Optimal. Leaf size=54 \[ \frac {p x}{a}-\frac {2 f p \sqrt {b+a x}}{a^2}-\frac {2 \left (b p-f^2 p-a q\right ) \log \left (f+\sqrt {b+a x}\right )}{a^2} \]
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Rubi [A]
time = 0.26, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 1, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {711}
\begin {gather*} -\frac {2 \left (-a q+b p+f^2 (-p)\right ) \log \left (\sqrt {a x+b}+f\right )}{a^2}-\frac {2 f p \sqrt {a x+b}}{a^2}+\frac {p x}{a} \end {gather*}
Antiderivative was successfully verified.
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Rule 711
Rubi steps
\begin {align*} \int \frac {q+p x}{\sqrt {b+a x} \left (f+\sqrt {b+a x}\right )} \, dx &=\frac {2 \text {Subst}\left (\int \frac {-b p+a q+p x^2}{f+x} \, dx,x,\sqrt {b+a x}\right )}{a^2}\\ &=\frac {2 \text {Subst}\left (\int \left (-f p+p x+\frac {-b p+f^2 p+a q}{f+x}\right ) \, dx,x,\sqrt {b+a x}\right )}{a^2}\\ &=\frac {p x}{a}-\frac {2 f p \sqrt {b+a x}}{a^2}-\frac {2 \left (b p-f^2 p-a q\right ) \log \left (f+\sqrt {b+a x}\right )}{a^2}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 51, normalized size = 0.94 \begin {gather*} \frac {p \left (b+a x-2 f \sqrt {b+a x}\right )+2 \left (-b p+f^2 p+a q\right ) \log \left (f+\sqrt {b+a x}\right )}{a^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.36, size = 50, normalized size = 0.93
method | result | size |
derivativedivides | \(\frac {-2 f p \sqrt {a x +b}+p \left (a x +b \right )+2 \left (f^{2} p +a q -p b \right ) \ln \left (f +\sqrt {a x +b}\right )}{a^{2}}\) | \(50\) |
default | \(\frac {-2 f p \sqrt {a x +b}+p \left (a x +b \right )+2 \left (f^{2} p +a q -p b \right ) \ln \left (f +\sqrt {a x +b}\right )}{a^{2}}\) | \(50\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 58, normalized size = 1.07 \begin {gather*} \frac {\frac {2 \, {\left ({\left (f^{2} - b\right )} p + a q\right )} \log \left (f + \sqrt {a x + b}\right )}{a} - \frac {2 \, \sqrt {a x + b} f p - {\left (a x + b\right )} p}{a}}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 45, normalized size = 0.83 \begin {gather*} \frac {a p x - 2 \, \sqrt {a x + b} f p + 2 \, {\left ({\left (f^{2} - b\right )} p + a q\right )} \log \left (f + \sqrt {a x + b}\right )}{a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 16.59, size = 99, normalized size = 1.83 \begin {gather*} - \frac {2 f p \sqrt {a x + b}}{a^{2}} - \frac {2 f \left (- a q + b p - f^{2} p\right ) \left (\begin {cases} \frac {1}{\sqrt {a x + b}} & \text {for}\: f = 0 \\\frac {\log {\left (\frac {f}{\sqrt {a x + b}} + 1 \right )}}{f} & \text {otherwise} \end {cases}\right )}{a^{2}} + \frac {p \left (a x + b\right )}{a^{2}} + \frac {2 \left (- a q + b p - f^{2} p\right ) \log {\left (\frac {1}{\sqrt {a x + b}} \right )}}{a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.44, size = 61, normalized size = 1.13 \begin {gather*} \frac {2 \, {\left (f^{2} p - b p + a q\right )} \log \left ({\left | f + \sqrt {a x + b} \right |}\right )}{a^{2}} - \frac {2 \, \sqrt {a x + b} a^{2} f p - {\left (a x + b\right )} a^{2} p}{a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.10, size = 50, normalized size = 0.93 \begin {gather*} \frac {\ln \left (f+\sqrt {b+a\,x}\right )\,\left (2\,p\,f^2+2\,a\,q-2\,b\,p\right )}{a^2}+\frac {p\,x}{a}-\frac {2\,f\,p\,\sqrt {b+a\,x}}{a^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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