Optimal. Leaf size=340 \[ \frac{\sqrt{a+b x} \sqrt{c+d x} \left (61 a^2 b c d^2-35 a^3 d^3-9 a b^2 c^2 d+15 b^3 c^3\right )}{960 a^2 c^3 x^2}-\frac{\sqrt{a+b x} \sqrt{c+d x} \left (-36 a^2 b^2 c^2 d^2+190 a^3 b c d^3-105 a^4 d^4-30 a b^3 c^3 d+45 b^4 c^4\right )}{1920 a^3 c^4 x}+\frac{\left (7 a^2 d^2+6 a b c d+3 b^2 c^2\right ) (b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{128 a^{7/2} c^{9/2}}-\frac{\sqrt{a+b x} \sqrt{c+d x} \left (\frac{3 b^2 c}{a}-\frac{7 a d^2}{c}+12 b d\right )}{240 c x^3}-\frac{(a+b x)^{3/2} \sqrt{c+d x}}{5 x^5}-\frac{\sqrt{a+b x} \sqrt{c+d x} (a d+3 b c)}{40 c x^4} \]
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Rubi [A] time = 0.319725, antiderivative size = 340, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {97, 149, 151, 12, 93, 208} \[ \frac{\sqrt{a+b x} \sqrt{c+d x} \left (61 a^2 b c d^2-35 a^3 d^3-9 a b^2 c^2 d+15 b^3 c^3\right )}{960 a^2 c^3 x^2}-\frac{\sqrt{a+b x} \sqrt{c+d x} \left (-36 a^2 b^2 c^2 d^2+190 a^3 b c d^3-105 a^4 d^4-30 a b^3 c^3 d+45 b^4 c^4\right )}{1920 a^3 c^4 x}+\frac{\left (7 a^2 d^2+6 a b c d+3 b^2 c^2\right ) (b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{128 a^{7/2} c^{9/2}}-\frac{\sqrt{a+b x} \sqrt{c+d x} \left (\frac{3 b^2 c}{a}-\frac{7 a d^2}{c}+12 b d\right )}{240 c x^3}-\frac{(a+b x)^{3/2} \sqrt{c+d x}}{5 x^5}-\frac{\sqrt{a+b x} \sqrt{c+d x} (a d+3 b c)}{40 c x^4} \]
Antiderivative was successfully verified.
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Rule 97
Rule 149
Rule 151
Rule 12
Rule 93
Rule 208
Rubi steps
\begin{align*} \int \frac{(a+b x)^{3/2} \sqrt{c+d x}}{x^6} \, dx &=-\frac{(a+b x)^{3/2} \sqrt{c+d x}}{5 x^5}+\frac{1}{5} \int \frac{\sqrt{a+b x} \left (\frac{1}{2} (3 b c+a d)+2 b d x\right )}{x^5 \sqrt{c+d x}} \, dx\\ &=-\frac{(3 b c+a d) \sqrt{a+b x} \sqrt{c+d x}}{40 c x^4}-\frac{(a+b x)^{3/2} \sqrt{c+d x}}{5 x^5}+\frac{\int \frac{\frac{1}{4} \left (3 b^2 c^2+12 a b c d-7 a^2 d^2\right )+\frac{1}{2} b d (7 b c-3 a d) x}{x^4 \sqrt{a+b x} \sqrt{c+d x}} \, dx}{20 c}\\ &=-\frac{(3 b c+a d) \sqrt{a+b x} \sqrt{c+d x}}{40 c x^4}-\frac{\left (\frac{3 b^2 c}{a}+12 b d-\frac{7 a d^2}{c}\right ) \sqrt{a+b x} \sqrt{c+d x}}{240 c x^3}-\frac{(a+b x)^{3/2} \sqrt{c+d x}}{5 x^5}-\frac{\int \frac{\frac{1}{8} \left (15 b^3 c^3-9 a b^2 c^2 d+61 a^2 b c d^2-35 a^3 d^3\right )+\frac{1}{2} b d \left (3 b^2 c^2+12 a b c d-7 a^2 d^2\right ) x}{x^3 \sqrt{a+b x} \sqrt{c+d x}} \, dx}{60 a c^2}\\ &=-\frac{(3 b c+a d) \sqrt{a+b x} \sqrt{c+d x}}{40 c x^4}-\frac{\left (\frac{3 b^2 c}{a}+12 b d-\frac{7 a d^2}{c}\right ) \sqrt{a+b x} \sqrt{c+d x}}{240 c x^3}+\frac{\left (15 b^3 c^3-9 a b^2 c^2 d+61 a^2 b c d^2-35 a^3 d^3\right ) \sqrt{a+b x} \sqrt{c+d x}}{960 a^2 c^3 x^2}-\frac{(a+b x)^{3/2} \sqrt{c+d x}}{5 x^5}+\frac{\int \frac{\frac{1}{16} \left (45 b^4 c^4-30 a b^3 c^3 d-36 a^2 b^2 c^2 d^2+190 a^3 b c d^3-105 a^4 d^4\right )+\frac{1}{8} b d \left (15 b^3 c^3-9 a b^2 c^2 d+61 a^2 b c d^2-35 a^3 d^3\right ) x}{x^2 \sqrt{a+b x} \sqrt{c+d x}} \, dx}{120 a^2 c^3}\\ &=-\frac{(3 b c+a d) \sqrt{a+b x} \sqrt{c+d x}}{40 c x^4}-\frac{\left (\frac{3 