Optimal. Leaf size=100 \[ \frac {32 c^3 \left (b x+c x^2\right )^{5/2}}{1155 b^4 x^5}-\frac {16 c^2 \left (b x+c x^2\right )^{5/2}}{231 b^3 x^6}+\frac {4 c \left (b x+c x^2\right )^{5/2}}{33 b^2 x^7}-\frac {2 \left (b x+c x^2\right )^{5/2}}{11 b x^8} \]
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Rubi [A] time = 0.04, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {658, 650} \begin {gather*} \frac {32 c^3 \left (b x+c x^2\right )^{5/2}}{1155 b^4 x^5}-\frac {16 c^2 \left (b x+c x^2\right )^{5/2}}{231 b^3 x^6}+\frac {4 c \left (b x+c x^2\right )^{5/2}}{33 b^2 x^7}-\frac {2 \left (b x+c x^2\right )^{5/2}}{11 b x^8} \end {gather*}
Antiderivative was successfully verified.
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Rule 650
Rule 658
Rubi steps
\begin {align*} \int \frac {\left (b x+c x^2\right )^{3/2}}{x^8} \, dx &=-\frac {2 \left (b x+c x^2\right )^{5/2}}{11 b x^8}-\frac {(6 c) \int \frac {\left (b x+c x^2\right )^{3/2}}{x^7} \, dx}{11 b}\\ &=-\frac {2 \left (b x+c x^2\right )^{5/2}}{11 b x^8}+\frac {4 c \left (b x+c x^2\right )^{5/2}}{33 b^2 x^7}+\frac {\left (8 c^2\right ) \int \frac {\left (b x+c x^2\right )^{3/2}}{x^6} \, dx}{33 b^2}\\ &=-\frac {2 \left (b x+c x^2\right )^{5/2}}{11 b x^8}+\frac {4 c \left (b x+c x^2\right )^{5/2}}{33 b^2 x^7}-\frac {16 c^2 \left (b x+c x^2\right )^{5/2}}{231 b^3 x^6}-\frac {\left (16 c^3\right ) \int \frac {\left (b x+c x^2\right )^{3/2}}{x^5} \, dx}{231 b^3}\\ &=-\frac {2 \left (b x+c x^2\right )^{5/2}}{11 b x^8}+\frac {4 c \left (b x+c x^2\right )^{5/2}}{33 b^2 x^7}-\frac {16 c^2 \left (b x+c x^2\right )^{5/2}}{231 b^3 x^6}+\frac {32 c^3 \left (b x+c x^2\right )^{5/2}}{1155 b^4 x^5}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 51, normalized size = 0.51 \begin {gather*} \frac {2 (x (b+c x))^{5/2} \left (-105 b^3+70 b^2 c x-40 b c^2 x^2+16 c^3 x^3\right )}{1155 b^4 x^8} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.27, size = 75, normalized size = 0.75 \begin {gather*} \frac {2 \sqrt {b x+c x^2} \left (-105 b^5-140 b^4 c x-5 b^3 c^2 x^2+6 b^2 c^3 x^3-8 b c^4 x^4+16 c^5 x^5\right )}{1155 b^4 x^6} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 71, normalized size = 0.71 \begin {gather*} \frac {2 \, {\left (16 \, c^{5} x^{5} - 8 \, b c^{4} x^{4} + 6 \, b^{2} c^{3} x^{3} - 5 \, b^{3} c^{2} x^{2} - 140 \, b^{4} c x - 105 \, b^{5}\right )} \sqrt {c x^{2} + b x}}{1155 \, b^{4} x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.21, size = 223, normalized size = 2.23 \begin {gather*} \frac {2 \, {\left (2310 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{7} c^{\frac {7}{2}} + 10164 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{6} b c^{3} + 19635 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{5} b^{2} c^{\frac {5}{2}} + 21285 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} b^{3} c^{2} + 13860 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} b^{4} c^{\frac {3}{2}} + 5390 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} b^{5} c + 1155 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} b^{6} \sqrt {c} + 105 \, b^{7}\right )}}{1155 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{11}} \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 = 0.55 \begin {gather*} -\frac {2 \left (c x +b \right ) \left (-16 x^{3} c^{3}+40 b \,x^{2} c^{2}-70 b^{2} x c +105 b^{3}\right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{1155 b^{4} x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.40, size = 139, normalized size = 1.39 \begin {gather*} \frac {32 \, \sqrt {c x^{2} + b x} c^{5}}{1155 \, b^{4} x} - \frac {16 \, \sqrt {c x^{2} + b x} c^{4}}{1155 \, b^{3} x^{2}} + \frac {4 \, \sqrt {c x^{2} + b x} c^{3}}{385 \, b^{2} x^{3}} - \frac {2 \, \sqrt {c x^{2} + b x} c^{2}}{231 \, b x^{4}} + \frac {\sqrt {c x^{2} + b x} c}{132 \, x^{5}} + \frac {3 \, \sqrt {c x^{2} + b x} b}{44 \, x^{6}} - \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}}}{4 \, x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.01, size = 123, normalized size = 1.23 \begin {gather*} \frac {4\,c^3\,\sqrt {c\,x^2+b\,x}}{385\,b^2\,x^3}-\frac {8\,c\,\sqrt {c\,x^2+b\,x}}{33\,x^5}-\frac {2\,c^2\,\sqrt {c\,x^2+b\,x}}{231\,b\,x^4}-\frac {2\,b\,\sqrt {c\,x^2+b\,x}}{11\,x^6}-\frac {16\,c^4\,\sqrt {c\,x^2+b\,x}}{1155\,b^3\,x^2}+\frac {32\,c^5\,\sqrt {c\,x^2+b\,x}}{1155\,b^4\,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}}}{x^{8}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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