Optimal. Leaf size=218 \[ -\frac{7 d^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{128 b^{3/2} (b c-a d)^{9/2}}+\frac{7 d^4 \sqrt{c+d x}}{128 b (a+b x) (b c-a d)^4}-\frac{7 d^3 \sqrt{c+d x}}{192 b (a+b x)^2 (b c-a d)^3}+\frac{7 d^2 \sqrt{c+d x}}{240 b (a+b x)^3 (b c-a d)^2}-\frac{d \sqrt{c+d x}}{40 b (a+b x)^4 (b c-a d)}-\frac{\sqrt{c+d x}}{5 b (a+b x)^5} \]
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Rubi [A] time = 0.15143, antiderivative size = 218, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {47, 51, 63, 208} \[ -\frac{7 d^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{128 b^{3/2} (b c-a d)^{9/2}}+\frac{7 d^4 \sqrt{c+d x}}{128 b (a+b x) (b c-a d)^4}-\frac{7 d^3 \sqrt{c+d x}}{192 b (a+b x)^2 (b c-a d)^3}+\frac{7 d^2 \sqrt{c+d x}}{240 b (a+b x)^3 (b c-a d)^2}-\frac{d \sqrt{c+d x}}{40 b (a+b x)^4 (b c-a d)}-\frac{\sqrt{c+d x}}{5 b (a+b x)^5} \]
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)^6} \, dx &=-\frac{\sqrt{c+d x}}{5 b (a+b x)^5}+\frac{d \int \frac{1}{(a+b x)^5 \sqrt{c+d x}} \, dx}{10 b}\\ &=-\frac{\sqrt{c+d x}}{5 b (a+b x)^5}-\frac{d \sqrt{c+d x}}{40 b (b c-a d) (a+b x)^4}-\frac{\left (7 d^2\right ) \int \frac{1}{(a+b x)^4 \sqrt{c+d x}} \, dx}{80 b (b c-a d)}\\ &=-\frac{\sqrt{c+d x}}{5 b (a+b x)^5}-\frac{d \sqrt{c+d x}}{40 b (b c-a d) (a+b x)^4}+\frac{7 d^2 \sqrt{c+d x}}{240 b (b c-a d)^2 (a+b x)^3}+\frac{\left (7 d^3\right ) \int \frac{1}{(a+b x)^3 \sqrt{c+d x}} \, dx}{96 b (b c-a d)^2}\\ &=-\frac{\sqrt{c+d x}}{5 b (a+b x)^5}-\frac{d \sqrt{c+d x}}{40 b (b c-a d) (a+b x)^4}+\frac{7 d^2 \sqrt{c+d x}}{240 b (b c-a d)^2 (a+b x)^3}-\frac{7 d^3 \sqrt{c+d x}}{192 b (b c-a d)^3 (a+b x)^2}-\frac{\left (7 d^4\right ) \int \frac{1}{(a+b x)^2 \sqrt{c+d x}} \, dx}{128 b (b c-a d)^3}\\ &=-\frac{\sqrt{c+d x}}{5 b (a+b x)^5}-\frac{d \sqrt{c+d x}}{40 b (b c-a d) (a+b x)^4}+\frac{7 d^2 \sqrt{c+d x}}{240 b (b c-a d)^2 (a+b x)^3}-\frac{7 d^3 \sqrt{c+d x}}{192 b (b c-a d)^3 (a+b x)^2}+\frac{7 d^4 \sqrt{c+d x}}{128 b (b c-a d)^4 (a+b x)}+\frac{\left (7 d^5\right ) \int \frac{1}{(a+b x) \sqrt{c+d x}} \, dx}{256 b (b c-a d)^4}\\ &=-\frac{\sqrt{c+d x}}{5 b (a+b x)^5}-\frac{d \sqrt{c+d x}}{40 b (b c-a d) (a+b x)^4}+\frac{7 d^2 \sqrt{c+d x}}{240 b (b c-a d)^2 (a+b x)^3}-\frac{7 d^3 \sqrt{c+d x}}{192 b (b c-a d)^3 (a+b x)^2}+\frac{7 d^4 \sqrt{c+d x}}{128 b (b c-a d)^4 (a+b x)}+\frac{\left (7 d^4\right ) \operatorname{Subst}\left (\int \frac{1}{a-\frac{b c}{d}+\frac{b x^2}{d}} \, dx,x,\sqrt{c+d x}\right )}{128 b (b c-a d)^4}\\ &=-\frac{\sqrt{c+d x}}{5 b (a+b x)^5}-\frac{d \sqrt{c+d x}}{40 b (b c-a d) (a+b x)^4}+\frac{7 d^2 \sqrt{c+d x}}{240 b (b c-a d)^2 (a+b x)^3}-\frac{7 d^3 \sqrt{c+d x}}{192 b (b c-a d)^3 (a+b x)^2}+\frac{7 d^4 \sqrt{c+d x}}{128 b (b c-a d)^4 (a+b x)}-\frac{7 d^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{128 b^{3/2} (b c-a d)^{9/2}}\\ \end{align*}
