Optimal. Leaf size=114 \[ -\frac{3 d^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{4 \sqrt{b} (b c-a d)^{5/2}}+\frac{3 d \sqrt{c+d x}}{4 (a+b x) (b c-a d)^2}-\frac{\sqrt{c+d x}}{2 (a+b x)^2 (b c-a d)} \]
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Rubi [A] time = 0.0372377, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {51, 63, 208} \[ -\frac{3 d^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{4 \sqrt{b} (b c-a d)^{5/2}}+\frac{3 d \sqrt{c+d x}}{4 (a+b x) (b c-a d)^2}-\frac{\sqrt{c+d x}}{2 (a+b x)^2 (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{(a+b x)^3 \sqrt{c+d x}} \, dx &=-\frac{\sqrt{c+d x}}{2 (b c-a d) (a+b x)^2}-\frac{(3 d) \int \frac{1}{(a+b x)^2 \sqrt{c+d x}} \, dx}{4 (b c-a d)}\\ &=-\frac{\sqrt{c+d x}}{2 (b c-a d) (a+b x)^2}+\frac{3 d \sqrt{c+d x}}{4 (b c-a d)^2 (a+b x)}+\frac{\left (3 d^2\right ) \int \frac{1}{(a+b x) \sqrt{c+d x}} \, dx}{8 (b c-a d)^2}\\ &=-\frac{\sqrt{c+d x}}{2 (b c-a d) (a+b x)^2}+\frac{3 d \sqrt{c+d x}}{4 (b c-a d)^2 (a+b x)}+\frac{(3 d) \operatorname{Subst}\left (\int \frac{1}{a-\frac{b c}{d}+\frac{b x^2}{d}} \, dx,x,\sqrt{c+d x}\right )}{4 (b c-a d)^2}\\ &=-\frac{\sqrt{c+d x}}{2 (b c-a d) (a+b x)^2}+\frac{3 d \sqrt{c+d x}}{4 (b c-a d)^2 (a+b x)}-\frac{3 d^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{4 \sqrt{b} (b c-a d)^{5/2}}\\ \end{align*}
Mathematica [C] time = 0.0110318, size = 50, normalized size = 0.44 \[ \frac{2 d^2 \sqrt{c+d x} \, _2F_1\left (\frac{1}{2},3;\frac{3}{2};-\frac{b (c+d x)}{a d-b c}\right )}{(a d-b c)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 115, normalized size = 1. \begin{align*}{\frac{{d}^{2}}{ \left ( 2\,ad-2\,bc \right ) \left ( bdx+ad \right ) ^{2}}\sqrt{dx+c}}+{\frac{3\,{d}^{2}}{4\, \left ( ad-bc \right ) ^{2} \left ( bdx+ad \right ) }\sqrt{dx+c}}+{\frac{3\,{d}^{2}}{4\, \left ( ad-bc \right ) ^{2}}\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 = 2.35511, size = 1119, normalized size = 9.82 \begin{align*} \left [\frac{3 \,{\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\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 (2 \, b^{3} c^{2} - 7 \, a b^{2} c d + 5 \, a^{2} b d^{2} - 3 \,{\left (b^{3} c d - a b^{2} d^{2}\right )} x\right )} \sqrt{d x + c}}{8 \,{\left (a^{2} b^{4} c^{3} - 3 \, a^{3} b^{3} c^{2} d + 3 \, a^{4} b^{2} c d^{2} - a^{5} b d^{3} +{\left (b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}\right )} x^{2} + 2 \,{\left (a b^{5} c^{3} - 3 \, a^{2} b^{4} c^{2} d + 3 \, a^{3} b^{3} c d^{2} - a^{4} b^{2} d^{3}\right )} x\right )}}, \frac{3 \,{\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\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 (2 \, b^{3} c^{2} - 7 \, a b^{2} c d + 5 \, a^{2} b d^{2} - 3 \,{\left (b^{3} c d - a b^{2} d^{2}\right )} x\right )} \sqrt{d x + c}}{4 \,{\left (a^{2} b^{4} c^{3} - 3 \, a^{3} b^{3} c^{2} d + 3 \, a^{4} b^{2} c d^{2} - a^{5} b d^{3} +{\left (b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}\right )} x^{2} + 2 \,{\left (a b^{5} c^{3} - 3 \, a^{2} b^{4} c^{2} d + 3 \, a^{3} b^{3} c d^{2} - a^{4} b^{2} d^{3}\right )} x\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [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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Giac [A] time = 1.0649, size = 200, normalized size = 1.75 \begin{align*} \frac{3 \, d^{2} \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{4 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{-b^{2} c + a b d}} + \frac{3 \,{\left (d x + c\right )}^{\frac{3}{2}} b d^{2} - 5 \, \sqrt{d x + c} b c d^{2} + 5 \, \sqrt{d x + c} a d^{3}}{4 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}{\left ({\left (d x + c\right )} b - b c + a d\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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