Optimal. Leaf size=99 \[ -\frac {x (3 A+4 B x)}{3 c^2 \sqrt {a+c x^2}}-\frac {x^3 (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}+\frac {A \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{c^{5/2}}+\frac {8 B \sqrt {a+c x^2}}{3 c^3} \]
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Rubi [A] time = 0.06, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {819, 641, 217, 206} \begin {gather*} -\frac {x (3 A+4 B x)}{3 c^2 \sqrt {a+c x^2}}-\frac {x^3 (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}+\frac {A \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{c^{5/2}}+\frac {8 B \sqrt {a+c x^2}}{3 c^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 641
Rule 819
Rubi steps
\begin {align*} \int \frac {x^4 (A+B x)}{\left (a+c x^2\right )^{5/2}} \, dx &=-\frac {x^3 (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}+\frac {\int \frac {x^2 (3 a A+4 a B x)}{\left (a+c x^2\right )^{3/2}} \, dx}{3 a c}\\ &=-\frac {x^3 (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}-\frac {x (3 A+4 B x)}{3 c^2 \sqrt {a+c x^2}}+\frac {\int \frac {3 a^2 A+8 a^2 B x}{\sqrt {a+c x^2}} \, dx}{3 a^2 c^2}\\ &=-\frac {x^3 (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}-\frac {x (3 A+4 B x)}{3 c^2 \sqrt {a+c x^2}}+\frac {8 B \sqrt {a+c x^2}}{3 c^3}+\frac {A \int \frac {1}{\sqrt {a+c x^2}} \, dx}{c^2}\\ &=-\frac {x^3 (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}-\frac {x (3 A+4 B x)}{3 c^2 \sqrt {a+c x^2}}+\frac {8 B \sqrt {a+c x^2}}{3 c^3}+\frac {A \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{c^2}\\ &=-\frac {x^3 (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}-\frac {x (3 A+4 B x)}{3 c^2 \sqrt {a+c x^2}}+\frac {8 B \sqrt {a+c x^2}}{3 c^3}+\frac {A \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 = 0.06, size = 89, normalized size = 0.90 \begin {gather*} \frac {8 a^2 B-3 a c x (A-4 B x)+3 A \sqrt {c} \left (a+c x^2\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )+c^2 x^3 (3 B x-4 A)}{3 c^3 \left (a+c x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.52, size = 87, normalized size = 0.88 \begin {gather*} \frac {8 a^2 B-3 a A c x+12 a B c x^2-4 A c^2 x^3+3 B c^2 x^4}{3 c^3 \left (a+c x^2\right )^{3/2}}-\frac {A \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 = 0.45, size = 258, normalized size = 2.61 \begin {gather*} \left [\frac {3 \, {\left (A c^{2} x^{4} + 2 \, A a c x^{2} + A a^{2}\right )} \sqrt {c} \log \left (-2 \, c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + 2 \, {\left (3 \, B c^{2} x^{4} - 4 \, A c^{2} x^{3} + 12 \, B a c x^{2} - 3 \, A a c x + 8 \, B a^{2}\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 (A c^{2} x^{4} + 2 \, A a c x^{2} + A a^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - {\left (3 \, B c^{2} x^{4} - 4 \, A c^{2} x^{3} + 12 \, B a c x^{2} - 3 \, A a c x + 8 \, B a^{2}\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 = 0.21, size = 82, normalized size = 0.83 \begin {gather*} \frac {{\left ({\left ({\left (\frac {3 \, B x}{c} - \frac {4 \, A}{c}\right )} x + \frac {12 \, B a}{c^{2}}\right )} x - \frac {3 \, A a}{c^{2}}\right )} x + \frac {8 \, B a^{2}}{c^{3}}}{3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}}} - \frac {A \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 = 111, normalized size = 1.12 \begin {gather*} \frac {B \,x^{4}}{\left (c \,x^{2}+a \right )^{\frac {3}{2}} c}-\frac {A \,x^{3}}{3 \left (c \,x^{2}+a \right )^{\frac {3}{2}} c}+\frac {4 B a \,x^{2}}{\left (c \,x^{2}+a \right )^{\frac {3}{2}} c^{2}}-\frac {A x}{\sqrt {c \,x^{2}+a}\, c^{2}}+\frac {8 B \,a^{2}}{3 \left (c \,x^{2}+a \right )^{\frac {3}{2}} c^{3}}+\frac {A \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 = 122, normalized size = 1.23 \begin {gather*} -\frac {1}{3} \, A 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 {B x^{4}}{{\left (c x^{2} + a\right )}^{\frac {3}{2}} c} + \frac {4 \, B a x^{2}}{{\left (c x^{2} + a\right )}^{\frac {3}{2}} c^{2}} - \frac {A x}{3 \, \sqrt {c x^{2} + a} c^{2}} + \frac {A \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{c^{\frac {5}{2}}} + \frac {8 \, B a^{2}}{3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} 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 )}{{\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 [B] time = 29.11, size = 445, normalized size = 4.49 \begin {gather*} A \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 ) + B \left (\begin {cases} \frac {8 a^{2}}{3 a c^{3} \sqrt {a + c x^{2}} + 3 c^{4} x^{2} \sqrt {a + c x^{2}}} + \frac {12 a c x^{2}}{3 a c^{3} \sqrt {a + c x^{2}} + 3 c^{4} x^{2} \sqrt {a + c x^{2}}} + \frac {3 c^{2} x^{4}}{3 a c^{3} \sqrt {a + c x^{2}} + 3 c^{4} x^{2} \sqrt {a + c x^{2}}} & \text {for}\: c \neq 0 \\\frac {x^{6}}{6 a^{\frac {5}{2}}} & \text {otherwise} \end {cases}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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