3.505 \(\int \frac{\sqrt{x} (A+B x)}{\sqrt{a+b x}} \, dx\)

Optimal. Leaf size=93 \[ -\frac{a (4 A b-3 a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{4 b^{5/2}}+\frac{\sqrt{x} \sqrt{a+b x} (4 A b-3 a B)}{4 b^2}+\frac{B x^{3/2} \sqrt{a+b x}}{2 b} \]

[Out]

((4*A*b - 3*a*B)*Sqrt[x]*Sqrt[a + b*x])/(4*b^2) + (B*x^(3/2)*Sqrt[a + b*x])/(2*b
) - (a*(4*A*b - 3*a*B)*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a + b*x]])/(4*b^(5/2))

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Rubi [A]  time = 0.103734, antiderivative size = 93, 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 \[ -\frac{a (4 A b-3 a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{4 b^{5/2}}+\frac{\sqrt{x} \sqrt{a+b x} (4 A b-3 a B)}{4 b^2}+\frac{B x^{3/2} \sqrt{a+b x}}{2 b} \]

Antiderivative was successfully verified.

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

[Out]

((4*A*b - 3*a*B)*Sqrt[x]*Sqrt[a + b*x])/(4*b^2) + (B*x^(3/2)*Sqrt[a + b*x])/(2*b
) - (a*(4*A*b - 3*a*B)*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a + b*x]])/(4*b^(5/2))

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Rubi in Sympy [A]  time = 8.87174, size = 85, normalized size = 0.91 \[ \frac{B x^{\frac{3}{2}} \sqrt{a + b x}}{2 b} - \frac{a \left (4 A b - 3 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{b} \sqrt{x}} \right )}}{4 b^{\frac{5}{2}}} + \frac{\sqrt{x} \sqrt{a + b x} \left (4 A b - 3 B a\right )}{4 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*x**(3/2)*sqrt(a + b*x)/(2*b) - a*(4*A*b - 3*B*a)*atanh(sqrt(a + b*x)/(sqrt(b)*
sqrt(x)))/(4*b**(5/2)) + sqrt(x)*sqrt(a + b*x)*(4*A*b - 3*B*a)/(4*b**2)

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Mathematica [A]  time = 0.0843091, size = 79, normalized size = 0.85 \[ \frac{\sqrt{b} \sqrt{x} \sqrt{a+b x} (-3 a B+4 A b+2 b B x)+a (3 a B-4 A b) \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )}{4 b^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[b]*Sqrt[x]*Sqrt[a + b*x]*(4*A*b - 3*a*B + 2*b*B*x) + a*(-4*A*b + 3*a*B)*Lo
g[b*Sqrt[x] + Sqrt[b]*Sqrt[a + b*x]])/(4*b^(5/2))

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Maple [A]  time = 0.017, size = 136, normalized size = 1.5 \[ -{\frac{1}{8}\sqrt{x}\sqrt{bx+a} \left ( -4\,Bx{b}^{3/2}\sqrt{x \left ( bx+a \right ) }+4\,A\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) ab-8\,A{b}^{3/2}\sqrt{x \left ( bx+a \right ) }-3\,B\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ){a}^{2}+6\,Ba\sqrt{b}\sqrt{x \left ( bx+a \right ) } \right ){b}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{x \left ( bx+a \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*x^(1/2)*(b*x+a)^(1/2)/b^(5/2)*(-4*B*x*b^(3/2)*(x*(b*x+a))^(1/2)+4*A*ln(1/2*
(2*(x*(b*x+a))^(1/2)*b^(1/2)+2*b*x+a)/b^(1/2))*a*b-8*A*b^(3/2)*(x*(b*x+a))^(1/2)
-3*B*ln(1/2*(2*(x*(b*x+a))^(1/2)*b^(1/2)+2*b*x+a)/b^(1/2))*a^2+6*B*a*b^(1/2)*(x*
(b*x+a))^(1/2))/(x*(b*x+a))^(1/2)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.237557, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (2 \, B b x - 3 \, B a + 4 \, A b\right )} \sqrt{b x + a} \sqrt{b} \sqrt{x} -{\left (3 \, B a^{2} - 4 \, A a b\right )} \log \left (-2 \, \sqrt{b x + a} b \sqrt{x} +{\left (2 \, b x + a\right )} \sqrt{b}\right )}{8 \, b^{\frac{5}{2}}}, \frac{{\left (2 \, B b x - 3 \, B a + 4 \, A b\right )} \sqrt{b x + a} \sqrt{-b} \sqrt{x} +{\left (3 \, B a^{2} - 4 \, A a b\right )} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right )}{4 \, \sqrt{-b} b^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/8*(2*(2*B*b*x - 3*B*a + 4*A*b)*sqrt(b*x + a)*sqrt(b)*sqrt(x) - (3*B*a^2 - 4*A
*a*b)*log(-2*sqrt(b*x + a)*b*sqrt(x) + (2*b*x + a)*sqrt(b)))/b^(5/2), 1/4*((2*B*
b*x - 3*B*a + 4*A*b)*sqrt(b*x + a)*sqrt(-b)*sqrt(x) + (3*B*a^2 - 4*A*a*b)*arctan
(sqrt(b*x + a)*sqrt(-b)/(b*sqrt(x))))/(sqrt(-b)*b^2)]

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Sympy [A]  time = 20.9745, size = 156, normalized size = 1.68 \[ \frac{A \sqrt{a} \sqrt{x} \sqrt{1 + \frac{b x}{a}}}{b} - \frac{A a \operatorname{asinh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{b^{\frac{3}{2}}} - \frac{3 B a^{\frac{3}{2}} \sqrt{x}}{4 b^{2} \sqrt{1 + \frac{b x}{a}}} - \frac{B \sqrt{a} x^{\frac{3}{2}}}{4 b \sqrt{1 + \frac{b x}{a}}} + \frac{3 B a^{2} \operatorname{asinh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{4 b^{\frac{5}{2}}} + \frac{B x^{\frac{5}{2}}}{2 \sqrt{a} \sqrt{1 + \frac{b x}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*sqrt(a)*sqrt(x)*sqrt(1 + b*x/a)/b - A*a*asinh(sqrt(b)*sqrt(x)/sqrt(a))/b**(3/2
) - 3*B*a**(3/2)*sqrt(x)/(4*b**2*sqrt(1 + b*x/a)) - B*sqrt(a)*x**(3/2)/(4*b*sqrt
(1 + b*x/a)) + 3*B*a**2*asinh(sqrt(b)*sqrt(x)/sqrt(a))/(4*b**(5/2)) + B*x**(5/2)
/(2*sqrt(a)*sqrt(1 + b*x/a))

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GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out