3.575 \(\int \frac{\sqrt{a+b x}}{(c+d x)^{3/2}} \, dx\)

Optimal. Leaf size=66 \[ \frac{2 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{d^{3/2}}-\frac{2 \sqrt{a+b x}}{d \sqrt{c+d x}} \]

[Out]

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

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Rubi [A]  time = 0.0720375, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{2 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{d^{3/2}}-\frac{2 \sqrt{a+b x}}{d \sqrt{c+d x}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b*x]/(c + d*x)^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 10.6235, size = 60, normalized size = 0.91 \[ \frac{2 \sqrt{b} \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{d^{\frac{3}{2}}} - \frac{2 \sqrt{a + b x}}{d \sqrt{c + d x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(1/2)/(d*x+c)**(3/2),x)

[Out]

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

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Mathematica [A]  time = 0.0565608, size = 78, normalized size = 1.18 \[ \frac{\sqrt{b} \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{d^{3/2}}-\frac{2 \sqrt{a+b x}}{d \sqrt{c+d x}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + b*x]/(c + d*x)^(3/2),x]

[Out]

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

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Maple [F]  time = 0., size = 0, normalized size = 0. \[ \int{1\sqrt{bx+a} \left ( dx+c \right ) ^{-{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(1/2)/(d*x+c)^(3/2),x)

[Out]

int((b*x+a)^(1/2)/(d*x+c)^(3/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x + a)/(d*x + c)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.283808, size = 1, normalized size = 0.02 \[ \left [\frac{{\left (d x + c\right )} \sqrt{\frac{b}{d}} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \,{\left (2 \, b d^{2} x + b c d + a d^{2}\right )} \sqrt{b x + a} \sqrt{d x + c} \sqrt{\frac{b}{d}} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right ) - 4 \, \sqrt{b x + a} \sqrt{d x + c}}{2 \,{\left (d^{2} x + c d\right )}}, \frac{{\left (d x + c\right )} \sqrt{-\frac{b}{d}} \arctan \left (\frac{2 \, b d x + b c + a d}{2 \, \sqrt{b x + a} \sqrt{d x + c} d \sqrt{-\frac{b}{d}}}\right ) - 2 \, \sqrt{b x + a} \sqrt{d x + c}}{d^{2} x + c d}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x + a)/(d*x + c)^(3/2),x, algorithm="fricas")

[Out]

[1/2*((d*x + c)*sqrt(b/d)*log(8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 4*
(2*b*d^2*x + b*c*d + a*d^2)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(b/d) + 8*(b^2*c*d +
a*b*d^2)*x) - 4*sqrt(b*x + a)*sqrt(d*x + c))/(d^2*x + c*d), ((d*x + c)*sqrt(-b/
d)*arctan(1/2*(2*b*d*x + b*c + a*d)/(sqrt(b*x + a)*sqrt(d*x + c)*d*sqrt(-b/d)))
- 2*sqrt(b*x + a)*sqrt(d*x + c))/(d^2*x + c*d)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**(1/2)/(d*x+c)**(3/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.237695, size = 130, normalized size = 1.97 \[ -\frac{2 \, b^{2}{\rm ln}\left ({\left | -\sqrt{b d} \sqrt{b x + a} + \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt{b d} d{\left | b \right |}} - \frac{2 \, \sqrt{b x + a} b^{2}}{\sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} d{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x + a)/(d*x + c)^(3/2),x, algorithm="giac")

[Out]

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