Optimal. Leaf size=126 \[ -\frac {2 \left (b x+c x^2\right )^{5/2}}{13 b x^9}+\frac {16 c \left (b x+c x^2\right )^{5/2}}{143 b^2 x^8}-\frac {32 c^2 \left (b x+c x^2\right )^{5/2}}{429 b^3 x^7}+\frac {128 c^3 \left (b x+c x^2\right )^{5/2}}{3003 b^4 x^6}-\frac {256 c^4 \left (b x+c x^2\right )^{5/2}}{15015 b^5 x^5} \]
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Rubi [A]
time = 0.04, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {672, 664}
\begin {gather*} -\frac {256 c^4 \left (b x+c x^2\right )^{5/2}}{15015 b^5 x^5}+\frac {128 c^3 \left (b x+c x^2\right )^{5/2}}{3003 b^4 x^6}-\frac {32 c^2 \left (b x+c x^2\right )^{5/2}}{429 b^3 x^7}+\frac {16 c \left (b x+c x^2\right )^{5/2}}{143 b^2 x^8}-\frac {2 \left (b x+c x^2\right )^{5/2}}{13 b x^9} \end {gather*}
Antiderivative was successfully verified.
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Rule 664
Rule 672
Rubi steps
\begin {align*} \int \frac {\left (b x+c x^2\right )^{3/2}}{x^9} \, dx &=-\frac {2 \left (b x+c x^2\right )^{5/2}}{13 b x^9}-\frac {(8 c) \int \frac {\left (b x+c x^2\right )^{3/2}}{x^8} \, dx}{13 b}\\ &=-\frac {2 \left (b x+c x^2\right )^{5/2}}{13 b x^9}+\frac {16 c \left (b x+c x^2\right )^{5/2}}{143 b^2 x^8}+\frac {\left (48 c^2\right ) \int \frac {\left (b x+c x^2\right )^{3/2}}{x^7} \, dx}{143 b^2}\\ &=-\frac {2 \left (b x+c x^2\right )^{5/2}}{13 b x^9}+\frac {16 c \left (b x+c x^2\right )^{5/2}}{143 b^2 x^8}-\frac {32 c^2 \left (b x+c x^2\right )^{5/2}}{429 b^3 x^7}-\frac {\left (64 c^3\right ) \int \frac {\left (b x+c x^2\right )^{3/2}}{x^6} \, dx}{429 b^3}\\ &=-\frac {2 \left (b x+c x^2\right )^{5/2}}{13 b x^9}+\frac {16 c \left (b x+c x^2\right )^{5/2}}{143 b^2 x^8}-\frac {32 c^2 \left (b x+c x^2\right )^{5/2}}{429 b^3 x^7}+\frac {128 c^3 \left (b x+c x^2\right )^{5/2}}{3003 b^4 x^6}+\frac {\left (128 c^4\right ) \int \frac {\left (b x+c x^2\right )^{3/2}}{x^5} \, dx}{3003 b^4}\\ &=-\frac {2 \left (b x+c x^2\right )^{5/2}}{13 b x^9}+\frac {16 c \left (b x+c x^2\right )^{5/2}}{143 b^2 x^8}-\frac {32 c^2 \left (b x+c x^2\right )^{5/2}}{429 b^3 x^7}+\frac {128 c^3 \left (b x+c x^2\right )^{5/2}}{3003 b^4 x^6}-\frac {256 c^4 \left (b x+c x^2\right )^{5/2}}{15015 b^5 x^5}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 62, normalized size = 0.49 \begin {gather*} -\frac {2 (x (b+c x))^{5/2} \left (1155 b^4-840 b^3 c x+560 b^2 c^2 x^2-320 b c^3 x^3+128 c^4 x^4\right )}{15015 b^5 x^9} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.42, size = 119, normalized size = 0.94
method | result | size |
gosper | \(-\frac {2 \left (c x +b \right ) \left (128 c^{4} x^{4}-320 b \,c^{3} x^{3}+560 b^{2} c^{2} x^{2}-840 b^{3} c x +1155 b^{4}\right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{15015 b^{5} x^{8}}\) | \(66\) |
trager | \(-\frac {2 \left (128 c^{6} x^{6}-64 b \,c^{5} x^{5}+48 b^{2} c^{4} x^{4}-40 b^{3} c^{3} x^{3}+35 b^{4} c^{2} x^{2}+1470 b^{5} c x +1155 b^{6}\right ) \sqrt {c \,x^{2}+b x}}{15015 b^{5} x^{7}}\) | \(83\) |
