Optimal. Leaf size=98 \[ \frac{3 b \sqrt{a+b x} \sqrt{c+d x}}{d^2}-\frac{3 \sqrt{b} (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{d^{5/2}}-\frac{2 (a+b x)^{3/2}}{d \sqrt{c+d x}} \]
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Rubi [A] time = 0.0488385, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {47, 50, 63, 217, 206} \[ \frac{3 b \sqrt{a+b x} \sqrt{c+d x}}{d^2}-\frac{3 \sqrt{b} (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{d^{5/2}}-\frac{2 (a+b x)^{3/2}}{d \sqrt{c+d x}} \]
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 63
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{(a+b x)^{3/2}}{(c+d x)^{3/2}} \, dx &=-\frac{2 (a+b x)^{3/2}}{d \sqrt{c+d x}}+\frac{(3 b) \int \frac{\sqrt{a+b x}}{\sqrt{c+d x}} \, dx}{d}\\ &=-\frac{2 (a+b x)^{3/2}}{d \sqrt{c+d x}}+\frac{3 b \sqrt{a+b x} \sqrt{c+d x}}{d^2}-\frac{(3 b (b c-a d)) \int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx}{2 d^2}\\ &=-\frac{2 (a+b x)^{3/2}}{d \sqrt{c+d x}}+\frac{3 b \sqrt{a+b x} \sqrt{c+d x}}{d^2}-\frac{(3 (b c-a d)) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c-\frac{a d}{b}+\frac{d x^2}{b}}} \, dx,x,\sqrt{a+b x}\right )}{d^2}\\ &=-\frac{2 (a+b x)^{3/2}}{d \sqrt{c+d x}}+\frac{3 b \sqrt{a+b x} \sqrt{c+d x}}{d^2}-\frac{(3 (b c-a d)) \operatorname{Subst}\left (\int \frac{1}{1-\frac{d x^2}{b}} \, dx,x,\frac{\sqrt{a+b x}}{\sqrt{c+d x}}\right )}{d^2}\\ &=-\frac{2 (a+b x)^{3/2}}{d \sqrt{c+d x}}+\frac{3 b \sqrt{a+b x} \sqrt{c+d x}}{d^2}-\frac{3 \sqrt{b} (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{d^{5/2}}\\ \end{align*}
Mathematica [C] time = 0.0485602, size = 73, normalized size = 0.74 \[ \frac{2 (a+b x)^{5/2} \left (\frac{b (c+d x)}{b c-a d}\right )^{3/2} \, _2F_1\left (\frac{3}{2},\frac{5}{2};\frac{7}{2};\frac{d (a+b x)}{a d-b c}\right )}{5 b (c+d x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{ \left ( bx+a \right ) ^{{\frac{3}{2}}} \left ( dx+c \right ) ^{-{\frac{3}{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 [A] time = 3.1339, size = 702, normalized size = 7.16 \begin{align*} \left [-\frac{3 \,{\left (b c^{2} - a c d +{\left (b c d - a d^{2}\right )} x\right )} \sqrt{\frac{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^{2} x + b c d + a d^{2}\right )} \sqrt{b x + a} \sqrt{d x + c} \sqrt{\frac{b}{d}} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right ) - 4 \,{\left (b d x + 3 \, b c - 2 \, a d\right )} \sqrt{b x + a} \sqrt{d x + c}}{4 \,{\left (d^{3} x + c d^{2}\right )}}, \frac{3 \,{\left (b c^{2} - a c d +{\left (b c d - a d^{2}\right )} x\right )} \sqrt{-\frac{b}{d}} \arctan \left (\frac{{\left (2 \, b d x + b c + a d\right )} \sqrt{b x + a} \sqrt{d x + c} \sqrt{-\frac{b}{d}}}{2 \,{\left (b^{2} d x^{2} + a b c +{\left (b^{2} c + a b d\right )} x\right )}}\right ) + 2 \,{\left (b d x + 3 \, b c - 2 \, a d\right )} \sqrt{b x + a} \sqrt{d x + c}}{2 \,{\left (d^{3} x + c d^{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 (a + b x\right )^{\frac{3}{2}}}{\left (c + d x\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.34742, size = 207, normalized size = 2.11 \begin{align*} \frac{{\left (\frac{{\left (b x + a\right )} b^{2} d^{2}}{b^{6} c d^{4} - a b^{5} d^{5}} + \frac{3 \,{\left (b^{3} c d - a b^{2} d^{2}\right )}}{b^{6} c d^{4} - a b^{5} d^{5}}\right )} \sqrt{b x + a}}{32 \, \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}} + \frac{3 \, \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 )}{32 \, \sqrt{b d} b^{3} d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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