Optimal. Leaf size=230 \[ \frac{2 \left (-3 a^2 b^2 c+a^4+3 b^4 c^2\right ) (c+d x)^{3/2}}{3 b^5 d^4}-\frac{a x \left (-3 a^2 b^2 c+a^4+3 b^4 c^2\right )}{b^6 d^3}+\frac{2 \left (a^2-3 b^2 c\right ) (c+d x)^{5/2}}{5 b^3 d^4}-\frac{a \left (a^2-3 b^2 c\right ) (c+d x)^2}{2 b^4 d^4}+\frac{2 \left (a^2-b^2 c\right )^3 \sqrt{c+d x}}{b^7 d^4}-\frac{2 a \left (a^2-b^2 c\right )^3 \log \left (a+b \sqrt{c+d x}\right )}{b^8 d^4}-\frac{a (c+d x)^3}{3 b^2 d^4}+\frac{2 (c+d x)^{7/2}}{7 b d^4} \]
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Rubi [A] time = 0.258577, antiderivative size = 230, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {371, 1398, 772} \[ \frac{2 \left (-3 a^2 b^2 c+a^4+3 b^4 c^2\right ) (c+d x)^{3/2}}{3 b^5 d^4}-\frac{a x \left (-3 a^2 b^2 c+a^4+3 b^4 c^2\right )}{b^6 d^3}+\frac{2 \left (a^2-3 b^2 c\right ) (c+d x)^{5/2}}{5 b^3 d^4}-\frac{a \left (a^2-3 b^2 c\right ) (c+d x)^2}{2 b^4 d^4}+\frac{2 \left (a^2-b^2 c\right )^3 \sqrt{c+d x}}{b^7 d^4}-\frac{2 a \left (a^2-b^2 c\right )^3 \log \left (a+b \sqrt{c+d x}\right )}{b^8 d^4}-\frac{a (c+d x)^3}{3 b^2 d^4}+\frac{2 (c+d x)^{7/2}}{7 b d^4} \]
Antiderivative was successfully verified.
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Rule 371
Rule 1398
Rule 772
Rubi steps
\begin{align*} \int \frac{x^3}{a+b \sqrt{c+d x}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(-c+x)^3}{a+b \sqrt{x}} \, dx,x,c+d x\right )}{d^4}\\ &=\frac{2 \operatorname{Subst}\left (\int \frac{x \left (-c+x^2\right )^3}{a+b x} \, dx,x,\sqrt{c+d x}\right )}{d^4}\\ &=\frac{2 \operatorname{Subst}\left (\int \left (-\frac{\left (-a^2+b^2 c\right )^3}{b^7}-\frac{a \left (a^4-3 a^2 b^2 c+3 b^4 c^2\right ) x}{b^6}+\frac{\left (a^4-3 a^2 b^2 c+3 b^4 c^2\right ) x^2}{b^5}-\frac{a \left (a^2-3 b^2 c\right ) x^3}{b^4}-\frac{\left (-a^2+3 b^2 c\right ) x^4}{b^3}-\frac{a x^5}{b^2}+\frac{x^6}{b}-\frac{a \left (a^2-b^2 c\right )^3}{b^7 (a+b x)}\right ) \, dx,x,\sqrt{c+d x}\right )}{d^4}\\ &=-\frac{a \left (a^4-3 a^2 b^2 c+3 b^4 c^2\right ) x}{b^6 d^3}+\frac{2 \left (a^2-b^2 c\right )^3 \sqrt{c+d x}}{b^7 d^4}+\frac{2 \left (a^4-3 a^2 b^2 c+3 b^4 c^2\right ) (c+d x)^{3/2}}{3 b^5 d^4}-\frac{a \left (a^2-3 b^2 c\right ) (c+d x)^2}{2 b^4 d^4}+\frac{2 \left (a^2-3 b^2 c\right ) (c+d x)^{5/2}}{5 b^3 d^4}-\frac{a (c+d x)^3}{3 b^2 d^4}+\frac{2 (c+d x)^{7/2}}{7 b d^4}-\frac{2 a \left (a^2-b^2 c\right )^3 \log \left (a+b \sqrt{c+d x}\right )}{b^8 d^4}\\ \end{align*}
