Optimal. Leaf size=106 \[ \frac {256 b^3 (a+2 b x)}{21 a^6 \sqrt {a x+b x^2}}-\frac {32 b^2 (a+2 b x)}{21 a^4 \left (a x+b x^2\right )^{3/2}}+\frac {4 b}{7 a^2 x \left (a x+b x^2\right )^{3/2}}-\frac {2}{7 a x^2 \left (a x+b x^2\right )^{3/2}} \]
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Rubi [A] time = 0.04, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {658, 614, 613} \begin {gather*} \frac {256 b^3 (a+2 b x)}{21 a^6 \sqrt {a x+b x^2}}-\frac {32 b^2 (a+2 b x)}{21 a^4 \left (a x+b x^2\right )^{3/2}}+\frac {4 b}{7 a^2 x \left (a x+b x^2\right )^{3/2}}-\frac {2}{7 a x^2 \left (a x+b x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 613
Rule 614
Rule 658
Rubi steps
\begin {align*} \int \frac {1}{x^2 \left (a x+b x^2\right )^{5/2}} \, dx &=-\frac {2}{7 a x^2 \left (a x+b x^2\right )^{3/2}}-\frac {(10 b) \int \frac {1}{x \left (a x+b x^2\right )^{5/2}} \, dx}{7 a}\\ &=-\frac {2}{7 a x^2 \left (a x+b x^2\right )^{3/2}}+\frac {4 b}{7 a^2 x \left (a x+b x^2\right )^{3/2}}+\frac {\left (16 b^2\right ) \int \frac {1}{\left (a x+b x^2\right )^{5/2}} \, dx}{7 a^2}\\ &=-\frac {2}{7 a x^2 \left (a x+b x^2\right )^{3/2}}+\frac {4 b}{7 a^2 x \left (a x+b x^2\right )^{3/2}}-\frac {32 b^2 (a+2 b x)}{21 a^4 \left (a x+b x^2\right )^{3/2}}-\frac {\left (128 b^3\right ) \int \frac {1}{\left (a x+b x^2\right )^{3/2}} \, dx}{21 a^4}\\ &=-\frac {2}{7 a x^2 \left (a x+b x^2\right )^{3/2}}+\frac {4 b}{7 a^2 x \left (a x+b x^2\right )^{3/2}}-\frac {32 b^2 (a+2 b x)}{21 a^4 \left (a x+b x^2\right )^{3/2}}+\frac {256 b^3 (a+2 b x)}{21 a^6 \sqrt {a x+b x^2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 73, normalized size = 0.69 \begin {gather*} \frac {2 \left (-3 a^5+6 a^4 b x-16 a^3 b^2 x^2+96 a^2 b^3 x^3+384 a b^4 x^4+256 b^5 x^5\right )}{21 a^6 x^2 (x (a+b x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.37, size = 82, normalized size = 0.77 \begin {gather*} \frac {2 \sqrt {a x+b x^2} \left (-3 a^5+6 a^4 b x-16 a^3 b^2 x^2+96 a^2 b^3 x^3+384 a b^4 x^4+256 b^5 x^5\right )}{21 a^6 x^4 (a+b x)^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 94, normalized size = 0.89 \begin {gather*} \frac {2 \, {\left (256 \, b^{5} x^{5} + 384 \, a b^{4} x^{4} + 96 \, a^{2} b^{3} x^{3} - 16 \, a^{3} b^{2} x^{2} + 6 \, a^{4} b x - 3 \, a^{5}\right )} \sqrt {b x^{2} + a x}}{21 \, {\left (a^{6} b^{2} x^{6} + 2 \, a^{7} b x^{5} + a^{8} x^{4}\right )}} \end {gather*}
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 \begin {gather*} \int \frac {1}{{\left (b x^{2} + a x\right )}^{\frac {5}{2}} x^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 77, normalized size = 0.73 \begin {gather*} -\frac {2 \left (b x +a \right ) \left (-256 b^{5} x^{5}-384 a \,b^{4} x^{4}-96 a^{2} b^{3} x^{3}+16 a^{3} b^{2} x^{2}-6 a^{4} b x +3 a^{5}\right )}{21 \left (b \,x^{2}+a x \right )^{\frac {5}{2}} a^{6} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.38, size = 118, normalized size = 1.11 \begin {gather*} -\frac {64 \, b^{3} x}{21 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a^{4}} + \frac {512 \, b^{4} x}{21 \, \sqrt {b x^{2} + a x} a^{6}} - \frac {32 \, b^{2}}{21 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a^{3}} + \frac {256 \, b^{3}}{21 \, \sqrt {b x^{2} + a x} a^{5}} + \frac {4 \, b}{7 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a^{2} x} - \frac {2}{7 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.35, size = 121, normalized size = 1.14 \begin {gather*} \frac {\sqrt {b\,x^2+a\,x}\,\left (\frac {256\,b^3}{21\,a^5}+\frac {512\,b^4\,x}{21\,a^6}\right )}{x\,\left (a+b\,x\right )}-\frac {\sqrt {b\,x^2+a\,x}\,\left (\frac {74\,b^2}{21\,a^3}+\frac {88\,b^3\,x}{21\,a^4}\right )}{x^2\,{\left (a+b\,x\right )}^2}-\frac {2\,\sqrt {b\,x^2+a\,x}}{7\,a^3\,x^4}+\frac {8\,b\,\sqrt {b\,x^2+a\,x}}{7\,a^4\,x^3} \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^{2} \left (x \left (a + b x\right )\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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