Optimal. Leaf size=173 \[ \frac {3 a^5 B \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{256 c^{5/2}}+\frac {3 a^4 B x \sqrt {a+c x^2}}{256 c^2}+\frac {a^3 B x \left (a+c x^2\right )^{3/2}}{128 c^2}+\frac {a^2 B x \left (a+c x^2\right )^{5/2}}{160 c^2}-\frac {a \left (a+c x^2\right )^{7/2} (160 A+189 B x)}{5040 c^2}+\frac {A x^2 \left (a+c x^2\right )^{7/2}}{9 c}+\frac {B x^3 \left (a+c x^2\right )^{7/2}}{10 c} \]
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Rubi [A] time = 0.10, antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {833, 780, 195, 217, 206} \[ \frac {3 a^4 B x \sqrt {a+c x^2}}{256 c^2}+\frac {a^3 B x \left (a+c x^2\right )^{3/2}}{128 c^2}+\frac {a^2 B x \left (a+c x^2\right )^{5/2}}{160 c^2}+\frac {3 a^5 B \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{256 c^{5/2}}-\frac {a \left (a+c x^2\right )^{7/2} (160 A+189 B x)}{5040 c^2}+\frac {A x^2 \left (a+c x^2\right )^{7/2}}{9 c}+\frac {B x^3 \left (a+c x^2\right )^{7/2}}{10 c} \]
Antiderivative was successfully verified.
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Rule 195
Rule 206
Rule 217
Rule 780
Rule 833
Rubi steps
\begin {align*} \int x^3 (A+B x) \left (a+c x^2\right )^{5/2} \, dx &=\frac {B x^3 \left (a+c x^2\right )^{7/2}}{10 c}+\frac {\int x^2 (-3 a B+10 A c x) \left (a+c x^2\right )^{5/2} \, dx}{10 c}\\ &=\frac {A x^2 \left (a+c x^2\right )^{7/2}}{9 c}+\frac {B x^3 \left (a+c x^2\right )^{7/2}}{10 c}+\frac {\int x (-20 a A c-27 a B c x) \left (a+c x^2\right )^{5/2} \, dx}{90 c^2}\\ &=\frac {A x^2 \left (a+c x^2\right )^{7/2}}{9 c}+\frac {B x^3 \left (a+c x^2\right )^{7/2}}{10 c}-\frac {a (160 A+189 B x) \left (a+c x^2\right )^{7/2}}{5040 c^2}+\frac {\left (3 a^2 B\right ) \int \left (a+c x^2\right )^{5/2} \, dx}{80 c^2}\\ &=\frac {a^2 B x \left (a+c x^2\right )^{5/2}}{160 c^2}+\frac {A x^2 \left (a+c x^2\right )^{7/2}}{9 c}+\frac {B x^3 \left (a+c x^2\right )^{7/2}}{10 c}-\frac {a (160 A+189 B x) \left (a+c x^2\right )^{7/2}}{5040 c^2}+\frac {\left (a^3 B\right ) \int \left (a+c x^2\right )^{3/2} \, dx}{32 c^2}\\ &=\frac {a^3 B x \left (a+c x^2\right )^{3/2}}{128 c^2}+\frac {a^2 B x \left (a+c x^2\right )^{5/2}}{160 c^2}+\frac {A x^2 \left (a+c x^2\right )^{7/2}}{9 c}+\frac {B x^3 \left (a+c x^2\right )^{7/2}}{10 c}-\frac {a (160 A+189 B x) \left (a+c x^2\right )^{7/2}}{5040 c^2}+\frac {\left (3 a^4 B\right ) \int \sqrt {a+c x^2} \, dx}{128 c^2}\\ &=\frac {3 a^4 B x \sqrt {a+c x^2}}{256 c^2}+\frac {a^3 B x \left (a+c x^2\right )^{3/2}}{128 c^2}+\frac {a^2 B x \left (a+c x^2\right )^{5/2}}{160 c^2}+\frac {A x^2 \left (a+c x^2\right )^{7/2}}{9 c}+\frac {B x^3 \left (a+c x^2\right )^{7/2}}{10 c}-\frac {a (160 A+189 B x) \left (a+c x^2\right )^{7/2}}{5040 c^2}+\frac {\left (3 a^5 B\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{256 c^2}\\ &=\frac {3 a^4 B x \sqrt {a+c x^2}}{256 c^2}+\frac {a^3 B x \left (a+c x^2\right )^{3/2}}{128 