Optimal. Leaf size=108 \[ \frac {32 b^3 \sqrt {x}}{5 c^4 \sqrt {b x+c x^2}}+\frac {16 b^2 x^{3/2}}{5 c^3 \sqrt {b x+c x^2}}-\frac {4 b x^{5/2}}{5 c^2 \sqrt {b x+c x^2}}+\frac {2 x^{7/2}}{5 c \sqrt {b x+c x^2}} \]
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Rubi [A]
time = 0.03, 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 = {670, 662}
\begin {gather*} \frac {32 b^3 \sqrt {x}}{5 c^4 \sqrt {b x+c x^2}}+\frac {16 b^2 x^{3/2}}{5 c^3 \sqrt {b x+c x^2}}-\frac {4 b x^{5/2}}{5 c^2 \sqrt {b x+c x^2}}+\frac {2 x^{7/2}}{5 c \sqrt {b x+c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 662
Rule 670
Rubi steps
\begin {align*} \int \frac {x^{9/2}}{\left (b x+c x^2\right )^{3/2}} \, dx &=\frac {2 x^{7/2}}{5 c \sqrt {b x+c x^2}}-\frac {(6 b) \int \frac {x^{7/2}}{\left (b x+c x^2\right )^{3/2}} \, dx}{5 c}\\ &=-\frac {4 b x^{5/2}}{5 c^2 \sqrt {b x+c x^2}}+\frac {2 x^{7/2}}{5 c \sqrt {b x+c x^2}}+\frac {\left (8 b^2\right ) \int \frac {x^{5/2}}{\left (b x+c x^2\right )^{3/2}} \, dx}{5 c^2}\\ &=\frac {16 b^2 x^{3/2}}{5 c^3 \sqrt {b x+c x^2}}-\frac {4 b x^{5/2}}{5 c^2 \sqrt {b x+c x^2}}+\frac {2 x^{7/2}}{5 c \sqrt {b x+c x^2}}-\frac {\left (16 b^3\right ) \int \frac {x^{3/2}}{\left (b x+c x^2\right )^{3/2}} \, dx}{5 c^3}\\ &=\frac {32 b^3 \sqrt {x}}{5 c^4 \sqrt {b x+c x^2}}+\frac {16 b^2 x^{3/2}}{5 c^3 \sqrt {b x+c x^2}}-\frac {4 b x^{5/2}}{5 c^2 \sqrt {b x+c x^2}}+\frac {2 x^{7/2}}{5 c \sqrt {b x+c x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 52, normalized size = 0.48 \begin {gather*} \frac {2 \sqrt {x} \left (16 b^3+8 b^2 c x-2 b c^2 x^2+c^3 x^3\right )}{5 c^4 \sqrt {x (b+c x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.43, size = 54, normalized size = 0.50
method | result | size |
gosper | \(\frac {2 \left (c x +b \right ) \left (c^{3} x^{3}-2 b \,c^{2} x^{2}+8 b^{2} c x +16 b^{3}\right ) x^{\frac {3}{2}}}{5 c^{4} \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}\) | \(54\) |
default | \(\frac {2 \sqrt {x \left (c x +b \right )}\, \left (c^{3} x^{3}-2 b \,c^{2} x^{2}+8 b^{2} c x +16 b^{3}\right )}{5 \sqrt {x}\, \left (c x +b \right ) c^{4}}\) | \(54\) |
risch | \(\frac {2 \left (c^{2} x^{2}-3 b c x +11 b^{2}\right ) \left (c x +b \right ) \sqrt {x}}{5 c^{4} \sqrt {x \left (c x +b \right )}}+\frac {2 b^{3} \sqrt {x}}{c^{4} \sqrt {x \left (c x +b \right )}}\) | \(62\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.54, size = 61, normalized size = 0.56 \begin {gather*} \frac {2 \, {\left (c^{3} x^{3} - 2 \, b c^{2} x^{2} + 8 \, b^{2} c x + 16 \, b^{3}\right )} \sqrt {c x^{2} + b x} \sqrt {x}}{5 \, {\left (c^{5} x^{2} + b c^{4} x\right )}} \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 {9}{2}}}{\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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Giac [A]
time = 1.56, size = 69, normalized size = 0.64 \begin {gather*} -\frac {32 \, b^{\frac {5}{2}}}{5 \, c^{4}} + \frac {2 \, b^{3}}{\sqrt {c x + b} c^{4}} + \frac {2 \, {\left ({\left (c x + b\right )}^{\frac {5}{2}} c^{16} - 5 \, {\left (c x + b\right )}^{\frac {3}{2}} b c^{16} + 15 \, \sqrt {c x + b} b^{2} c^{16}\right )}}{5 \, c^{20}} \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^{9/2}}{{\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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