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

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

[Out]

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

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Rubi [A]  time = 0.105413, 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 \[ \frac{a (4 A b-a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{4 b^{3/2}}+\frac{\sqrt{x} \sqrt{a+b x} (4 A b-a B)}{4 b}+\frac{B \sqrt{x} (a+b x)^{3/2}}{2 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 8.70083, size = 80, normalized size = 0.86 \[ \frac{B \sqrt{x} \left (a + b x\right )^{\frac{3}{2}}}{2 b} + \frac{a \left (4 A b - B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a + b x}} \right )}}{4 b^{\frac{3}{2}}} + \frac{\sqrt{x} \sqrt{a + b x} \left (4 A b - B a\right )}{4 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0729657, size = 78, normalized size = 0.84 \[ \frac{\sqrt{b} \sqrt{x} \sqrt{a+b x} (B (a+2 b x)+4 A b)+a (4 A b-a B) \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )}{4 b^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[b]*Sqrt[x]*Sqrt[a + b*x]*(4*A*b + B*(a + 2*b*x)) + a*(4*A*b - a*B)*Log[b*S
qrt[x] + Sqrt[b]*Sqrt[a + b*x]])/(4*b^(3/2))

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Maple [A]  time = 0.015, size = 136, normalized size = 1.5 \[{\frac{1}{8}\sqrt{bx+a}\sqrt{x} \left ( 4\,Bx{b}^{3/2}\sqrt{x \left ( bx+a \right ) }+4\,A\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) ab+8\,A{b}^{3/2}\sqrt{x \left ( bx+a \right ) }-B\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a \right ){\frac{1}{\sqrt{b}}}} \right ){a}^{2}+2\,Ba\sqrt{b}\sqrt{x \left ( bx+a \right ) } \right ){b}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{x \left ( bx+a \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*(b*x+a)^(1/2)*x^(1/2)/b^(3/2)*(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)-B
*ln(1/2*(2*(x*(b*x+a))^(1/2)*b^(1/2)+2*b*x+a)/b^(1/2))*a^2+2*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(b*x + a)/sqrt(x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.240535, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (2 \, B b x + B a + 4 \, A b\right )} \sqrt{b x + a} \sqrt{b} \sqrt{x} -{\left (B a^{2} - 4 \, A a b\right )} \log \left (2 \, \sqrt{b x + a} b \sqrt{x} +{\left (2 \, b x + a\right )} \sqrt{b}\right )}{8 \, b^{\frac{3}{2}}}, \frac{{\left (2 \, B b x + B a + 4 \, A b\right )} \sqrt{b x + a} \sqrt{-b} \sqrt{x} -{\left (B a^{2} - 4 \, A a b\right )} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right )}{4 \, \sqrt{-b} b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 25.6781, size = 568, normalized size = 6.11 \[ \frac{2 A \left (\begin{cases} \frac{\sqrt{a} \sqrt{b} \sqrt{\frac{b x}{a}} \sqrt{a + b x}}{2} + \frac{a \sqrt{b} \operatorname{acosh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{2} & \text{for}\: \left |{1 + \frac{b x}{a}}\right | > 1 \\\frac{i \sqrt{a} \sqrt{b} \sqrt{a + b x}}{2 \sqrt{- \frac{b x}{a}}} - \frac{i a \sqrt{b} \operatorname{asin}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{2} - \frac{i \sqrt{b} \left (a + b x\right )^{\frac{3}{2}}}{2 \sqrt{a} \sqrt{- \frac{b x}{a}}} & \text{otherwise} \end{cases}\right )}{b} - \frac{2 B a \left (\begin{cases} \frac{\sqrt{a} \sqrt{b} \sqrt{\frac{b x}{a}} \sqrt{a + b x}}{2} + \frac{a \sqrt{b} \operatorname{acosh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{2} & \text{for}\: \left |{1 + \frac{b x}{a}}\right | > 1 \\\frac{i \sqrt{a} \sqrt{b} \sqrt{a + b x}}{2 \sqrt{- \frac{b x}{a}}} - \frac{i a \sqrt{b} \operatorname{asin}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{2} - \frac{i \sqrt{b} \left (a + b x\right )^{\frac{3}{2}}}{2 \sqrt{a} \sqrt{- \frac{b x}{a}}} & \text{otherwise} \end{cases}\right )}{b^{2}} + \frac{2 B \left (\begin{cases} - \frac{3 a^{\frac{3}{2}} \sqrt{b} \sqrt{a + b x}}{8 \sqrt{\frac{b x}{a}}} + \frac{\sqrt{a} \sqrt{b} \left (a + b x\right )^{\frac{3}{2}}}{8 \sqrt{\frac{b x}{a}}} + \frac{3 a^{2} \sqrt{b} \operatorname{acosh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{8} + \frac{\sqrt{b} \left (a + b x\right )^{\frac{5}{2}}}{4 \sqrt{a} \sqrt{\frac{b x}{a}}} & \text{for}\: \left |{1 + \frac{b x}{a}}\right | > 1 \\\frac{3 i a^{\frac{3}{2}} \sqrt{b} \sqrt{a + b x}}{8 \sqrt{- \frac{b x}{a}}} - \frac{i \sqrt{a} \sqrt{b} \left (a + b x\right )^{\frac{3}{2}}}{8 \sqrt{- \frac{b x}{a}}} - \frac{3 i a^{2} \sqrt{b} \operatorname{asin}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{8} - \frac{i \sqrt{b} \left (a + b x\right )^{\frac{5}{2}}}{4 \sqrt{a} \sqrt{- \frac{b x}{a}}} & \text{otherwise} \end{cases}\right )}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*A*Piecewise((sqrt(a)*sqrt(b)*sqrt(b*x/a)*sqrt(a + b*x)/2 + a*sqrt(b)*acosh(sqr
t(a + b*x)/sqrt(a))/2, Abs(1 + b*x/a) > 1), (I*sqrt(a)*sqrt(b)*sqrt(a + b*x)/(2*
sqrt(-b*x/a)) - I*a*sqrt(b)*asin(sqrt(a + b*x)/sqrt(a))/2 - I*sqrt(b)*(a + b*x)*
*(3/2)/(2*sqrt(a)*sqrt(-b*x/a)), True))/b - 2*B*a*Piecewise((sqrt(a)*sqrt(b)*sqr
t(b*x/a)*sqrt(a + b*x)/2 + a*sqrt(b)*acosh(sqrt(a + b*x)/sqrt(a))/2, Abs(1 + b*x
/a) > 1), (I*sqrt(a)*sqrt(b)*sqrt(a + b*x)/(2*sqrt(-b*x/a)) - I*a*sqrt(b)*asin(s
qrt(a + b*x)/sqrt(a))/2 - I*sqrt(b)*(a + b*x)**(3/2)/(2*sqrt(a)*sqrt(-b*x/a)), T
rue))/b**2 + 2*B*Piecewise((-3*a**(3/2)*sqrt(b)*sqrt(a + b*x)/(8*sqrt(b*x/a)) +
sqrt(a)*sqrt(b)*(a + b*x)**(3/2)/(8*sqrt(b*x/a)) + 3*a**2*sqrt(b)*acosh(sqrt(a +
b*x)/sqrt(a))/8 + sqrt(b)*(a + b*x)**(5/2)/(4*sqrt(a)*sqrt(b*x/a)), Abs(1 + b*x
/a) > 1), (3*I*a**(3/2)*sqrt(b)*sqrt(a + b*x)/(8*sqrt(-b*x/a)) - I*sqrt(a)*sqrt(
b)*(a + b*x)**(3/2)/(8*sqrt(-b*x/a)) - 3*I*a**2*sqrt(b)*asin(sqrt(a + b*x)/sqrt(
a))/8 - I*sqrt(b)*(a + b*x)**(5/2)/(4*sqrt(a)*sqrt(-b*x/a)), True))/b**2

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GIAC/XCAS [A]  time = 12.8024, size = 4, normalized size = 0.04 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x