Optimal. Leaf size=96 \[ \frac {4 b \sqrt {b x+c x^2} (4 b B-5 A c)}{15 c^3 \sqrt {x}}-\frac {2 \sqrt {x} \sqrt {b x+c x^2} (4 b B-5 A c)}{15 c^2}+\frac {2 B x^{3/2} \sqrt {b x+c x^2}}{5 c} \]
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Rubi [A] time = 0.08, antiderivative size = 96, 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 = {794, 656, 648} \begin {gather*} -\frac {2 \sqrt {x} \sqrt {b x+c x^2} (4 b B-5 A c)}{15 c^2}+\frac {4 b \sqrt {b x+c x^2} (4 b B-5 A c)}{15 c^3 \sqrt {x}}+\frac {2 B x^{3/2} \sqrt {b x+c x^2}}{5 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 648
Rule 656
Rule 794
Rubi steps
\begin {align*} \int \frac {x^{3/2} (A+B x)}{\sqrt {b x+c x^2}} \, dx &=\frac {2 B x^{3/2} \sqrt {b x+c x^2}}{5 c}+\frac {\left (2 \left (\frac {3}{2} (-b B+A c)+\frac {1}{2} (-b B+2 A c)\right )\right ) \int \frac {x^{3/2}}{\sqrt {b x+c x^2}} \, dx}{5 c}\\ &=-\frac {2 (4 b B-5 A c) \sqrt {x} \sqrt {b x+c x^2}}{15 c^2}+\frac {2 B x^{3/2} \sqrt {b x+c x^2}}{5 c}+\frac {(2 b (4 b B-5 A c)) \int \frac {\sqrt {x}}{\sqrt {b x+c x^2}} \, dx}{15 c^2}\\ &=\frac {4 b (4 b B-5 A c) \sqrt {b x+c x^2}}{15 c^3 \sqrt {x}}-\frac {2 (4 b B-5 A c) \sqrt {x} \sqrt {b x+c x^2}}{15 c^2}+\frac {2 B x^{3/2} \sqrt {b x+c x^2}}{5 c}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 55, normalized size = 0.57 \begin {gather*} \frac {2 \sqrt {x (b+c x)} \left (-2 b c (5 A+2 B x)+c^2 x (5 A+3 B x)+8 b^2 B\right )}{15 c^3 \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.10, size = 59, normalized size = 0.61 \begin {gather*} \frac {2 \sqrt {b x+c x^2} \left (-10 A b c+5 A c^2 x+8 b^2 B-4 b B c x+3 B c^2 x^2\right )}{15 c^3 \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 55, normalized size = 0.57 \begin {gather*} \frac {2 \, {\left (3 \, B c^{2} x^{2} + 8 \, B b^{2} - 10 \, A b c - {\left (4 \, B b c - 5 \, A c^{2}\right )} x\right )} \sqrt {c x^{2} + b x}}{15 \, c^{3} \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 81, normalized size = 0.84 \begin {gather*} \frac {2 \, {\left (B b^{2} - A b c\right )} \sqrt {c x + b}}{c^{3}} + \frac {2 \, {\left (3 \, {\left (c x + b\right )}^{\frac {5}{2}} B - 10 \, {\left (c x + b\right )}^{\frac {3}{2}} B b + 5 \, {\left (c x + b\right )}^{\frac {3}{2}} A c\right )}}{15 \, c^{3}} - \frac {4 \, {\left (4 \, B b^{\frac {5}{2}} - 5 \, A b^{\frac {3}{2}} c\right )}}{15 \, c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 59, normalized size = 0.61 \begin {gather*} -\frac {2 \left (c x +b \right ) \left (-3 B \,c^{2} x^{2}-5 A \,c^{2} x +4 B b c x +10 A b c -8 b^{2} B \right ) \sqrt {x}}{15 \sqrt {c \,x^{2}+b x}\, c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.57, size = 75, normalized size = 0.78 \begin {gather*} \frac {2 \, {\left (c^{2} x^{2} - b c x - 2 \, b^{2}\right )} A}{3 \, \sqrt {c x + b} c^{2}} + \frac {2 \, {\left (3 \, c^{3} x^{3} - b c^{2} x^{2} + 4 \, b^{2} c x + 8 \, b^{3}\right )} B}{15 \, \sqrt {c x + b} 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^{3/2}\,\left (A+B\,x\right )}{\sqrt {c\,x^2+b\,x}} \,d x \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 {x^{\frac {3}{2}} \left (A + B x\right )}{\sqrt {x \left (b + c x\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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