Optimal. Leaf size=57 \[ -\frac {2 A \left (b x+c x^2\right )^{5/2}}{7 b x^6}-\frac {2 (7 b B-2 A c) \left (b x+c x^2\right )^{5/2}}{35 b^2 x^5} \]
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Rubi [A]
time = 0.03, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {806, 664}
\begin {gather*} -\frac {2 \left (b x+c x^2\right )^{5/2} (7 b B-2 A c)}{35 b^2 x^5}-\frac {2 A \left (b x+c x^2\right )^{5/2}}{7 b x^6} \end {gather*}
Antiderivative was successfully verified.
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Rule 664
Rule 806
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{x^6} \, dx &=-\frac {2 A \left (b x+c x^2\right )^{5/2}}{7 b x^6}+\frac {\left (2 \left (-6 (-b B+A c)+\frac {5}{2} (-b B+2 A c)\right )\right ) \int \frac {\left (b x+c x^2\right )^{3/2}}{x^5} \, dx}{7 b}\\ &=-\frac {2 A \left (b x+c x^2\right )^{5/2}}{7 b x^6}-\frac {2 (7 b B-2 A c) \left (b x+c x^2\right )^{5/2}}{35 b^2 x^5}\\ \end {align*}
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Mathematica [A]
time = 0.18, size = 36, normalized size = 0.63 \begin {gather*} -\frac {2 (x (b+c x))^{5/2} (5 A b+7 b B x-2 A c x)}{35 b^2 x^6} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.52, size = 64, normalized size = 1.12
method | result | size |
gosper | \(-\frac {2 \left (c x +b \right ) \left (-2 A c x +7 b B x +5 A b \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{35 b^{2} x^{5}}\) | \(40\) |
default | \(A \left (-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{7 b \,x^{6}}+\frac {4 c \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{35 b^{2} x^{5}}\right )-\frac {2 B \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{5 b \,x^{5}}\) | \(64\) |
trager | \(-\frac {2 \left (-2 A \,c^{3} x^{3}+7 B b \,c^{2} x^{3}+A b \,c^{2} x^{2}+14 B \,b^{2} c \,x^{2}+8 A \,b^{2} c x +7 B \,b^{3} x +5 A \,b^{3}\right ) \sqrt {c \,x^{2}+b x}}{35 b^{2} x^{4}}\) | \(80\) |
risch | \(-\frac {2 \left (c x +b \right ) \left (-2 A \,c^{3} x^{3}+7 B b \,c^{2} x^{3}+A b \,c^{2} x^{2}+14 B \,b^{2} c \,x^{2}+8 A \,b^{2} c x +7 B \,b^{3} x +5 A \,b^{3}\right )}{35 x^{3} \sqrt {x \left (c x +b \right )}\, b^{2}}\) | \(83\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 176 vs.
\(2 (49) = 98\).
time = 0.26, size = 176, normalized size = 3.09 \begin {gather*} -\frac {2 \, \sqrt {c x^{2} + b x} B c^{2}}{5 \, b x} + \frac {4 \, \sqrt {c x^{2} + b x} A c^{3}}{35 \, b^{2} x} + \frac {\sqrt {c x^{2} + b x} B c}{5 \, x^{2}} - \frac {2 \, \sqrt {c x^{2} + b x} A c^{2}}{35 \, b x^{2}} + \frac {3 \, \sqrt {c x^{2} + b x} B b}{5 \, x^{3}} + \frac {3 \, \sqrt {c x^{2} + b x} A c}{70 \, x^{3}} - \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} B}{x^{4}} + \frac {3 \, \sqrt {c x^{2} + b x} A b}{14 \, x^{4}} - \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} A}{2 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.77, size = 78, normalized size = 1.37 \begin {gather*} -\frac {2 \, {\left (5 \, A b^{3} + {\left (7 \, B b c^{2} - 2 \, A c^{3}\right )} x^{3} + {\left (14 \, B b^{2} c + A b c^{2}\right )} x^{2} + {\left (7 \, B b^{3} + 8 \, A b^{2} c\right )} x\right )} \sqrt {c x^{2} + b x}}{35 \, b^{2} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x \left (b + c x\right )\right )^{\frac {3}{2}} \left (A + B x\right )}{x^{6}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 311 vs.
\(2 (49) = 98\).
time = 0.65, size = 311, normalized size = 5.46 \begin {gather*} \frac {2 \, {\left (35 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{6} B c^{2} + 70 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{5} B b c^{\frac {3}{2}} + 35 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{5} A c^{\frac {5}{2}} + 70 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} B b^{2} c + 105 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} A b c^{2} + 35 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} B b^{3} \sqrt {c} + 140 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} A b^{2} c^{\frac {3}{2}} + 7 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} B b^{4} + 98 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} A b^{3} c + 35 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} A b^{4} \sqrt {c} + 5 \, A b^{5}\right )}}{35 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.02, size = 142, normalized size = 2.49 \begin {gather*} \frac {4\,A\,c^3\,\sqrt {c\,x^2+b\,x}}{35\,b^2\,x}-\frac {16\,A\,c\,\sqrt {c\,x^2+b\,x}}{35\,x^3}-\frac {2\,B\,b\,\sqrt {c\,x^2+b\,x}}{5\,x^3}-\frac {4\,B\,c\,\sqrt {c\,x^2+b\,x}}{5\,x^2}-\frac {2\,A\,c^2\,\sqrt {c\,x^2+b\,x}}{35\,b\,x^2}-\frac {2\,A\,b\,\sqrt {c\,x^2+b\,x}}{7\,x^4}-\frac {2\,B\,c^2\,\sqrt {c\,x^2+b\,x}}{5\,b\,x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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