Optimal. Leaf size=209 \[ \frac{2 (c+d x)^{7/2} \left (a^2 d^2+a b c d+b^2 c^2\right )}{7 b^3 d^3}-\frac{2 a^3 (c+d x)^{5/2}}{5 b^4}-\frac{2 a^3 (c+d x)^{3/2} (b c-a d)}{3 b^5}-\frac{2 a^3 \sqrt{c+d x} (b c-a d)^2}{b^6}+\frac{2 a^3 (b c-a d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{b^{13/2}}-\frac{2 (c+d x)^{9/2} (a d+2 b c)}{9 b^2 d^3}+\frac{2 (c+d x)^{11/2}}{11 b d^3} \]
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Rubi [A] time = 0.171209, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {88, 50, 63, 208} \[ \frac{2 (c+d x)^{7/2} \left (a^2 d^2+a b c d+b^2 c^2\right )}{7 b^3 d^3}-\frac{2 a^3 (c+d x)^{5/2}}{5 b^4}-\frac{2 a^3 (c+d x)^{3/2} (b c-a d)}{3 b^5}-\frac{2 a^3 \sqrt{c+d x} (b c-a d)^2}{b^6}+\frac{2 a^3 (b c-a d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{b^{13/2}}-\frac{2 (c+d x)^{9/2} (a d+2 b c)}{9 b^2 d^3}+\frac{2 (c+d x)^{11/2}}{11 b d^3} \]
Antiderivative was successfully verified.
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Rule 88
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{x^3 (c+d x)^{5/2}}{a+b x} \, dx &=\int \left (\frac{\left (b^2 c^2+a b c d+a^2 d^2\right ) (c+d x)^{5/2}}{b^3 d^2}-\frac{a^3 (c+d x)^{5/2}}{b^3 (a+b x)}+\frac{(-2 b c-a d) (c+d x)^{7/2}}{b^2 d^2}+\frac{(c+d x)^{9/2}}{b d^2}\right ) \, dx\\ &=\frac{2 \left (b^2 c^2+a b c d+a^2 d^2\right ) (c+d x)^{7/2}}{7 b^3 d^3}-\frac{2 (2 b c+a d) (c+d x)^{9/2}}{9 b^2 d^3}+\frac{2 (c+d x)^{11/2}}{11 b d^3}-\frac{a^3 \int \frac{(c+d x)^{5/2}}{a+b x} \, dx}{b^3}\\ &=-\frac{2 a^3 (c+d x)^{5/2}}{5 b^4}+\frac{2 \left (b^2 c^2+a b c d+a^2 d^2\right ) (c+d x)^{7/2}}{7 b^3 d^3}-\frac{2 (2 b c+a d) (c+d x)^{9/2}}{9 b^2 d^3}+\frac{2 (c+d x)^{11/2}}{11 b d^3}-\frac{\left (a^3 (b c-a d)\right ) \int \frac{(c+d x)^{3/2}}{a+b x} \, dx}{b^4}\\ &=-\frac{2 a^3 (b c-a d) (c+d x)^{3/2}}{3 b^5}-\frac{2 a^3 (c+d x)^{5/2}}{5 b^4}+\frac{2 \left (b^2 c^2+a b c d+a^2 d^2\right ) (c+d x)^{7/2}}{7 b^3 d^3}-\frac{2 (2 b c+a d) (c+d x)^{9/2}}{9 b^2 d^3}+\frac{2 (c+d x)^{11/2}}{11 b d^3}-\frac{\left (a^3 (b c-a d)^2\right ) \int \frac{\sqrt{c+d x}}{a+b x} \, dx}{b^5}\\ &=-\frac{2 a^3 (b c-a d)^2 \sqrt{c+d x}}{b^6}-\frac{2 a^3 (b c-a d) (c+d x)^{3/2}}{3 b^5}-\frac{2 a^3 (c+d x)^{5/2}}{5 b^4}+\frac{2 \left (b^2 c^2+a b c d+a^2 d^2\right ) (c+d x)^{7/2}}{7 b^3 d^3}-\frac{2 (2 b c+a d) (c+d x)^{9/2}}{9 b^2 d^3}+\frac{2 (c+d x)^{11/2}}{11 b d^3}-\frac{\left (a^3 (b c-a d)^3\right ) \int \frac{1}{(a+b x) \sqrt{c+d x}} \, dx}{b^6}\\ &=-\frac{2 a^3 (b c-a d)^2 \sqrt{c+d x}}{b^6}-\frac{2 a^3 (b c-a d) (c+d x)^{3/2}}{3 b^5}-\frac{2 a^3 (c+d x)^{5/2}}{5 b^4}+\frac{2 \left (b^2 c^2+a b c d+a^2 d^2\right ) (c+d x)^{7/2}}{7 b^3 d^3}-\frac{2 (2 b c+a d) (c+d x)^{9/2}}{9 b^2 d^3}+\frac{2 (c+d x)^{11/2}}{11 b d^3}-\frac{\left (2 a^3 (b c-a d)^3\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{2 a^3 (b c-a d)^2 \sqrt{c+d x}}{b^6}-\frac{2 a^3 (b c-a d) (c+d x)^{3/2}}{3 b^5}-\frac{2 a^3 (c+d x)^{5/2}}{5 b^4}+\frac{2 \left (b^2 c^2+a b c d+a^2 d^2\right ) (c+d x)^{7/2}}{7 b^3 d^3}-\frac{2 (2 b c+a d) (c+d x)^{9/2}}{9 b^2 d^3}+\frac{2 (c+d x)^{11/2}}{11 b d^3}+\frac{2 a^3 (b c-a d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{b^{13/2}}\\ \end{align*}