b^2 c}{a}+12 b d-\frac{7 a d^2}{c}\right ) \sqrt{a+b x} \sqrt{c+d x}}{240 c x^3}+\frac{\left (15 b^3 c^3-9 a b^2 c^2 d+61 a^2 b c d^2-35 a^3 d^3\right ) \sqrt{a+b x} \sqrt{c+d x}}{960 a^2 c^3 x^2}-\frac{\left (45 b^4 c^4-30 a b^3 c^3 d-36 a^2 b^2 c^2 d^2+190 a^3 b c d^3-105 a^4 d^4\right ) \sqrt{a+b x} \sqrt{c+d x}}{1920 a^3 c^4 x}-\frac{(a+b x)^{3/2} \sqrt{c+d x}}{5 x^5}-\frac{\int \frac{15 (b c-a d)^3 \left (3 b^2 c^2+6 a b c d+7 a^2 d^2\right )}{32 x \sqrt{a+b x} \sqrt{c+d x}} \, dx}{120 a^3 c^4}\\ &=-\frac{(3 b c+a d) \sqrt{a+b x} \sqrt{c+d x}}{40 c x^4}-\frac{\left (\frac{3 b^2 c}{a}+12 b d-\frac{7 a d^2}{c}\right ) \sqrt{a+b x} \sqrt{c+d x}}{240 c x^3}+\frac{\left (15 b^3 c^3-9 a b^2 c^2 d+61 a^2 b c d^2-35 a^3 d^3\right ) \sqrt{a+b x} \sqrt{c+d x}}{960 a^2 c^3 x^2}-\frac{\left (45 b^4 c^4-30 a b^3 c^3 d-36 a^2 b^2 c^2 d^2+190 a^3 b c d^3-105 a^4 d^4\right ) \sqrt{a+b x} \sqrt{c+d x}}{1920 a^3 c^4 x}-\frac{(a+b x)^{3/2} \sqrt{c+d x}}{5 x^5}-\frac{\left ((b c-a d)^3 \left (3 b^2 c^2+6 a b c d+7 a^2 d^2\right )\right ) \int \frac{1}{x \sqrt{a+b x} \sqrt{c+d x}} \, dx}{256 a^3 c^4}\\ &=-\frac{(3 b c+a d) \sqrt{a+b x} \sqrt{c+d x}}{40 c x^4}-\frac{\left (\frac{3 b^2 c}{a}+12 b d-\frac{7 a d^2}{c}\right ) \sqrt{a+b x} \sqrt{c+d x}}{240 c x^3}+\frac{\left (15 b^3 c^3-9 a b^2 c^2 d+61 a^2 b c d^2-35 a^3 d^3\right ) \sqrt{a+b x} \sqrt{c+d x}}{960 a^2 c^3 x^2}-\frac{\left (45 b^4 c^4-30 a b^3 c^3 d-36 a^2 b^2 c^2 d^2+190 a^3 b c d^3-105 a^4 d^4\right ) \sqrt{a+b x} \sqrt{c+d x}}{1920 a^3 c^4 x}-\frac{(a+b x)^{3/2} \sqrt{c+d x}}{5 x^5}-\frac{\left ((b c-a d)^3 \left (3 b^2 c^2+6 a b c d+7 a^2 d^2\right )\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^3 c^4}\\ &=-\frac{(3 b c+a d) \sqrt{a+b x} \sqrt{c+d x}}{40 c x^4}-\frac{\left (\frac{3 b^2 c}{a}+12 b d-\frac{7 a d^2}{c}\right ) \sqrt{a+b x} \sqrt{c+d x}}{240 c x^3}+\frac{\left (15 b^3 c^3-9 a b^2 c^2 d+61 a^2 b c d^2-35 a^3 d^3\right ) \sqrt{a+b x} \sqrt{c+d x}}{960 a^2 c^3 x^2}-\frac{\left (45 b^4 c^4-30 a b^3 c^3 d-36 a^2 b^2 c^2 d^2+190 a^3 b c d^3-105 a^4 d^4\right ) \sqrt{a+b x} \sqrt{c+d x}}{1920 a^3 c^4 x}-\frac{(a+b x)^{3/2} \sqrt{c+d x}}{5 x^5}+\frac{(b c-a d)^3 \left (3 b^2 c^2+6 a b c d+7 a^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{128 a^{7/2} c^{9/2}}\\ \end{align*}
Mathematica [A] time = 0.291498, size = 231, normalized size = 0.68 \[ \frac{\frac{5 \left (7 a^2 d^2+6 a b c d+3 b^2 c^2\right ) \left (\frac{3 x (b c-a d) \left (x^2 (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )-\sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x} (2 a c+a d x+b c x)\right )}{a^{3/2} c^{3/2}}-8 (a+b x)^{3/2} (c+d x)^{3/2}\right )}{24 c x^3}+\frac{2 (a+b x)^{5/2} (c+d x)^{3/2} (7 a d+5 b c)}{x^4}-\frac{16 a c (a+b x)^{5/2} (c+d x)^{3/2}}{x^5}}{80 a^2 c^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.018, size = 967, normalized size = 2.8 \begin{align*} -{\frac{1}{3840\,{a}^{3}{c}^{4}{x}^{5}}\sqrt{bx+a}\sqrt{dx+c} \left ( 