c^2}+\frac {a^2 B x \left (a+c x^2\right )^{5/2}}{160 c^2}+\frac {A x^2 \left (a+c x^2\right )^{7/2}}{9 c}+\frac {B x^3 \left (a+c x^2\right )^{7/2}}{10 c}-\frac {a (160 A+189 B x) \left (a+c x^2\right )^{7/2}}{5040 c^2}+\frac {\left (3 a^5 B\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 B x \sqrt {a+c x^2}}{256 c^2}+\frac {a^3 B x \left (a+c x^2\right )^{3/2}}{128 c^2}+\frac {a^2 B x \left (a+c x^2\right )^{5/2}}{160 c^2}+\frac {A x^2 \left (a+c x^2\right )^{7/2}}{9 c}+\frac {B x^3 \left (a+c x^2\right )^{7/2}}{10 c}-\frac {a (160 A+189 B x) \left (a+c x^2\right )^{7/2}}{5040 c^2}+\frac {3 a^5 B \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{256 c^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.27, size = 145, normalized size = 0.84 \[ \frac {\sqrt {a+c x^2} \left (\frac {945 a^{9/2} B \sinh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {\frac {c x^2}{a}+1}}+\sqrt {c} \left (-5 a^4 (512 A+189 B x)+10 a^3 c x^2 (128 A+63 B x)+24 a^2 c^2 x^4 (800 A+651 B x)+16 a c^3 x^6 (1520 A+1323 B x)+896 c^4 x^8 (10 A+9 B x)\right )\right )}{80640 c^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.01, size = 302, normalized size = 1.75 \[ \left [\frac {945 \, B a^{5} \sqrt {c} \log \left (-2 \, c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + 2 \, {\left (8064 \, B c^{5} x^{9} + 8960 \, A c^{5} x^{8} + 21168 \, B a c^{4} x^{7} + 24320 \, A a c^{4} x^{6} + 15624 \, B a^{2} c^{3} x^{5} + 19200 \, A a^{2} c^{3} x^{4} + 630 \, B a^{3} c^{2} x^{3} + 1280 \, A a^{3} c^{2} x^{2} - 945 \, B a^{4} c x - 2560 \, A a^{4} c\right )} \sqrt {c x^{2} + a}}{161280 \, c^{3}}, -\frac {945 \, B a^{5} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - {\left (8064 \, B c^{5} x^{9} + 8960 \, A c^{5} x^{8} + 21168 \, B a c^{4} x^{7} + 24320 \, A a c^{4} x^{6} + 15624 \, B a^{2} c^{3} x^{5} + 19200 \, A a^{2} c^{3} x^{4} + 630 \, B a^{3} c^{2} x^{3} + 1280 \, A a^{3} c^{2} x^{2} - 945 \, B a^{4} c x - 2560 \, A a^{4} c\right )} \sqrt {c x^{2} + a}}{80640 \, c^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 140, normalized size = 0.81 \[ -\frac {3 \, B a^{5} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{256 \, c^{\frac {5}{2}}} - \frac {1}{80640} \, {\left (\frac {2560 \, A a^{4}}{c^{2}} + {\left (\frac {945 \, B a^{4}}{c^{2}} - 2 \, {\left (\frac {640 \, A a^{3}}{c} + {\left (\frac {315 \, B a^{3}}{c} + 4 \, {\left (2400 \, A a^{2} + {\left (1953 \, B a^{2} + 2 \, {\left (1520 \, A a c + 7 \, {\left (189 \, B a c + 8 \, {\left (9 \, B c^{2} x + 10 \, A c^{2}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt {c x^{2} + a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 153, normalized size = 0.88 \[ \frac {3 B \,a^{5} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{256 c^{\frac {5}{2}}}+\frac {3 \sqrt {c \,x^{2}+a}\, B \,a^{4} x}{256 c^{2}}+\frac {\left (c \,x^{2}+a \right )^{\frac {3}{2}} B \,a^{3} x}{128 