Optimal. Leaf size=198 \[ \frac{3 a^4 A x \sqrt{a+c x^2}}{256 c^2}+\frac{a^3 A x \left (a+c x^2\right )^{3/2}}{128 c^2}+\frac{a^2 A x \left (a+c x^2\right )^{5/2}}{160 c^2}+\frac{3 a^5 A \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{256 c^{5/2}}+\frac{a \left (a+c x^2\right )^{7/2} (640 a B-2079 A c x)}{55440 c^3}+\frac{A x^3 \left (a+c x^2\right )^{7/2}}{10 c}-\frac{4 a B x^2 \left (a+c x^2\right )^{7/2}}{99 c^2}+\frac{B x^4 \left (a+c x^2\right )^{7/2}}{11 c} \]
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Rubi [A] time = 0.151294, antiderivative size = 198, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {833, 780, 195, 217, 206} \[ \frac{3 a^4 A x \sqrt{a+c x^2}}{256 c^2}+\frac{a^3 A x \left (a+c x^2\right )^{3/2}}{128 c^2}+\frac{a^2 A x \left (a+c x^2\right )^{5/2}}{160 c^2}+\frac{3 a^5 A \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{256 c^{5/2}}+\frac{a \left (a+c x^2\right )^{7/2} (640 a B-2079 A c x)}{55440 c^3}+\frac{A x^3 \left (a+c x^2\right )^{7/2}}{10 c}-\frac{4 a B x^2 \left (a+c x^2\right )^{7/2}}{99 c^2}+\frac{B x^4 \left (a+c x^2\right )^{7/2}}{11 c} \]
Antiderivative was successfully verified.
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Rule 833
Rule 780
Rule 195
Rule 217
Rule 206
Rubi steps
\begin{align*} \int x^4 (A+B x) \left (a+c x^2\right )^{5/2} \, dx &=\frac{B x^4 \left (a+c x^2\right )^{7/2}}{11 c}+\frac{\int x^3 (-4 a B+11 A c x) \left (a+c x^2\right )^{5/2} \, dx}{11 c}\\ &=\frac{A x^3 \left (a+c x^2\right )^{7/2}}{10 c}+\frac{B x^4 \left (a+c x^2\right )^{7/2}}{11 c}+\frac{\int x^2 (-33 a A c-40 a B c x) \left (a+c x^2\right )^{5/2} \, dx}{110 c^2}\\ &=-\frac{4 a B x^2 \left (a+c x^2\right )^{7/2}}{99 c^2}+\frac{A x^3 \left (a+c x^2\right )^{7/2}}{10 c}+\frac{B x^4 \left (a+c x^2\right )^{7/2}}{11 c}+\frac{\int x \left (80 a^2 B c-297 a A c^2 x\right ) \left (a+c x^2\right )^{5/2} \, dx}{990 c^3}\\ &=-\frac{4 a B x^2 \left (a+c x^2\right )^{7/2}}{99 c^2}+\frac{A x^3 \left (a+c x^2\right )^{7/2}}{10 c}+\frac{B x^4 \left (a+c x^2\right )^{7/2}}{11 c}+\frac{a (640 a B-2079 A c x) \left (a+c x^2\right )^{7/2}}{55440 c^3}+\frac{\left (3 a^2 A\right ) \int \left (a+c x^2\right )^{5/2} \, dx}{80 c^2}\\ &=\frac{a^2 A x \left (a+c x^2\right )^{5/2}}{160 c^2}-\frac{4 a B x^2 \left (a+c x^2\right )^{7/2}}{99 c^2}+\frac{A x^3 \left (a+c x^2\right )^{7/2}}{10 c}+\frac{B x^4 \left (a+c x^2\right )^{7/2}}{11 c}+\frac{a (640 a B-2079 A c x) \left (a+c x^2\right )^{7/2}}{55440 c^3}+\frac{\left (a^3 A\right ) \int \left (a+c x^2\right )^{3/2} \, dx}{32 c^2}\\ &=\frac{a^3 A x \left (a+c x^2\right )^{3/2}}{128 c^2}+\frac{a^2 A x \left (a+c x^2\right )^{5/2}}{160 c^2}-\frac{4 a B x^2 \left (a+c x^2\right )^{7/2}}{99 c^2}+\frac{A x^3 \left (a+c x^2\right )^{7/2}}{10 c}+\frac{B x^4 \left (a+c x^2\right )^{7/2}}{11 c}+\frac{a (640 a B-2079 A c x) \left (a+c x^2\right )^{7/2}}{55440 c^3}+\frac{\left (3 a^4 A\right ) \int \sqrt{a+c x^2} \, dx}{128 c^2}\\ &=\frac{3 a^4 A x \sqrt{a+c x^2}}{256 c^2}+\frac{a^3 A x \left (a+c x^2\right )^{3/2}}{128 c^2}+\frac{a^2 A x \left (a+c x^2\right )^{5/2}}{160 c^2}-\frac{4 a B x^2 \left (a+c x^2\right )^{7/2}}{99 c^2}+\frac{A x^3 \left (a+c x^2\right )^{7/2}}{10 c}+\frac{B x^4 \left (a+c x^2\right )^{7/2}}{11 c}+\frac{a (640 a B-2079 A c x) \left (a+c x^2\right )^{7/2}}{55440 c^3}+\frac{\left (3 a^5 A\right ) \int \frac{1}{\sqrt{a+c x^2}} \, dx}{256 c^2}\\ &=\frac{3 a^4 A x \sqrt{a+c x^2}}{256 c^2}+\frac{a^3 A x \left (a+c x^2\right )^{3/2}}{128 c^2}+\frac{a^2 A x \left (a+c x^2\right )^{5/2}}{160 c^2}-\frac{4 a B x^2 \left (a+c x^2\right )^{7/2}}{99 c^2}+\frac{A x^3 \left (a+c x^2\right )^{7/2}}{10 c}+\frac{B x^4 \left (a+c x^2\right )^{7/2}}{11 c}+\frac{a (640 a B-2079 A c x) \left (a+c x^2\right )^{7/2}}{55440 c^3}+\frac{\left (3 a^5 A\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{a+c x^2}}\right )}{256 c^2}\\ &=\frac{3 a^4 A x \sqrt{a+c x^2}}{256 c^2}+\frac{a^3 A x \left (a+c x^2\right )^{3/2}}{128 c^2}+\frac{a^2 A x \left (a+c x^2\right )^{5/2}}{160 c^2}-\frac{4 a B x^2 \left (a+c x^2\right )^{7/2}}{99 c^2}+\frac{A x^3 \left (a+c x^2\right )^{7/2}}{10 c}+\frac{B x^4 \left (a+c x^2\right )^{7/2}}{11 c}+\frac{a (640 a B-2079 A c x) \left (a+c x^2\right )^{7/2}}{55440 c^3}+\frac{3 a^5 A \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{256 c^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.292198, size = 151, normalized size = 0.76 \[ \frac{\sqrt{a+c x^2} \left (8 a^2 c^3 x^5 (21483 A+18080 B x)+30 a^3 c^2 x^3 (231 A+128 B x)-5 a^4 c x (2079 A+1024 B x)+\frac{10395 a^{9/2} A \sqrt{c} \sinh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{\frac{c x^2}{a}+1}}+10240 a^5 B+112 a c^4 x^7 (2079 A+1840 B x)+8064 c^5 x^9 (11 A+10 B x)\right )}{887040 c^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 174, normalized size = 0.9 \begin{align*}{\frac{B{x}^{4}}{11\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{4\,aB{x}^{2}}{99\,{c}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{8\,B{a}^{2}}{693\,{c}^{3}} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{A{x}^{3}}{10\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{3\,aAx}{80\,{c}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{{a}^{2}Ax}{160\,{c}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{{a}^{3}Ax}{128\,{c}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{3\,{a}^{4}Ax}{256\,{c}^{2}}\sqrt{c{x}^{2}+a}}+{\frac{3\,A{a}^{5}}{256}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{5}{2}}}} \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 = 1.80274, size = 859, normalized size = 4.34 \begin{align*} \left [\frac{10395 \, A a^{5} \sqrt{c} \log \left (-2 \, c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{c} x - a\right ) + 2 \,{\left (80640 \, B c^{5} x^{10} + 88704 \, A c^{5} x^{9} + 206080 \, B a c^{4} x^{8} + 232848 \, A a c^{4} x^{7} + 144640 \, B a^{2} c^{3} x^{6} + 171864 \, A a^{2} c^{3} x^{5} + 3840 \, B a^{3} c^{2} x^{4} + 6930 \, A a^{3} c^{2} x^{3} - 5120 \, B a^{4} c x^{2} - 10395 \, A a^{4} c x + 