Mathematica [A] time = 0.335301, size = 197, normalized size = 0.94 \[ \frac{2 (c+d x)^{7/2} \left (a^2 d^2+a b c d+b^2 c^2\right )}{7 b^3 d^3}-\frac{2 a^3 (c+d x)^{5/2}}{5 b^4}-\frac{2 a^3 (a d-b c) \left (3 (b c-a d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )-\sqrt{b} \sqrt{c+d x} (-3 a d+4 b c+b d x)\right )}{3 b^{13/2}}-\frac{2 (c+d x)^{9/2} (a d+2 b c)}{9 b^2 d^3}+\frac{2 (c+d x)^{11/2}}{11 b d^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.012, size = 384, normalized size = 1.8 \begin{align*}{\frac{2}{11\,b{d}^{3}} \left ( dx+c \right ) ^{{\frac{11}{2}}}}-{\frac{2\,a}{9\,{b}^{2}{d}^{2}} \left ( dx+c \right ) ^{{\frac{9}{2}}}}-{\frac{4\,c}{9\,b{d}^{3}} \left ( dx+c \right ) ^{{\frac{9}{2}}}}+{\frac{2\,{a}^{2}}{7\,d{b}^{3}} \left ( dx+c \right ) ^{{\frac{7}{2}}}}+{\frac{2\,ac}{7\,{b}^{2}{d}^{2}} \left ( dx+c \right ) ^{{\frac{7}{2}}}}+{\frac{2\,{c}^{2}}{7\,b{d}^{3}} \left ( dx+c \right ) ^{{\frac{7}{2}}}}-{\frac{2\,{a}^{3}}{5\,{b}^{4}} \left ( dx+c \right ) ^{{\frac{5}{2}}}}+{\frac{2\,d{a}^{4}}{3\,{b}^{5}} \left ( dx+c \right ) ^{{\frac{3}{2}}}}-{\frac{2\,{a}^{3}c}{3\,{b}^{4}} \left ( dx+c \right ) ^{{\frac{3}{2}}}}-2\,{\frac{{d}^{2}{a}^{5}\sqrt{dx+c}}{{b}^{6}}}+4\,{\frac{d{a}^{4}c\sqrt{dx+c}}{{b}^{5}}}-2\,{\frac{{a}^{3}{c}^{2}\sqrt{dx+c}}{{b}^{4}}}+2\,{\frac{{d}^{3}{a}^{6}}{{b}^{6}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{b\sqrt{dx+c}}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }-6\,{\frac{{d}^{2}{a}^{5}c}{{b}^{5}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{b\sqrt{dx+c}}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }+6\,{\frac{d{a}^{4}{c}^{2}}{{b}^{4}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{b\sqrt{dx+c}}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }-2\,{\frac{{a}^{3}{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 [A] time = 2.51873, size = 1580, normalized size = 7.56 \begin{align*} \left [\frac{3465 \,{\left (a^{3} b^{2} c^{2} d^{3} - 2 \, a^{4} b c d^{4} + a^{5} d^{5}\right )} \sqrt{\frac{b c - a d}{b}} \log \left (\frac{b d x + 2 \, b c - a d + 2 \, \sqrt{d x + c} b \sqrt{\frac{b c - a d}{b}}}{b x + a}\right ) + 2 \,{\left (315 \, b^{5} d^{5} x^{5} + 40 \, b^{5} c^{5} + 110 \, a b^{4} c^{4} d + 495 \, a^{2} b^{3} c^{3} d^{2} - 5313 \, a^{3} b^{2} c^{2} d^{3} + 8085 \, a^{4} b c d^{4} - 3465 \, a^{5} d^{5} + 35 \,{\left (23 \, b^{5} c d^{4} - 11 \, a b^{4} d^{5}\right )} x^{4} + 5 \,{\left (113 \, b^{5} c^{2} d^{3} - 209 \, a b^{4} c d^{4} + 99 \, a^{2} b^{3} d^{5}\right )} x^{3} + 3 \,{\left (5 \, b^{5} c^{3} d^{2} - 275 \, a b^{4} c^{2} d^{3} + 495 \, a^{2} b^{3} c d^{4} - 231 \, a^{3} b^{2} d^{5}\right )} x^{2} -{\left (20 \, b^{5} c^{4} d + 55 \, a b^{4} c^{3} d^{2} - 1485 \, a^{2} b^{3} c^{2} d^{3} + 2541 \, a^{3} b^{2} c d^{4} - 1155 \, a^{4} b d^{5}\right )} x\right )} \sqrt{d x + c}}{3465 \, b^{6} d^{3}}, \frac{2 \,{\left (3465 \,{\left (a^{3} b^{2} c^{2} d^{3} - 2 \, a^{4} b c d^{4} + a^{5} d^{5}\right )} \sqrt{-\frac{b c - a d}{b}} \arctan \left (-\frac{\sqrt{d x + c} b \sqrt{-\frac{b c - a d}{b}}}{b c - a d}\right ) +{\left (315 \, b^{5} d^{5} x^{5} + 40 \, b^{5} c^{5} + 110 \, a b^{4} c^{4} d + 495 \, a^{2} b^{3} c^{3} d^{2} - 5313 \, a^{3} b^{2} c^{2} d^{3} + 8085 \, a^{4} b c d^{4} - 3465 \, a^{5} d^{5} + 35 \,{\left (23 \, b^{5} c d^{4} - 11 \, a b^{4} d^{5}\right )} x^{4} + 5 \,{\left (113 \, b^{5} c^{2} d^{3} - 209 \, a b^{4} c d^{4} + 99 \, a^{2} b^{3} d^{5}\right )} x^{3} + 3 \,{\left (5 \, b^{5} c^{3} d^{2} - 275 \, a b^{4} c^{2} d^{3} + 495 \, a^{2} b^{3} c d^{4} - 231 \, a^{3} b^{2} d^{5}\right )} x^{2} -{\left (20 \, b^{5} c^{4} d + 55 \, a b^{4} c^{3} d^{2} - 1485 \, a^{2} b^{3} c^{2} d^{3} + 2541 \, a^{3} b^{2} c d^{4} - 1155 \, a^{4} b d^{5}\right )} x\right )} \sqrt{d x + c}\right )}}{3465 \, b^{6} d^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 95.9039, size = 228, normalized size = 1.09 \begin{align*} - \frac{2 a^{3} \left (c + d x\right )^{\frac{5}{2}}}{5 b^{4}} + \frac{2 a^{3} \left (a d - b c\right )^{3} \operatorname{atan}{\left (\frac{\sqrt{c + d x}}{\sqrt{\frac{a d - b c}{b}}} \right )}}{b^{7} \sqrt{\frac{a d - b c}{b}}} + \frac{2 \left (c + d x\right )^{\frac{11}{2}}}{11 b d^{3}} + \frac{\left (c + d x\right )^{\frac{9}{2}} \left (- 2 a d - 4 b c\right )}{9 b^{2} d^{3}} + \frac{\left (c + d x\right )^{\frac{7}{2}} \left (2 a^{2} d^{2} + 2 a b c d + 2 b^{2} c^{2}\right )}{7 b^{3} d^{3}} + \frac{\left (c + d x\right )^{\frac{3}{2}} \left (2 a^{4} d - 2 a^{3} b c\right )}{3 b^{5}} + \frac{\sqrt{c + d x} \left (- 2 a^{5} d^{2} + 4 a^{4} b c d - 2 a^{3} b^{2} c^{2}\right )}{b^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22394, size = 412, normalized size = 1.97 \begin{align*} -\frac{2 \,{\left (a^{3} b^{3} c^{3} - 3 \, a^{4} b^{2} c^{2} d + 3 \, a^{5} b c d^{2} - a^{6} 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{2 \,{\left (315 \,{\left (d x + c\right )}^{\frac{11}{2}} b^{10} d^{30} - 770 \,{\left (d x + c\right )}^{\frac{9}{2}} b^{10} c d^{30} + 495 \,{\left (d x + c\right )}^{\frac{7}{2}} b^{10} c^{2} d^{30} - 385 \,{\left (d x + c\right )}^{\frac{9}{2}} a b^{9} d^{31} + 495 \,{\left (d x + c\right )}^{\frac{7}{2}} a b^{9} c d^{31} + 495 \,{\left (d x + c\right )}^{\frac{7}{2}} a^{2} b^{8} d^{32} - 693 \,{\left (d x + c\right )}^{\frac{5}{2}} a^{3} b^{7} d^{33} - 1155 \,{\left (d x + c\right )}^{\frac{3}{2}} a^{3} b^{7} c d^{33} - 3465 \, \sqrt{d x + c} a^{3} b^{7} c^{2} d^{33} + 1155 \,{\left (d x + c\right )}^{\frac{3}{2}} a^{4} b^{6} d^{34} + 6930 \, \sqrt{d x + c} a^{4} b^{6} c d^{34} - 3465 \, \sqrt{d x + c} a^{5} b^{5} d^{35}\right )}}{3465 \, b^{11} d^{33}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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