64 a c^3 x^2}-\frac{(b c-a d)^2 \sqrt{a+b x} (c+d x)^{5/2}}{16 c^3 x^3}-\frac{(b c-a d) (a+b x)^{3/2} (c+d x)^{5/2}}{8 c^2 x^4}-\frac{(a+b x)^{5/2} (c+d x)^{5/2}}{5 c x^5}+\frac{\left (3 (b c-a d)^5\right ) \operatorname{Subst}\left (\int \frac{1}{-a+c x^2} \, dx,x,\frac{\sqrt{a+b x}}{\sqrt{c+d x}}\right )}{128 a^2 c^3}\\ &=\frac{3 (b c-a d)^4 \sqrt{a+b x} \sqrt{c+d x}}{128 a^2 c^3 x}-\frac{(b c-a d)^3 \sqrt{a+b x} (c+d x)^{3/2}}{64 a c^3 x^2}-\frac{(b c-a d)^2 \sqrt{a+b x} (c+d x)^{5/2}}{16 c^3 x^3}-\frac{(b c-a d) (a+b x)^{3/2} (c+d x)^{5/2}}{8 c^2 x^4}-\frac{(a+b x)^{5/2} (c+d x)^{5/2}}{5 c x^5}-\frac{3 (b c-a d)^5 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{128 a^{5/2} c^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.914359, size = 208, normalized size = 0.87 \[ -\frac{5 x (b c-a d) \left (x (b c-a d) \left (\frac{x (b c-a d) \left (3 x^2 (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )+\sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x} (2 a c+5 a d x-3 b c x)\right )}{a^{5/2} \sqrt{c}}+8 \sqrt{a+b x} (c+d x)^{5/2}\right )+16 c (a+b x)^{3/2} (c+d x)^{5/2}\right )+128 c^2 (a+b x)^{5/2} (c+d x)^{5/2}}{640 c^3 x^5} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.018, size = 967, normalized size = 4.1 \begin{align*}{\frac{1}{1280\,{a}^{2}{c}^{3}{x}^{5}}\sqrt{bx+a}\sqrt{dx+c} \left ( 15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{5}{a}^{5}{d}^{5}-75\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{5}{a}^{4}bc{d}^{4}+150\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{5}{a}^{3}{b}^{2}{c}^{2}{d}^{3}-150\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{5}{a}^{2}{b}^{3}{c}^{3}{d}^{2}+75\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{5}a{b}^{4}{c}^{4}d-15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{5}{b}^{5}{c}^{5}-30\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{4}{a}^{4}{d}^{4}+140\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{4}{a}^{3}bc{d}^{3}-256\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{4}{a}^{2}{b}^{2}{c}^{2}{d}^{2}-140\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{4}a{b}^{3}{c}^{3}d+30\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{4}{b}^{4}{c}^{4}+20\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{3}{a}^{4}c{d}^{3}-92\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{3}{a}^{3}b{c}^{2}{d}^{2}-932\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{3}{a}^{2}{b}^{2}{c}^{3}d-20\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{3}a{b}^{3}{c}^{4}-16\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{2}{a}^{4}{c}^{2}{d}^{2}-1024\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{2}{a}^{3}b{c}^{3}d-496\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{2}{a}^{2}{b}^{2}{c}^{4}-352\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}x{a}^{4}{c}^{3}d-672\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}x{a}^{3}b{c}^{4}-256\,\sqrt{d{x}^{2}b+adx+bcx+ac}{a}^{4}{c}^{4}\sqrt{ac} \right ){\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{d{x}^{2}b+adx+bcx+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 = 65.2685, size = 1590, normalized size = 6.65 \begin{align*} \left [-\frac{15 \,{\left (b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}\right )} \sqrt{a c} x^{5} \log \left (\frac{8 \, a^{2} c^{2} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 4 \,{\left (2 \, a c +{\left (b c + a d\right )} x\right )} \sqrt{a c} \sqrt{b x + a} \sqrt{d x + c} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) + 4 \,{\left (128 \, a^{5} c^{5} -{\left (15 \, a b^{4} c^{5} - 70 \, a^{2} b^{3} c^{4} d - 128 \, a^{3} b^{2} c^{3} d^{2} + 70 \, a^{4} b c^{2} d^{3} - 15 \, a^{5} c d^{4}\right )} x^{4} + 2 \,{\left (5 \, a^{2} b^{3} c^{5} + 233 \, a^{3} b^{2} c^{4} d + 23 \, a^{4} b c^{3} d^{2} - 5 \, a^{5} c^{2} d^{3}\right )} x^{3} + 8 \,{\left (31 \, a^{3} b^{2} c^{5} + 64 \, a^{4} b c^{4} d + a^{5} c^{3} d^{2}\right )} x^{2} + 16 \,{\left (21 \, a^{4} b c^{5} + 11 \, a^{5} c^{4} d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{2560 \, a^{3} c^{4} x^{5}}, \frac{15 \,{\left (b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}\right )} \sqrt{-a c} x^{5} \arctan \left (\frac{{\left (2 \, a c +{\left (b c + a d\right )} x\right )} \sqrt{-a c} \sqrt{b x + a} \sqrt{d x + c}}{2 \,{\left (a b c d x^{2} + a^{2} c^{2} +{\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) - 2 \,{\left (128 \, a^{5} c^{5} -{\left (15 \, a b^{4} c^{5} - 70 \, a^{2} b^{3} c^{4} d - 128 \, a^{3} b^{2} c^{3} d^{2} + 70 \, a^{4} b c^{2} d^{3} - 15 \, a^{5} c d^{4}\right )} x^{4} + 2 \,{\left (5 \, a^{2} b^{3} c^{5} + 233 \, a^{3} b^{2} c^{4} d + 23 \, a^{4} b c^{3} d^{2} - 5 \, a^{5} c^{2} d^{3}\right )} x^{3} + 8 \,{\left (31 \, a^{3} b^{2} c^{5} + 64 \, a^{4} b c^{4} d + a^{5} c^{3} d^{2}\right )} x^{2} + 16 \,{\left (21 \, a^{4} b c^{5} + 11 \, a^{5} c^{4} d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{1280 \, a^{3} c^{4} x^{5}}\right ] \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 [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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