Mathematica [C] time = 0.0153353, size = 52, normalized size = 0.24 \[ \frac{2 d^5 (c+d x)^{3/2} \, _2F_1\left (\frac{3}{2},6;\frac{5}{2};-\frac{b (c+d x)}{a d-b c}\right )}{3 (a d-b c)^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 337, normalized size = 1.6 \begin{align*}{\frac{7\,{d}^{5}{b}^{3}}{128\, \left ( bdx+ad \right ) ^{5} \left ({a}^{4}{d}^{4}-4\,{a}^{3}bc{d}^{3}+6\,{a}^{2}{b}^{2}{c}^{2}{d}^{2}-4\,a{b}^{3}{c}^{3}d+{b}^{4}{c}^{4} \right ) } \left ( dx+c \right ) ^{{\frac{9}{2}}}}+{\frac{49\,{d}^{5}{b}^{2}}{192\, \left ( bdx+ad \right ) ^{5} \left ({a}^{3}{d}^{3}-3\,{a}^{2}bc{d}^{2}+3\,a{b}^{2}{c}^{2}d-{b}^{3}{c}^{3} \right ) } \left ( dx+c \right ) ^{{\frac{7}{2}}}}+{\frac{7\,{d}^{5}b}{15\, \left ( bdx+ad \right ) ^{5} \left ({a}^{2}{d}^{2}-2\,abcd+{b}^{2}{c}^{2} \right ) } \left ( dx+c \right ) ^{{\frac{5}{2}}}}+{\frac{79\,{d}^{5}}{192\, \left ( bdx+ad \right ) ^{5} \left ( ad-bc \right ) } \left ( dx+c \right ) ^{{\frac{3}{2}}}}-{\frac{7\,{d}^{5}}{128\, \left ( bdx+ad \right ) ^{5}b}\sqrt{dx+c}}+{\frac{7\,{d}^{5}}{128\,b \left ({a}^{4}{d}^{4}-4\,{a}^{3}bc{d}^{3}+6\,{a}^{2}{b}^{2}{c}^{2}{d}^{2}-4\,a{b}^{3}{c}^{3}d+{b}^{4}{c}^{4} \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 = 2.43228, size = 3452, normalized size = 15.83 \begin{align*} \text{result too large to display} \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 [B] time = 1.10144, size = 583, normalized size = 2.67 \begin{align*} \frac{7 \, d^{5} \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{128 \,{\left (b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}\right )} \sqrt{-b^{2} c + a b d}} + \frac{105 \,{\left (d x + c\right )}^{\frac{9}{2}} b^{4} d^{5} - 490 \,{\left (d x + c\right )}^{\frac{7}{2}} b^{4} c d^{5} + 896 \,{\left (d x + c\right )}^{\frac{5}{2}} b^{4} c^{2} d^{5} - 790 \,{\left (d x + c\right )}^{\frac{3}{2}} b^{4} c^{3} d^{5} - 105 \, \sqrt{d x + c} b^{4} c^{4} d^{5} + 490 \,{\left (d x + c\right )}^{\frac{7}{2}} a b^{3} d^{6} - 1792 \,{\left (d x + c\right )}^{\frac{5}{2}} a b^{3} c d^{6} + 2370 \,{\left (d x + c\right )}^{\frac{3}{2}} a b^{3} c^{2} d^{6} + 420 \, \sqrt{d x + c} a b^{3} c^{3} d^{6} + 896 \,{\left (d x + c\right )}^{\frac{5}{2}} a^{2} b^{2} d^{7} - 2370 \,{\left (d x + c\right )}^{\frac{3}{2}} a^{2} b^{2} c d^{7} - 630 \, \sqrt{d x + c} a^{2} b^{2} c^{2} d^{7} + 790 \,{\left (d x + c\right )}^{\frac{3}{2}} a^{3} b d^{8} + 420 \, \sqrt{d x + c} a^{3} b c d^{8} - 105 \, \sqrt{d x + c} a^{4} d^{9}}{1920 \,{\left (b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}\right )}{\left ({\left (d x + c\right )} b - b c + a d\right )}^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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