Optimal. Leaf size=116 \[ -\frac{(b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{4 b^{3/2} d^{3/2}}+\frac{\sqrt{a+b x} \sqrt{c+d x} (b c-a d)}{4 b d}+\frac{(a+b x)^{3/2} \sqrt{c+d x}}{2 b} \]
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Rubi [A] time = 0.0682431, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {50, 63, 217, 206} \[ -\frac{(b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{4 b^{3/2} d^{3/2}}+\frac{\sqrt{a+b x} \sqrt{c+d x} (b c-a d)}{4 b d}+\frac{(a+b x)^{3/2} \sqrt{c+d x}}{2 b} \]
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} \sqrt{c+d x} \, dx &=\frac{(a+b x)^{3/2} \sqrt{c+d x}}{2 b}+\frac{(b c-a d) \int \frac{\sqrt{a+b x}}{\sqrt{c+d x}} \, dx}{4 b}\\ &=\frac{(b c-a d) \sqrt{a+b x} \sqrt{c+d x}}{4 b d}+\frac{(a+b x)^{3/2} \sqrt{c+d x}}{2 b}-\frac{(b c-a d)^2 \int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx}{8 b d}\\ &=\frac{(b c-a d) \sqrt{a+b x} \sqrt{c+d x}}{4 b d}+\frac{(a+b x)^{3/2} \sqrt{c+d x}}{2 b}-\frac{(b c-a d)^2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{c-\frac{a d}{b}+\frac{d x^2}{b}}} \, dx,x,\sqrt{a+b x}\right )}{4 b^2 d}\\ &=\frac{(b c-a d) \sqrt{a+b x} \sqrt{c+d x}}{4 b d}+\frac{(a+b x)^{3/2} \sqrt{c+d x}}{2 b}-\frac{(b c-a d)^2 \operatorname{Subst}\left (\int \frac{1}{1-\frac{d x^2}{b}} \, dx,x,\frac{\sqrt{a+b x}}{\sqrt{c+d x}}\right )}{4 b^2 d}\\ &=\frac{(b c-a d) \sqrt{a+b x} \sqrt{c+d x}}{4 b d}+\frac{(a+b x)^{3/2} \sqrt{c+d x}}{2 b}-\frac{(b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{4 b^{3/2} d^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.291796, size = 118, normalized size = 1.02 \[ \frac{b \sqrt{d} \sqrt{a+b x} (c+d x) (a d+b (c+2 d x))-(b c-a d)^{5/2} \sqrt{\frac{b (c+d x)}{b c-a d}} \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b c-a d}}\right )}{4 b^2 d^{3/2} \sqrt{c+d x}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.011, size = 305, normalized size = 2.6 \begin{align*}{\frac{1}{2\,d}\sqrt{bx+a} \left ( dx+c \right ) ^{{\frac{3}{2}}}}+{\frac{a}{4\,b}\sqrt{bx+a}\sqrt{dx+c}}-{\frac{c}{4\,d}\sqrt{bx+a}\sqrt{dx+c}}-{\frac{d{a}^{2}}{8\,b}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\ln \left ({ \left ({\frac{ad}{2}}+{\frac{bc}{2}}+bdx \right ){\frac{1}{\sqrt{bd}}}}+\sqrt{bd{x}^{2}+ \left ( ad+bc \right ) x+ac} \right ){\frac{1}{\sqrt{bx+a}}}{\frac{1}{\sqrt{dx+c}}}{\frac{1}{\sqrt{bd}}}}+{\frac{ac}{4}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\ln \left ({ \left ({\frac{ad}{2}}+{\frac{bc}{2}}+bdx \right ){\frac{1}{\sqrt{bd}}}}+\sqrt{bd{x}^{2}+ \left ( ad+bc \right ) x+ac} \right ){\frac{1}{\sqrt{bx+a}}}{\frac{1}{\sqrt{dx+c}}}{\frac{1}{\sqrt{bd}}}}-{\frac{{c}^{2}b}{8\,d}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\ln \left ({ \left ({\frac{ad}{2}}+{\frac{bc}{2}}+bdx \right ){\frac{1}{\sqrt{bd}}}}+\sqrt{bd{x}^{2}+ \left ( ad+bc \right ) x+ac} \right ){\frac{1}{\sqrt{bx+a}}}{\frac{1}{\sqrt{dx+c}}}{\frac{1}{\sqrt{bd}}}} \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 = 1.93164, size = 699, normalized size = 6.03 \begin{align*} \left [\frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\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 (2 \, b^{2} d^{2} x + b^{2} c d + a b d^{2}\right )} \sqrt{b x + a} \sqrt{d x + c}}{16 \, b^{2} d^{2}}, \frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\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 (2 \, b^{2} d^{2} x + b^{2} c d + a b d^{2}\right )} \sqrt{b x + a} \sqrt{d x + c}}{8 \, b^{2} d^{2}}\right ] \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 \sqrt{a + b x} \sqrt{c + d x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1314, size = 189, normalized size = 1.63 \begin{align*} \frac{{\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 )}{\left | b \right |}}{96 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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