Optimal. Leaf size=103 \[ \frac {128 c^3 (b+2 c x)}{35 b^5 \sqrt {b x+c x^2}}-\frac {32 c^2}{35 b^3 x \sqrt {b x+c x^2}}+\frac {16 c}{35 b^2 x^2 \sqrt {b x+c x^2}}-\frac {2}{7 b x^3 \sqrt {b x+c x^2}} \]
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Rubi [A] time = 0.04, antiderivative size = 103, 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, 613} \[ \frac {128 c^3 (b+2 c x)}{35 b^5 \sqrt {b x+c x^2}}-\frac {32 c^2}{35 b^3 x \sqrt {b x+c x^2}}+\frac {16 c}{35 b^2 x^2 \sqrt {b x+c x^2}}-\frac {2}{7 b x^3 \sqrt {b x+c x^2}} \]
Antiderivative was successfully verified.
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Rule 613
Rule 658
Rubi steps
\begin {align*} \int \frac {1}{x^3 \left (b x+c x^2\right )^{3/2}} \, dx &=-\frac {2}{7 b x^3 \sqrt {b x+c x^2}}-\frac {(8 c) \int \frac {1}{x^2 \left (b x+c x^2\right )^{3/2}} \, dx}{7 b}\\ &=-\frac {2}{7 b x^3 \sqrt {b x+c x^2}}+\frac {16 c}{35 b^2 x^2 \sqrt {b x+c x^2}}+\frac {\left (48 c^2\right ) \int \frac {1}{x \left (b x+c x^2\right )^{3/2}} \, dx}{35 b^2}\\ &=-\frac {2}{7 b x^3 \sqrt {b x+c x^2}}+\frac {16 c}{35 b^2 x^2 \sqrt {b x+c x^2}}-\frac {32 c^2}{35 b^3 x \sqrt {b x+c x^2}}-\frac {\left (64 c^3\right ) \int \frac {1}{\left (b x+c x^2\right )^{3/2}} \, dx}{35 b^3}\\ &=-\frac {2}{7 b x^3 \sqrt {b x+c x^2}}+\frac {16 c}{35 b^2 x^2 \sqrt {b x+c x^2}}-\frac {32 c^2}{35 b^3 x \sqrt {b x+c x^2}}+\frac {128 c^3 (b+2 c x)}{35 b^5 \sqrt {b x+c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 62, normalized size = 0.60 \[ \frac {2 \left (-5 b^4+8 b^3 c x-16 b^2 c^2 x^2+64 b c^3 x^3+128 c^4 x^4\right )}{35 b^5 x^3 \sqrt {x (b+c x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.87, size = 72, normalized size = 0.70 \[ \frac {2 \, {\left (128 \, c^{4} x^{4} + 64 \, b c^{3} x^{3} - 16 \, b^{2} c^{2} x^{2} + 8 \, b^{3} c x - 5 \, b^{4}\right )} \sqrt {c x^{2} + b x}}{35 \, {\left (b^{5} c x^{5} + b^{6} x^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 66, normalized size = 0.64 \[ -\frac {2 \left (c x +b \right ) \left (-128 c^{4} x^{4}-64 x^{3} c^{3} b +16 c^{2} x^{2} b^{2}-8 c x \,b^{3}+5 b^{4}\right )}{35 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} b^{5} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.31, size = 101, normalized size = 0.98 \[ \frac {256 \, c^{4} x}{35 \, \sqrt {c x^{2} + b x} b^{5}} + \frac {128 \, c^{3}}{35 \, \sqrt {c x^{2} + b x} b^{4}} - \frac {32 \, c^{2}}{35 \, \sqrt {c x^{2} + b x} b^{3} x} + \frac {16 \, c}{35 \, \sqrt {c x^{2} + b x} b^{2} x^{2}} - \frac {2}{7 \, \sqrt {c x^{2} + b x} b x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.32, size = 102, normalized size = 0.99 \[ \frac {\sqrt {c\,x^2+b\,x}\,\left (\frac {186\,c^3}{35\,b^4}+\frac {256\,c^4\,x}{35\,b^5}\right )}{x\,\left (b+c\,x\right )}-\frac {58\,c^2\,\sqrt {c\,x^2+b\,x}}{35\,b^4\,x^2}-\frac {2\,\sqrt {c\,x^2+b\,x}}{7\,b^2\,x^4}+\frac {26\,c\,\sqrt {c\,x^2+b\,x}}{35\,b^3\,x^3} \]
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 \[ \int \frac {1}{x^{3} \left (x \left (b + c x\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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