3.177 \(\int \frac{a+b \sqrt{x}}{c+d \sqrt{x}} \, dx\)

Optimal. Leaf size=49 \[ \frac{2 c (b c-a d) \log \left (c+d \sqrt{x}\right )}{d^3}-\frac{2 \sqrt{x} (b c-a d)}{d^2}+\frac{b x}{d} \]

[Out]

(-2*(b*c - a*d)*Sqrt[x])/d^2 + (b*x)/d + (2*c*(b*c - a*d)*Log[c + d*Sqrt[x]])/d^
3

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Rubi [A]  time = 0.122266, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ \frac{2 c (b c-a d) \log \left (c+d \sqrt{x}\right )}{d^3}-\frac{2 \sqrt{x} (b c-a d)}{d^2}+\frac{b x}{d} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*Sqrt[x])/(c + d*Sqrt[x]),x]

[Out]

(-2*(b*c - a*d)*Sqrt[x])/d^2 + (b*x)/d + (2*c*(b*c - a*d)*Log[c + d*Sqrt[x]])/d^
3

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{2 b \int ^{\sqrt{x}} x\, dx}{d} - \frac{2 c \left (a d - b c\right ) \log{\left (c + d \sqrt{x} \right )}}{d^{3}} + \left (2 a d - 2 b c\right ) \int ^{\sqrt{x}} \frac{1}{d^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b*x**(1/2))/(c+d*x**(1/2)),x)

[Out]

2*b*Integral(x, (x, sqrt(x)))/d - 2*c*(a*d - b*c)*log(c + d*sqrt(x))/d**3 + (2*a
*d - 2*b*c)*Integral(d**(-2), (x, sqrt(x)))

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Mathematica [A]  time = 0.0436009, size = 51, normalized size = 1.04 \[ \frac{2 \left (b c^2-a c d\right ) \log \left (c+d \sqrt{x}\right )}{d^3}+\frac{2 \sqrt{x} (a d-b c)}{d^2}+\frac{b x}{d} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*Sqrt[x])/(c + d*Sqrt[x]),x]

[Out]

(2*(-(b*c) + a*d)*Sqrt[x])/d^2 + (b*x)/d + (2*(b*c^2 - a*c*d)*Log[c + d*Sqrt[x]]
)/d^3

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Maple [A]  time = 0.007, size = 59, normalized size = 1.2 \[{\frac{bx}{d}}+2\,{\frac{a\sqrt{x}}{d}}-2\,{\frac{b\sqrt{x}c}{{d}^{2}}}-2\,{\frac{c\ln \left ( c+d\sqrt{x} \right ) a}{{d}^{2}}}+2\,{\frac{{c}^{2}\ln \left ( c+d\sqrt{x} \right ) b}{{d}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^(1/2))/(c+d*x^(1/2)),x)

[Out]

b*x/d+2/d*a*x^(1/2)-2/d^2*b*x^(1/2)*c-2*c/d^2*ln(c+d*x^(1/2))*a+2*c^2/d^3*ln(c+d
*x^(1/2))*b

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Maxima [A]  time = 1.70584, size = 63, normalized size = 1.29 \[ \frac{b d x - 2 \,{\left (b c - a d\right )} \sqrt{x}}{d^{2}} + \frac{2 \,{\left (b c^{2} - a c d\right )} \log \left (d \sqrt{x} + c\right )}{d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)/(d*sqrt(x) + c),x, algorithm="maxima")

[Out]

(b*d*x - 2*(b*c - a*d)*sqrt(x))/d^2 + 2*(b*c^2 - a*c*d)*log(d*sqrt(x) + c)/d^3

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Fricas [A]  time = 0.239749, size = 65, normalized size = 1.33 \[ \frac{b d^{2} x + 2 \,{\left (b c^{2} - a c d\right )} \log \left (d \sqrt{x} + c\right ) - 2 \,{\left (b c d - a d^{2}\right )} \sqrt{x}}{d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)/(d*sqrt(x) + c),x, algorithm="fricas")

[Out]

(b*d^2*x + 2*(b*c^2 - a*c*d)*log(d*sqrt(x) + c) - 2*(b*c*d - a*d^2)*sqrt(x))/d^3

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Sympy [A]  time = 0.817719, size = 82, normalized size = 1.67 \[ \begin{cases} - \frac{2 a c \log{\left (\frac{c}{d} + \sqrt{x} \right )}}{d^{2}} + \frac{2 a \sqrt{x}}{d} + \frac{2 b c^{2} \log{\left (\frac{c}{d} + \sqrt{x} \right )}}{d^{3}} - \frac{2 b c \sqrt{x}}{d^{2}} + \frac{b x}{d} & \text{for}\: d \neq 0 \\\frac{a x + \frac{2 b x^{\frac{3}{2}}}{3}}{c} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b*x**(1/2))/(c+d*x**(1/2)),x)

[Out]

Piecewise((-2*a*c*log(c/d + sqrt(x))/d**2 + 2*a*sqrt(x)/d + 2*b*c**2*log(c/d + s
qrt(x))/d**3 - 2*b*c*sqrt(x)/d**2 + b*x/d, Ne(d, 0)), ((a*x + 2*b*x**(3/2)/3)/c,
True))

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GIAC/XCAS [A]  time = 0.215998, size = 66, normalized size = 1.35 \[ \frac{b d x - 2 \, b c \sqrt{x} + 2 \, a d \sqrt{x}}{d^{2}} + \frac{2 \,{\left (b c^{2} - a c d\right )}{\rm ln}\left ({\left | d \sqrt{x} + c \right |}\right )}{d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)/(d*sqrt(x) + c),x, algorithm="giac")

[Out]

(b*d*x - 2*b*c*sqrt(x) + 2*a*d*sqrt(x))/d^2 + 2*(b*c^2 - a*c*d)*ln(abs(d*sqrt(x)
+ c))/d^3