risch | \(-\frac {2 \left (c x +b \right ) \left (128 c^{6} x^{6}-64 b \,c^{5} x^{5}+48 b^{2} c^{4} x^{4}-40 b^{3} c^{3} x^{3}+35 b^{4} c^{2} x^{2}+1470 b^{5} c x +1155 b^{6}\right )}{15015 x^{6} \sqrt {x \left (c x +b \right )}\, b^{5}}\) | \(86\) |
default | \(-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{13 b \,x^{9}}-\frac {8 c \left (-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{11 b \,x^{8}}-\frac {6 c \left (-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{9 b \,x^{7}}-\frac {4 c \left (-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{7 b \,x^{6}}+\frac {4 c \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{35 b^{2} x^{5}}\right )}{9 b}\right )}{11 b}\right )}{13 b}\) | \(119\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 161, normalized size = 1.28 \begin {gather*} -\frac {256 \, \sqrt {c x^{2} + b x} c^{6}}{15015 \, b^{5} x} + \frac {128 \, \sqrt {c x^{2} + b x} c^{5}}{15015 \, b^{4} x^{2}} - \frac {32 \, \sqrt {c x^{2} + b x} c^{4}}{5005 \, b^{3} x^{3}} + \frac {16 \, \sqrt {c x^{2} + b x} c^{3}}{3003 \, b^{2} x^{4}} - \frac {2 \, \sqrt {c x^{2} + b x} c^{2}}{429 \, b x^{5}} + \frac {3 \, \sqrt {c x^{2} + b x} c}{715 \, x^{6}} + \frac {3 \, \sqrt {c x^{2} + b x} b}{65 \, x^{7}} - \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}}}{5 \, x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.50, size = 82, normalized size = 0.65 \begin {gather*} -\frac {2 \, {\left (128 \, c^{6} x^{6} - 64 \, b c^{5} x^{5} + 48 \, b^{2} c^{4} x^{4} - 40 \, b^{3} c^{3} x^{3} + 35 \, b^{4} c^{2} x^{2} + 1470 \, b^{5} c x + 1155 \, b^{6}\right )} \sqrt {c x^{2} + b x}}{15015 \, b^{5} x^{7}} \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}}}{x^{9}}\, 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. 252 vs.
\(2 (106) = 212\).
time = 0.89, size = 252, normalized size = 2.00 \begin {gather*} \frac {2 \, {\left (48048 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{8} c^{4} + 240240 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{7} b c^{\frac {7}{2}} + 531960 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{6} b^{2} c^{3} + 675675 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{5} b^{3} c^{\frac {5}{2}} + 535535 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} b^{4} c^{2} + 270270 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} b^{5} c^{\frac {3}{2}} + 84630 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} b^{6} c + 15015 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} b^{7} \sqrt {c} + 1155 \, b^{8}\right )}}{15015 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{13}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.21, size = 145, normalized size = 1.15 \begin {gather*} \frac {16\,c^3\,\sqrt {c\,x^2+b\,x}}{3003\,b^2\,x^4}-\frac {28\,c\,\sqrt {c\,x^2+b\,x}}{143\,x^6}-\frac {2\,c^2\,\sqrt {c\,x^2+b\,x}}{429\,b\,x^5}-\frac {2\,b\,\sqrt {c\,x^2+b\,x}}{13\,x^7}-\frac {32\,c^4\,\sqrt {c\,x^2+b\,x}}{5005\,b^3\,x^3}+\frac {128\,c^5\,\sqrt {c\,x^2+b\,x}}{15015\,b^4\,x^2}-\frac {256\,c^6\,\sqrt {c\,x^2+b\,x}}{15015\,b^5\,x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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