Optimal. Leaf size=80 \[ -\frac {2}{5 a x \left (a x+b x^2\right )^{3/2}}+\frac {16 b (a+2 b x)}{15 a^3 \left (a x+b x^2\right )^{3/2}}-\frac {128 b^2 (a+2 b x)}{15 a^5 \sqrt {a x+b x^2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {672, 628, 627}
\begin {gather*} -\frac {128 b^2 (a+2 b x)}{15 a^5 \sqrt {a x+b x^2}}+\frac {16 b (a+2 b x)}{15 a^3 \left (a x+b x^2\right )^{3/2}}-\frac {2}{5 a x \left (a x+b x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 627
Rule 628
Rule 672
Rubi steps
\begin {align*} \int \frac {1}{x \left (a x+b x^2\right )^{5/2}} \, dx &=-\frac {2}{5 a x \left (a x+b x^2\right )^{3/2}}-\frac {(8 b) \int \frac {1}{\left (a x+b x^2\right )^{5/2}} \, dx}{5 a}\\ &=-\frac {2}{5 a x \left (a x+b x^2\right )^{3/2}}+\frac {16 b (a+2 b x)}{15 a^3 \left (a x+b x^2\right )^{3/2}}+\frac {\left (64 b^2\right ) \int \frac {1}{\left (a x+b x^2\right )^{3/2}} \, dx}{15 a^3}\\ &=-\frac {2}{5 a x \left (a x+b x^2\right )^{3/2}}+\frac {16 b (a+2 b x)}{15 a^3 \left (a x+b x^2\right )^{3/2}}-\frac {128 b^2 (a+2 b x)}{15 a^5 \sqrt {a x+b x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 64, normalized size = 0.80 \begin {gather*} -\frac {2 (a+b x) \left (3 a^4-8 a^3 b x+48 a^2 b^2 x^2+192 a b^3 x^3+128 b^4 x^4\right )}{15 a^5 (x (a+b x))^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.43, size = 73, normalized size = 0.91
method | result | size |
gosper | \(-\frac {2 \left (b x +a \right ) \left (128 b^{4} x^{4}+192 a \,b^{3} x^{3}+48 a^{2} b^{2} x^{2}-8 a^{3} b x +3 a^{4}\right )}{15 a^{5} \left (b \,x^{2}+a x \right )^{\frac {5}{2}}}\) | \(63\) |
trager | \(-\frac {2 \left (128 b^{4} x^{4}+192 a \,b^{3} x^{3}+48 a^{2} b^{2} x^{2}-8 a^{3} b x +3 a^{4}\right ) \sqrt {b \,x^{2}+a x}}{15 a^{5} x^{3} \left (b x +a \right )^{2}}\) | \(68\) |
default | \(-\frac {2}{5 a x \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}-\frac {8 b \left (-\frac {2 \left (2 b x +a \right )}{3 a^{2} \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}+\frac {16 b \left (2 b x +a \right )}{3 a^{4} \sqrt {b \,x^{2}+a x}}\right )}{5 a}\) | \(73\) |
risch | \(-\frac {2 \left (b x +a \right ) \left (73 b^{2} x^{2}-14 a b x +3 a^{2}\right )}{15 a^{5} x^{2} \sqrt {x \left (b x +a \right )}}-\frac {2 b^{3} \left (11 b x +12 a \right ) x}{3 \sqrt {x \left (b x +a \right )}\, \left (b x +a \right ) a^{5}}\) | \(76\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 96, normalized size = 1.20 \begin {gather*} \frac {32 \, b^{2} x}{15 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a^{3}} - \frac {256 \, b^{3} x}{15 \, \sqrt {b x^{2} + a x} a^{5}} + \frac {16 \, b}{15 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a^{2}} - \frac {128 \, b^{2}}{15 \, \sqrt {b x^{2} + a x} a^{4}} - \frac {2}{5 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.95, size = 83, normalized size = 1.04 \begin {gather*} -\frac {2 \, {\left (128 \, b^{4} x^{4} + 192 \, a b^{3} x^{3} + 48 \, a^{2} b^{2} x^{2} - 8 \, a^{3} b x + 3 \, a^{4}\right )} \sqrt {b x^{2} + a x}}{15 \, {\left (a^{5} b^{2} x^{5} + 2 \, a^{6} b x^{4} + a^{7} x^{3}\right )}} \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 \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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.30, size = 67, normalized size = 0.84 \begin {gather*} -\frac {2\,\sqrt {b\,x^2+a\,x}\,\left (3\,a^4-8\,a^3\,b\,x+48\,a^2\,b^2\,x^2+192\,a\,b^3\,x^3+128\,b^4\,x^4\right )}{15\,a^5\,x^3\,{\left (a+b\,x\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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