Optimal. Leaf size=186 \[ -\frac{5 (b c-a d)^4 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{64 b^{7/2} d^{3/2}}+\frac{5 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^3}{64 b^3 d}+\frac{5 (a+b x)^{3/2} \sqrt{c+d x} (b c-a d)^2}{32 b^3}+\frac{5 (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)}{24 b^2}+\frac{(a+b x)^{3/2} (c+d x)^{5/2}}{4 b} \]
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Rubi [A] time = 0.0931561, antiderivative size = 186, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {50, 63, 217, 206} \[ -\frac{5 (b c-a d)^4 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{64 b^{7/2} d^{3/2}}+\frac{5 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^3}{64 b^3 d}+\frac{5 (a+b x)^{3/2} \sqrt{c+d x} (b c-a d)^2}{32 b^3}+\frac{5 (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)}{24 b^2}+\frac{(a+b x)^{3/2} (c+d x)^{5/2}}{4 b} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \sqrt{a+b x} (c+d x)^{5/2} \, dx &=\frac{(a+b x)^{3/2} (c+d x)^{5/2}}{4 b}+\frac{(5 (b c-a d)) \int \sqrt{a+b x} (c+d x)^{3/2} \, dx}{8 b}\\ &=\frac{5 (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}{24 b^2}+\frac{(a+b x)^{3/2} (c+d x)^{5/2}}{4 b}+\frac{\left (5 (b c-a d)^2\right ) \int \sqrt{a+b x} \sqrt{c+d x} \, dx}{16 b^2}\\ &=\frac{5 (b c-a d)^2 (a+b x)^{3/2} \sqrt{c+d x}}{32 b^3}+\frac{5 (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}{24 b^2}+\frac{(a+b x)^{3/2} (c+d x)^{5/2}}{4 b}+\frac{\left (5 (b c-a d)^3\right ) \int \frac{\sqrt{a+b x}}{\sqrt{c+d x}} \, dx}{64 b^3}\\ &=\frac{5 (b c-a d)^3 \sqrt{a+b x} \sqrt{c+d x}}{64 b^3 d}+\frac{5 (b c-a d)^2 (a+b x)^{3/2} \sqrt{c+d x}}{32 b^3}+\frac{5 (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}{24 b^2}+\frac{(a+b x)^{3/2} (c+d x)^{5/2}}{4 b}-\frac{\left (5 (b c-a d)^4\right ) \int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx}{128 b^3 d}\\ &=\frac{5 (b c-a d)^3 \sqrt{a+b x} \sqrt{c+d x}}{64 b^3 d}+\frac{5 (b c-a d)^2 (a+b x)^{3/2} \sqrt{c+d x}}{32 b^3}+\frac{5 (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}{24 b^2}+\frac{(a+b x)^{3/2} (c+d x)^{5/2}}{4 b}-\frac{\left (5 (b c-a d)^4\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 )}{64 b^4 d}\\ &=\frac{5 (b c-a d)^3 \sqrt{a+b x} \sqrt{c+d x}}{64 b^3 d}+\frac{5 (b c-a d)^2 (a+b x)^{3/2} \sqrt{c+d x}}{32 b^3}+\frac{5 (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}{24 b^2}+\frac{(a+b x)^{3/2} (c+d x)^{5/2}}{4 b}-\frac{\left (5 (b c-a d)^4\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 )}{64 b^4 d}\\ &=\frac{5 (b c-a d)^3 \sqrt{a+b x} \sqrt{c+d x}}{64 b^3 d}+\frac{5 (b c-a d)^2 (a+b x)^{3/2} \sqrt{c+d x}}{32 b^3}+\frac{5 (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}{24 b^2}+\frac{(a+b x)^{3/2} (c+d x)^{5/2}}{4 b}-\frac{5 (b c-a d)^4 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{64 b^{7/2} d^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.569371, size = 191, normalized size = 1.03 \[ \frac{b \sqrt{d} \sqrt{a+b x} (c+d x) \left (-5 a^2 b d^2 (11 c+2 d x)+15 a^3 d^3+a b^2 d \left (73 c^2+36 c d x+8 d^2 x^2\right )+b^3 \left (118 c^2 d x+15 c^3+136 c d^2 x^2+48 d^3 x^3\right )\right )-15 (b c-a d)^{9/2} \sqrt{\frac{b (c+d x)}{b c-a d}} \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b c-a d}}\right )}{192 b^4 d^{3/2} \sqrt{c+d x}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.006, size = 641, normalized size = 3.5 \begin{align*} \text{result too large to display} \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 = 2.88168, size = 1203, normalized size = 6.47 \begin{align*} \left [\frac{15 \,{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \sqrt{b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} - 4 \,{\left (2 \, b d x + b c + a d\right )} \sqrt{b d} \sqrt{b x + a} \sqrt{d x + c} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right ) + 4 \,{\left (48 \, b^{4} d^{4} x^{3} + 15 \, b^{4} c^{3} d + 73 \, a b^{3} c^{2} d^{2} - 55 \, a^{2} b^{2} c d^{3} + 15 \, a^{3} b d^{4} + 8 \,{\left (17 \, b^{4} c d^{3} + a b^{3} d^{4}\right )} x^{2} + 2 \,{\left (59 \, b^{4} c^{2} d^{2} + 18 \, a b^{3} c d^{3} - 5 \, a^{2} b^{2} d^{4}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{768 \, b^{4} d^{2}}, \frac{15 \,{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \sqrt{-b d} \arctan \left (\frac{{\left (2 \, b d x + b c + a d\right )} \sqrt{-b d} \sqrt{b x + a} \sqrt{d x + c}}{2 \,{\left (b^{2} d^{2} x^{2} + a b c d +{\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) + 2 \,{\left (48 \, b^{4} d^{4} x^{3} + 15 \, b^{4} c^{3} d + 73 \, a b^{3} c^{2} d^{2} - 55 \, a^{2} b^{2} c d^{3} + 15 \, a^{3} b d^{4} + 8 \,{\left (17 \, b^{4} c d^{3} + a b^{3} d^{4}\right )} x^{2} + 2 \,{\left (59 \, b^{4} c^{2} d^{2} + 18 \, a b^{3} c d^{3} - 5 \, a^{2} b^{2} d^{4}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{384 \, b^{4} d^{2}}\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 [B] time = 1.4592, size = 852, normalized size = 4.58 \begin{align*} \frac{\frac{10 \,{\left (\sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \sqrt{b x + a}{\left (\frac{2 \,{\left (b x + a\right )}}{b^{4} d^{2}} + \frac{b c d - a d^{2}}{b^{4} d^{4}}\right )} + \frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\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 |}\right )}{\sqrt{b d} b^{3} d^{3}}\right )} c^{2}{\left | b \right |}}{b^{2}} + \frac{5 \,{\left (\sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}{\left (2 \,{\left (b x + a\right )}{\left (4 \,{\left (b x + a\right )}{\left (\frac{6 \,{\left (b x + a\right )}}{b^{2}} + \frac{b^{7} c d^{5} - 17 \, a b^{6} d^{6}}{b^{8} d^{6}}\right )} - \frac{5 \, b^{8} c^{2} d^{4} + 6 \, a b^{7} c d^{5} - 59 \, a^{2} b^{6} d^{6}}{b^{8} d^{6}}\right )} + \frac{3 \,{\left (5 \, b^{9} c^{3} d^{3} + a b^{8} c^{2} d^{4} - a^{2} b^{7} c d^{5} - 5 \, a^{3} b^{6} d^{6}\right )}}{b^{8} d^{6}}\right )} \sqrt{b x + a} + \frac{3 \,{\left (5 \, b^{4} c^{4} - 4 \, a b^{3} c^{3} d - 2 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + 5 \, a^{4} d^{4}\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 |}\right )}{\sqrt{b d} b d^{3}}\right )} d^{2}{\left | b \right |}}{b^{2}} + \frac{{\left (\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 )}}{b^{6} d^{2}} + \frac{b c d^{3} - 7 \, a d^{4}}{b^{6} d^{6}}\right )} - \frac{3 \,{\left (b^{2} c^{2} d^{2} - a^{2} d^{4}\right )}}{b^{6} d^{6}}\right )} - \frac{3 \,{\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\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 |}\right )}{\sqrt{b d} b^{5} d^{4}}\right )} c d{\left | b \right |}}{b^{3}}}{960 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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