c^{2}}+\frac {\left (c \,x^{2}+a \right )^{\frac {7}{2}} B \,x^{3}}{10 c}+\frac {\left (c \,x^{2}+a \right )^{\frac {7}{2}} A \,x^{2}}{9 c}+\frac {\left (c \,x^{2}+a \right )^{\frac {5}{2}} B \,a^{2} x}{160 c^{2}}-\frac {3 \left (c \,x^{2}+a \right )^{\frac {7}{2}} B a x}{80 c^{2}}-\frac {2 \left (c \,x^{2}+a \right )^{\frac {7}{2}} A a}{63 c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.59, size = 145, normalized size = 0.84 \[ \frac {{\left (c x^{2} + a\right )}^{\frac {7}{2}} B x^{3}}{10 \, c} + \frac {{\left (c x^{2} + a\right )}^{\frac {7}{2}} A x^{2}}{9 \, c} - \frac {3 \, {\left (c x^{2} + a\right )}^{\frac {7}{2}} B a x}{80 \, c^{2}} + \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} B a^{2} x}{160 \, c^{2}} + \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} B a^{3} x}{128 \, c^{2}} + \frac {3 \, \sqrt {c x^{2} + a} B a^{4} x}{256 \, c^{2}} + \frac {3 \, B a^{5} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{256 \, c^{\frac {5}{2}}} - \frac {2 \, {\left (c x^{2} + a\right )}^{\frac {7}{2}} A a}{63 \, c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^3\,{\left (c\,x^2+a\right )}^{5/2}\,\left (A+B\,x\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 34.72, size = 469, normalized size = 2.71 \[ A a^{2} \left (\begin {cases} - \frac {2 a^{2} \sqrt {a + c x^{2}}}{15 c^{2}} + \frac {a x^{2} \sqrt {a + c x^{2}}}{15 c} + \frac {x^{4} \sqrt {a + c x^{2}}}{5} & \text {for}\: c \neq 0 \\\frac {\sqrt {a} x^{4}}{4} & \text {otherwise} \end {cases}\right ) + 2 A a c \left (\begin {cases} \frac {8 a^{3} \sqrt {a + c x^{2}}}{105 c^{3}} - \frac {4 a^{2} x^{2} \sqrt {a + c x^{2}}}{105 c^{2}} + \frac {a x^{4} \sqrt {a + c x^{2}}}{35 c} + \frac {x^{6} \sqrt {a + c x^{2}}}{7} & \text {for}\: c \neq 0 \\\frac {\sqrt {a} x^{6}}{6} & \text {otherwise} \end {cases}\right ) + A c^{2} \left (\begin {cases} - \frac {16 a^{4} \sqrt {a + c x^{2}}}{315 c^{4}} + \frac {8 a^{3} x^{2} \sqrt {a + c x^{2}}}{315 c^{3}} - \frac {2 a^{2} x^{4} \sqrt {a + c x^{2}}}{105 c^{2}} + \frac {a x^{6} \sqrt {a + c x^{2}}}{63 c} + \frac {x^{8} \sqrt {a + c x^{2}}}{9} & \text {for}\: c \neq 0 \\\frac {\sqrt {a} x^{8}}{8} & \text {otherwise} \end {cases}\right ) - \frac {3 B a^{\frac {9}{2}} x}{256 c^{2} \sqrt {1 + \frac {c x^{2}}{a}}} - \frac {B a^{\frac {7}{2}} x^{3}}{256 c \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {129 B a^{\frac {5}{2}} x^{5}}{640 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {73 B a^{\frac {3}{2}} c x^{7}}{160 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {29 B \sqrt {a} c^{2} x^{9}}{80 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {3 B a^{5} \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {a}} \right )}}{256 c^{\frac {5}{2}}} + \frac {B c^{3} x^{11}}{10 \sqrt {a} \sqrt {1 + \frac {c x^{2}}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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