Optimal. Leaf size=66 \[ -\frac {a^2}{4 c x^3 \sqrt {c x^2}}-\frac {2 a b}{3 c x^2 \sqrt {c x^2}}-\frac {b^2}{2 c x \sqrt {c x^2}} \]
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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}{4 c x^3 \sqrt {c x^2}}-\frac {2 a b}{3 c x^2 \sqrt {c x^2}}-\frac {b^2}{2 c x \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 )^{3/2}} \, dx &=\frac {x \int \frac {(a+b x)^2}{x^5} \, dx}{c \sqrt {c x^2}}\\ &=\frac {x \int \left (\frac {a^2}{x^5}+\frac {2 a b}{x^4}+\frac {b^2}{x^3}\right ) \, dx}{c \sqrt {c x^2}}\\ &=-\frac {a^2}{4 c x^3 \sqrt {c x^2}}-\frac {2 a b}{3 c x^2 \sqrt {c x^2}}-\frac {b^2}{2 c x \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 (3 a^2+8 a b x+6 b^2 x^2\right )}{12 c^2 x^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 32, normalized size = 0.48
method | result | size |
gosper | \(-\frac {6 x^{2} b^{2}+8 a b x +3 a^{2}}{12 x \left (c \,x^{2}\right )^{\frac {3}{2}}}\) | \(32\) |
default | \(-\frac {6 x^{2} b^{2}+8 a b x +3 a^{2}}{12 x \left (c \,x^{2}\right )^{\frac {3}{2}}}\) | \(32\) |
risch | \(\frac {-\frac {1}{2} x^{2} b^{2}-\frac {2}{3} a b x -\frac {1}{4} a^{2}}{c \,x^{3} \sqrt {c \,x^{2}}}\) | \(34\) |
trager | \(\frac {\left (-1+x \right ) \left (3 a^{2} x^{3}+8 a b \,x^{3}+6 b^{2} x^{3}+3 a^{2} x^{2}+8 a b \,x^{2}+6 x^{2} b^{2}+3 a^{2} x +8 a b x +3 a^{2}\right ) \sqrt {c \,x^{2}}}{12 c^{2} x^{5}}\) | \(82\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 33, normalized size = 0.50 \begin {gather*} -\frac {b^{2}}{2 \, c^{\frac {3}{2}} x^{2}} - \frac {2 \, a b}{3 \, c^{\frac {3}{2}} x^{3}} - \frac {a^{2}}{4 \, c^{\frac {3}{2}} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.43, size = 34, normalized size = 0.52 \begin {gather*} -\frac {{\left (6 \, b^{2} x^{2} + 8 \, a b x + 3 \, a^{2}\right )} \sqrt {c x^{2}}}{12 \, c^{2} x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.25, size = 46, normalized size = 0.70 \begin {gather*} - \frac {a^{2}}{4 x \left (c x^{2}\right )^{\frac {3}{2}}} - \frac {2 a b}{3 \left (c x^{2}\right )^{\frac {3}{2}}} - \frac {b^{2} x}{2 \left (c x^{2}\right )^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.56, size = 40, normalized size = 0.61 \begin {gather*} -\frac {6 \, b^{2} \sqrt {c} x^{2} + 8 \, a b \sqrt {c} x + 3 \, a^{2} \sqrt {c}}{12 \, c^{2} x^{4} \mathrm {sgn}\left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.19, size = 42, normalized size = 0.64 \begin {gather*} -\frac {3\,a^2\,\sqrt {x^2}+6\,b^2\,x^2\,\sqrt {x^2}+8\,a\,b\,x\,\sqrt {x^2}}{12\,c^{3/2}\,x^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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