10240 \, B a^{5}\right )} \sqrt{c x^{2} + a}}{1774080 \, c^{3}}, -\frac{10395 \, A a^{5} \sqrt{-c} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right ) -{\left (80640 \, B c^{5} x^{10} + 88704 \, A c^{5} x^{9} + 206080 \, B a c^{4} x^{8} + 232848 \, A a c^{4} x^{7} + 144640 \, B a^{2} c^{3} x^{6} + 171864 \, A a^{2} c^{3} x^{5} + 3840 \, B a^{3} c^{2} x^{4} + 6930 \, A a^{3} c^{2} x^{3} - 5120 \, B a^{4} c x^{2} - 10395 \, A a^{4} c x + 10240 \, B a^{5}\right )} \sqrt{c x^{2} + a}}{887040 \, c^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 46.5287, size = 541, normalized size = 2.73 \begin{align*} - \frac{3 A a^{\frac{9}{2}} x}{256 c^{2} \sqrt{1 + \frac{c x^{2}}{a}}} - \frac{A a^{\frac{7}{2}} x^{3}}{256 c \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{129 A a^{\frac{5}{2}} x^{5}}{640 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{73 A a^{\frac{3}{2}} c x^{7}}{160 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{29 A \sqrt{a} c^{2} x^{9}}{80 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{3 A a^{5} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{256 c^{\frac{5}{2}}} + \frac{A c^{3} x^{11}}{10 \sqrt{a} \sqrt{1 + \frac{c x^{2}}{a}}} + B a^{2} \left (\begin{cases} \frac{8 a^{3} \sqrt{a + c x^{2}}}{105 c^{3}} - \frac{4 a^{2} x^{2} \sqrt{a + c x^{2}}}{105 c^{2}} + \frac{a x^{4} \sqrt{a + c x^{2}}}{35 c} + \frac{x^{6} \sqrt{a + c x^{2}}}{7} & \text{for}\: c \neq 0 \\\frac{\sqrt{a} x^{6}}{6} & \text{otherwise} \end{cases}\right ) + 2 B a c \left (\begin{cases} - \frac{16 a^{4} \sqrt{a + c x^{2}}}{315 c^{4}} + \frac{8 a^{3} x^{2} \sqrt{a + c x^{2}}}{315 c^{3}} - \frac{2 a^{2} x^{4} \sqrt{a + c x^{2}}}{105 c^{2}} + \frac{a x^{6} \sqrt{a + c x^{2}}}{63 c} + \frac{x^{8} \sqrt{a + c x^{2}}}{9} & \text{for}\: c \neq 0 \\\frac{\sqrt{a} x^{8}}{8} & \text{otherwise} \end{cases}\right ) + B c^{2} \left (\begin{cases} \frac{128 a^{5} \sqrt{a + c x^{2}}}{3465 c^{5}} - \frac{64 a^{4} x^{2} \sqrt{a + c x^{2}}}{3465 c^{4}} + \frac{16 a^{3} x^{4} \sqrt{a + c x^{2}}}{1155 c^{3}} - \frac{8 a^{2} x^{6} \sqrt{a + c x^{2}}}{693 c^{2}} + \frac{a x^{8} \sqrt{a + c x^{2}}}{99 c} + \frac{x^{10} \sqrt{a + c x^{2}}}{11} & \text{for}\: c \neq 0 \\\frac{\sqrt{a} x^{10}}{10} & \text{otherwise} \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14589, size = 209, normalized size = 1.06 \begin{align*} -\frac{3 \, A a^{5} \log \left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{256 \, c^{\frac{5}{2}}} + \frac{1}{887040} \,{\left (\frac{10240 \, B a^{5}}{c^{3}} -{\left (\frac{10395 \, A a^{4}}{c^{2}} + 2 \,{\left (\frac{2560 \, B a^{4}}{c^{2}} -{\left (\frac{3465 \, A a^{3}}{c} + 4 \,{\left (\frac{480 \, B a^{3}}{c} +{\left (21483 \, A a^{2} + 2 \,{\left (9040 \, B a^{2} + 7 \,{\left (2079 \, A a c + 8 \,{\left (230 \, B a c + 9 \,{\left (10 \, B c^{2} x + 11 \, A c^{2}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt{c x^{2} + a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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