Optimal. Leaf size=283 \[ -\frac{\sqrt{a+b x} (c+d x)^{3/2} (3 a d+7 b c) (b c-a d)^2}{192 a^3 c^2 x^2}+\frac{\sqrt{a+b x} (c+d x)^{5/2} (3 a d+7 b c) (b c-a d)}{240 a^2 c^2 x^3}+\frac{\sqrt{a+b x} \sqrt{c+d x} (3 a d+7 b c) (b c-a d)^3}{128 a^4 c^2 x}-\frac{(3 a d+7 b c) (b c-a d)^4 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{128 a^{9/2} c^{5/2}}+\frac{\sqrt{a+b x} (c+d x)^{7/2} (3 a d+7 b c)}{40 a c^2 x^4}-\frac{(a+b x)^{3/2} (c+d x)^{7/2}}{5 a c x^5} \]
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Rubi [A] time = 0.161629, antiderivative size = 283, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {96, 94, 93, 208} \[ -\frac{\sqrt{a+b x} (c+d x)^{3/2} (3 a d+7 b c) (b c-a d)^2}{192 a^3 c^2 x^2}+\frac{\sqrt{a+b x} (c+d x)^{5/2} (3 a d+7 b c) (b c-a d)}{240 a^2 c^2 x^3}+\frac{\sqrt{a+b x} \sqrt{c+d x} (3 a d+7 b c) (b c-a d)^3}{128 a^4 c^2 x}-\frac{(3 a d+7 b c) (b c-a d)^4 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{128 a^{9/2} c^{5/2}}+\frac{\sqrt{a+b x} (c+d x)^{7/2} (3 a d+7 b c)}{40 a c^2 x^4}-\frac{(a+b x)^{3/2} (c+d x)^{7/2}}{5 a c x^5} \]
Antiderivative was successfully verified.
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Rule 96
Rule 94
Rule 93
Rule 208
Rubi steps
\begin{align*} \int \frac{\sqrt{a+b x} (c+d x)^{5/2}}{x^6} \, dx &=-\frac{(a+b x)^{3/2} (c+d x)^{7/2}}{5 a c x^5}-\frac{\left (\frac{7 b c}{2}+\frac{3 a d}{2}\right ) \int \frac{\sqrt{a+b x} (c+d x)^{5/2}}{x^5} \, dx}{5 a c}\\ &=\frac{(7 b c+3 a d) \sqrt{a+b x} (c+d x)^{7/2}}{40 a c^2 x^4}-\frac{(a+b x)^{3/2} (c+d x)^{7/2}}{5 a c x^5}-\frac{((b c-a d) (7 b c+3 a d)) \int \frac{(c+d x)^{5/2}}{x^4 \sqrt{a+b x}} \, dx}{80 a c^2}\\ &=\frac{(b c-a d) (7 b c+3 a d) \sqrt{a+b x} (c+d x)^{5/2}}{240 a^2 c^2 x^3}+\frac{(7 b c+3 a d) \sqrt{a+b x} (c+d x)^{7/2}}{40 a c^2 x^4}-\frac{(a+b x)^{3/2} (c+d x)^{7/2}}{5 a c x^5}+\frac{\left ((b c-a d)^2 (7 b c+3 a d)\right ) \int \frac{(c+d x)^{3/2}}{x^3 \sqrt{a+b x}} \, dx}{96 a^2 c^2}\\ &=-\frac{(b c-a d)^2 (7 b c+3 a d) \sqrt{a+b x} (c+d x)^{3/2}}{192 a^3 c^2 x^2}+\frac{(b c-a d) (7 b c+3 a d) \sqrt{a+b x} (c+d x)^{5/2}}{240 a^2 c^2 x^3}+\frac{(7 b c+3 a d) \sqrt{a+b x} (c+d x)^{7/2}}{40 a c^2 x^4}-\frac{(a+b x)^{3/2} (c+d x)^{7/2}}{5 a c x^5}-\frac{\left ((b c-a d)^3 (7 b c+3 a d)\right ) \int \frac{\sqrt{c+d x}}{x^2 \sqrt{a+b x}} \, dx}{128 a^3 c^2}\\ &=\frac{(b c-a d)^3 (7 b c+3 a d) \sqrt{a+b x} \sqrt{c+d x}}{128 a^4 c^2 x}-\frac{(b c-a d)^2 (7 b c+3 a d) \sqrt{a+b x} (c+d x)^{3/2}}{192 a^3 c^2 x^2}+\frac{(b c-a d) (7 b c+3 a d) \sqrt{a+b x} (c+d x)^{5/2}}{240 a^2 c^2 x^3}+\frac{(7 b c+3 a d) \sqrt{a+b x} (c+d x)^{7/2}}{40 a c^2 x^4}-\frac{(a+b x)^{3/2} (c+d x)^{7/2}}{5 a c x^5}+\frac{\left ((b c-a d)^4 (7 b c+3 a d)\right ) \int \frac{1}{x \sqrt{a+b x} \sqrt{c+d x}} \, dx}{256 a^4 c^2}\\ &=\frac{(b c-a d)^3 (7 b c+3 a d) \sqrt{a+b x} \sqrt{c+d x}}{128 a^4 c^2 x}-\frac{(b c-a d)^2 (7 b c+3 a d) \sqrt{a+b x} (c+d x)^{3/2}}{192 a^3 c^2 x^2}+\frac{(b c-a d) (7 b c+3 a d) \sqrt{a+b x} (c+d x)^{5/2}}{240 a^2 c^2 x^3}+\frac{(7 b c+3 a d) \sqrt{a+b x} (c+d x)^{7/2}}{40 a c^2 x^4}-\frac{(a+b x)^{3/2} (c+d x)^{7/2}}{5 a c x^5}+\frac{\left ((b c-a d)^4 (7 b c+3 a d)\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^4 c^2}\\ &=\frac{(b c-a d)^3 (7 b c+3 a d) \sqrt{a+b x} \sqrt{c+d x}}{128 a^4 c^2 x}-\frac{(b c-a d)^2 (7 b c+3 a d) \sqrt{a+b x} (c+d x)^{3/2}}{192 a^3 c^2 x^2}+\frac{(b c-a d) (7 b c+3 a d) \sqrt{a+b x} (c+d x)^{5/2}}{240 a^2 c^2 x^3}+\frac{(7 b c+3 a d) \sqrt{a+b x} (c+d x)^{7/2}}{40 a c^2 x^4}-\frac{(a+b x)^{3/2} (c+d x)^{7/2}}{5 a c x^5}-\frac{(b c-a d)^4 (7 b c+3 a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{128 