3.471 \(\int \frac{\sqrt{a+b x} (A+B x)}{x^{3/2}} \, dx\)

Optimal. Leaf size=82 \[ \frac{\sqrt{x} \sqrt{a+b x} (a B+2 A b)}{a}+\frac{(a B+2 A b) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{\sqrt{b}}-\frac{2 A (a+b x)^{3/2}}{a \sqrt{x}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.108, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{\sqrt{x} \sqrt{a+b x} (a B+2 A b)}{a}+\frac{(a B+2 A b) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{\sqrt{b}}-\frac{2 A (a+b x)^{3/2}}{a \sqrt{x}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 8.96259, size = 78, normalized size = 0.95 \[ - \frac{2 A \left (a + b x\right )^{\frac{3}{2}}}{a \sqrt{x}} + \frac{2 \left (A b + \frac{B a}{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{b} \sqrt{x}} \right )}}{\sqrt{b}} + \frac{2 \sqrt{x} \sqrt{a + b x} \left (A b + \frac{B a}{2}\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*A*(a + b*x)**(3/2)/(a*sqrt(x)) + 2*(A*b + B*a/2)*atanh(sqrt(a + b*x)/(sqrt(b)
*sqrt(x)))/sqrt(b) + 2*sqrt(x)*sqrt(a + b*x)*(A*b + B*a/2)/a

_______________________________________________________________________________________

Mathematica [A]  time = 0.0685122, size = 61, normalized size = 0.74 \[ \frac{\sqrt{a+b x} (B x-2 A)}{\sqrt{x}}+\frac{(a B+2 A b) \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )}{\sqrt{b}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b*x]*(-2*A + B*x))/Sqrt[x] + ((2*A*b + a*B)*Log[b*Sqrt[x] + Sqrt[b]*Sq
rt[a + b*x]])/Sqrt[b]

_______________________________________________________________________________________

Maple [A]  time = 0.019, size = 118, normalized size = 1.4 \[{\frac{1}{2}\sqrt{bx+a} \left ( 2\,A\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) xb+B\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a \right ){\frac{1}{\sqrt{b}}}} \right ) xa+2\,Bx\sqrt{x \left ( bx+a \right ) }\sqrt{b}-4\,A\sqrt{x \left ( bx+a \right ) }\sqrt{b} \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{x \left ( bx+a \right ) }}}{\frac{1}{\sqrt{b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(b*x+a)^(1/2)*(2*A*ln(1/2*(2*(x*(b*x+a))^(1/2)*b^(1/2)+2*b*x+a)/b^(1/2))*x*b
+B*ln(1/2*(2*(x*(b*x+a))^(1/2)*b^(1/2)+2*b*x+a)/b^(1/2))*x*a+2*B*x*(x*(b*x+a))^(
1/2)*b^(1/2)-4*A*(x*(b*x+a))^(1/2)*b^(1/2))/x^(1/2)/(x*(b*x+a))^(1/2)/b^(1/2)

_______________________________________________________________________________________

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((B*x + A)*sqrt(b*x + a)/x^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.236892, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (B a + 2 \, A b\right )} x \log \left (2 \, \sqrt{b x + a} b \sqrt{x} +{\left (2 \, b x + a\right )} \sqrt{b}\right ) + 2 \,{\left (B x - 2 \, A\right )} \sqrt{b x + a} \sqrt{b} \sqrt{x}}{2 \, \sqrt{b} x}, \frac{{\left (B a + 2 \, A b\right )} x \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right ) +{\left (B x - 2 \, A\right )} \sqrt{b x + a} \sqrt{-b} \sqrt{x}}{\sqrt{-b} x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2*((B*a + 2*A*b)*x*log(2*sqrt(b*x + a)*b*sqrt(x) + (2*b*x + a)*sqrt(b)) + 2*(
B*x - 2*A)*sqrt(b*x + a)*sqrt(b)*sqrt(x))/(sqrt(b)*x), ((B*a + 2*A*b)*x*arctan(s
qrt(b*x + a)*sqrt(-b)/(b*sqrt(x))) + (B*x - 2*A)*sqrt(b*x + a)*sqrt(-b)*sqrt(x))
/(sqrt(-b)*x)]

_______________________________________________________________________________________

Sympy [A]  time = 22.4121, size = 116, normalized size = 1.41 \[ A \left (- \frac{2 \sqrt{a}}{\sqrt{x} \sqrt{1 + \frac{b x}{a}}} + 2 \sqrt{b} \operatorname{asinh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )} - \frac{2 b \sqrt{x}}{\sqrt{a} \sqrt{1 + \frac{b x}{a}}}\right ) + B \left (\sqrt{a} \sqrt{x} \sqrt{1 + \frac{b x}{a}} + \frac{a \operatorname{asinh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{\sqrt{b}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*(-2*sqrt(a)/(sqrt(x)*sqrt(1 + b*x/a)) + 2*sqrt(b)*asinh(sqrt(b)*sqrt(x)/sqrt(a
)) - 2*b*sqrt(x)/(sqrt(a)*sqrt(1 + b*x/a))) + B*(sqrt(a)*sqrt(x)*sqrt(1 + b*x/a)
+ a*asinh(sqrt(b)*sqrt(x)/sqrt(a))/sqrt(b))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 12.6577, size = 4, normalized size = 0.05 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x