Optimal. Leaf size=47 \[ \frac {2 a}{b^2 d \left (a+b \sqrt {c+d x}\right )}+\frac {2 \log \left (a+b \sqrt {c+d x}\right )}{b^2 d} \]
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Rubi [A]
time = 0.02, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {253, 196, 45}
\begin {gather*} \frac {2 a}{b^2 d \left (a+b \sqrt {c+d x}\right )}+\frac {2 \log \left (a+b \sqrt {c+d x}\right )}{b^2 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 196
Rule 253
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b \sqrt {c+d x}\right )^2} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\left (a+b \sqrt {x}\right )^2} \, dx,x,c+d x\right )}{d}\\ &=\frac {2 \text {Subst}\left (\int \frac {x}{(a+b x)^2} \, dx,x,\sqrt {c+d x}\right )}{d}\\ &=\frac {2 \text {Subst}\left (\int \left (-\frac {a}{b (a+b x)^2}+\frac {1}{b (a+b x)}\right ) \, dx,x,\sqrt {c+d x}\right )}{d}\\ &=\frac {2 a}{b^2 d \left (a+b \sqrt {c+d x}\right )}+\frac {2 \log \left (a+b \sqrt {c+d x}\right )}{b^2 d}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 43, normalized size = 0.91 \begin {gather*} \frac {2 \left (\frac {a}{a+b \sqrt {c+d x}}+\log \left (b d \left (a+b \sqrt {c+d x}\right )\right )\right )}{b^2 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(218\) vs.
\(2(43)=86\).
time = 0.03, size = 219, normalized size = 4.66
method | result | size |
derivativedivides | \(\frac {\frac {2 \ln \left (a +b \sqrt {d x +c}\right )}{b^{2}}+\frac {2 a}{b^{2} \left (a +b \sqrt {d x +c}\right )}}{d}\) | \(41\) |
default | \(\frac {a^{2}}{\left (-b^{2} d x -b^{2} c +a^{2}\right ) b^{2} d}+\frac {c}{\left (-b^{2} d x -b^{2} c +a^{2}\right ) d}+\frac {\ln \left (b^{2} d x +b^{2} c -a^{2}\right )}{b^{2} d}+\frac {c}{d \left (b^{2} d x +b^{2} c -a^{2}\right )}-\frac {a^{2}}{b^{2} d \left (b^{2} d x +b^{2} c -a^{2}\right )}+\frac {a}{b^{2} d \left (a +b \sqrt {d x +c}\right )}+\frac {\ln \left (a +b \sqrt {d x +c}\right )}{b^{2} d}+\frac {a}{b^{2} d \left (-a +b \sqrt {d x +c}\right )}-\frac {\ln \left (-a +b \sqrt {d x +c}\right )}{b^{2} d}\) | \(219\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 43, normalized size = 0.91 \begin {gather*} \frac {2 \, {\left (\frac {a}{\sqrt {d x + c} b^{3} + a b^{2}} + \frac {\log \left (\sqrt {d x + c} b + a\right )}{b^{2}}\right )}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 75, normalized size = 1.60 \begin {gather*} \frac {2 \, {\left (\sqrt {d x + c} a b - a^{2} + {\left (b^{2} d x + b^{2} c - a^{2}\right )} \log \left (\sqrt {d x + c} b + a\right )\right )}}{b^{4} d^{2} x + {\left (b^{4} c - a^{2} b^{2}\right )} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 124 vs.
\(2 (39) = 78\).
time = 0.62, size = 124, normalized size = 2.64 \begin {gather*} \begin {cases} \frac {x}{a^{2}} & \text {for}\: b = 0 \wedge \left (b = 0 \vee d = 0\right ) \\\frac {x}{\left (a + b \sqrt {c}\right )^{2}} & \text {for}\: d = 0 \\\frac {2 a \log {\left (\frac {a}{b} + \sqrt {c + d x} \right )}}{a b^{2} d + b^{3} d \sqrt {c + d x}} + \frac {2 a}{a b^{2} d + b^{3} d \sqrt {c + d x}} + \frac {2 b \sqrt {c + d x} \log {\left (\frac {a}{b} + \sqrt {c + d x} \right )}}{a b^{2} d + b^{3} d \sqrt {c + d x}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.60, size = 44, normalized size = 0.94 \begin {gather*} \frac {2 \, \log \left ({\left | \sqrt {d x + c} b + a \right |}\right )}{b^{2} d} + \frac {2 \, a}{{\left (\sqrt {d x + c} b + a\right )} b^{2} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 43, normalized size = 0.91 \begin {gather*} \frac {2\,\ln \left (a+b\,\sqrt {c+d\,x}\right )}{b^2\,d}+\frac {2\,a}{b^2\,\left (a\,d+b\,d\,\sqrt {c+d\,x}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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