Optimal. Leaf size=66 \[ -\frac {a^2}{6 c^2 x^5 \sqrt {c x^2}}-\frac {2 a b}{5 c^2 x^4 \sqrt {c x^2}}-\frac {b^2}{4 c^2 x^3 \sqrt {c x^2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {15, 45}
\begin {gather*} -\frac {a^2}{6 c^2 x^5 \sqrt {c x^2}}-\frac {2 a b}{5 c^2 x^4 \sqrt {c x^2}}-\frac {b^2}{4 c^2 x^3 \sqrt {c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 45
Rubi steps
\begin {align*} \int \frac {(a+b x)^2}{x^2 \left (c x^2\right )^{5/2}} \, dx &=\frac {x \int \frac {(a+b x)^2}{x^7} \, dx}{c^2 \sqrt {c x^2}}\\ &=\frac {x \int \left (\frac {a^2}{x^7}+\frac {2 a b}{x^6}+\frac {b^2}{x^5}\right ) \, dx}{c^2 \sqrt {c x^2}}\\ &=-\frac {a^2}{6 c^2 x^5 \sqrt {c x^2}}-\frac {2 a b}{5 c^2 x^4 \sqrt {c x^2}}-\frac {b^2}{4 c^2 x^3 \sqrt {c x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 38, normalized size = 0.58 \begin {gather*} -\frac {\sqrt {c x^2} \left (10 a^2+24 a b x+15 b^2 x^2\right )}{60 c^3 x^7} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 32, normalized size = 0.48
method | result | size |
gosper | \(-\frac {15 x^{2} b^{2}+24 a b x +10 a^{2}}{60 x \left (c \,x^{2}\right )^{\frac {5}{2}}}\) | \(32\) |
default | \(-\frac {15 x^{2} b^{2}+24 a b x +10 a^{2}}{60 x \left (c \,x^{2}\right )^{\frac {5}{2}}}\) | \(32\) |
risch | \(\frac {-\frac {1}{4} x^{2} b^{2}-\frac {2}{5} a b x -\frac {1}{6} a^{2}}{c^{2} x^{5} \sqrt {c \,x^{2}}}\) | \(34\) |
trager | \(\frac {\left (-1+x \right ) \left (10 a^{2} x^{5}+24 a b \,x^{5}+15 b^{2} x^{5}+10 a^{2} x^{4}+24 a b \,x^{4}+15 b^{2} x^{4}+10 a^{2} x^{3}+24 a b \,x^{3}+15 b^{2} x^{3}+10 a^{2} x^{2}+24 a b \,x^{2}+15 x^{2} b^{2}+10 a^{2} x +24 a b x +10 a^{2}\right ) \sqrt {c \,x^{2}}}{60 c^{3} x^{7}}\) | \(128\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 33, normalized size = 0.50 \begin {gather*} -\frac {b^{2}}{4 \, c^{\frac {5}{2}} x^{4}} - \frac {2 \, a b}{5 \, c^{\frac {5}{2}} x^{5}} - \frac {a^{2}}{6 \, c^{\frac {5}{2}} x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.47, size = 34, normalized size = 0.52 \begin {gather*} -\frac {{\left (15 \, b^{2} x^{2} + 24 \, a b x + 10 \, a^{2}\right )} \sqrt {c x^{2}}}{60 \, c^{3} x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.35, size = 46, normalized size = 0.70 \begin {gather*} - \frac {a^{2}}{6 x \left (c x^{2}\right )^{\frac {5}{2}}} - \frac {2 a b}{5 \left (c x^{2}\right )^{\frac {5}{2}}} - \frac {b^{2} x}{4 \left (c x^{2}\right )^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.99, size = 40, normalized size = 0.61 \begin {gather*} -\frac {15 \, b^{2} \sqrt {c} x^{2} + 24 \, a b \sqrt {c} x + 10 \, a^{2} \sqrt {c}}{60 \, c^{3} x^{6} \mathrm {sgn}\left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.18, size = 42, normalized size = 0.64 \begin {gather*} -\frac {10\,a^2\,\sqrt {x^2}+15\,b^2\,x^2\,\sqrt {x^2}+24\,a\,b\,x\,\sqrt {x^2}}{60\,c^{5/2}\,x^7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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