Optimal. Leaf size=108 \[ -\frac {32 b^3 \left (b x+c x^2\right )^{3/2}}{315 c^4 x^{3/2}}+\frac {16 b^2 \left (b x+c x^2\right )^{3/2}}{105 c^3 \sqrt {x}}-\frac {4 b \sqrt {x} \left (b x+c x^2\right )^{3/2}}{21 c^2}+\frac {2 x^{3/2} \left (b x+c x^2\right )^{3/2}}{9 c} \]
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Rubi [A] time = 0.04, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {656, 648} \begin {gather*} -\frac {32 b^3 \left (b x+c x^2\right )^{3/2}}{315 c^4 x^{3/2}}+\frac {16 b^2 \left (b x+c x^2\right )^{3/2}}{105 c^3 \sqrt {x}}-\frac {4 b \sqrt {x} \left (b x+c x^2\right )^{3/2}}{21 c^2}+\frac {2 x^{3/2} \left (b x+c x^2\right )^{3/2}}{9 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 648
Rule 656
Rubi steps
\begin {align*} \int x^{5/2} \sqrt {b x+c x^2} \, dx &=\frac {2 x^{3/2} \left (b x+c x^2\right )^{3/2}}{9 c}-\frac {(2 b) \int x^{3/2} \sqrt {b x+c x^2} \, dx}{3 c}\\ &=-\frac {4 b \sqrt {x} \left (b x+c x^2\right )^{3/2}}{21 c^2}+\frac {2 x^{3/2} \left (b x+c x^2\right )^{3/2}}{9 c}+\frac {\left (8 b^2\right ) \int \sqrt {x} \sqrt {b x+c x^2} \, dx}{21 c^2}\\ &=\frac {16 b^2 \left (b x+c x^2\right )^{3/2}}{105 c^3 \sqrt {x}}-\frac {4 b \sqrt {x} \left (b x+c x^2\right )^{3/2}}{21 c^2}+\frac {2 x^{3/2} \left (b x+c x^2\right )^{3/2}}{9 c}-\frac {\left (16 b^3\right ) \int \frac {\sqrt {b x+c x^2}}{\sqrt {x}} \, dx}{105 c^3}\\ &=-\frac {32 b^3 \left (b x+c x^2\right )^{3/2}}{315 c^4 x^{3/2}}+\frac {16 b^2 \left (b x+c x^2\right )^{3/2}}{105 c^3 \sqrt {x}}-\frac {4 b \sqrt {x} \left (b x+c x^2\right )^{3/2}}{21 c^2}+\frac {2 x^{3/2} \left (b x+c x^2\right )^{3/2}}{9 c}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 53, normalized size = 0.49 \begin {gather*} \frac {2 (x (b+c x))^{3/2} \left (-16 b^3+24 b^2 c x-30 b c^2 x^2+35 c^3 x^3\right )}{315 c^4 x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.09, size = 66, normalized size = 0.61 \begin {gather*} \frac {2 \sqrt {b x+c x^2} \left (-16 b^4+8 b^3 c x-6 b^2 c^2 x^2+5 b c^3 x^3+35 c^4 x^4\right )}{315 c^4 \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 60, normalized size = 0.56 \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^{2} + b x}}{315 \, c^{4} \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 58, normalized size = 0.54 \begin {gather*} \frac {32 \, b^{\frac {9}{2}}}{315 \, c^{4}} + \frac {2 \, {\left (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}\right )}}{315 \, c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 55, normalized size = 0.51 \begin {gather*} -\frac {2 \left (c x +b \right ) \left (-35 x^{3} c^{3}+30 b \,x^{2} c^{2}-24 b^{2} x c +16 b^{3}\right ) \sqrt {c \,x^{2}+b x}}{315 c^{4} \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.49, size = 53, normalized size = 0.49 \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}}{315 \, 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 x^{5/2}\,\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 x^{\frac {5}{2}} \sqrt {x \left (b + c x\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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