Optimal. Leaf size=110 \[ \frac{5 d \sqrt{c+d x} (b c-a d)}{b^3}-\frac{5 d (b c-a d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{b^{7/2}}-\frac{(c+d x)^{5/2}}{b (a+b x)}+\frac{5 d (c+d x)^{3/2}}{3 b^2} \]
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Rubi [A] time = 0.0569942, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {47, 50, 63, 208} \[ \frac{5 d \sqrt{c+d x} (b c-a d)}{b^3}-\frac{5 d (b c-a d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{b^{7/2}}-\frac{(c+d x)^{5/2}}{b (a+b x)}+\frac{5 d (c+d x)^{3/2}}{3 b^2} \]
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{(c+d x)^{5/2}}{(a+b x)^2} \, dx &=-\frac{(c+d x)^{5/2}}{b (a+b x)}+\frac{(5 d) \int \frac{(c+d x)^{3/2}}{a+b x} \, dx}{2 b}\\ &=\frac{5 d (c+d x)^{3/2}}{3 b^2}-\frac{(c+d x)^{5/2}}{b (a+b x)}+\frac{(5 d (b c-a d)) \int \frac{\sqrt{c+d x}}{a+b x} \, dx}{2 b^2}\\ &=\frac{5 d (b c-a d) \sqrt{c+d x}}{b^3}+\frac{5 d (c+d x)^{3/2}}{3 b^2}-\frac{(c+d x)^{5/2}}{b (a+b x)}+\frac{\left (5 d (b c-a d)^2\right ) \int \frac{1}{(a+b x) \sqrt{c+d x}} \, dx}{2 b^3}\\ &=\frac{5 d (b c-a d) \sqrt{c+d x}}{b^3}+\frac{5 d (c+d x)^{3/2}}{3 b^2}-\frac{(c+d x)^{5/2}}{b (a+b x)}+\frac{\left (5 (b c-a d)^2\right ) \operatorname{Subst}\left (\int \frac{1}{a-\frac{b c}{d}+\frac{b x^2}{d}} \, dx,x,\sqrt{c+d x}\right )}{b^3}\\ &=\frac{5 d (b c-a d) \sqrt{c+d x}}{b^3}+\frac{5 d (c+d x)^{3/2}}{3 b^2}-\frac{(c+d x)^{5/2}}{b (a+b x)}-\frac{5 d (b c-a d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{b^{7/2}}\\ \end{align*}
Mathematica [C] time = 0.0144742, size = 50, normalized size = 0.45 \[ \frac{2 d (c+d x)^{7/2} \, _2F_1\left (2,\frac{7}{2};\frac{9}{2};-\frac{b (c+d x)}{a d-b c}\right )}{7 (a d-b c)^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.013, size = 258, normalized size = 2.4 \begin{align*}{\frac{2\,d}{3\,{b}^{2}} \left ( dx+c \right ) ^{{\frac{3}{2}}}}-4\,{\frac{a{d}^{2}\sqrt{dx+c}}{{b}^{3}}}+4\,{\frac{d\sqrt{dx+c}c}{{b}^{2}}}-{\frac{{a}^{2}{d}^{3}}{{b}^{3} \left ( bdx+ad \right ) }\sqrt{dx+c}}+2\,{\frac{\sqrt{dx+c}ac{d}^{2}}{{b}^{2} \left ( bdx+ad \right ) }}-{\frac{d{c}^{2}}{b \left ( bdx+ad \right ) }\sqrt{dx+c}}+5\,{\frac{{a}^{2}{d}^{3}}{{b}^{3}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{b\sqrt{dx+c}}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }-10\,{\frac{ac{d}^{2}}{{b}^{2}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{b\sqrt{dx+c}}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }+5\,{\frac{d{c}^{2}}{b\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{b\sqrt{dx+c}}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) } \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 = 2.67534, size = 707, normalized size = 6.43 \begin{align*} \left [-\frac{15 \,{\left (a b c d - a^{2} d^{2} +{\left (b^{2} c d - a b d^{2}\right )} x\right )} \sqrt{\frac{b c - a d}{b}} \log \left (\frac{b d x + 2 \, b c - a d + 2 \, \sqrt{d x + c} b \sqrt{\frac{b c - a d}{b}}}{b x + a}\right ) - 2 \,{\left (2 \, b^{2} d^{2} x^{2} - 3 \, b^{2} c^{2} + 20 \, a b c d - 15 \, a^{2} d^{2} + 2 \,{\left (7 \, b^{2} c d - 5 \, a b d^{2}\right )} x\right )} \sqrt{d x + c}}{6 \,{\left (b^{4} x + a b^{3}\right )}}, -\frac{15 \,{\left (a b c d - a^{2} d^{2} +{\left (b^{2} c d - a b d^{2}\right )} x\right )} \sqrt{-\frac{b c - a d}{b}} \arctan \left (-\frac{\sqrt{d x + c} b \sqrt{-\frac{b c - a d}{b}}}{b c - a d}\right ) -{\left (2 \, b^{2} d^{2} x^{2} - 3 \, b^{2} c^{2} + 20 \, a b c d - 15 \, a^{2} d^{2} + 2 \,{\left (7 \, b^{2} c d - 5 \, a b d^{2}\right )} x\right )} \sqrt{d x + c}}{3 \,{\left (b^{4} x + a b^{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 [A] time = 1.196, size = 244, normalized size = 2.22 \begin{align*} \frac{5 \,{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} b^{3}} - \frac{\sqrt{d x + c} b^{2} c^{2} d - 2 \, \sqrt{d x + c} a b c d^{2} + \sqrt{d x + c} a^{2} d^{3}}{{\left ({\left (d x + c\right )} b - b c + a d\right )} b^{3}} + \frac{2 \,{\left ({\left (d x + c\right )}^{\frac{3}{2}} b^{4} d + 6 \, \sqrt{d x + c} b^{4} c d - 6 \, \sqrt{d x + c} a b^{3} d^{2}\right )}}{3 \, b^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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