Optimal. Leaf size=103 \[ -\frac {2 a^3 A}{5 x^{5/2}}-\frac {2 a^3 B}{3 x^{3/2}}-\frac {6 a^2 A c}{\sqrt {x}}+6 a^2 B c \sqrt {x}+2 a A c^2 x^{3/2}+\frac {6}{5} a B c^2 x^{5/2}+\frac {2}{7} A c^3 x^{7/2}+\frac {2}{9} B c^3 x^{9/2} \]
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Rubi [A] time = 0.04, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {766} \begin {gather*} -\frac {6 a^2 A c}{\sqrt {x}}-\frac {2 a^3 A}{5 x^{5/2}}+6 a^2 B c \sqrt {x}-\frac {2 a^3 B}{3 x^{3/2}}+2 a A c^2 x^{3/2}+\frac {6}{5} a B c^2 x^{5/2}+\frac {2}{7} A c^3 x^{7/2}+\frac {2}{9} B c^3 x^{9/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 766
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a+c x^2\right )^3}{x^{7/2}} \, dx &=\int \left (\frac {a^3 A}{x^{7/2}}+\frac {a^3 B}{x^{5/2}}+\frac {3 a^2 A c}{x^{3/2}}+\frac {3 a^2 B c}{\sqrt {x}}+3 a A c^2 \sqrt {x}+3 a B c^2 x^{3/2}+A c^3 x^{5/2}+B c^3 x^{7/2}\right ) \, dx\\ &=-\frac {2 a^3 A}{5 x^{5/2}}-\frac {2 a^3 B}{3 x^{3/2}}-\frac {6 a^2 A c}{\sqrt {x}}+6 a^2 B c \sqrt {x}+2 a A c^2 x^{3/2}+\frac {6}{5} a B c^2 x^{5/2}+\frac {2}{7} A c^3 x^{7/2}+\frac {2}{9} B c^3 x^{9/2}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 72, normalized size = 0.70 \begin {gather*} \frac {2 \left (-21 a^3 (3 A+5 B x)+945 a^2 c x^2 (B x-A)+63 a c^2 x^4 (5 A+3 B x)+5 c^3 x^6 (9 A+7 B x)\right )}{315 x^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.06, size = 81, normalized size = 0.79 \begin {gather*} \frac {2 \left (-63 a^3 A-105 a^3 B x-945 a^2 A c x^2+945 a^2 B c x^3+315 a A c^2 x^4+189 a B c^2 x^5+45 A c^3 x^6+35 B c^3 x^7\right )}{315 x^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 77, normalized size = 0.75 \begin {gather*} \frac {2 \, {\left (35 \, B c^{3} x^{7} + 45 \, A c^{3} x^{6} + 189 \, B a c^{2} x^{5} + 315 \, A a c^{2} x^{4} + 945 \, B a^{2} c x^{3} - 945 \, A a^{2} c x^{2} - 105 \, B a^{3} x - 63 \, A a^{3}\right )}}{315 \, x^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 78, normalized size = 0.76 \begin {gather*} \frac {2}{9} \, B c^{3} x^{\frac {9}{2}} + \frac {2}{7} \, A c^{3} x^{\frac {7}{2}} + \frac {6}{5} \, B a c^{2} x^{\frac {5}{2}} + 2 \, A a c^{2} x^{\frac {3}{2}} + 6 \, B a^{2} c \sqrt {x} - \frac {2 \, {\left (45 \, A a^{2} c x^{2} + 5 \, B a^{3} x + 3 \, A a^{3}\right )}}{15 \, x^{\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 = 78, normalized size = 0.76 \begin {gather*} -\frac {2 \left (-35 B \,c^{3} x^{7}-45 A \,c^{3} x^{6}-189 B a \,c^{2} x^{5}-315 A a \,c^{2} x^{4}-945 B \,a^{2} c \,x^{3}+945 A \,a^{2} c \,x^{2}+105 B \,a^{3} x +63 A \,a^{3}\right )}{315 x^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 78, normalized size = 0.76 \begin {gather*} \frac {2}{9} \, B c^{3} x^{\frac {9}{2}} + \frac {2}{7} \, A c^{3} x^{\frac {7}{2}} + \frac {6}{5} \, B a c^{2} x^{\frac {5}{2}} + 2 \, A a c^{2} x^{\frac {3}{2}} + 6 \, B a^{2} c \sqrt {x} - \frac {2 \, {\left (45 \, A a^{2} c x^{2} + 5 \, B a^{3} x + 3 \, A a^{3}\right )}}{15 \, x^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 78, normalized size = 0.76 \begin {gather*} \frac {2\,A\,c^3\,x^{7/2}}{7}-\frac {\frac {2\,B\,a^3\,x}{3}+\frac {2\,A\,a^3}{5}+6\,A\,c\,a^2\,x^2}{x^{5/2}}+\frac {2\,B\,c^3\,x^{9/2}}{9}+2\,A\,a\,c^2\,x^{3/2}+6\,B\,a^2\,c\,\sqrt {x}+\frac {6\,B\,a\,c^2\,x^{5/2}}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.49, size = 109, normalized size = 1.06 \begin {gather*} - \frac {2 A a^{3}}{5 x^{\frac {5}{2}}} - \frac {6 A a^{2} c}{\sqrt {x}} + 2 A a c^{2} x^{\frac {3}{2}} + \frac {2 A c^{3} x^{\frac {7}{2}}}{7} - \frac {2 B a^{3}}{3 x^{\frac {3}{2}}} + 6 B a^{2} c \sqrt {x} + \frac {6 B a c^{2} x^{\frac {5}{2}}}{5} + \frac {2 B c^{3} x^{\frac {9}{2}}}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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