Optimal. Leaf size=124 \[ \frac{5 b^{3/2} d \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{(b c-a d)^{7/2}}-\frac{5 b d}{\sqrt{c+d x} (b c-a d)^3}-\frac{1}{(a+b x) (c+d x)^{3/2} (b c-a d)}-\frac{5 d}{3 (c+d x)^{3/2} (b c-a d)^2} \]
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Rubi [A] time = 0.0497228, antiderivative size = 124, 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 = {51, 63, 208} \[ \frac{5 b^{3/2} d \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{(b c-a d)^{7/2}}-\frac{5 b d}{\sqrt{c+d x} (b c-a d)^3}-\frac{1}{(a+b x) (c+d x)^{3/2} (b c-a d)}-\frac{5 d}{3 (c+d x)^{3/2} (b c-a d)^2} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{(a+b x)^2 (c+d x)^{5/2}} \, dx &=-\frac{1}{(b c-a d) (a+b x) (c+d x)^{3/2}}-\frac{(5 d) \int \frac{1}{(a+b x) (c+d x)^{5/2}} \, dx}{2 (b c-a d)}\\ &=-\frac{5 d}{3 (b c-a d)^2 (c+d x)^{3/2}}-\frac{1}{(b c-a d) (a+b x) (c+d x)^{3/2}}-\frac{(5 b d) \int \frac{1}{(a+b x) (c+d x)^{3/2}} \, dx}{2 (b c-a d)^2}\\ &=-\frac{5 d}{3 (b c-a d)^2 (c+d x)^{3/2}}-\frac{1}{(b c-a d) (a+b x) (c+d x)^{3/2}}-\frac{5 b d}{(b c-a d)^3 \sqrt{c+d x}}-\frac{\left (5 b^2 d\right ) \int \frac{1}{(a+b x) \sqrt{c+d x}} \, dx}{2 (b c-a d)^3}\\ &=-\frac{5 d}{3 (b c-a d)^2 (c+d x)^{3/2}}-\frac{1}{(b c-a d) (a+b x) (c+d x)^{3/2}}-\frac{5 b d}{(b c-a d)^3 \sqrt{c+d x}}-\frac{\left (5 b^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 c-a d)^3}\\ &=-\frac{5 d}{3 (b c-a d)^2 (c+d x)^{3/2}}-\frac{1}{(b c-a d) (a+b x) (c+d x)^{3/2}}-\frac{5 b d}{(b c-a d)^3 \sqrt{c+d x}}+\frac{5 b^{3/2} d \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{(b c-a d)^{7/2}}\\ \end{align*}
Mathematica [C] time = 0.0143284, size = 50, normalized size = 0.4 \[ -\frac{2 d \, _2F_1\left (-\frac{3}{2},2;-\frac{1}{2};-\frac{b (c+d x)}{a d-b c}\right )}{3 (c+d x)^{3/2} (a d-b c)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 125, normalized size = 1. \begin{align*} -{\frac{2\,d}{3\, \left ( ad-bc \right ) ^{2}} \left ( dx+c \right ) ^{-{\frac{3}{2}}}}+4\,{\frac{bd}{ \left ( ad-bc \right ) ^{3}\sqrt{dx+c}}}+{\frac{{b}^{2}d}{ \left ( ad-bc \right ) ^{3} \left ( bdx+ad \right ) }\sqrt{dx+c}}+5\,{\frac{{b}^{2}d}{ \left ( ad-bc \right ) ^{3}\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 [B] time = 2.29637, size = 1592, normalized size = 12.84 \begin{align*} \left [-\frac{15 \,{\left (b^{2} d^{3} x^{3} + a b c^{2} d +{\left (2 \, b^{2} c d^{2} + a b d^{3}\right )} x^{2} +{\left (b^{2} c^{2} d + 2 \, a b c d^{2}\right )} x\right )} \sqrt{\frac{b}{b c - a d}} \log \left (\frac{b d x + 2 \, b c - a d - 2 \,{\left (b c - a d\right )} \sqrt{d x + c} \sqrt{\frac{b}{b c - a d}}}{b x + a}\right ) + 2 \,{\left (15 \, b^{2} d^{2} x^{2} + 3 \, b^{2} c^{2} + 14 \, a b c d - 2 \, a^{2} d^{2} + 10 \,{\left (2 \, b^{2} c d + a b d^{2}\right )} x\right )} \sqrt{d x + c}}{6 \,{\left (a b^{3} c^{5} - 3 \, a^{2} b^{2} c^{4} d + 3 \, a^{3} b c^{3} d^{2} - a^{4} c^{2} d^{3} +{\left (b^{4} c^{3} d^{2} - 3 \, a b^{3} c^{2} d^{3} + 3 \, a^{2} b^{2} c d^{4} - a^{3} b d^{5}\right )} x^{3} +{\left (2 \, b^{4} c^{4} d - 5 \, a b^{3} c^{3} d^{2} + 3 \, a^{2} b^{2} c^{2} d^{3} + a^{3} b c d^{4} - a^{4} d^{5}\right )} x^{2} +{\left (b^{4} c^{5} - a b^{3} c^{4} d - 3 \, a^{2} b^{2} c^{3} d^{2} + 5 \, a^{3} b c^{2} d^{3} - 2 \, a^{4} c d^{4}\right )} x\right )}}, \frac{15 \,{\left (b^{2} d^{3} x^{3} + a b c^{2} d +{\left (2 \, b^{2} c d^{2} + a b d^{3}\right )} x^{2} +{\left (b^{2} c^{2} d + 2 \, a b c d^{2}\right )} x\right )} \sqrt{-\frac{b}{b c - a d}} \arctan \left (-\frac{{\left (b c - a d\right )} \sqrt{d x + c} \sqrt{-\frac{b}{b c - a d}}}{b d x + b c}\right ) -{\left (15 \, b^{2} d^{2} x^{2} + 3 \, b^{2} c^{2} + 14 \, a b c d - 2 \, a^{2} d^{2} + 10 \,{\left (2 \, b^{2} c d + a b d^{2}\right )} x\right )} \sqrt{d x + c}}{3 \,{\left (a b^{3} c^{5} - 3 \, a^{2} b^{2} c^{4} d + 3 \, a^{3} b c^{3} d^{2} - a^{4} c^{2} d^{3} +{\left (b^{4} c^{3} d^{2} - 3 \, a b^{3} c^{2} d^{3} + 3 \, a^{2} b^{2} c d^{4} - a^{3} b d^{5}\right )} x^{3} +{\left (2 \, b^{4} c^{4} d - 5 \, a b^{3} c^{3} d^{2} + 3 \, a^{2} b^{2} c^{2} d^{3} + a^{3} b c d^{4} - a^{4} d^{5}\right )} x^{2} +{\left (b^{4} c^{5} - a b^{3} c^{4} d - 3 \, a^{2} b^{2} c^{3} d^{2} + 5 \, a^{3} b c^{2} d^{3} - 2 \, a^{4} c d^{4}\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 [B] time = 1.06954, size = 292, normalized size = 2.35 \begin{align*} -\frac{5 \, b^{2} d \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt{-b^{2} c + a b d}} - \frac{\sqrt{d x + c} b^{2} d}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )}{\left ({\left (d x + c\right )} b - b c + a d\right )}} - \frac{2 \,{\left (6 \,{\left (d x + c\right )} b d + b c d - a d^{2}\right )}}{3 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )}{\left (d x + c\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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