Optimal. Leaf size=128 \[ \frac{5 b^2 \sqrt{a+b x} \sqrt{c+d x}}{d^3}-\frac{5 b^{3/2} (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{d^{7/2}}-\frac{10 b (a+b x)^{3/2}}{3 d^2 \sqrt{c+d x}}-\frac{2 (a+b x)^{5/2}}{3 d (c+d x)^{3/2}} \]
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Rubi [A] time = 0.0667664, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 6, 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} \[ \frac{5 b^2 \sqrt{a+b x} \sqrt{c+d x}}{d^3}-\frac{5 b^{3/2} (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{d^{7/2}}-\frac{10 b (a+b x)^{3/2}}{3 d^2 \sqrt{c+d x}}-\frac{2 (a+b x)^{5/2}}{3 d (c+d x)^{3/2}} \]
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)^{5/2}}{(c+d x)^{5/2}} \, dx &=-\frac{2 (a+b x)^{5/2}}{3 d (c+d x)^{3/2}}+\frac{(5 b) \int \frac{(a+b x)^{3/2}}{(c+d x)^{3/2}} \, dx}{3 d}\\ &=-\frac{2 (a+b x)^{5/2}}{3 d (c+d x)^{3/2}}-\frac{10 b (a+b x)^{3/2}}{3 d^2 \sqrt{c+d x}}+\frac{\left (5 b^2\right ) \int \frac{\sqrt{a+b x}}{\sqrt{c+d x}} \, dx}{d^2}\\ &=-\frac{2 (a+b x)^{5/2}}{3 d (c+d x)^{3/2}}-\frac{10 b (a+b x)^{3/2}}{3 d^2 \sqrt{c+d x}}+\frac{5 b^2 \sqrt{a+b x} \sqrt{c+d x}}{d^3}-\frac{\left (5 b^2 (b c-a d)\right ) \int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx}{2 d^3}\\ &=-\frac{2 (a+b x)^{5/2}}{3 d (c+d x)^{3/2}}-\frac{10 b (a+b x)^{3/2}}{3 d^2 \sqrt{c+d x}}+\frac{5 b^2 \sqrt{a+b x} \sqrt{c+d x}}{d^3}-\frac{(5 b (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^3}\\ &=-\frac{2 (a+b x)^{5/2}}{3 d (c+d x)^{3/2}}-\frac{10 b (a+b x)^{3/2}}{3 d^2 \sqrt{c+d x}}+\frac{5 b^2 \sqrt{a+b x} \sqrt{c+d x}}{d^3}-\frac{(5 b (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^3}\\ &=-\frac{2 (a+b x)^{5/2}}{3 d (c+d x)^{3/2}}-\frac{10 b (a+b x)^{3/2}}{3 d^2 \sqrt{c+d x}}+\frac{5 b^2 \sqrt{a+b x} \sqrt{c+d x}}{d^3}-\frac{5 b^{3/2} (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{d^{7/2}}\\ \end{align*}
Mathematica [C] time = 0.0704367, size = 73, normalized size = 0.57 \[ \frac{2 (a+b x)^{7/2} \left (\frac{b (c+d x)}{b c-a d}\right )^{5/2} \, _2F_1\left (\frac{5}{2},\frac{7}{2};\frac{9}{2};\frac{d (a+b x)}{a d-b c}\right )}{7 b (c+d x)^{5/2}} \]
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{5}{2}}} \left ( dx+c \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 = 5.77857, size = 1033, normalized size = 8.07 \begin{align*} \left [-\frac{15 \,{\left (b^{2} c^{3} - a b c^{2} d +{\left (b^{2} c d^{2} - a b d^{3}\right )} x^{2} + 2 \,{\left (b^{2} c^{2} d - a b c 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 (3 \, b^{2} d^{2} x^{2} + 15 \, b^{2} c^{2} - 10 \, a b c d - 2 \, a^{2} d^{2} + 2 \,{\left (10 \, b^{2} c d - 7 \, a b d^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{12 \,{\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}}, \frac{15 \,{\left (b^{2} c^{3} - a b c^{2} d +{\left (b^{2} c d^{2} - a b d^{3}\right )} x^{2} + 2 \,{\left (b^{2} c^{2} d - a b c 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 (3 \, b^{2} d^{2} x^{2} + 15 \, b^{2} c^{2} - 10 \, a b c d - 2 \, a^{2} d^{2} + 2 \,{\left (10 \, b^{2} c d - 7 \, a b d^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{6 \,{\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}}\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 = 2.32651, size = 373, normalized size = 2.91 \begin{align*} \frac{{\left ({\left (b x + a\right )}{\left (\frac{3 \,{\left (b^{6} c d^{4} - a b^{5} d^{5}\right )}{\left (b x + a\right )}}{b^{2} c d^{5}{\left | b \right |} - a b d^{6}{\left | b \right |}} + \frac{20 \,{\left (b^{7} c^{2} d^{3} - 2 \, a b^{6} c d^{4} + a^{2} b^{5} d^{5}\right )}}{b^{2} c d^{5}{\left | b \right |} - a b d^{6}{\left | b \right |}}\right )} + \frac{15 \,{\left (b^{8} c^{3} d^{2} - 3 \, a b^{7} c^{2} d^{3} + 3 \, a^{2} b^{6} c d^{4} - a^{3} b^{5} d^{5}\right )}}{b^{2} c d^{5}{\left | b \right |} - a b d^{6}{\left | b \right |}}\right )} \sqrt{b x + a}}{3 \,{\left (b^{2} c +{\left (b x + a\right )} b d - a b d\right )}^{\frac{3}{2}}} + \frac{5 \,{\left (b^{4} c - a b^{3} d\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} d^{3}{\left | b \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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