Optimal. Leaf size=127 \[ -\frac {a^3 A \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{16 c^{3/2}}-\frac {a^2 A x \sqrt {a+c x^2}}{16 c}-\frac {\left (a+c x^2\right )^{5/2} (12 a B-35 A c x)}{210 c^2}-\frac {a A x \left (a+c x^2\right )^{3/2}}{24 c}+\frac {B x^2 \left (a+c x^2\right )^{5/2}}{7 c} \]
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Rubi [A] time = 0.06, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 6, 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} \begin {gather*} -\frac {a^3 A \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{16 c^{3/2}}-\frac {a^2 A x \sqrt {a+c x^2}}{16 c}-\frac {\left (a+c x^2\right )^{5/2} (12 a B-35 A c x)}{210 c^2}-\frac {a A x \left (a+c x^2\right )^{3/2}}{24 c}+\frac {B x^2 \left (a+c x^2\right )^{5/2}}{7 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 195
Rule 206
Rule 217
Rule 780
Rule 833
Rubi steps
\begin {align*} \int x^2 (A+B x) \left (a+c x^2\right )^{3/2} \, dx &=\frac {B x^2 \left (a+c x^2\right )^{5/2}}{7 c}+\frac {\int x (-2 a B+7 A c x) \left (a+c x^2\right )^{3/2} \, dx}{7 c}\\ &=\frac {B x^2 \left (a+c x^2\right )^{5/2}}{7 c}-\frac {(12 a B-35 A c x) \left (a+c x^2\right )^{5/2}}{210 c^2}-\frac {(a A) \int \left (a+c x^2\right )^{3/2} \, dx}{6 c}\\ &=-\frac {a A x \left (a+c x^2\right )^{3/2}}{24 c}+\frac {B x^2 \left (a+c x^2\right )^{5/2}}{7 c}-\frac {(12 a B-35 A c x) \left (a+c x^2\right )^{5/2}}{210 c^2}-\frac {\left (a^2 A\right ) \int \sqrt {a+c x^2} \, dx}{8 c}\\ &=-\frac {a^2 A x \sqrt {a+c x^2}}{16 c}-\frac {a A x \left (a+c x^2\right )^{3/2}}{24 c}+\frac {B x^2 \left (a+c x^2\right )^{5/2}}{7 c}-\frac {(12 a B-35 A c x) \left (a+c x^2\right )^{5/2}}{210 c^2}-\frac {\left (a^3 A\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{16 c}\\ &=-\frac {a^2 A x \sqrt {a+c x^2}}{16 c}-\frac {a A x \left (a+c x^2\right )^{3/2}}{24 c}+\frac {B x^2 \left (a+c x^2\right )^{5/2}}{7 c}-\frac {(12 a B-35 A c x) \left (a+c x^2\right )^{5/2}}{210 c^2}-\frac {\left (a^3 A\right ) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{16 c}\\ &=-\frac {a^2 A x \sqrt {a+c x^2}}{16 c}-\frac {a A x \left (a+c x^2\right )^{3/2}}{24 c}+\frac {B x^2 \left (a+c x^2\right )^{5/2}}{7 c}-\frac {(12 a B-35 A c x) \left (a+c x^2\right )^{5/2}}{210 c^2}-\frac {a^3 A \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{16 c^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 113, normalized size = 0.89 \begin {gather*} \frac {\sqrt {a+c x^2} \left (-\frac {105 a^{5/2} A \sqrt {c} \sinh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {\frac {c x^2}{a}+1}}-96 a^3 B+3 a^2 c x (35 A+16 B x)+2 a c^2 x^3 (245 A+192 B x)+40 c^3 x^5 (7 A+6 B x)\right )}{1680 c^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.42, size = 116, normalized size = 0.91 \begin {gather*} \frac {a^3 A \log \left (\sqrt {a+c x^2}-\sqrt {c} x\right )}{16 c^{3/2}}+\frac {\sqrt {a+c x^2} \left (-96 a^3 B+105 a^2 A c x+48 a^2 B c x^2+490 a A c^2 x^3+384 a B c^2 x^4+280 A c^3 x^5+240 B c^3 x^6\right )}{1680 c^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 223, normalized size = 1.76 \begin {gather*} \left [\frac {105 \, A a^{3} \sqrt {c} \log \left (-2 \, c x^{2} + 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + 2 \, {\left (240 \, B c^{3} x^{6} + 280 \, A c^{3} x^{5} + 384 \, B a c^{2} x^{4} + 490 \, A a c^{2} x^{3} + 48 \, B a^{2} c x^{2} + 105 \, A a^{2} c x - 96 \, B a^{3}\right )} \sqrt {c x^{2} + a}}{3360 \, c^{2}}, \frac {105 \, A a^{3} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) + {\left (240 \, B c^{3} x^{6} + 280 \, A c^{3} x^{5} + 384 \, B a c^{2} x^{4} + 490 \, A a c^{2} x^{3} + 48 \, B a^{2} c x^{2} + 105 \, A a^{2} c x - 96 \, B a^{3}\right )} \sqrt {c x^{2} + a}}{1680 \, c^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 103, normalized size = 0.81 \begin {gather*} \frac {A a^{3} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{16 \, c^{\frac {3}{2}}} - \frac {1}{1680} \, \sqrt {c x^{2} + a} {\left (\frac {96 \, B a^{3}}{c^{2}} - {\left (\frac {105 \, A a^{2}}{c} + 2 \, {\left (\frac {24 \, B a^{2}}{c} + {\left (245 \, A a + 4 \, {\left (48 \, B a + 5 \, {\left (6 \, B c x + 7 \, A c\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 113, normalized size = 0.89 \begin {gather*} -\frac {A \,a^{3} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{16 c^{\frac {3}{2}}}-\frac {\sqrt {c \,x^{2}+a}\, A \,a^{2} x}{16 c}-\frac {\left (c \,x^{2}+a \right )^{\frac {3}{2}} A a x}{24 c}+\frac {\left (c \,x^{2}+a \right )^{\frac {5}{2}} B \,x^{2}}{7 c}+\frac {\left (c \,x^{2}+a \right )^{\frac {5}{2}} A x}{6 c}-\frac {2 \left (c \,x^{2}+a \right )^{\frac {5}{2}} B a}{35 c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 105, normalized size = 0.83 \begin {gather*} \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} B x^{2}}{7 \, c} + \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} A x}{6 \, c} - \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} A a x}{24 \, c} - \frac {\sqrt {c x^{2} + a} A a^{2} x}{16 \, c} - \frac {A a^{3} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{16 \, c^{\frac {3}{2}}} - \frac {2 \, {\left (c x^{2} + a\right )}^{\frac {5}{2}} B a}{35 \, c^{2}} \end {gather*}
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 \begin {gather*} \int x^2\,{\left (c\,x^2+a\right )}^{3/2}\,\left (A+B\,x\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 12.45, size = 287, normalized size = 2.26 \begin {gather*} \frac {A a^{\frac {5}{2}} x}{16 c \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {17 A a^{\frac {3}{2}} x^{3}}{48 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {11 A \sqrt {a} c x^{5}}{24 \sqrt {1 + \frac {c x^{2}}{a}}} - \frac {A a^{3} \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {a}} \right )}}{16 c^{\frac {3}{2}}} + \frac {A c^{2} x^{7}}{6 \sqrt {a} \sqrt {1 + \frac {c x^{2}}{a}}} + B a \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 ) + B 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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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