Optimal. Leaf size=92 \[ \frac{2 d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{b^{5/2}}-\frac{2 d \sqrt{c+d x}}{b^2 \sqrt{a+b x}}-\frac{2 (c+d x)^{3/2}}{3 b (a+b x)^{3/2}} \]
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Rubi [A] time = 0.0397725, antiderivative size = 92, 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 = {47, 63, 217, 206} \[ \frac{2 d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{b^{5/2}}-\frac{2 d \sqrt{c+d x}}{b^2 \sqrt{a+b x}}-\frac{2 (c+d x)^{3/2}}{3 b (a+b x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{(c+d x)^{3/2}}{(a+b x)^{5/2}} \, dx &=-\frac{2 (c+d x)^{3/2}}{3 b (a+b x)^{3/2}}+\frac{d \int \frac{\sqrt{c+d x}}{(a+b x)^{3/2}} \, dx}{b}\\ &=-\frac{2 d \sqrt{c+d x}}{b^2 \sqrt{a+b x}}-\frac{2 (c+d x)^{3/2}}{3 b (a+b x)^{3/2}}+\frac{d^2 \int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx}{b^2}\\ &=-\frac{2 d \sqrt{c+d x}}{b^2 \sqrt{a+b x}}-\frac{2 (c+d x)^{3/2}}{3 b (a+b x)^{3/2}}+\frac{\left (2 d^2\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 )}{b^3}\\ &=-\frac{2 d \sqrt{c+d x}}{b^2 \sqrt{a+b x}}-\frac{2 (c+d x)^{3/2}}{3 b (a+b x)^{3/2}}+\frac{\left (2 d^2\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 )}{b^3}\\ &=-\frac{2 d \sqrt{c+d x}}{b^2 \sqrt{a+b x}}-\frac{2 (c+d x)^{3/2}}{3 b (a+b x)^{3/2}}+\frac{2 d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{b^{5/2}}\\ \end{align*}
Mathematica [C] time = 0.0446405, size = 73, normalized size = 0.79 \[ -\frac{2 (c+d x)^{3/2} \, _2F_1\left (-\frac{3}{2},-\frac{3}{2};-\frac{1}{2};\frac{d (a+b x)}{a d-b c}\right )}{3 b (a+b x)^{3/2} \left (\frac{b (c+d x)}{b c-a d}\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.033, size = 0, normalized size = 0. \begin{align*} \int{ \left ( dx+c \right ) ^{{\frac{3}{2}}} \left ( bx+a \right ) ^{-{\frac{5}{2}}}}\, dx \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 [B] time = 3.5806, size = 734, normalized size = 7.98 \begin{align*} \left [\frac{3 \,{\left (b^{2} d x^{2} + 2 \, a b d x + a^{2} d\right )} \sqrt{\frac{d}{b}} \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^{2} d x + b^{2} c + a b d\right )} \sqrt{b x + a} \sqrt{d x + c} \sqrt{\frac{d}{b}} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right ) - 4 \,{\left (4 \, b d x + b c + 3 \, a d\right )} \sqrt{b x + a} \sqrt{d x + c}}{6 \,{\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )}}, -\frac{3 \,{\left (b^{2} d x^{2} + 2 \, a b d x + a^{2} d\right )} \sqrt{-\frac{d}{b}} \arctan \left (\frac{{\left (2 \, b d x + b c + a d\right )} \sqrt{b x + a} \sqrt{d x + c} \sqrt{-\frac{d}{b}}}{2 \,{\left (b d^{2} x^{2} + a c d +{\left (b c d + a d^{2}\right )} x\right )}}\right ) + 2 \,{\left (4 \, b d x + b c + 3 \, a d\right )} \sqrt{b x + a} \sqrt{d x + c}}{3 \,{\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )}}\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 \frac{\left (c + d x\right )^{\frac{3}{2}}}{\left (a + b x\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.30569, size = 614, normalized size = 6.67 \begin{align*} -\frac{\sqrt{b d} d{\left | b \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 )}^{2}\right )}{b^{4}} - \frac{8 \,{\left (2 \, \sqrt{b d} b^{5} c^{3} d{\left | b \right |} - 6 \, \sqrt{b d} a b^{4} c^{2} d^{2}{\left | b \right |} + 6 \, \sqrt{b d} a^{2} b^{3} c d^{3}{\left | b \right |} - 2 \, \sqrt{b d} a^{3} b^{2} d^{4}{\left | b \right |} - 3 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2} b^{3} c^{2} d{\left | b \right |} + 6 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2} a b^{2} c d^{2}{\left | b \right |} - 3 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2} a^{2} b d^{3}{\left | b \right |} + 3 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{4} b c d{\left | b \right |} - 3 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{4} a d^{2}{\left | b \right |}\right )}}{3 \,{\left (b^{2} c - a b d -{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}^{3} b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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