Optimal. Leaf size=136 \[ -\frac{d^2 \sqrt{c+d x}}{8 b^2 (a+b x) (b c-a d)}+\frac{d^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{8 b^{5/2} (b c-a d)^{3/2}}-\frac{d \sqrt{c+d x}}{4 b^2 (a+b x)^2}-\frac{(c+d x)^{3/2}}{3 b (a+b x)^3} \]
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Rubi [A] time = 0.0571202, antiderivative size = 136, 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} \[ -\frac{d^2 \sqrt{c+d x}}{8 b^2 (a+b x) (b c-a d)}+\frac{d^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{8 b^{5/2} (b c-a d)^{3/2}}-\frac{d \sqrt{c+d x}}{4 b^2 (a+b x)^2}-\frac{(c+d x)^{3/2}}{3 b (a+b x)^3} \]
Antiderivative was successfully verified.
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Rule 47
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{(c+d x)^{3/2}}{(a+b x)^4} \, dx &=-\frac{(c+d x)^{3/2}}{3 b (a+b x)^3}+\frac{d \int \frac{\sqrt{c+d x}}{(a+b x)^3} \, dx}{2 b}\\ &=-\frac{d \sqrt{c+d x}}{4 b^2 (a+b x)^2}-\frac{(c+d x)^{3/2}}{3 b (a+b x)^3}+\frac{d^2 \int \frac{1}{(a+b x)^2 \sqrt{c+d x}} \, dx}{8 b^2}\\ &=-\frac{d \sqrt{c+d x}}{4 b^2 (a+b x)^2}-\frac{d^2 \sqrt{c+d x}}{8 b^2 (b c-a d) (a+b x)}-\frac{(c+d x)^{3/2}}{3 b (a+b x)^3}-\frac{d^3 \int \frac{1}{(a+b x) \sqrt{c+d x}} \, dx}{16 b^2 (b c-a d)}\\ &=-\frac{d \sqrt{c+d x}}{4 b^2 (a+b x)^2}-\frac{d^2 \sqrt{c+d x}}{8 b^2 (b c-a d) (a+b x)}-\frac{(c+d x)^{3/2}}{3 b (a+b x)^3}-\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^2 (b c-a d)}\\ &=-\frac{d \sqrt{c+d x}}{4 b^2 (a+b x)^2}-\frac{d^2 \sqrt{c+d x}}{8 b^2 (b c-a d) (a+b x)}-\frac{(c+d x)^{3/2}}{3 b (a+b x)^3}+\frac{d^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{8 b^{5/2} (b c-a d)^{3/2}}\\ \end{align*}
Mathematica [C] time = 0.0172427, size = 52, normalized size = 0.38 \[ \frac{2 d^3 (c+d x)^{5/2} \, _2F_1\left (\frac{5}{2},4;\frac{7}{2};-\frac{b (c+d x)}{a d-b c}\right )}{5 (a d-b c)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 163, normalized size = 1.2 \begin{align*}{\frac{{d}^{3}}{8\, \left ( bdx+ad \right ) ^{3} \left ( ad-bc \right ) } \left ( dx+c \right ) ^{{\frac{5}{2}}}}-{\frac{{d}^{3}}{3\, \left ( bdx+ad \right ) ^{3}b} \left ( dx+c \right ) ^{{\frac{3}{2}}}}-{\frac{{d}^{4}a}{8\, \left ( bdx+ad \right ) ^{3}{b}^{2}}\sqrt{dx+c}}+{\frac{{d}^{3}c}{8\, \left ( bdx+ad \right ) ^{3}b}\sqrt{dx+c}}+{\frac{{d}^{3}}{ \left ( 8\,ad-8\,bc \right ){b}^{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 = 1.85628, size = 1368, normalized size = 10.06 \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} - 10 \, a b^{3} c^{2} d - 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 (7 \, b^{4} c^{2} d - 11 \, 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^{2} - 2 \, a^{4} b^{4} c d + a^{5} b^{3} d^{2} +{\left (b^{8} c^{2} - 2 \, a b^{7} c d + a^{2} b^{6} d^{2}\right )} x^{3} + 3 \,{\left (a b^{7} c^{2} - 2 \, a^{2} b^{6} c d + a^{3} b^{5} d^{2}\right )} x^{2} + 3 \,{\left (a^{2} b^{6} c^{2} - 2 \, a^{3} b^{5} c d + a^{4} b^{4} d^{2}\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} - 10 \, a b^{3} c^{2} d - 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 (7 \, b^{4} c^{2} d - 11 \, 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^{2} - 2 \, a^{4} b^{4} c d + a^{5} b^{3} d^{2} +{\left (b^{8} c^{2} - 2 \, a b^{7} c d + a^{2} b^{6} d^{2}\right )} x^{3} + 3 \,{\left (a b^{7} c^{2} - 2 \, a^{2} b^{6} c d + a^{3} b^{5} d^{2}\right )} x^{2} + 3 \,{\left (a^{2} b^{6} c^{2} - 2 \, a^{3} b^{5} c d + a^{4} b^{4} d^{2}\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 = 1.08899, size = 250, normalized size = 1.84 \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 - a b^{2} d\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 - a b^{2} d\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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