3.1465 \(\int \frac{\sqrt{c+d x}}{(a+b x)^{3/2}} \, dx\)

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

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0680277, 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 \[ \frac{2 \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{b^{3/2}}-\frac{2 \sqrt{c+d x}}{b \sqrt{a+b x}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 10.2039, size = 60, normalized size = 0.91 \[ - \frac{2 \sqrt{c + d x}}{b \sqrt{a + b x}} + \frac{2 \sqrt{d} \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0545014, size = 78, normalized size = 1.18 \[ \frac{\sqrt{d} \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{b^{3/2}}-\frac{2 \sqrt{c+d x}}{b \sqrt{a+b x}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [F]  time = 0.043, size = 0, normalized size = 0. \[ \int{1\sqrt{dx+c} \left ( bx+a \right ) ^{-{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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(d*x + c)/(b*x + a)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.292375, size = 1, normalized size = 0.02 \[ \left [\frac{{\left (b x + a\right )} \sqrt{\frac{d}{b}} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \,{\left (2 \, b^{2} d x + b^{2} c + a b d\right )} \sqrt{b x + a} \sqrt{d x + c} \sqrt{\frac{d}{b}} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right ) - 4 \, \sqrt{b x + a} \sqrt{d x + c}}{2 \,{\left (b^{2} x + a b\right )}}, \frac{{\left (b x + a\right )} \sqrt{-\frac{d}{b}} \arctan \left (\frac{2 \, b d x + b c + a d}{2 \, \sqrt{b x + a} \sqrt{d x + c} b \sqrt{-\frac{d}{b}}}\right ) - 2 \, \sqrt{b x + a} \sqrt{d x + c}}{b^{2} x + a b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.547128, size = 4, normalized size = 0.06 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x