Optimal. Leaf size=133 \[ -\frac {16 b^2 \sqrt {b x+c x^2} (6 b B-7 A c)}{105 c^4 \sqrt {x}}+\frac {8 b \sqrt {x} \sqrt {b x+c x^2} (6 b B-7 A c)}{105 c^3}-\frac {2 x^{3/2} \sqrt {b x+c x^2} (6 b B-7 A c)}{35 c^2}+\frac {2 B x^{5/2} \sqrt {b x+c x^2}}{7 c} \]
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Rubi [A] time = 0.11, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 4, 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 {16 b^2 \sqrt {b x+c x^2} (6 b B-7 A c)}{105 c^4 \sqrt {x}}-\frac {2 x^{3/2} \sqrt {b x+c x^2} (6 b B-7 A c)}{35 c^2}+\frac {8 b \sqrt {x} \sqrt {b x+c x^2} (6 b B-7 A c)}{105 c^3}+\frac {2 B x^{5/2} \sqrt {b x+c x^2}}{7 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 648
Rule 656
Rule 794
Rubi steps
\begin {align*} \int \frac {x^{5/2} (A+B x)}{\sqrt {b x+c x^2}} \, dx &=\frac {2 B x^{5/2} \sqrt {b x+c x^2}}{7 c}+\frac {\left (2 \left (\frac {5}{2} (-b B+A c)+\frac {1}{2} (-b B+2 A c)\right )\right ) \int \frac {x^{5/2}}{\sqrt {b x+c x^2}} \, dx}{7 c}\\ &=-\frac {2 (6 b B-7 A c) x^{3/2} \sqrt {b x+c x^2}}{35 c^2}+\frac {2 B x^{5/2} \sqrt {b x+c x^2}}{7 c}+\frac {(4 b (6 b B-7 A c)) \int \frac {x^{3/2}}{\sqrt {b x+c x^2}} \, dx}{35 c^2}\\ &=\frac {8 b (6 b B-7 A c) \sqrt {x} \sqrt {b x+c x^2}}{105 c^3}-\frac {2 (6 b B-7 A c) x^{3/2} \sqrt {b x+c x^2}}{35 c^2}+\frac {2 B x^{5/2} \sqrt {b x+c x^2}}{7 c}-\frac {\left (8 b^2 (6 b B-7 A c)\right ) \int \frac {\sqrt {x}}{\sqrt {b x+c x^2}} \, dx}{105 c^3}\\ &=-\frac {16 b^2 (6 b B-7 A c) \sqrt {b x+c x^2}}{105 c^4 \sqrt {x}}+\frac {8 b (6 b B-7 A c) \sqrt {x} \sqrt {b x+c x^2}}{105 c^3}-\frac {2 (6 b B-7 A c) x^{3/2} \sqrt {b x+c x^2}}{35 c^2}+\frac {2 B x^{5/2} \sqrt {b x+c x^2}}{7 c}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 75, normalized size = 0.56 \begin {gather*} \frac {2 \sqrt {x (b+c x)} \left (8 b^2 c (7 A+3 B x)-2 b c^2 x (14 A+9 B x)+3 c^3 x^2 (7 A+5 B x)-48 b^3 B\right )}{105 c^4 \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.12, size = 83, normalized size = 0.62 \begin {gather*} \frac {2 \sqrt {b x+c x^2} \left (56 A b^2 c-28 A b c^2 x+21 A c^3 x^2-48 b^3 B+24 b^2 B c x-18 b B c^2 x^2+15 B c^3 x^3\right )}{105 c^4 \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 79, normalized size = 0.59 \begin {gather*} \frac {2 \, {\left (15 \, B c^{3} x^{3} - 48 \, B b^{3} + 56 \, A b^{2} c - 3 \, {\left (6 \, B b c^{2} - 7 \, A c^{3}\right )} x^{2} + 4 \, {\left (6 \, B b^{2} c - 7 \, A b c^{2}\right )} x\right )} \sqrt {c x^{2} + b x}}{105 \, c^{4} \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 108, normalized size = 0.81 \begin {gather*} -\frac {2 \, {\left (B b^{3} - A b^{2} c\right )} \sqrt {c x + b}}{c^{4}} + \frac {2 \, {\left (15 \, {\left (c x + b\right )}^{\frac {7}{2}} B - 63 \, {\left (c x + b\right )}^{\frac {5}{2}} B b + 105 \, {\left (c x + b\right )}^{\frac {3}{2}} B b^{2} + 21 \, {\left (c x + b\right )}^{\frac {5}{2}} A c - 70 \, {\left (c x + b\right )}^{\frac {3}{2}} A b c\right )}}{105 \, c^{4}} + \frac {16 \, {\left (6 \, B b^{\frac {7}{2}} - 7 \, A b^{\frac {5}{2}} c\right )}}{105 \, c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 83, normalized size = 0.62 \begin {gather*} \frac {2 \left (c x +b \right ) \left (15 B \,c^{3} x^{3}+21 A \,c^{3} x^{2}-18 B b \,c^{2} x^{2}-28 A b \,c^{2} x +24 B \,b^{2} c x +56 A \,b^{2} c -48 b^{3} B \right ) \sqrt {x}}{105 \sqrt {c \,x^{2}+b x}\, c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.64, size = 98, normalized size = 0.74 \begin {gather*} \frac {2 \, {\left (3 \, c^{3} x^{3} - b c^{2} x^{2} + 4 \, b^{2} c x + 8 \, b^{3}\right )} A}{15 \, \sqrt {c x + b} c^{3}} + \frac {2 \, {\left (5 \, c^{4} x^{4} - b c^{3} x^{3} + 2 \, b^{2} c^{2} x^{2} - 8 \, b^{3} c x - 16 \, b^{4}\right )} B}{35 \, \sqrt {c x + b} c^{4}} \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^{5/2}\,\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 {5}{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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