Optimal. Leaf size=108 \[ \frac {4 \sqrt {x} (4 b B-A c)}{3 c^3 \sqrt {b x+c x^2}}+\frac {2 x^{3/2} (4 b B-A c)}{3 b c^2 \sqrt {b x+c x^2}}-\frac {2 x^{7/2} (b B-A c)}{3 b c \left (b x+c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.09, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {788, 656, 648} \begin {gather*} \frac {2 x^{3/2} (4 b B-A c)}{3 b c^2 \sqrt {b x+c x^2}}+\frac {4 \sqrt {x} (4 b B-A c)}{3 c^3 \sqrt {b x+c x^2}}-\frac {2 x^{7/2} (b B-A c)}{3 b c \left (b x+c x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 648
Rule 656
Rule 788
Rubi steps
\begin {align*} \int \frac {x^{7/2} (A+B x)}{\left (b x+c x^2\right )^{5/2}} \, dx &=-\frac {2 (b B-A c) x^{7/2}}{3 b c \left (b x+c x^2\right )^{3/2}}-\frac {\left (2 \left (\frac {7}{2} (-b B+A c)-\frac {3}{2} (-b B+2 A c)\right )\right ) \int \frac {x^{5/2}}{\left (b x+c x^2\right )^{3/2}} \, dx}{3 b c}\\ &=-\frac {2 (b B-A c) x^{7/2}}{3 b c \left (b x+c x^2\right )^{3/2}}+\frac {2 (4 b B-A c) x^{3/2}}{3 b c^2 \sqrt {b x+c x^2}}-\frac {(2 (4 b B-A c)) \int \frac {x^{3/2}}{\left (b x+c x^2\right )^{3/2}} \, dx}{3 c^2}\\ &=-\frac {2 (b B-A c) x^{7/2}}{3 b c \left (b x+c x^2\right )^{3/2}}+\frac {4 (4 b B-A c) \sqrt {x}}{3 c^3 \sqrt {b x+c x^2}}+\frac {2 (4 b B-A c) x^{3/2}}{3 b c^2 \sqrt {b x+c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 53, normalized size = 0.49 \begin {gather*} \frac {2 x^{3/2} \left (-2 b c (A-6 B x)+3 c^2 x (B x-A)+8 b^2 B\right )}{3 c^3 (x (b+c x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.82, size = 59, normalized size = 0.55 \begin {gather*} \frac {2 x^{3/2} \left (-2 A b c-3 A c^2 x+8 b^2 B+12 b B c x+3 B c^2 x^2\right )}{3 c^3 \left (b x+c x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 79, normalized size = 0.73 \begin {gather*} \frac {2 \, {\left (3 \, B c^{2} x^{2} + 8 \, B b^{2} - 2 \, A b c + 3 \, {\left (4 \, B b c - A c^{2}\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {x}}{3 \, {\left (c^{5} x^{3} + 2 \, b c^{4} x^{2} + b^{2} c^{3} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 72, normalized size = 0.67 \begin {gather*} \frac {2 \, \sqrt {c x + b} B}{c^{3}} - \frac {4 \, {\left (4 \, B b - A c\right )}}{3 \, \sqrt {b} c^{3}} + \frac {2 \, {\left (6 \, {\left (c x + b\right )} B b - B b^{2} - 3 \, {\left (c x + b\right )} A c + A b c\right )}}{3 \, {\left (c x + b\right )}^{\frac {3}{2}} c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 59, normalized size = 0.55 \begin {gather*} -\frac {2 \left (c x +b \right ) \left (-3 B \,c^{2} x^{2}+3 A \,c^{2} x -12 B b c x +2 A b c -8 b^{2} B \right ) x^{\frac {5}{2}}}{3 \left (c \,x^{2}+b x \right )^{\frac {5}{2}} c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (B x + A\right )} x^{\frac {7}{2}}}{{\left (c x^{2} + b x\right )}^{\frac {5}{2}}}\,{d x} \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^{7/2}\,\left (A+B\,x\right )}{{\left (c\,x^2+b\,x\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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