Optimal. Leaf size=220 \[ -\frac{(c+d x)^{5/2} \left (-693 a^2 d^2-5 b d x (10 b c-99 a d)+180 a b c d+20 b^2 c^2\right )}{315 b^4 d^2}+\frac{a^2 (c+d x)^{3/2} (6 b c-11 a d)}{3 b^5}+\frac{a^2 \sqrt{c+d x} (6 b c-11 a d) (b c-a d)}{b^6}-\frac{a^2 (6 b c-11 a d) (b c-a d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{b^{13/2}}-\frac{x^3 (c+d x)^{5/2}}{b (a+b x)}+\frac{11 x^2 (c+d x)^{5/2}}{9 b^2} \]
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Rubi [A] time = 0.221488, antiderivative size = 220, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {97, 153, 147, 50, 63, 208} \[ -\frac{(c+d x)^{5/2} \left (-693 a^2 d^2-5 b d x (10 b c-99 a d)+180 a b c d+20 b^2 c^2\right )}{315 b^4 d^2}+\frac{a^2 (c+d x)^{3/2} (6 b c-11 a d)}{3 b^5}+\frac{a^2 \sqrt{c+d x} (6 b c-11 a d) (b c-a d)}{b^6}-\frac{a^2 (6 b c-11 a d) (b c-a d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{b^{13/2}}-\frac{x^3 (c+d x)^{5/2}}{b (a+b x)}+\frac{11 x^2 (c+d x)^{5/2}}{9 b^2} \]
Antiderivative was successfully verified.
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Rule 97
Rule 153
Rule 147
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{x^3 (c+d x)^{5/2}}{(a+b x)^2} \, dx &=-\frac{x^3 (c+d x)^{5/2}}{b (a+b x)}+\frac{\int \frac{x^2 (c+d x)^{3/2} \left (3 c+\frac{11 d x}{2}\right )}{a+b x} \, dx}{b}\\ &=\frac{11 x^2 (c+d x)^{5/2}}{9 b^2}-\frac{x^3 (c+d x)^{5/2}}{b (a+b x)}+\frac{2 \int \frac{x (c+d x)^{3/2} \left (-11 a c d+\frac{1}{4} d (10 b c-99 a d) x\right )}{a+b x} \, dx}{9 b^2 d}\\ &=\frac{11 x^2 (c+d x)^{5/2}}{9 b^2}-\frac{x^3 (c+d x)^{5/2}}{b (a+b x)}-\frac{(c+d x)^{5/2} \left (20 b^2 c^2+180 a b c d-693 a^2 d^2-5 b d (10 b c-99 a d) x\right )}{315 b^4 d^2}+\frac{\left (a^2 (6 b c-11 a d)\right ) \int \frac{(c+d x)^{3/2}}{a+b x} \, dx}{2 b^4}\\ &=\frac{a^2 (6 b c-11 a d) (c+d x)^{3/2}}{3 b^5}+\frac{11 x^2 (c+d x)^{5/2}}{9 b^2}-\frac{x^3 (c+d x)^{5/2}}{b (a+b x)}-\frac{(c+d x)^{5/2} \left (20 b^2 c^2+180 a b c d-693 a^2 d^2-5 b d (10 b c-99 a d) x\right )}{315 b^4 d^2}+\frac{\left (a^2 (6 b c-11 a d) (b c-a d)\right ) \int \frac{\sqrt{c+d x}}{a+b x} \, dx}{2 b^5}\\ &=\frac{a^2 (6 b c-11 a d) (b c-a d) \sqrt{c+d x}}{b^6}+\frac{a^2 (6 b c-11 a d) (c+d x)^{3/2}}{3 b^5}+\frac{11 x^2 (c+d x)^{5/2}}{9 b^2}-\frac{x^3 (c+d x)^{5/2}}{b (a+b x)}-\frac{(c+d x)^{5/2} \left (20 b^2 c^2+180 a b c d-693 a^2 d^2-5 b d (10 b c-99 a d) x\right )}{315 b^4 d^2}+\frac{\left (a^2 (6 b c-11 a d) (b c-a d)^2\right ) \int \frac{1}{(a+b x) \sqrt{c+d x}} \, dx}{2 b^6}\\ &=\frac{a^2 (6 b c-11 a d) (b c-a d) \sqrt{c+d x}}{b^6}+\frac{a^2 (6 b c-11 a d) (c+d x)^{3/2}}{3 b^5}+\frac{11 x^2 (c+d x)^{5/2}}{9 b^2}-\frac{x^3 (c+d x)^{5/2}}{b (a+b x)}-\frac{(c+d x)^{5/2} \left (20 b^2 c^2+180 a b c d-693 a^2 d^2-5 b d (10 b c-99 a d) x\right )}{315 b^4 d^2}+\frac{\left (a^2 (6 b c-11 a d) (b c-a d)^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^6 d}\\ &=\frac{a^2 (6 b c-11 a d) (b c-a d) \sqrt{c+d x}}{b^6}+\frac{a^2 (6 b c-11 a d) (c+d x)^{3/2}}{3 b^5}+\frac{11 x^2 (c+d x)^{5/2}}{9 b^2}-\frac{x^3 (c+d x)^{5/2}}{b (a+b x)}-\frac{(c+d x)^{5/2} \left (20 b^2 c^2+180 a b c d-693 a^2 d^2-5 b d (10 b c-99 a d) x\right )}{315 b^4 d^2}-\frac{a^2 (6 b c-11 a d) (b c-a d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{b^{13/2}}\\ \end{align*}
