Optimal. Leaf size=170 \[ \frac {32 b^3 \left (b x+c x^2\right )^{3/2} (8 b B-11 A c)}{3465 c^5 x^{3/2}}-\frac {16 b^2 \left (b x+c x^2\right )^{3/2} (8 b B-11 A c)}{1155 c^4 \sqrt {x}}+\frac {4 b \sqrt {x} \left (b x+c x^2\right )^{3/2} (8 b B-11 A c)}{231 c^3}-\frac {2 x^{3/2} \left (b x+c x^2\right )^{3/2} (8 b B-11 A c)}{99 c^2}+\frac {2 B x^{5/2} \left (b x+c x^2\right )^{3/2}}{11 c} \]
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Rubi [A] time = 0.16, antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 5, 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 {32 b^3 \left (b x+c x^2\right )^{3/2} (8 b B-11 A c)}{3465 c^5 x^{3/2}}-\frac {16 b^2 \left (b x+c x^2\right )^{3/2} (8 b B-11 A c)}{1155 c^4 \sqrt {x}}+\frac {4 b \sqrt {x} \left (b x+c x^2\right )^{3/2} (8 b B-11 A c)}{231 c^3}-\frac {2 x^{3/2} \left (b x+c x^2\right )^{3/2} (8 b B-11 A c)}{99 c^2}+\frac {2 B x^{5/2} \left (b x+c x^2\right )^{3/2}}{11 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 648
Rule 656
Rule 794
Rubi steps
\begin {align*} \int x^{5/2} (A+B x) \sqrt {b x+c x^2} \, dx &=\frac {2 B x^{5/2} \left (b x+c x^2\right )^{3/2}}{11 c}+\frac {\left (2 \left (\frac {5}{2} (-b B+A c)+\frac {3}{2} (-b B+2 A c)\right )\right ) \int x^{5/2} \sqrt {b x+c x^2} \, dx}{11 c}\\ &=-\frac {2 (8 b B-11 A c) x^{3/2} \left (b x+c x^2\right )^{3/2}}{99 c^2}+\frac {2 B x^{5/2} \left (b x+c x^2\right )^{3/2}}{11 c}+\frac {(2 b (8 b B-11 A c)) \int x^{3/2} \sqrt {b x+c x^2} \, dx}{33 c^2}\\ &=\frac {4 b (8 b B-11 A c) \sqrt {x} \left (b x+c x^2\right )^{3/2}}{231 c^3}-\frac {2 (8 b B-11 A c) x^{3/2} \left (b x+c x^2\right )^{3/2}}{99 c^2}+\frac {2 B x^{5/2} \left (b x+c x^2\right )^{3/2}}{11 c}-\frac {\left (8 b^2 (8 b B-11 A c)\right ) \int \sqrt {x} \sqrt {b x+c x^2} \, dx}{231 c^3}\\ &=-\frac {16 b^2 (8 b B-11 A c) \left (b x+c x^2\right )^{3/2}}{1155 c^4 \sqrt {x}}+\frac {4 b (8 b B-11 A c) \sqrt {x} \left (b x+c x^2\right )^{3/2}}{231 c^3}-\frac {2 (8 b B-11 A c) x^{3/2} \left (b x+c x^2\right )^{3/2}}{99 c^2}+\frac {2 B x^{5/2} \left (b x+c x^2\right )^{3/2}}{11 c}+\frac {\left (16 b^3 (8 b B-11 A c)\right ) \int \frac {\sqrt {b x+c x^2}}{\sqrt {x}} \, dx}{1155 c^4}\\ &=\frac {32 b^3 (8 b B-11 A c) \left (b x+c x^2\right )^{3/2}}{3465 c^5 x^{3/2}}-\frac {16 b^2 (8 b B-11 A c) \left (b x+c x^2\right )^{3/2}}{1155 c^4 \sqrt {x}}+\frac {4 b (8 b B-11 A c) \sqrt {x} \left (b x+c x^2\right )^{3/2}}{231 c^3}-\frac {2 (8 b B-11 A c) x^{3/2} \left (b x+c x^2\right )^{3/2}}{99 c^2}+\frac {2 B x^{5/2} \left (b x+c x^2\right )^{3/2}}{11 c}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 94, normalized size = 0.55 \begin {gather*} \frac {2 (x (b+c x))^{3/2} \left (-16 b^3 c (11 A+12 B x)+24 b^2 c^2 x (11 A+10 B x)-10 b c^3 x^2 (33 A+28 B