105\,\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}-225\,\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}+90\,\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}+30\,\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}+45\,\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-45\,\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}-210\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{4}{a}^{4}{d}^{4}+380\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{4}{a}^{3}bc{d}^{3}-72\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{4}{a}^{2}{b}^{2}{c}^{2}{d}^{2}-60\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{4}a{b}^{3}{c}^{3}d+90\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{4}{b}^{4}{c}^{4}+140\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{3}{a}^{4}c{d}^{3}-244\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{3}{a}^{3}b{c}^{2}{d}^{2}+36\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{3}{a}^{2}{b}^{2}{c}^{3}d-60\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{3}a{b}^{3}{c}^{4}-112\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{2}{a}^{4}{c}^{2}{d}^{2}+192\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{2}{a}^{3}b{c}^{3}d+48\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{2}{a}^{2}{b}^{2}{c}^{4}+96\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}x{a}^{4}{c}^{3}d+1056\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}x{a}^{3}b{c}^{4}+768\,\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 = 76.1617, size = 1597, normalized size = 4.7 \begin{align*} \left [-\frac{15 \,{\left (3 \, b^{5} c^{5} - 3 \, a b^{4} c^{4} d - 2 \, a^{2} b^{3} c^{3} d^{2} - 6 \, a^{3} b^{2} c^{2} d^{3} + 15 \, a^{4} b c d^{4} - 7 \, 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 (384 \, a^{5} c^{5} +{\left (45 \, a b^{4} c^{5} - 30 \, a^{2} b^{3} c^{4} d - 36 \, a^{3} b^{2} c^{3} d^{2} + 190 \, a^{4} b c^{2} d^{3} - 105 \, a^{5} c d^{4}\right )} x^{4} - 2 \,{\left (15 \, a^{2} b^{3} c^{5} - 9 \, a^{3} b^{2} c^{4} d + 61 \, a^{4} b c^{3} d^{2} - 35 \, a^{5} c^{2} d^{3}\right )} x^{3} + 8 \,{\left (3 \, a^{3} b^{2} c^{5} + 12 \, a^{4} b c^{4} d - 7 \, a^{5} c^{3} d^{2}\right )} x^{2} + 48 \,{\left (11 \, a^{4} b c^{5} + a^{5} c^{4} d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{7680 \, a^{4} c^{5} x^{5}}, -\frac{15 \,{\left (3 \, b^{5} c^{5} - 3 \, a b^{4} c^{4} d - 2 \, a^{2} b^{3} c^{3} d^{2} - 6 \, a^{3} b^{2} c^{2} d^{3} + 15 \, a^{4} b c d^{4} - 7 \, 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 (384 \, a^{5} c^{5} +{\left (45 \, a b^{4} c^{5} - 30 \, a^{2} b^{3} c^{4} d - 36 \, a^{3} b^{2} c^{3} d^{2} + 190 \, a^{4} b c^{2} d^{3} - 105 \, a^{5} c d^{4}\right )} x^{4} - 2 \,{\left (15 \, a^{2} b^{3} c^{5} - 9 \, a^{3} b^{2} c^{4} d + 61 \, a^{4} b c^{3} d^{2} - 35 \, a^{5} c^{2} d^{3}\right )} x^{3} + 8 \,{\left (3 \, a^{3} b^{2} c^{5} + 12 \, a^{4} b c^{4} d - 7 \, a^{5} c^{3} d^{2}\right )} x^{2} + 48 \,{\left (11 \, a^{4} b c^{5} + a^{5} c^{4} d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{3840 \, a^{4} c^{5} 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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