Optimal. Leaf size=52 \[ \frac {2 a b x^2}{\sqrt {c x^2}}+\frac {b^2 x^3}{2 \sqrt {c x^2}}+\frac {a^2 x \log (x)}{\sqrt {c x^2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {15, 45}
\begin {gather*} \frac {a^2 x \log (x)}{\sqrt {c x^2}}+\frac {2 a b x^2}{\sqrt {c x^2}}+\frac {b^2 x^3}{2 \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}{\sqrt {c x^2}} \, dx &=\frac {x \int \frac {(a+b x)^2}{x} \, dx}{\sqrt {c x^2}}\\ &=\frac {x \int \left (2 a b+\frac {a^2}{x}+b^2 x\right ) \, dx}{\sqrt {c x^2}}\\ &=\frac {2 a b x^2}{\sqrt {c x^2}}+\frac {b^2 x^3}{2 \sqrt {c x^2}}+\frac {a^2 x \log (x)}{\sqrt {c x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 32, normalized size = 0.62 \begin {gather*} \frac {x \left (b x (4 a+b x)+2 a^2 \log (x)\right )}{2 \sqrt {c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 31, normalized size = 0.60
method | result | size |
default | \(\frac {x \left (x^{2} b^{2}+2 a^{2} \ln \left (x \right )+4 a b x \right )}{2 \sqrt {c \,x^{2}}}\) | \(31\) |
risch | \(\frac {x b \left (\frac {1}{2} x^{2} b +2 a x \right )}{\sqrt {c \,x^{2}}}+\frac {a^{2} x \ln \left (x \right )}{\sqrt {c \,x^{2}}}\) | \(37\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 35, normalized size = 0.67 \begin {gather*} \frac {b^{2} x^{2}}{2 \, \sqrt {c}} + \frac {a^{2} \log \left (x\right )}{\sqrt {c}} + \frac {2 \, \sqrt {c x^{2}} a b}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 35, normalized size = 0.67 \begin {gather*} \frac {{\left (b^{2} x^{2} + 4 \, a b x + 2 \, a^{2} \log \left (x\right )\right )} \sqrt {c x^{2}}}{2 \, c x} \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 {\left (a + b x\right )^{2}}{\sqrt {c x^{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.99, size = 43, normalized size = 0.83 \begin {gather*} \frac {a^{2} \log \left ({\left | x \right |}\right )}{\sqrt {c} \mathrm {sgn}\left (x\right )} + \frac {b^{2} c^{\frac {3}{2}} x^{2} \mathrm {sgn}\left (x\right ) + 4 \, a b c^{\frac {3}{2}} x \mathrm {sgn}\left (x\right )}{2 \, c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\left (a+b\,x\right )}^2}{\sqrt {c\,x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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