Optimal. Leaf size=126 \[ -\frac {256 c^4 \sqrt {b x+c x^2}}{315 b^5 x}+\frac {128 c^3 \sqrt {b x+c x^2}}{315 b^4 x^2}-\frac {32 c^2 \sqrt {b x+c x^2}}{105 b^3 x^3}+\frac {16 c \sqrt {b x+c x^2}}{63 b^2 x^4}-\frac {2 \sqrt {b x+c x^2}}{9 b x^5} \]
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Rubi [A] time = 0.05, 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 = {658, 650} \begin {gather*} -\frac {256 c^4 \sqrt {b x+c x^2}}{315 b^5 x}+\frac {128 c^3 \sqrt {b x+c x^2}}{315 b^4 x^2}-\frac {32 c^2 \sqrt {b x+c x^2}}{105 b^3 x^3}+\frac {16 c \sqrt {b x+c x^2}}{63 b^2 x^4}-\frac {2 \sqrt {b x+c x^2}}{9 b x^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 650
Rule 658
Rubi steps
\begin {align*} \int \frac {1}{x^5 \sqrt {b x+c x^2}} \, dx &=-\frac {2 \sqrt {b x+c x^2}}{9 b x^5}-\frac {(8 c) \int \frac {1}{x^4 \sqrt {b x+c x^2}} \, dx}{9 b}\\ &=-\frac {2 \sqrt {b x+c x^2}}{9 b x^5}+\frac {16 c \sqrt {b x+c x^2}}{63 b^2 x^4}+\frac {\left (16 c^2\right ) \int \frac {1}{x^3 \sqrt {b x+c x^2}} \, dx}{21 b^2}\\ &=-\frac {2 \sqrt {b x+c x^2}}{9 b x^5}+\frac {16 c \sqrt {b x+c x^2}}{63 b^2 x^4}-\frac {32 c^2 \sqrt {b x+c x^2}}{105 b^3 x^3}-\frac {\left (64 c^3\right ) \int \frac {1}{x^2 \sqrt {b x+c x^2}} \, dx}{105 b^3}\\ &=-\frac {2 \sqrt {b x+c x^2}}{9 b x^5}+\frac {16 c \sqrt {b x+c x^2}}{63 b^2 x^4}-\frac {32 c^2 \sqrt {b x+c x^2}}{105 b^3 x^3}+\frac {128 c^3 \sqrt {b x+c x^2}}{315 b^4 x^2}+\frac {\left (128 c^4\right ) \int \frac {1}{x \sqrt {b x+c x^2}} \, dx}{315 b^4}\\ &=-\frac {2 \sqrt {b x+c x^2}}{9 b x^5}+\frac {16 c \sqrt {b x+c x^2}}{63 b^2 x^4}-\frac {32 c^2 \sqrt {b x+c x^2}}{105 b^3 x^3}+\frac {128 c^3 \sqrt {b x+c x^2}}{315 b^4 x^2}-\frac {256 c^4 \sqrt {b x+c x^2}}{315 b^5 x}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 62, normalized size = 0.49 \begin {gather*} -\frac {2 \sqrt {x (b+c x)} \left (35 b^4-40 b^3 c x+48 b^2 c^2 x^2-64 b c^3 x^3+128 c^4 x^4\right )}{315 b^5 x^5} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.17, size = 64, normalized size = 0.51 \begin {gather*} -\frac {2 \sqrt {b x+c x^2} \left (35 b^4-40 b^3 c x+48 b^2 c^2 x^2-64 b c^3 x^3+128 c^4 x^4\right )}{315 b^5 x^5} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 60, normalized size = 0.48 \begin {gather*} -\frac {2 \, {\left (128 \, c^{4} x^{4} - 64 \, b c^{3} x^{3} + 48 \, b^{2} c^{2} x^{2} - 40 \, b^{3} c x + 35 \, b^{4}\right )} \sqrt {c x^{2} + b x}}{315 \, b^{5} x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 136, normalized size = 1.08 \begin {gather*} \frac {2 \, {\left (1008 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} c^{2} + 1680 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} b c^{\frac {3}{2}} + 1080 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} b^{2} c + 315 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} b^{3} \sqrt {c} + 35 \, b^{4}\right )}}{315 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 66, normalized size = 0.52 \begin {gather*} -\frac {2 \left (c x +b \right ) \left (128 c^{4} x^{4}-64 x^{3} c^{3} b +48 c^{2} x^{2} b^{2}-40 c x \,b^{3}+35 b^{4}\right )}{315 \sqrt {c \,x^{2}+b x}\, b^{5} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.40, size = 106, normalized size = 0.84 \begin {gather*} -\frac {256 \, \sqrt {c x^{2} + b x} c^{4}}{315 \, b^{5} x} + \frac {128 \, \sqrt {c x^{2} + b x} c^{3}}{315 \, b^{4} x^{2}} - \frac {32 \, \sqrt {c x^{2} + b x} c^{2}}{105 \, b^{3} x^{3}} + \frac {16 \, \sqrt {c x^{2} + b x} c}{63 \, b^{2} x^{4}} - \frac {2 \, \sqrt {c x^{2} + b x}}{9 \, b x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.18, size = 106, normalized size = 0.84 \begin {gather*} \frac {128\,c^3\,\sqrt {c\,x^2+b\,x}}{315\,b^4\,x^2}-\frac {32\,c^2\,\sqrt {c\,x^2+b\,x}}{105\,b^3\,x^3}-\frac {2\,\sqrt {c\,x^2+b\,x}}{9\,b\,x^5}-\frac {256\,c^4\,\sqrt {c\,x^2+b\,x}}{315\,b^5\,x}+\frac {16\,c\,\sqrt {c\,x^2+b\,x}}{63\,b^2\,x^4} \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 {1}{x^{5} \sqrt {x \left (b + c x\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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