Optimal. Leaf size=154 \[ \frac{(b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 b^{3/2} d^{5/2}}-\frac{\sqrt{a+b x} \sqrt{c+d x} (b c-a d)^2}{8 b d^2}+\frac{(a+b x)^{3/2} \sqrt{c+d x} (b c-a d)}{12 b d}+\frac{(a+b x)^{5/2} \sqrt{c+d x}}{3 b} \]
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Rubi [A] time = 0.0740076, antiderivative size = 154, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {50, 63, 217, 206} \[ \frac{(b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 b^{3/2} d^{5/2}}-\frac{\sqrt{a+b x} \sqrt{c+d x} (b c-a d)^2}{8 b d^2}+\frac{(a+b x)^{3/2} \sqrt{c+d x} (b c-a d)}{12 b d}+\frac{(a+b x)^{5/2} \sqrt{c+d x}}{3 b} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 217
Rule 206
Rubi steps
\begin{align*} \int (a+b x)^{3/2} \sqrt{c+d x} \, dx &=\frac{(a+b x)^{5/2} \sqrt{c+d x}}{3 b}+\frac{(b c-a d) \int \frac{(a+b x)^{3/2}}{\sqrt{c+d x}} \, dx}{6 b}\\ &=\frac{(b c-a d) (a+b x)^{3/2} \sqrt{c+d x}}{12 b d}+\frac{(a+b x)^{5/2} \sqrt{c+d x}}{3 b}-\frac{(b c-a d)^2 \int \frac{\sqrt{a+b x}}{\sqrt{c+d x}} \, dx}{8 b d}\\ &=-\frac{(b c-a d)^2 \sqrt{a+b x} \sqrt{c+d x}}{8 b d^2}+\frac{(b c-a d) (a+b x)^{3/2} \sqrt{c+d x}}{12 b d}+\frac{(a+b x)^{5/2} \sqrt{c+d x}}{3 b}+\frac{(b c-a d)^3 \int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx}{16 b d^2}\\ &=-\frac{(b c-a d)^2 \sqrt{a+b x} \sqrt{c+d x}}{8 b d^2}+\frac{(b c-a d) (a+b x)^{3/2} \sqrt{c+d x}}{12 b d}+\frac{(a+b x)^{5/2} \sqrt{c+d x}}{3 b}+\frac{(b c-a d)^3 \operatorname{Subst}\left (\int \frac{1}{\sqrt{c-\frac{a d}{b}+\frac{d x^2}{b}}} \, dx,x,\sqrt{a+b x}\right )}{8 b^2 d^2}\\ &=-\frac{(b c-a d)^2 \sqrt{a+b x} \sqrt{c+d x}}{8 b d^2}+\frac{(b c-a d) (a+b x)^{3/2} \sqrt{c+d x}}{12 b d}+\frac{(a+b x)^{5/2} \sqrt{c+d x}}{3 b}+\frac{(b c-a d)^3 \operatorname{Subst}\left (\int \frac{1}{1-\frac{d x^2}{b}} \, dx,x,\frac{\sqrt{a+b x}}{\sqrt{c+d x}}\right )}{8 b^2 d^2}\\ &=-\frac{(b c-a d)^2 \sqrt{a+b x} \sqrt{c+d x}}{8 b d^2}+\frac{(b c-a d) (a+b x)^{3/2} \sqrt{c+d x}}{12 b d}+\frac{(a+b x)^{5/2} \sqrt{c+d x}}{3 b}+\frac{(b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 b^{3/2} d^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.424652, size = 151, normalized size = 0.98 \[ \frac{b \sqrt{d} \sqrt{a+b x} (c+d x) \left (3 a^2 d^2+2 a b d (4 c+7 d x)+b^2 \left (-3 c^2+2 c d x+8 d^2 x^2\right )\right )+3 (b c-a d)^{7/2} \sqrt{\frac{b (c+d x)}{b c-a d}} \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b c-a d}}\right )}{24 b^2 d^{5/2} \sqrt{c+d x}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.006, size = 460, normalized size = 3. \begin{align*}{\frac{1}{3\,d} \left ( bx+a \right ) ^{{\frac{3}{2}}} \left ( dx+c \right ) ^{{\frac{3}{2}}}}+{\frac{a}{4\,d}\sqrt{bx+a} \left ( dx+c \right ) ^{{\frac{3}{2}}}}-{\frac{bc}{4\,{d}^{2}}\sqrt{bx+a} \left ( dx+c \right ) ^{{\frac{3}{2}}}}+{\frac{{a}^{2}}{8\,b}\sqrt{bx+a}\sqrt{dx+c}}-{\frac{ac}{4\,d}\sqrt{bx+a}\sqrt{dx+c}}+{\frac{{c}^{2}b}{8\,{d}^{2}}\sqrt{bx+a}\sqrt{dx+c}}-{\frac{d{a}^{3}}{16\,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{d{x}^{2}b+ \left ( ad+bc \right ) x+ac} \right ){\frac{1}{\sqrt{bx+a}}}{\frac{1}{\sqrt{dx+c}}}{\frac{1}{\sqrt{bd}}}}+{\frac{3\,{a}^{2}c}{16}\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{d{x}^{2}b+ \left ( ad+bc \right ) x+ac} \right ){\frac{1}{\sqrt{bx+a}}}{\frac{1}{\sqrt{dx+c}}}{\frac{1}{\sqrt{bd}}}}-{\frac{3\,a{c}^{2}b}{16\,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{d{x}^{2}b+ \left ( ad+bc \right ) x+ac} \right ){\frac{1}{\sqrt{bx+a}}}{\frac{1}{\sqrt{dx+c}}}{\frac{1}{\sqrt{bd}}}}+{\frac{{c}^{3}{b}^{2}}{16\,{d}^{2}}\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{d{x}^{2}b+ \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 = 2.22837, size = 919, normalized size = 5.97 \begin{align*} \left [-\frac{3 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\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 (8 \, b^{3} d^{3} x^{2} - 3 \, b^{3} c^{2} d + 8 \, a b^{2} c d^{2} + 3 \, a^{2} b d^{3} + 2 \,{\left (b^{3} c d^{2} + 7 \, a b^{2} d^{3}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{96 \, b^{2} d^{3}}, -\frac{3 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\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 (8 \, b^{3} d^{3} x^{2} - 3 \, b^{3} c^{2} d + 8 \, a b^{2} c d^{2} + 3 \, a^{2} b d^{3} + 2 \,{\left (b^{3} c d^{2} + 7 \, a b^{2} d^{3}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{48 \, b^{2} d^{3}}\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 = 1.35242, size = 456, normalized size = 2.96 \begin{align*} \frac{\frac{20 \,{\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 )} a{\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 )}{\left | b \right |}}{b^{2}}}{1920 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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