3.566 \(\int \frac{q+p x}{\sqrt{b+a x} \left (f+\sqrt{b+a x}\right )} \, dx\)

Optimal. Leaf size=54 \[ -\frac{2 \left (-a q+b p+f^2 (-p)\right ) \log \left (\sqrt{a x+b}+f\right )}{a^2}-\frac{2 f p \sqrt{a x+b}}{a^2}+\frac{p x}{a} \]

[Out]

(p*x)/a - (2*f*p*Sqrt[b + a*x])/a^2 - (2*(b*p - f^2*p - a*q)*Log[f + Sqrt[b + a*
x]])/a^2

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Rubi [A]  time = 0.602673, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 1, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036 \[ -\frac{2 \left (-a q+b p+f^2 (-p)\right ) \log \left (\sqrt{a x+b}+f\right )}{a^2}-\frac{2 f p \sqrt{a x+b}}{a^2}+\frac{p x}{a} \]

Antiderivative was successfully verified.

[In]  Int[(q + p*x)/(Sqrt[b + a*x]*(f + Sqrt[b + a*x])),x]

[Out]

(p*x)/a - (2*f*p*Sqrt[b + a*x])/a^2 - (2*(b*p - f^2*p - a*q)*Log[f + Sqrt[b + a*
x]])/a^2

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{2 q \log{\left (f + \sqrt{a x + b} \right )}}{a} - \frac{2 p \left (b - f^{2}\right ) \log{\left (f + \sqrt{a x + b} \right )}}{a^{2}} - \frac{2 p \int ^{\sqrt{a x + b}} f\, dx}{a^{2}} + \frac{2 p \int ^{\sqrt{a x + b}} x\, dx}{a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((p*x+q)/(a*x+b)**(1/2)/(f+(a*x+b)**(1/2)),x)

[Out]

2*q*log(f + sqrt(a*x + b))/a - 2*p*(b - f**2)*log(f + sqrt(a*x + b))/a**2 - 2*p*
Integral(f, (x, sqrt(a*x + b)))/a**2 + 2*p*Integral(x, (x, sqrt(a*x + b)))/a**2

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Mathematica [A]  time = 0.129915, size = 77, normalized size = 1.43 \[ \frac{\left (a q-b p+f^2 p\right ) \log \left (a x+b-f^2\right )+2 \left (a q-b p+f^2 p\right ) \tanh ^{-1}\left (\frac{\sqrt{a x+b}}{f}\right )+p \left (a x-2 f \sqrt{a x+b}\right )}{a^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(q + p*x)/(Sqrt[b + a*x]*(f + Sqrt[b + a*x])),x]

[Out]

(p*(a*x - 2*f*Sqrt[b + a*x]) + 2*(-(b*p) + f^2*p + a*q)*ArcTanh[Sqrt[b + a*x]/f]
+ (-(b*p) + f^2*p + a*q)*Log[b - f^2 + a*x])/a^2

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Maple [A]  time = 0.007, size = 80, normalized size = 1.5 \[{\frac{px}{a}}+{\frac{pb}{{a}^{2}}}-2\,{\frac{fp\sqrt{ax+b}}{{a}^{2}}}+2\,{\frac{\ln \left ( f+\sqrt{ax+b} \right ){f}^{2}p}{{a}^{2}}}+2\,{\frac{\ln \left ( f+\sqrt{ax+b} \right ) q}{a}}-2\,{\frac{\ln \left ( f+\sqrt{ax+b} \right ) bp}{{a}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((p*x+q)/(a*x+b)^(1/2)/(f+(a*x+b)^(1/2)),x)

[Out]

p*x/a+1/a^2*p*b-2*f*p*(a*x+b)^(1/2)/a^2+2/a^2*ln(f+(a*x+b)^(1/2))*f^2*p+2/a*ln(f
+(a*x+b)^(1/2))*q-2/a^2*ln(f+(a*x+b)^(1/2))*b*p

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Maxima [A]  time = 0.715266, size = 78, normalized size = 1.44 \[ \frac{\frac{2 \,{\left ({\left (f^{2} - b\right )} p + a q\right )} \log \left (f + \sqrt{a x + b}\right )}{a} - \frac{2 \, \sqrt{a x + b} f p -{\left (a x + b\right )} p}{a}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((p*x + q)/(sqrt(a*x + b)*(f + sqrt(a*x + b))),x, algorithm="maxima")

[Out]

(2*((f^2 - b)*p + a*q)*log(f + sqrt(a*x + b))/a - (2*sqrt(a*x + b)*f*p - (a*x +
b)*p)/a)/a

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Fricas [A]  time = 0.264479, size = 61, normalized size = 1.13 \[ \frac{a p x - 2 \, \sqrt{a x + b} f p + 2 \,{\left ({\left (f^{2} - b\right )} p + a q\right )} \log \left (f + \sqrt{a x + b}\right )}{a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((p*x + q)/(sqrt(a*x + b)*(f + sqrt(a*x + b))),x, algorithm="fricas")

[Out]

(a*p*x - 2*sqrt(a*x + b)*f*p + 2*((f^2 - b)*p + a*q)*log(f + sqrt(a*x + b)))/a^2

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Sympy [A]  time = 11.7899, size = 99, normalized size = 1.83 \[ - \frac{2 f p \sqrt{a x + b}}{a^{2}} - \frac{2 f \left (- a q + b p - f^{2} p\right ) \left (\begin{cases} \frac{1}{\sqrt{a x + b}} & \text{for}\: f = 0 \\\frac{\log{\left (\frac{f}{\sqrt{a x + b}} + 1 \right )}}{f} & \text{otherwise} \end{cases}\right )}{a^{2}} + \frac{p \left (a x + b\right )}{a^{2}} + \frac{2 \left (- a q + b p - f^{2} p\right ) \log{\left (\frac{1}{\sqrt{a x + b}} \right )}}{a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((p*x+q)/(a*x+b)**(1/2)/(f+(a*x+b)**(1/2)),x)

[Out]

-2*f*p*sqrt(a*x + b)/a**2 - 2*f*(-a*q + b*p - f**2*p)*Piecewise((1/sqrt(a*x + b)
, Eq(f, 0)), (log(f/sqrt(a*x + b) + 1)/f, True))/a**2 + p*(a*x + b)/a**2 + 2*(-a
*q + b*p - f**2*p)*log(1/sqrt(a*x + b))/a**2

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GIAC/XCAS [A]  time = 0.275324, size = 119, normalized size = 2.2 \[ \frac{2 \,{\left (f^{2} p - b p + a q\right )}{\rm ln}\left ({\left | f + \sqrt{a x + b} \right |}\right )}{a^{2}} - \frac{2 \,{\left (f^{2} p{\rm ln}\left ({\left | f \right |}\right ) - b p{\rm ln}\left ({\left | f \right |}\right ) + a q{\rm ln}\left ({\left | f \right |}\right )\right )}}{a^{2}} - \frac{2 \, \sqrt{a x + b} a^{2} f p -{\left (a x + b\right )} a^{2} p}{a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((p*x + q)/(sqrt(a*x + b)*(f + sqrt(a*x + b))),x, algorithm="giac")

[Out]

2*(f^2*p - b*p + a*q)*ln(abs(f + sqrt(a*x + b)))/a^2 - 2*(f^2*p*ln(abs(f)) - b*p
*ln(abs(f)) + a*q*ln(abs(f)))/a^2 - (2*sqrt(a*x + b)*a^2*f*p - (a*x + b)*a^2*p)/
a^4