Optimal. Leaf size=126 \[ -\frac{5 d^2 \sqrt{c+d x}}{8 b^3 (a+b x)}-\frac{5 d^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{8 b^{7/2} \sqrt{b c-a d}}-\frac{5 d (c+d x)^{3/2}}{12 b^2 (a+b x)^2}-\frac{(c+d x)^{5/2}}{3 b (a+b x)^3} \]
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Rubi [A] time = 0.0501402, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {47, 63, 208} \[ -\frac{5 d^2 \sqrt{c+d x}}{8 b^3 (a+b x)}-\frac{5 d^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{8 b^{7/2} \sqrt{b c-a d}}-\frac{5 d (c+d x)^{3/2}}{12 b^2 (a+b x)^2}-\frac{(c+d x)^{5/2}}{3 b (a+b x)^3} \]
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{(c+d x)^{5/2}}{(a+b x)^4} \, dx &=-\frac{(c+d x)^{5/2}}{3 b (a+b x)^3}+\frac{(5 d) \int \frac{(c+d x)^{3/2}}{(a+b x)^3} \, dx}{6 b}\\ &=-\frac{5 d (c+d x)^{3/2}}{12 b^2 (a+b x)^2}-\frac{(c+d x)^{5/2}}{3 b (a+b x)^3}+\frac{\left (5 d^2\right ) \int \frac{\sqrt{c+d x}}{(a+b x)^2} \, dx}{8 b^2}\\ &=-\frac{5 d^2 \sqrt{c+d x}}{8 b^3 (a+b x)}-\frac{5 d (c+d x)^{3/2}}{12 b^2 (a+b x)^2}-\frac{(c+d x)^{5/2}}{3 b (a+b x)^3}+\frac{\left (5 d^3\right ) \int \frac{1}{(a+b x) \sqrt{c+d x}} \, dx}{16 b^3}\\ &=-\frac{5 d^2 \sqrt{c+d x}}{8 b^3 (a+b x)}-\frac{5 d (c+d x)^{3/2}}{12 b^2 (a+b x)^2}-\frac{(c+d x)^{5/2}}{3 b (a+b x)^3}+\frac{\left (5 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 )}{8 b^3}\\ &=-\frac{5 d^2 \sqrt{c+d x}}{8 b^3 (a+b x)}-\frac{5 d (c+d x)^{3/2}}{12 b^2 (a+b x)^2}-\frac{(c+d x)^{5/2}}{3 b (a+b x)^3}-\frac{5 d^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{8 b^{7/2} \sqrt{b c-a d}}\\ \end{align*}
Mathematica [A] time = 0.147758, size = 119, normalized size = 0.94 \[ \frac{5 d^3 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{a d-b c}}\right )}{8 b^{7/2} \sqrt{a d-b c}}-\frac{\sqrt{c+d x} \left (15 a^2 d^2+10 a b d (c+4 d x)+b^2 \left (8 c^2+26 c d x+33 d^2 x^2\right )\right )}{24 b^3 (a+b x)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 204, normalized size = 1.6 \begin{align*} -{\frac{11\,{d}^{3}}{8\, \left ( bdx+ad \right ) ^{3}b} \left ( dx+c \right ) ^{{\frac{5}{2}}}}-{\frac{5\,{d}^{4}a}{3\, \left ( bdx+ad \right ) ^{3}{b}^{2}} \left ( dx+c \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{d}^{3}c}{3\, \left ( bdx+ad \right ) ^{3}b} \left ( dx+c \right ) ^{{\frac{3}{2}}}}-{\frac{5\,{d}^{5}{a}^{2}}{8\, \left ( bdx+ad \right ) ^{3}{b}^{3}}\sqrt{dx+c}}+{\frac{5\,{d}^{4}ac}{4\, \left ( bdx+ad \right ) ^{3}{b}^{2}}\sqrt{dx+c}}-{\frac{5\,{d}^{3}{c}^{2}}{8\, \left ( bdx+ad \right ) ^{3}b}\sqrt{dx+c}}+{\frac{5\,{d}^{3}}{8\,{b}^{3}}\arctan \left ({b\sqrt{dx+c}{\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}} \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 = 1.91429, size = 1162, normalized size = 9.22 \begin{align*} \left [\frac{15 \,{\left (b^{3} d^{3} x^{3} + 3 \, a b^{2} d^{3} x^{2} + 3 \, a^{2} b d^{3} x + a^{3} d^{3}\right )} \sqrt{b^{2} c - a b d} \log \left (\frac{b d x + 2 \, b c - a d - 2 \, \sqrt{b^{2} c - a b d} \sqrt{d x + c}}{b x + a}\right ) - 2 \,{\left (8 \, b^{4} c^{3} + 2 \, a b^{3} c^{2} d + 5 \, a^{2} b^{2} c d^{2} - 15 \, a^{3} b d^{3} + 33 \,{\left (b^{4} c d^{2} - a b^{3} d^{3}\right )} x^{2} + 2 \,{\left (13 \, b^{4} c^{2} d + 7 \, a b^{3} c d^{2} - 20 \, a^{2} b^{2} d^{3}\right )} x\right )} \sqrt{d x + c}}{48 \,{\left (a^{3} b^{5} c - a^{4} b^{4} d +{\left (b^{8} c - a b^{7} d\right )} x^{3} + 3 \,{\left (a b^{7} c - a^{2} b^{6} d\right )} x^{2} + 3 \,{\left (a^{2} b^{6} c - a^{3} b^{5} d\right )} x\right )}}, \frac{15 \,{\left (b^{3} d^{3} x^{3} + 3 \, a b^{2} d^{3} x^{2} + 3 \, a^{2} b d^{3} x + a^{3} d^{3}\right )} \sqrt{-b^{2} c + a b d} \arctan \left (\frac{\sqrt{-b^{2} c + a b d} \sqrt{d x + c}}{b d x + b c}\right ) -{\left (8 \, b^{4} c^{3} + 2 \, a b^{3} c^{2} d + 5 \, a^{2} b^{2} c d^{2} - 15 \, a^{3} b d^{3} + 33 \,{\left (b^{4} c d^{2} - a b^{3} d^{3}\right )} x^{2} + 2 \,{\left (13 \, b^{4} c^{2} d + 7 \, a b^{3} c d^{2} - 20 \, a^{2} b^{2} d^{3}\right )} x\right )} \sqrt{d x + c}}{24 \,{\left (a^{3} b^{5} c - a^{4} b^{4} d +{\left (b^{8} c - a b^{7} d\right )} x^{3} + 3 \,{\left (a b^{7} c - a^{2} b^{6} d\right )} x^{2} + 3 \,{\left (a^{2} b^{6} c - a^{3} b^{5} d\right )} x\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 = 1.0882, size = 217, normalized size = 1.72 \begin{align*} \frac{5 \, d^{3} \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{8 \, \sqrt{-b^{2} c + a b d} b^{3}} - \frac{33 \,{\left (d x + c\right )}^{\frac{5}{2}} b^{2} d^{3} - 40 \,{\left (d x + c\right )}^{\frac{3}{2}} b^{2} c d^{3} + 15 \, \sqrt{d x + c} b^{2} c^{2} d^{3} + 40 \,{\left (d x + c\right )}^{\frac{3}{2}} a b d^{4} - 30 \, \sqrt{d x + c} a b c d^{4} + 15 \, \sqrt{d x + c} a^{2} d^{5}}{24 \,{\left ({\left (d x + c\right )} b - b c + a d\right )}^{3} b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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