a^{9/2} c^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.389108, size = 218, normalized size = 0.77 \[ -\frac{\frac{(3 a d+7 b c) \left (\frac{x (b c-a d) \left (\frac{5 x (b c-a d) \left (3 x^2 (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )+\sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x} (2 a c+5 a d x-3 b c x)\right )}{a^{5/2} \sqrt{c}}-8 \sqrt{a+b x} (c+d x)^{5/2}\right )}{a}-48 \sqrt{a+b x} (c+d x)^{7/2}\right )}{384 c x^4}+\frac{(a+b x)^{3/2} (c+d x)^{7/2}}{x^5}}{5 a c} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.019, size = 967, normalized size = 3.4 \begin{align*} -{\frac{1}{3840\,{a}^{4}{c}^{2}{x}^{5}}\sqrt{bx+a}\sqrt{dx+c} \left ( 45\,\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}+450\,\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}-375\,\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+105\,\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}-90\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{4}{a}^{4}{d}^{4}+120\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{4}{a}^{3}bc{d}^{3}-692\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{4}{a}^{2}{b}^{2}{c}^{2}{d}^{2}+680\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{4}a{b}^{3}{c}^{3}d-210\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{4}{b}^{4}{c}^{4}+60\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{3}{a}^{4}c{d}^{3}+436\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{3}{a}^{3}b{c}^{2}{d}^{2}-444\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{3}{a}^{2}{b}^{2}{c}^{3}d+140\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{3}a{b}^{3}{c}^{4}+1488\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{2}{a}^{4}{c}^{2}{d}^{2}+352\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{2}{a}^{3}b{c}^{3}d-112\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{2}{a}^{2}{b}^{2}{c}^{4}+2016\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}x{a}^{4}{c}^{3}d+96\,\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 = 79.7085, size = 1613, normalized size = 5.7 \begin{align*} \left [\frac{15 \,{\left (7 \, b^{5} c^{5} - 25 \, a b^{4} c^{4} d + 30 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} - 5 \, a^{4} b c d^{4} + 3 \, 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 (105 \, a b^{4} c^{5} - 340 \, a^{2} b^{3} c^{4} d + 346 \, a^{3} b^{2} c^{3} d^{2} - 60 \, a^{4} b c^{2} d^{3} + 45 \, a^{5} c d^{4}\right )} x^{4} + 2 \,{\left (35 \, a^{2} b^{3} c^{5} - 111 \, a^{3} b^{2} c^{4} d + 109 \, a^{4} b c^{3} d^{2} + 15 \, a^{5} c^{2} d^{3}\right )} x^{3} - 8 \,{\left (7 \, a^{3} b^{2} c^{5} - 22 \, a^{4} b c^{4} d - 93 \, a^{5} c^{3} d^{2}\right )} x^{2} + 48 \,{\left (a^{4} b c^{5} + 21 \, a^{5} c^{4} d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{7680 \, a^{5} c^{3} x^{5}}, \frac{15 \,{\left (7 \, b^{5} c^{5} - 25 \, a b^{4} c^{4} d + 30 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} - 5 \, a^{4} b c d^{4} + 3 \, 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 (105 \, a b^{4} c^{5} - 340 \, a^{2} b^{3} c^{4} d + 346 \, a^{3} b^{2} c^{3} d^{2} - 60 \, a^{4} b c^{2} d^{3} + 45 \, a^{5} c d^{4}\right )} x^{4} + 2 \,{\left (35 \, a^{2} b^{3} c^{5} - 111 \, a^{3} b^{2} c^{4} d + 109 \, a^{4} b c^{3} d^{2} + 15 \, a^{5} c^{2} d^{3}\right )} x^{3} - 8 \,{\left (7 \, a^{3} b^{2} c^{5} - 22 \, a^{4} b c^{4} d - 93 \, a^{5} c^{3} d^{2}\right )} x^{2} + 48 \,{\left (a^{4} b c^{5} + 21 \, a^{5} c^{4} d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{3840 \, a^{5} c^{3} 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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