Optimal. Leaf size=52 \[ -\frac {4 b \left (b x+c x^2\right )^{5/2}}{35 c^2 x^{5/2}}+\frac {2 \left (b x+c x^2\right )^{5/2}}{7 c x^{3/2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {670, 662}
\begin {gather*} \frac {2 \left (b x+c x^2\right )^{5/2}}{7 c x^{3/2}}-\frac {4 b \left (b x+c x^2\right )^{5/2}}{35 c^2 x^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 662
Rule 670
Rubi steps
\begin {align*} \int \frac {\left (b x+c x^2\right )^{3/2}}{\sqrt {x}} \, dx &=\frac {2 \left (b x+c x^2\right )^{5/2}}{7 c x^{3/2}}-\frac {(2 b) \int \frac {\left (b x+c x^2\right )^{3/2}}{x^{3/2}} \, dx}{7 c}\\ &=-\frac {4 b \left (b x+c x^2\right )^{5/2}}{35 c^2 x^{5/2}}+\frac {2 \left (b x+c x^2\right )^{5/2}}{7 c x^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 31, normalized size = 0.60 \begin {gather*} \frac {2 (x (b+c x))^{5/2} (-2 b+5 c x)}{35 c^2 x^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.40, size = 33, normalized size = 0.63
method | result | size |
gosper | \(-\frac {2 \left (c x +b \right ) \left (-5 c x +2 b \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{35 c^{2} x^{\frac {3}{2}}}\) | \(33\) |
default | \(-\frac {2 \sqrt {x \left (c x +b \right )}\, \left (c x +b \right )^{2} \left (-5 c x +2 b \right )}{35 \sqrt {x}\, c^{2}}\) | \(33\) |
risch | \(-\frac {2 \left (c x +b \right ) \sqrt {x}\, \left (-5 c^{3} x^{3}-8 b \,c^{2} x^{2}-b^{2} c x +2 b^{3}\right )}{35 \sqrt {x \left (c x +b \right )}\, c^{2}}\) | \(53\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 77, normalized size = 1.48 \begin {gather*} \frac {2 \, {\left ({\left (15 \, c^{3} x^{3} + 3 \, b c^{2} x^{2} - 4 \, b^{2} c x + 8 \, b^{3}\right )} x^{2} + 7 \, {\left (3 \, b c^{2} x^{3} + b^{2} c x^{2} - 2 \, b^{3} x\right )} x\right )} \sqrt {c x + b}}{105 \, c^{2} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.38, size = 48, normalized size = 0.92 \begin {gather*} \frac {2 \, {\left (5 \, c^{3} x^{3} + 8 \, b c^{2} x^{2} + b^{2} c x - 2 \, b^{3}\right )} \sqrt {c x^{2} + b x}}{35 \, c^{2} \sqrt {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 {\left (x \left (b + c x\right )\right )^{\frac {3}{2}}}{\sqrt {x}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 86 vs.
\(2 (40) = 80\).
time = 1.17, size = 86, normalized size = 1.65 \begin {gather*} -\frac {2}{105} \, c {\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 )} + \frac {2}{15} \, b {\left (\frac {2 \, b^{\frac {5}{2}}}{c^{2}} + \frac {3 \, {\left (c x + b\right )}^{\frac {5}{2}} - 5 \, {\left (c x + b\right )}^{\frac {3}{2}} b}{c^{2}}\right )} \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 = -0.02 \begin {gather*} \int \frac {{\left (c\,x^2+b\,x\right )}^{3/2}}{\sqrt {x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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