Optimal. Leaf size=105 \[ -\frac {\sqrt {a+c x^2} (16 a B-9 A c x)}{6 c^3}-\frac {x^3 (A+B x)}{c \sqrt {a+c x^2}}-\frac {3 a A \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 c^{5/2}}+\frac {4 B x^2 \sqrt {a+c x^2}}{3 c^2} \]
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Rubi [A] time = 0.07, antiderivative size = 105, 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 = {819, 833, 780, 217, 206} \begin {gather*} -\frac {\sqrt {a+c x^2} (16 a B-9 A c x)}{6 c^3}-\frac {x^3 (A+B x)}{c \sqrt {a+c x^2}}-\frac {3 a A \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 c^{5/2}}+\frac {4 B x^2 \sqrt {a+c x^2}}{3 c^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 780
Rule 819
Rule 833
Rubi steps
\begin {align*} \int \frac {x^4 (A+B x)}{\left (a+c x^2\right )^{3/2}} \, dx &=-\frac {x^3 (A+B x)}{c \sqrt {a+c x^2}}+\frac {\int \frac {x^2 (3 a A+4 a B x)}{\sqrt {a+c x^2}} \, dx}{a c}\\ &=-\frac {x^3 (A+B x)}{c \sqrt {a+c x^2}}+\frac {4 B x^2 \sqrt {a+c x^2}}{3 c^2}+\frac {\int \frac {x \left (-8 a^2 B+9 a A c x\right )}{\sqrt {a+c x^2}} \, dx}{3 a c^2}\\ &=-\frac {x^3 (A+B x)}{c \sqrt {a+c x^2}}+\frac {4 B x^2 \sqrt {a+c x^2}}{3 c^2}-\frac {(16 a B-9 A c x) \sqrt {a+c x^2}}{6 c^3}-\frac {(3 a A) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{2 c^2}\\ &=-\frac {x^3 (A+B x)}{c \sqrt {a+c x^2}}+\frac {4 B x^2 \sqrt {a+c x^2}}{3 c^2}-\frac {(16 a B-9 A c x) \sqrt {a+c x^2}}{6 c^3}-\frac {(3 a A) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{2 c^2}\\ &=-\frac {x^3 (A+B x)}{c \sqrt {a+c x^2}}+\frac {4 B x^2 \sqrt {a+c x^2}}{3 c^2}-\frac {(16 a B-9 A c x) \sqrt {a+c x^2}}{6 c^3}-\frac {3 a A \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 c^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 91, normalized size = 0.87 \begin {gather*} \frac {-16 a^2 B+a c x (9 A-8 B x)-9 a A \sqrt {c} \sqrt {a+c x^2} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )+c^2 x^3 (3 A+2 B x)}{6 c^3 \sqrt {a+c x^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.41, size = 90, normalized size = 0.86 \begin {gather*} \frac {-16 a^2 B+9 a A c x-8 a B c x^2+3 A c^2 x^3+2 B c^2 x^4}{6 c^3 \sqrt {a+c x^2}}+\frac {3 a A \log \left (\sqrt {a+c x^2}-\sqrt {c} x\right )}{2 c^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 217, normalized size = 2.07 \begin {gather*} \left [\frac {9 \, {\left (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 (2 \, B c^{2} x^{4} + 3 \, A c^{2} x^{3} - 8 \, B a c x^{2} + 9 \, A a c x - 16 \, B a^{2}\right )} \sqrt {c x^{2} + a}}{12 \, {\left (c^{4} x^{2} + a c^{3}\right )}}, \frac {9 \, {\left (A a c x^{2} + A a^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) + {\left (2 \, B c^{2} x^{4} + 3 \, A c^{2} x^{3} - 8 \, B a c x^{2} + 9 \, A a c x - 16 \, B a^{2}\right )} \sqrt {c x^{2} + a}}{6 \, {\left (c^{4} x^{2} + a c^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 83, normalized size = 0.79 \begin {gather*} \frac {{\left ({\left ({\left (\frac {2 \, B x}{c} + \frac {3 \, A}{c}\right )} x - \frac {8 \, B a}{c^{2}}\right )} x + \frac {9 \, A a}{c^{2}}\right )} x - \frac {16 \, B a^{2}}{c^{3}}}{6 \, \sqrt {c x^{2} + a}} + \frac {3 \, A a \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{2 \, 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 = 115, normalized size = 1.10 \begin {gather*} \frac {B \,x^{4}}{3 \sqrt {c \,x^{2}+a}\, c}+\frac {A \,x^{3}}{2 \sqrt {c \,x^{2}+a}\, c}-\frac {4 B a \,x^{2}}{3 \sqrt {c \,x^{2}+a}\, c^{2}}+\frac {3 A a x}{2 \sqrt {c \,x^{2}+a}\, c^{2}}-\frac {3 A a \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{2 c^{\frac {5}{2}}}-\frac {8 B \,a^{2}}{3 \sqrt {c \,x^{2}+a}\, c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 107, normalized size = 1.02 \begin {gather*} \frac {B x^{4}}{3 \, \sqrt {c x^{2} + a} c} + \frac {A x^{3}}{2 \, \sqrt {c x^{2} + a} c} - \frac {4 \, B a x^{2}}{3 \, \sqrt {c x^{2} + a} c^{2}} + \frac {3 \, A a x}{2 \, \sqrt {c x^{2} + a} c^{2}} - \frac {3 \, A a \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{2 \, c^{\frac {5}{2}}} - \frac {8 \, B a^{2}}{3 \, \sqrt {c x^{2} + a} 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 )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 12.31, size = 144, normalized size = 1.37 \begin {gather*} A \left (\frac {3 \sqrt {a} x}{2 c^{2} \sqrt {1 + \frac {c x^{2}}{a}}} - \frac {3 a \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {a}} \right )}}{2 c^{\frac {5}{2}}} + \frac {x^{3}}{2 \sqrt {a} c \sqrt {1 + \frac {c x^{2}}{a}}}\right ) + B \left (\begin {cases} - \frac {8 a^{2}}{3 c^{3} \sqrt {a + c x^{2}}} - \frac {4 a x^{2}}{3 c^{2} \sqrt {a + c x^{2}}} + \frac {x^{4}}{3 c \sqrt {a + c x^{2}}} & \text {for}\: c \neq 0 \\\frac {x^{6}}{6 a^{\frac {3}{2}}} & \text {otherwise} \end {cases}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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