Mathematica [A] time = 0.652182, size = 246, normalized size = 1.12 \[ \frac{21 a^2 d^2 (a+b x) (6 b c-11 a d) \left (\sqrt{b} \sqrt{c+d x} \left (15 a^2 d^2-5 a b d (7 c+d x)+b^2 \left (23 c^2+11 c d x+3 d^2 x^2\right )\right )-15 (b c-a d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )\right )+5 b^{7/2} (c+d x)^{7/2} \left (2 a^2 b d (11 d x-16 c)+99 a^3 d^2-2 a b^2 c (2 c+9 d x)-4 b^3 c^2 x\right )+70 b^{11/2} d x^2 (c+d x)^{7/2} (b c-a d)}{315 b^{13/2} d^2 (a+b x) (b c-a d)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.018, size = 415, normalized size = 1.9 \begin{align*}{\frac{2}{9\,{b}^{2}{d}^{2}} \left ( dx+c \right ) ^{{\frac{9}{2}}}}-{\frac{4\,a}{7\,d{b}^{3}} \left ( dx+c \right ) ^{{\frac{7}{2}}}}-{\frac{2\,c}{7\,{b}^{2}{d}^{2}} \left ( dx+c \right ) ^{{\frac{7}{2}}}}+{\frac{6\,{a}^{2}}{5\,{b}^{4}} \left ( dx+c \right ) ^{{\frac{5}{2}}}}-{\frac{8\,d{a}^{3}}{3\,{b}^{5}} \left ( dx+c \right ) ^{{\frac{3}{2}}}}+2\,{\frac{ \left ( dx+c \right ) ^{3/2}{a}^{2}c}{{b}^{4}}}+10\,{\frac{{d}^{2}{a}^{4}\sqrt{dx+c}}{{b}^{6}}}-16\,{\frac{d{a}^{3}c\sqrt{dx+c}}{{b}^{5}}}+6\,{\frac{{a}^{2}{c}^{2}\sqrt{dx+c}}{{b}^{4}}}+{\frac{{d}^{3}{a}^{5}}{{b}^{6} \left ( bdx+ad \right ) }\sqrt{dx+c}}-2\,{\frac{{d}^{2}{a}^{4}\sqrt{dx+c}c}{{b}^{5} \left ( bdx+ad \right ) }}+{\frac{d{a}^{3}{c}^{2}}{{b}^{4} \left ( bdx+ad \right ) }\sqrt{dx+c}}-11\,{\frac{{d}^{3}{a}^{5}}{{b}^{6}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{b\sqrt{dx+c}}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }+28\,{\frac{{d}^{2}{a}^{4}c}{{b}^{5}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{b\sqrt{dx+c}}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }-23\,{\frac{d{a}^{3}{c}^{2}}{{b}^{4}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{b\sqrt{dx+c}}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }+6\,{\frac{{a}^{2}{c}^{3}}{{b}^{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 = 2.78741, size = 1721, normalized size = 7.82 \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 [A] time = 1.21316, size = 436, normalized size = 1.98 \begin{align*} \frac{{\left (6 \, a^{2} b^{3} c^{3} - 23 \, a^{3} b^{2} c^{2} d + 28 \, a^{4} b c d^{2} - 11 \, a^{5} d^{3}\right )} \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} b^{6}} + \frac{\sqrt{d x + c} a^{3} b^{2} c^{2} d - 2 \, \sqrt{d x + c} a^{4} b c d^{2} + \sqrt{d x + c} a^{5} d^{3}}{{\left ({\left (d x + c\right )} b - b c + a d\right )} b^{6}} + \frac{2 \,{\left (35 \,{\left (d x + c\right )}^{\frac{9}{2}} b^{16} d^{16} - 45 \,{\left (d x + c\right )}^{\frac{7}{2}} b^{16} c d^{16} - 90 \,{\left (d x + c\right )}^{\frac{7}{2}} a b^{15} d^{17} + 189 \,{\left (d x + c\right )}^{\frac{5}{2}} a^{2} b^{14} d^{18} + 315 \,{\left (d x + c\right )}^{\frac{3}{2}} a^{2} b^{14} c d^{18} + 945 \, \sqrt{d x + c} a^{2} b^{14} c^{2} d^{18} - 420 \,{\left (d x + c\right )}^{\frac{3}{2}} a^{3} b^{13} d^{19} - 2520 \, \sqrt{d x + c} a^{3} b^{13} c d^{19} + 1575 \, \sqrt{d x + c} a^{4} b^{12} d^{20}\right )}}{315 \, b^{18} d^{18}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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