Optimal. Leaf size=146 \[ -\frac{d^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{8 b^{3/2} (b c-a d)^{5/2}}+\frac{d^2 \sqrt{c+d x}}{8 b (a+b x) (b c-a d)^2}-\frac{d \sqrt{c+d x}}{12 b (a+b x)^2 (b c-a d)}-\frac{\sqrt{c+d x}}{3 b (a+b x)^3} \]
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Rubi [A] time = 0.0978502, antiderivative size = 146, 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, 51, 63, 208} \[ -\frac{d^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{8 b^{3/2} (b c-a d)^{5/2}}+\frac{d^2 \sqrt{c+d x}}{8 b (a+b x) (b c-a d)^2}-\frac{d \sqrt{c+d x}}{12 b (a+b x)^2 (b c-a d)}-\frac{\sqrt{c+d x}}{3 b (a+b x)^3} \]
Antiderivative was successfully verified.
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Rule 47
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\sqrt{c+d x}}{(a+b x)^4} \, dx &=-\frac{\sqrt{c+d x}}{3 b (a+b x)^3}+\frac{d \int \frac{1}{(a+b x)^3 \sqrt{c+d x}} \, dx}{6 b}\\ &=-\frac{\sqrt{c+d x}}{3 b (a+b x)^3}-\frac{d \sqrt{c+d x}}{12 b (b c-a d) (a+b x)^2}-\frac{d^2 \int \frac{1}{(a+b x)^2 \sqrt{c+d x}} \, dx}{8 b (b c-a d)}\\ &=-\frac{\sqrt{c+d x}}{3 b (a+b x)^3}-\frac{d \sqrt{c+d x}}{12 b (b c-a d) (a+b x)^2}+\frac{d^2 \sqrt{c+d x}}{8 b (b c-a d)^2 (a+b x)}+\frac{d^3 \int \frac{1}{(a+b x) \sqrt{c+d x}} \, dx}{16 b (b c-a d)^2}\\ &=-\frac{\sqrt{c+d x}}{3 b (a+b x)^3}-\frac{d \sqrt{c+d x}}{12 b (b c-a d) (a+b x)^2}+\frac{d^2 \sqrt{c+d x}}{8 b (b c-a d)^2 (a+b x)}+\frac{d^2 \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 (b c-a d)^2}\\ &=-\frac{\sqrt{c+d x}}{3 b (a+b x)^3}-\frac{d \sqrt{c+d x}}{12 b (b c-a d) (a+b x)^2}+\frac{d^2 \sqrt{c+d x}}{8 b (b c-a d)^2 (a+b x)}-\frac{d^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{8 b^{3/2} (b c-a d)^{5/2}}\\ \end{align*}
Mathematica [C] time = 0.0133593, size = 52, normalized size = 0.36 \[ \frac{2 d^3 (c+d x)^{3/2} \, _2F_1\left (\frac{3}{2},4;\frac{5}{2};-\frac{b (c+d x)}{a d-b c}\right )}{3 (a d-b c)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 170, normalized size = 1.2 \begin{align*}{\frac{{d}^{3}b}{8\, \left ( bdx+ad \right ) ^{3} \left ({a}^{2}{d}^{2}-2\,abcd+{b}^{2}{c}^{2} \right ) } \left ( dx+c \right ) ^{{\frac{5}{2}}}}+{\frac{{d}^{3}}{3\, \left ( bdx+ad \right ) ^{3} \left ( ad-bc \right ) } \left ( dx+c \right ) ^{{\frac{3}{2}}}}-{\frac{{d}^{3}}{8\, \left ( bdx+ad \right ) ^{3}b}\sqrt{dx+c}}+{\frac{{d}^{3}}{8\,b \left ({a}^{2}{d}^{2}-2\,abcd+{b}^{2}{c}^{2} \right ) }\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 = 2.27726, size = 1581, normalized size = 10.83 \begin{align*} \left [\frac{3 \,{\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} - 22 \, a b^{3} c^{2} d + 17 \, a^{2} b^{2} c d^{2} - 3 \, a^{3} b d^{3} - 3 \,{\left (b^{4} c d^{2} - a b^{3} d^{3}\right )} x^{2} + 2 \,{\left (b^{4} c^{2} d - 5 \, a b^{3} c d^{2} + 4 \, a^{2} b^{2} d^{3}\right )} x\right )} \sqrt{d x + c}}{48 \,{\left (a^{3} b^{5} c^{3} - 3 \, a^{4} b^{4} c^{2} d + 3 \, a^{5} b^{3} c d^{2} - a^{6} b^{2} d^{3} +{\left (b^{8} c^{3} - 3 \, a b^{7} c^{2} d + 3 \, a^{2} b^{6} c d^{2} - a^{3} b^{5} d^{3}\right )} x^{3} + 3 \,{\left (a b^{7} c^{3} - 3 \, a^{2} b^{6} c^{2} d + 3 \, a^{3} b^{5} c d^{2} - a^{4} b^{4} d^{3}\right )} x^{2} + 3 \,{\left (a^{2} b^{6} c^{3} - 3 \, a^{3} b^{5} c^{2} d + 3 \, a^{4} b^{4} c d^{2} - a^{5} b^{3} d^{3}\right )} x\right )}}, \frac{3 \,{\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} - 22 \, a b^{3} c^{2} d + 17 \, a^{2} b^{2} c d^{2} - 3 \, a^{3} b d^{3} - 3 \,{\left (b^{4} c d^{2} - a b^{3} d^{3}\right )} x^{2} + 2 \,{\left (b^{4} c^{2} d - 5 \, a b^{3} c d^{2} + 4 \, a^{2} b^{2} d^{3}\right )} x\right )} \sqrt{d x + c}}{24 \,{\left (a^{3} b^{5} c^{3} - 3 \, a^{4} b^{4} c^{2} d + 3 \, a^{5} b^{3} c d^{2} - a^{6} b^{2} d^{3} +{\left (b^{8} c^{3} - 3 \, a b^{7} c^{2} d + 3 \, a^{2} b^{6} c d^{2} - a^{3} b^{5} d^{3}\right )} x^{3} + 3 \,{\left (a b^{7} c^{3} - 3 \, a^{2} b^{6} c^{2} d + 3 \, a^{3} b^{5} c d^{2} - a^{4} b^{4} d^{3}\right )} x^{2} + 3 \,{\left (a^{2} b^{6} c^{3} - 3 \, a^{3} b^{5} c^{2} d + 3 \, a^{4} b^{4} c d^{2} - a^{5} b^{3} d^{3}\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 = 1.08041, size = 279, normalized size = 1.91 \begin{align*} \frac{d^{3} \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{8 \,{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} \sqrt{-b^{2} c + a b d}} + \frac{3 \,{\left (d x + c\right )}^{\frac{5}{2}} b^{2} d^{3} - 8 \,{\left (d x + c\right )}^{\frac{3}{2}} b^{2} c d^{3} - 3 \, \sqrt{d x + c} b^{2} c^{2} d^{3} + 8 \,{\left (d x + c\right )}^{\frac{3}{2}} a b d^{4} + 6 \, \sqrt{d x + c} a b c d^{4} - 3 \, \sqrt{d x + c} a^{2} d^{5}}{24 \,{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )}{\left ({\left (d x + c\right )} b - b c + a d\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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