Optimal. Leaf size=79 \[ -\frac {2 A+3 B x}{3 c^2 \sqrt {a+c x^2}}-\frac {x^2 (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}+\frac {B \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{c^{5/2}} \]
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Rubi [A] time = 0.04, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {819, 778, 217, 206} \begin {gather*} -\frac {2 A+3 B x}{3 c^2 \sqrt {a+c x^2}}-\frac {x^2 (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}+\frac {B \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{c^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 778
Rule 819
Rubi steps
\begin {align*} \int \frac {x^3 (A+B x)}{\left (a+c x^2\right )^{5/2}} \, dx &=-\frac {x^2 (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}+\frac {\int \frac {x (2 a A+3 a B x)}{\left (a+c x^2\right )^{3/2}} \, dx}{3 a c}\\ &=-\frac {x^2 (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}-\frac {2 A+3 B x}{3 c^2 \sqrt {a+c x^2}}+\frac {B \int \frac {1}{\sqrt {a+c x^2}} \, dx}{c^2}\\ &=-\frac {x^2 (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}-\frac {2 A+3 B x}{3 c^2 \sqrt {a+c x^2}}+\frac {B \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{c^2}\\ &=-\frac {x^2 (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}-\frac {2 A+3 B x}{3 c^2 \sqrt {a+c x^2}}+\frac {B \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{c^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 69, normalized size = 0.87 \begin {gather*} \frac {B \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{c^{5/2}}-\frac {a (2 A+3 B x)+c x^2 (3 A+4 B x)}{3 c^2 \left (a+c x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.51, size = 72, normalized size = 0.91 \begin {gather*} \frac {-2 a A-3 a B x-3 A c x^2-4 B c x^3}{3 c^2 \left (a+c x^2\right )^{3/2}}-\frac {B \log \left (\sqrt {a+c x^2}-\sqrt {c} x\right )}{c^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 239, normalized size = 3.03 \begin {gather*} \left [\frac {3 \, {\left (B c^{2} x^{4} + 2 \, B a c x^{2} + B a^{2}\right )} \sqrt {c} \log \left (-2 \, c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) - 2 \, {\left (4 \, B c^{2} x^{3} + 3 \, A c^{2} x^{2} + 3 \, B a c x + 2 \, A a c\right )} \sqrt {c x^{2} + a}}{6 \, {\left (c^{5} x^{4} + 2 \, a c^{4} x^{2} + a^{2} c^{3}\right )}}, -\frac {3 \, {\left (B c^{2} x^{4} + 2 \, B a c x^{2} + B a^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) + {\left (4 \, B c^{2} x^{3} + 3 \, A c^{2} x^{2} + 3 \, B a c x + 2 \, A a c\right )} \sqrt {c x^{2} + a}}{3 \, {\left (c^{5} x^{4} + 2 \, a c^{4} x^{2} + a^{2} c^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 70, normalized size = 0.89 \begin {gather*} -\frac {{\left ({\left (\frac {4 \, B x}{c} + \frac {3 \, A}{c}\right )} x + \frac {3 \, B a}{c^{2}}\right )} x + \frac {2 \, A a}{c^{2}}}{3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}}} - \frac {B \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{c^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 91, normalized size = 1.15 \begin {gather*} -\frac {B \,x^{3}}{3 \left (c \,x^{2}+a \right )^{\frac {3}{2}} c}-\frac {A \,x^{2}}{\left (c \,x^{2}+a \right )^{\frac {3}{2}} c}-\frac {2 A a}{3 \left (c \,x^{2}+a \right )^{\frac {3}{2}} c^{2}}-\frac {B x}{\sqrt {c \,x^{2}+a}\, c^{2}}+\frac {B \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{c^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.62, size = 102, normalized size = 1.29 \begin {gather*} -\frac {1}{3} \, B x {\left (\frac {3 \, x^{2}}{{\left (c x^{2} + a\right )}^{\frac {3}{2}} c} + \frac {2 \, a}{{\left (c x^{2} + a\right )}^{\frac {3}{2}} c^{2}}\right )} - \frac {A x^{2}}{{\left (c x^{2} + a\right )}^{\frac {3}{2}} c} - \frac {B x}{3 \, \sqrt {c x^{2} + a} c^{2}} + \frac {B \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{c^{\frac {5}{2}}} - \frac {2 \, A a}{3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} 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 \frac {x^3\,\left (A+B\,x\right )}{{\left (c\,x^2+a\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 15.98, size = 400, normalized size = 5.06 \begin {gather*} A \left (\begin {cases} - \frac {2 a}{3 a c^{2} \sqrt {a + c x^{2}} + 3 c^{3} x^{2} \sqrt {a + c x^{2}}} - \frac {3 c x^{2}}{3 a c^{2} \sqrt {a + c x^{2}} + 3 c^{3} x^{2} \sqrt {a + c x^{2}}} & \text {for}\: c \neq 0 \\\frac {x^{4}}{4 a^{\frac {5}{2}}} & \text {otherwise} \end {cases}\right ) + B \left (\frac {3 a^{\frac {39}{2}} c^{11} \sqrt {1 + \frac {c x^{2}}{a}} \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {a}} \right )}}{3 a^{\frac {39}{2}} c^{\frac {27}{2}} \sqrt {1 + \frac {c x^{2}}{a}} + 3 a^{\frac {37}{2}} c^{\frac {29}{2}} x^{2} \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {3 a^{\frac {37}{2}} c^{12} x^{2} \sqrt {1 + \frac {c x^{2}}{a}} \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {a}} \right )}}{3 a^{\frac {39}{2}} c^{\frac {27}{2}} \sqrt {1 + \frac {c x^{2}}{a}} + 3 a^{\frac {37}{2}} c^{\frac {29}{2}} x^{2} \sqrt {1 + \frac {c x^{2}}{a}}} - \frac {3 a^{19} c^{\frac {23}{2}} x}{3 a^{\frac {39}{2}} c^{\frac {27}{2}} \sqrt {1 + \frac {c x^{2}}{a}} + 3 a^{\frac {37}{2}} c^{\frac {29}{2}} x^{2} \sqrt {1 + \frac {c x^{2}}{a}}} - \frac {4 a^{18} c^{\frac {25}{2}} x^{3}}{3 a^{\frac {39}{2}} c^{\frac {27}{2}} \sqrt {1 + \frac {c x^{2}}{a}} + 3 a^{\frac {37}{2}} c^{\frac {29}{2}} x^{2} \sqrt {1 + \frac {c x^{2}}{a}}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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