Mathematica [A] time = 0.204044, size = 213, normalized size = 0.93 \[ \frac{b \left (84 a^2 b^4 \sqrt{c+d x} \left (11 c^2-3 c d x+d^2 x^2\right )-140 a^4 b^2 (8 c-d x) \sqrt{c+d x}-105 a^3 b^3 d x (d x-4 c)-210 a^5 b d x+420 a^6 \sqrt{c+d x}-35 a b^5 d x \left (6 c^2-3 c d x+2 d^2 x^2\right )+12 b^6 \sqrt{c+d x} \left (8 c^2 d x-16 c^3-6 c d^2 x^2+5 d^3 x^3\right )\right )-420 a \left (a^2-b^2 c\right )^3 \log \left (a+b \sqrt{c+d x}\right )}{210 b^8 d^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 394, normalized size = 1.7 \begin{align*} -{\frac{{a}^{5}c}{{d}^{4}{b}^{6}}}+{\frac{5\,{a}^{3}{c}^{2}}{2\,{d}^{4}{b}^{4}}}-{\frac{11\,a{c}^{3}}{6\,{d}^{4}{b}^{2}}}-{\frac{6\,c}{5\,b{d}^{4}} \left ( dx+c \right ) ^{{\frac{5}{2}}}}+{\frac{2\,{a}^{2}}{5\,{d}^{4}{b}^{3}} \left ( dx+c \right ) ^{{\frac{5}{2}}}}+2\,{\frac{{c}^{2} \left ( dx+c \right ) ^{3/2}}{b{d}^{4}}}-2\,{\frac{{c}^{3}\sqrt{dx+c}}{b{d}^{4}}}+{\frac{2\,{a}^{4}}{3\,{d}^{4}{b}^{5}} \left ( dx+c \right ) ^{{\frac{3}{2}}}}+2\,{\frac{{a}^{3}xc}{{b}^{4}{d}^{3}}}-{\frac{ax{c}^{2}}{{d}^{3}{b}^{2}}}+{\frac{a{x}^{2}c}{2\,{b}^{2}{d}^{2}}}-{\frac{x{a}^{5}}{{d}^{3}{b}^{6}}}-{\frac{{x}^{2}{a}^{3}}{2\,{b}^{4}{d}^{2}}}-6\,{\frac{{a}^{4}c\sqrt{dx+c}}{{d}^{4}{b}^{5}}}-{\frac{a{x}^{3}}{3\,{b}^{2}d}}-2\,{\frac{ \left ( dx+c \right ) ^{3/2}{a}^{2}c}{{d}^{4}{b}^{3}}}+6\,{\frac{{a}^{2}{c}^{2}\sqrt{dx+c}}{{d}^{4}{b}^{3}}}+{\frac{2}{7\,b{d}^{4}} \left ( dx+c \right ) ^{{\frac{7}{2}}}}+2\,{\frac{{a}^{6}\sqrt{dx+c}}{{d}^{4}{b}^{7}}}+2\,{\frac{a\ln \left ( a+b\sqrt{dx+c} \right ){c}^{3}}{{d}^{4}{b}^{2}}}-6\,{\frac{{a}^{3}\ln \left ( a+b\sqrt{dx+c} \right ){c}^{2}}{{d}^{4}{b}^{4}}}+6\,{\frac{{a}^{5}\ln \left ( a+b\sqrt{dx+c} \right ) c}{{d}^{4}{b}^{6}}}-2\,{\frac{{a}^{7}\ln \left ( a+b\sqrt{dx+c} \right ) }{{d}^{4}{b}^{8}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.39428, size = 328, normalized size = 1.43 \begin{align*} \frac{\frac{60 \,{\left (d x + c\right )}^{\frac{7}{2}} b^{6} - 70 \,{\left (d x + c\right )}^{3} a b^{5} - 84 \,{\left (3 \, b^{6} c - a^{2} b^{4}\right )}{\left (d x + c\right )}^{\frac{5}{2}} + 105 \,{\left (3 \, a b^{5} c - a^{3} b^{3}\right )}{\left (d x + c\right )}^{2} + 140 \,{\left (3 \, b^{6} c^{2} - 3 \, a^{2} b^{4} c + a^{4} b^{2}\right )}{\left (d x + c\right )}^{\frac{3}{2}} - 210 \,{\left (3 \, a b^{5} c^{2} - 3 \, a^{3} b^{3} c + a^{5} b\right )}{\left (d x + c\right )} - 420 \,{\left (b^{6} c^{3} - 3 \, a^{2} b^{4} c^{2} + 3 \, a^{4} b^{2} c - a^{6}\right )} \sqrt{d x + c}}{b^{7}} + \frac{420 \,{\left (a b^{6} c^{3} - 3 \, a^{3} b^{4} c^{2} + 3 \, a^{5} b^{2} c - a^{7}\right )} \log \left (\sqrt{d x + c} b + a\right )}{b^{8}}}{210 \, d^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.8614, size = 501, normalized size = 2.18 \begin{align*} -\frac{70 \, a b^{6} d^{3} x^{3} - 105 \,{\left (a b^{6} c - a^{3} b^{4}\right )} d^{2} x^{2} + 210 \,{\left (a b^{6} c^{2} - 2 \, a^{3} b^{4} c + a^{5} b^{2}\right )} d x - 420 \,{\left (a b^{6} c^{3} - 3 \, a^{3} b^{4} c^{2} + 3 \, a^{5} b^{2} c - a^{7}\right )} \log \left (\sqrt{d x + c} b + a\right ) - 4 \,{\left (15 \, b^{7} d^{3} x^{3} - 48 \, b^{7} c^{3} + 231 \, a^{2} b^{5} c^{2} - 280 \, a^{4} b^{3} c + 105 \, a^{6} b - 3 \,{\left (6 \, b^{7} c - 7 \, a^{2} b^{5}\right )} d^{2} x^{2} +{\left (24 \, b^{7} c^{2} - 63 \, a^{2} b^{5} c + 35 \, a^{4} b^{3}\right )} d x\right )} \sqrt{d x + c}}{210 \, b^{8} d^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{a + b \sqrt{c + d x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.37316, size = 533, normalized size = 2.32 \begin{align*} \frac{2 \,{\left (a b^{6} c^{3} - 3 \, a^{3} b^{4} c^{2} + 3 \, a^{5} b^{2} c - a^{7}\right )} \log \left ({\left | \sqrt{d x + c} b + a \right |}\right )}{b^{8} d^{4}} - \frac{2 \,{\left (a b^{6} c^{3} \log \left ({\left | a \right |}\right ) - 3 \, a^{3} b^{4} c^{2} \log \left ({\left | a \right |}\right ) + 3 \, a^{5} b^{2} c \log \left ({\left | a \right |}\right ) - a^{7} \log \left ({\left | a \right |}\right )\right )}}{b^{8} d^{4}} + \frac{60 \,{\left (d x + c\right )}^{\frac{7}{2}} b^{6} d^{24} - 252 \,{\left (d x + c\right )}^{\frac{5}{2}} b^{6} c d^{24} + 420 \,{\left (d x + c\right )}^{\frac{3}{2}} b^{6} c^{2} d^{24} - 420 \, \sqrt{d x + c} b^{6} c^{3} d^{24} - 70 \,{\left (d x + c\right )}^{3} a b^{5} d^{24} + 315 \,{\left (d x + c\right )}^{2} a b^{5} c d^{24} - 630 \,{\left (d x + c\right )} a b^{5} c^{2} d^{24} + 84 \,{\left (d x + c\right )}^{\frac{5}{2}} a^{2} b^{4} d^{24} - 420 \,{\left (d x + c\right )}^{\frac{3}{2}} a^{2} b^{4} c d^{24} + 1260 \, \sqrt{d x + c} a^{2} b^{4} c^{2} d^{24} - 105 \,{\left (d x + c\right )}^{2} a^{3} b^{3} d^{24} + 630 \,{\left (d x + c\right )} a^{3} b^{3} c d^{24} + 140 \,{\left (d x + c\right )}^{\frac{3}{2}} a^{4} b^{2} d^{24} - 1260 \, \sqrt{d x + c} a^{4} b^{2} c d^{24} - 210 \,{\left (d x + c\right )} a^{5} b d^{24} + 420 \, \sqrt{d x + c} a^{6} d^{24}}{210 \, b^{7} d^{28}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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