Optimal. Leaf size=129 \[ \frac {3 a^2 A \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{8 c^{5/2}}+\frac {a \sqrt {a+c x^2} (64 a B-45 A c x)}{120 c^3}+\frac {A x^3 \sqrt {a+c x^2}}{4 c}-\frac {4 a B x^2 \sqrt {a+c x^2}}{15 c^2}+\frac {B x^4 \sqrt {a+c x^2}}{5 c} \]
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Rubi [A] time = 0.11, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {833, 780, 217, 206} \begin {gather*} \frac {3 a^2 A \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{8 c^{5/2}}+\frac {a \sqrt {a+c x^2} (64 a B-45 A c x)}{120 c^3}+\frac {A x^3 \sqrt {a+c x^2}}{4 c}-\frac {4 a B x^2 \sqrt {a+c x^2}}{15 c^2}+\frac {B x^4 \sqrt {a+c x^2}}{5 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 780
Rule 833
Rubi steps
\begin {align*} \int \frac {x^4 (A+B x)}{\sqrt {a+c x^2}} \, dx &=\frac {B x^4 \sqrt {a+c x^2}}{5 c}+\frac {\int \frac {x^3 (-4 a B+5 A c x)}{\sqrt {a+c x^2}} \, dx}{5 c}\\ &=\frac {A x^3 \sqrt {a+c x^2}}{4 c}+\frac {B x^4 \sqrt {a+c x^2}}{5 c}+\frac {\int \frac {x^2 (-15 a A c-16 a B c x)}{\sqrt {a+c x^2}} \, dx}{20 c^2}\\ &=-\frac {4 a B x^2 \sqrt {a+c x^2}}{15 c^2}+\frac {A x^3 \sqrt {a+c x^2}}{4 c}+\frac {B x^4 \sqrt {a+c x^2}}{5 c}+\frac {\int \frac {x \left (32 a^2 B c-45 a A c^2 x\right )}{\sqrt {a+c x^2}} \, dx}{60 c^3}\\ &=-\frac {4 a B x^2 \sqrt {a+c x^2}}{15 c^2}+\frac {A x^3 \sqrt {a+c x^2}}{4 c}+\frac {B x^4 \sqrt {a+c x^2}}{5 c}+\frac {a (64 a B-45 A c x) \sqrt {a+c x^2}}{120 c^3}+\frac {\left (3 a^2 A\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{8 c^2}\\ &=-\frac {4 a B x^2 \sqrt {a+c x^2}}{15 c^2}+\frac {A x^3 \sqrt {a+c x^2}}{4 c}+\frac {B x^4 \sqrt {a+c x^2}}{5 c}+\frac {a (64 a B-45 A c x) \sqrt {a+c x^2}}{120 c^3}+\frac {\left (3 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^2}\\ &=-\frac {4 a B x^2 \sqrt {a+c x^2}}{15 c^2}+\frac {A x^3 \sqrt {a+c x^2}}{4 c}+\frac {B x^4 \sqrt {a+c x^2}}{5 c}+\frac {a (64 a B-45 A c x) \sqrt {a+c x^2}}{120 c^3}+\frac {3 a^2 A \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{8 c^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 86, normalized size = 0.67 \begin {gather*} \frac {\sqrt {a+c x^2} \left (64 a^2 B-a c x (45 A+32 B x)+6 c^2 x^3 (5 A+4 B x)\right )+45 a^2 A \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{120 c^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.30, size = 92, normalized size = 0.71 \begin {gather*} \frac {\sqrt {a+c x^2} \left (64 a^2 B-45 a A c x-32 a B c x^2+30 A c^2 x^3+24 B c^2 x^4\right )}{120 c^3}-\frac {3 a^2 A \log \left (\sqrt {a+c x^2}-\sqrt {c} x\right )}{8 c^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 176, normalized size = 1.36 \begin {gather*} \left [\frac {45 \, 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} - 32 \, B a c x^{2} - 45 \, A a c x + 64 \, B a^{2}\right )} \sqrt {c x^{2} + a}}{240 \, c^{3}}, -\frac {45 \, 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} - 32 \, B a c x^{2} - 45 \, A a c x + 64 \, B a^{2}\right )} \sqrt {c x^{2} + a}}{120 \, c^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 87, normalized size = 0.67 \begin {gather*} \frac {1}{120} \, \sqrt {c x^{2} + a} {\left ({\left (2 \, {\left (3 \, {\left (\frac {4 \, B x}{c} + \frac {5 \, A}{c}\right )} x - \frac {16 \, B a}{c^{2}}\right )} x - \frac {45 \, A a}{c^{2}}\right )} x + \frac {64 \, B a^{2}}{c^{3}}\right )} - \frac {3 \, A a^{2} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{8 \, 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 = 117, normalized size = 0.91 \begin {gather*} \frac {\sqrt {c \,x^{2}+a}\, B \,x^{4}}{5 c}+\frac {\sqrt {c \,x^{2}+a}\, A \,x^{3}}{4 c}-\frac {4 \sqrt {c \,x^{2}+a}\, B a \,x^{2}}{15 c^{2}}+\frac {3 A \,a^{2} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{8 c^{\frac {5}{2}}}-\frac {3 \sqrt {c \,x^{2}+a}\, A a x}{8 c^{2}}+\frac {8 \sqrt {c \,x^{2}+a}\, B \,a^{2}}{15 c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.57, size = 109, normalized size = 0.84 \begin {gather*} \frac {\sqrt {c x^{2} + a} B x^{4}}{5 \, c} + \frac {\sqrt {c x^{2} + a} A x^{3}}{4 \, c} - \frac {4 \, \sqrt {c x^{2} + a} B a x^{2}}{15 \, c^{2}} - \frac {3 \, \sqrt {c x^{2} + a} A a x}{8 \, c^{2}} + \frac {3 \, A a^{2} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{8 \, c^{\frac {5}{2}}} + \frac {8 \, \sqrt {c x^{2} + a} B a^{2}}{15 \, c^{3}} \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^4\,\left (A+B\,x\right )}{\sqrt {c\,x^2+a}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.58, size = 173, normalized size = 1.34 \begin {gather*} - \frac {3 A a^{\frac {3}{2}} x}{8 c^{2} \sqrt {1 + \frac {c x^{2}}{a}}} - \frac {A \sqrt {a} x^{3}}{8 c \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {3 A a^{2} \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {a}} \right )}}{8 c^{\frac {5}{2}}} + \frac {A x^{5}}{4 \sqrt {a} \sqrt {1 + \frac {c x^{2}}{a}}} + B \left (\begin {cases} \frac {8 a^{2} \sqrt {a + c x^{2}}}{15 c^{3}} - \frac {4 a x^{2} \sqrt {a + c x^{2}}}{15 c^{2}} + \frac {x^{4} \sqrt {a + c x^{2}}}{5 c} & \text {for}\: c \neq 0 \\\frac {x^{6}}{6 \sqrt {a}} & \text {otherwise} \end {cases}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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