Optimal. Leaf size=104 \[ -\frac {a^2 A \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{8 c^{3/2}}-\frac {\left (a+c x^2\right )^{3/2} (8 a B-15 A c x)}{60 c^2}-\frac {a A x \sqrt {a+c x^2}}{8 c}+\frac {B x^2 \left (a+c x^2\right )^{3/2}}{5 c} \]
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Rubi [A] time = 0.05, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 5, 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^2 A \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{8 c^{3/2}}-\frac {\left (a+c x^2\right )^{3/2} (8 a B-15 A c x)}{60 c^2}-\frac {a A x \sqrt {a+c x^2}}{8 c}+\frac {B x^2 \left (a+c x^2\right )^{3/2}}{5 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) \sqrt {a+c x^2} \, dx &=\frac {B x^2 \left (a+c x^2\right )^{3/2}}{5 c}+\frac {\int x (-2 a B+5 A c x) \sqrt {a+c x^2} \, dx}{5 c}\\ &=\frac {B x^2 \left (a+c x^2\right )^{3/2}}{5 c}-\frac {(8 a B-15 A c x) \left (a+c x^2\right )^{3/2}}{60 c^2}-\frac {(a A) \int \sqrt {a+c x^2} \, dx}{4 c}\\ &=-\frac {a A x \sqrt {a+c x^2}}{8 c}+\frac {B x^2 \left (a+c x^2\right )^{3/2}}{5 c}-\frac {(8 a B-15 A c x) \left (a+c x^2\right )^{3/2}}{60 c^2}-\frac {\left (a^2 A\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{8 c}\\ &=-\frac {a A x \sqrt {a+c x^2}}{8 c}+\frac {B x^2 \left (a+c x^2\right )^{3/2}}{5 c}-\frac {(8 a B-15 A c x) \left (a+c x^2\right )^{3/2}}{60 c^2}-\frac {\left (a^2 A\right ) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{8 c}\\ &=-\frac {a A x \sqrt {a+c x^2}}{8 c}+\frac {B x^2 \left (a+c x^2\right )^{3/2}}{5 c}-\frac {(8 a B-15 A c x) \left (a+c x^2\right )^{3/2}}{60 c^2}-\frac {a^2 A \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{8 c^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 93, normalized size = 0.89 \begin {gather*} \frac {\sqrt {a+c x^2} \left (-\frac {15 a^{3/2} A \sqrt {c} \sinh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {\frac {c x^2}{a}+1}}-16 a^2 B+a c x (15 A+8 B x)+6 c^2 x^3 (5 A+4 B x)\right )}{120 c^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.23, size = 92, normalized size = 0.88 \begin {gather*} \frac {\sqrt {a+c x^2} \left (-16 a^2 B+15 a A c x+8 a B c x^2+30 A c^2 x^3+24 B c^2 x^4\right )}{120 c^2}+\frac {a^2 A \log \left (\sqrt {a+c x^2}-\sqrt {c} x\right )}{8 c^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 175, normalized size = 1.68 \begin {gather*} \left [\frac {15 \, A a^{2} \sqrt {c} \log \left (-2 \, c x^{2} + 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + 2 \, {\left (24 \, B c^{2} x^{4} + 30 \, A c^{2} x^{3} + 8 \, B a c x^{2} + 15 \, A a c x - 16 \, B a^{2}\right )} \sqrt {c x^{2} + a}}{240 \, c^{2}}, \frac {15 \, A a^{2} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) + {\left (24 \, B c^{2} x^{4} + 30 \, A c^{2} x^{3} + 8 \, B a c x^{2} + 15 \, A a c x - 16 \, B a^{2}\right )} \sqrt {c x^{2} + a}}{120 \, c^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.53, size = 81, normalized size = 0.78 \begin {gather*} \frac {A a^{2} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{8 \, c^{\frac {3}{2}}} + \frac {1}{120} \, \sqrt {c x^{2} + a} {\left ({\left (2 \, {\left (3 \, {\left (4 \, B x + 5 \, A\right )} x + \frac {4 \, B a}{c}\right )} x + \frac {15 \, A a}{c}\right )} x - \frac {16 \, B a^{2}}{c^{2}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 94, normalized size = 0.90 \begin {gather*} -\frac {A \,a^{2} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{8 c^{\frac {3}{2}}}-\frac {\sqrt {c \,x^{2}+a}\, A a x}{8 c}+\frac {\left (c \,x^{2}+a \right )^{\frac {3}{2}} B \,x^{2}}{5 c}+\frac {\left (c \,x^{2}+a \right )^{\frac {3}{2}} A x}{4 c}-\frac {2 \left (c \,x^{2}+a \right )^{\frac {3}{2}} B a}{15 c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 86, normalized size = 0.83 \begin {gather*} \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} B x^{2}}{5 \, c} + \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} A x}{4 \, c} - \frac {\sqrt {c x^{2} + a} A a x}{8 \, c} - \frac {A a^{2} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{8 \, c^{\frac {3}{2}}} - \frac {2 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} B a}{15 \, 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\,\sqrt {c\,x^2+a}\,\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 = 5.46, size = 165, normalized size = 1.59 \begin {gather*} \frac {A a^{\frac {3}{2}} x}{8 c \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {3 A \sqrt {a} x^{3}}{8 \sqrt {1 + \frac {c x^{2}}{a}}} - \frac {A a^{2} \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {a}} \right )}}{8 c^{\frac {3}{2}}} + \frac {A c x^{5}}{4 \sqrt {a} \sqrt {1 + \frac {c x^{2}}{a}}} + B \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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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