Optimal. Leaf size=80 \[ \frac {16 b^2 \left (b x+c x^2\right )^{5/2}}{315 c^3 x^{5/2}}-\frac {8 b \left (b x+c x^2\right )^{5/2}}{63 c^2 x^{3/2}}+\frac {2 \left (b x+c x^2\right )^{5/2}}{9 c \sqrt {x}} \]
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Rubi [A] time = 0.03, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 3, 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 {16 b^2 \left (b x+c x^2\right )^{5/2}}{315 c^3 x^{5/2}}-\frac {8 b \left (b x+c x^2\right )^{5/2}}{63 c^2 x^{3/2}}+\frac {2 \left (b x+c x^2\right )^{5/2}}{9 c \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 648
Rule 656
Rubi steps
\begin {align*} \int \sqrt {x} \left (b x+c x^2\right )^{3/2} \, dx &=\frac {2 \left (b x+c x^2\right )^{5/2}}{9 c \sqrt {x}}-\frac {(4 b) \int \frac {\left (b x+c x^2\right )^{3/2}}{\sqrt {x}} \, dx}{9 c}\\ &=-\frac {8 b \left (b x+c x^2\right )^{5/2}}{63 c^2 x^{3/2}}+\frac {2 \left (b x+c x^2\right )^{5/2}}{9 c \sqrt {x}}+\frac {\left (8 b^2\right ) \int \frac {\left (b x+c x^2\right )^{3/2}}{x^{3/2}} \, dx}{63 c^2}\\ &=\frac {16 b^2 \left (b x+c x^2\right )^{5/2}}{315 c^3 x^{5/2}}-\frac {8 b \left (b x+c x^2\right )^{5/2}}{63 c^2 x^{3/2}}+\frac {2 \left (b x+c x^2\right )^{5/2}}{9 c \sqrt {x}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 42, normalized size = 0.52 \begin {gather*} \frac {2 (x (b+c x))^{5/2} \left (8 b^2-20 b c x+35 c^2 x^2\right )}{315 c^3 x^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.33, size = 66, normalized size = 0.82 \begin {gather*} \frac {2 \sqrt {b x+c x^2} \left (8 b^4-4 b^3 c x+3 b^2 c^2 x^2+50 b c^3 x^3+35 c^4 x^4\right )}{315 c^3 \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 60, normalized size = 0.75 \begin {gather*} \frac {2 \, {\left (35 \, c^{4} x^{4} + 50 \, b c^{3} x^{3} + 3 \, b^{2} c^{2} x^{2} - 4 \, b^{3} c x + 8 \, b^{4}\right )} \sqrt {c x^{2} + b x}}{315 \, c^{3} \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 110, normalized size = 1.38 \begin {gather*} \frac {2}{315} \, c {\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 )} - \frac {2}{105} \, b {\left (\frac {8 \, b^{\frac {7}{2}}}{c^{3}} - \frac {15 \, {\left (c x + b\right )}^{\frac {7}{2}} - 42 \, {\left (c x + b\right )}^{\frac {5}{2}} b + 35 \, {\left (c x + b\right )}^{\frac {3}{2}} b^{2}}{c^{3}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 44, normalized size = 0.55 \begin {gather*} \frac {2 \left (c x +b \right ) \left (35 c^{2} x^{2}-20 b c x +8 b^{2}\right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{315 c^{3} x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.47, size = 102, normalized size = 1.28 \begin {gather*} \frac {2 \, {\left ({\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 )} x^{3} + 3 \, {\left (15 \, b c^{3} x^{4} + 3 \, b^{2} c^{2} x^{3} - 4 \, b^{3} c x^{2} + 8 \, b^{4} x\right )} x^{2}\right )} \sqrt {c x + b}}{315 \, c^{3} x^{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 \sqrt {x}\,{\left (c\,x^2+b\,x\right )}^{3/2} \,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 \sqrt {x} \left (x \left (b + c x\right )\right )^{\frac {3}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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