x)+35 c^4 x^3 (11 A+9 B x)+128 b^4 B\right )}{3465 c^5 x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.14, size = 107, normalized size = 0.63 \begin {gather*} \frac {2 \left (b x+c x^2\right )^{3/2} \left (-176 A b^3 c+264 A b^2 c^2 x-330 A b c^3 x^2+385 A c^4 x^3+128 b^4 B-192 b^3 B c x+240 b^2 B c^2 x^2-280 b B c^3 x^3+315 B c^4 x^4\right )}{3465 c^5 x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 126, normalized size = 0.74 \begin {gather*} \frac {2 \, {\left (315 \, B c^{5} x^{5} + 128 \, B b^{5} - 176 \, A b^{4} c + 35 \, {\left (B b c^{4} + 11 \, A c^{5}\right )} x^{4} - 5 \, {\left (8 \, B b^{2} c^{3} - 11 \, A b c^{4}\right )} x^{3} + 6 \, {\left (8 \, B b^{3} c^{2} - 11 \, A b^{2} c^{3}\right )} x^{2} - 8 \, {\left (8 \, B b^{4} c - 11 \, A b^{3} c^{2}\right )} x\right )} \sqrt {c x^{2} + b x}}{3465 \, c^{5} \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 134, normalized size = 0.79 \begin {gather*} -\frac {2}{3465} \, B {\left (\frac {128 \, b^{\frac {11}{2}}}{c^{5}} - \frac {315 \, {\left (c x + b\right )}^{\frac {11}{2}} - 1540 \, {\left (c x + b\right )}^{\frac {9}{2}} b + 2970 \, {\left (c x + b\right )}^{\frac {7}{2}} b^{2} - 2772 \, {\left (c x + b\right )}^{\frac {5}{2}} b^{3} + 1155 \, {\left (c x + b\right )}^{\frac {3}{2}} b^{4}}{c^{5}}\right )} + \frac {2}{315} \, A {\left (\frac {16 \, b^{\frac {9}{2}}}{c^{4}} + \frac {35 \, {\left (c x + b\right )}^{\frac {9}{2}} - 135 \, {\left (c x + b\right )}^{\frac {7}{2}} b + 189 \, {\left (c x + b\right )}^{\frac {5}{2}} b^{2} - 105 \, {\left (c x + b\right )}^{\frac {3}{2}} b^{3}}{c^{4}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 107, normalized size = 0.63 \begin {gather*} -\frac {2 \left (c x +b \right ) \left (-315 B \,x^{4} c^{4}-385 A \,c^{4} x^{3}+280 B b \,c^{3} x^{3}+330 A b \,c^{3} x^{2}-240 B \,b^{2} c^{2} x^{2}-264 A \,b^{2} c^{2} x +192 B \,b^{3} c x +176 A \,b^{3} c -128 b^{4} B \right ) \sqrt {c \,x^{2}+b x}}{3465 c^{5} \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.73, size = 120, normalized size = 0.71 \begin {gather*} \frac {2 \, {\left (35 \, c^{4} x^{4} + 5 \, b c^{3} x^{3} - 6 \, b^{2} c^{2} x^{2} + 8 \, b^{3} c x - 16 \, b^{4}\right )} \sqrt {c x + b} A}{315 \, c^{4}} + \frac {2 \, {\left (315 \, c^{5} x^{5} + 35 \, b c^{4} x^{4} - 40 \, b^{2} c^{3} x^{3} + 48 \, b^{3} c^{2} x^{2} - 64 \, b^{4} c x + 128 \, b^{5}\right )} \sqrt {c x + b} B}{3465 \, c^{5}} \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 x^{5/2}\,\sqrt {c\,x^2+b\,x}\,\left (A+B\,x\right ) \,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 x^{\frac {5}{2}} \sqrt {x \left (b + c x\right )} \left (A + B x\right )\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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