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

Optimal. Leaf size=98 \[ \frac{(2 A b-3 a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{b^{5/2}}-\frac{\sqrt{x} \sqrt{a+b x} (2 A b-3 a B)}{a b^2}+\frac{2 x^{3/2} (A b-a B)}{a b \sqrt{a+b x}} \]

[Out]

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

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Rubi [A]  time = 0.111622, antiderivative size = 98, 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 \[ \frac{(2 A b-3 a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{b^{5/2}}-\frac{\sqrt{x} \sqrt{a+b x} (2 A b-3 a B)}{a b^2}+\frac{2 x^{3/2} (A b-a B)}{a b \sqrt{a+b x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 10.8632, size = 92, normalized size = 0.94 \[ \frac{2 \left (A b - \frac{3 B a}{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a + b x}} \right )}}{b^{\frac{5}{2}}} + \frac{2 x^{\frac{3}{2}} \left (A b - B a\right )}{a b \sqrt{a + b x}} - \frac{2 \sqrt{x} \sqrt{a + b x} \left (A b - \frac{3 B a}{2}\right )}{a b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.114043, size = 71, normalized size = 0.72 \[ \frac{(2 A b-3 a B) \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )}{b^{5/2}}+\frac{\sqrt{x} (3 a B-2 A b+b B x)}{b^2 \sqrt{a+b x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.021, size = 201, normalized size = 2.1 \[{\frac{1}{2} \left ( 2\,A\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) x{b}^{2}-3\,B\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) xab+2\,Bx{b}^{3/2}\sqrt{x \left ( bx+a \right ) }+2\,A\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) ab-4\,A{b}^{3/2}\sqrt{x \left ( bx+a \right ) }-3\,B\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ){a}^{2}+6\,Ba\sqrt{b}\sqrt{x \left ( bx+a \right ) } \right ) \sqrt{x}{b}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{x \left ( bx+a \right ) }}}{\frac{1}{\sqrt{bx+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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^2-3*B*ln(1/2*
(2*(x*(b*x+a))^(1/2)*b^(1/2)+2*b*x+a)/b^(1/2))*x*a*b+2*B*x*b^(3/2)*(x*(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))*a*b-4*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))/b^(5/2)*x^(1/2)/(x*(b*x+a))^(1/2)/(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)/(b*x + a)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.250395, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (3 \, B a - 2 \, A b\right )} \sqrt{b x + a} \sqrt{x} \log \left (2 \, \sqrt{b x + a} b \sqrt{x} +{\left (2 \, b x + a\right )} \sqrt{b}\right ) - 2 \,{\left (B b x^{2} +{\left (3 \, B a - 2 \, A b\right )} x\right )} \sqrt{b}}{2 \, \sqrt{b x + a} b^{\frac{5}{2}} \sqrt{x}}, -\frac{{\left (3 \, B a - 2 \, A b\right )} \sqrt{b x + a} \sqrt{x} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right ) -{\left (B b x^{2} +{\left (3 \, B a - 2 \, A b\right )} x\right )} \sqrt{-b}}{\sqrt{b x + a} \sqrt{-b} b^{2} \sqrt{x}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 28.4063, size = 122, normalized size = 1.24 \[ A \left (\frac{2 \operatorname{asinh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{b^{\frac{3}{2}}} - \frac{2 \sqrt{x}}{\sqrt{a} b \sqrt{1 + \frac{b x}{a}}}\right ) + B \left (\frac{3 \sqrt{a} \sqrt{x}}{b^{2} \sqrt{1 + \frac{b x}{a}}} - \frac{3 a \operatorname{asinh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{b^{\frac{5}{2}}} + \frac{x^{\frac{3}{2}}}{\sqrt{a} b \sqrt{1 + \frac{b x}{a}}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.247414, size = 194, normalized size = 1.98 \[ \frac{\sqrt{{\left (b x + a\right )} b - a b} \sqrt{b x + a} B{\left | b \right |}}{b^{4}} + \frac{{\left (3 \, B a \sqrt{b}{\left | b \right |} - 2 \, A b^{\frac{3}{2}}{\left | b \right |}\right )}{\rm ln}\left ({\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2}\right )}{2 \, b^{4}} + \frac{4 \,{\left (B a^{2} \sqrt{b}{\left | b \right |} - A a b^{\frac{3}{2}}{\left | b \right |}\right )}}{{\left ({\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt((b*x + a)*b - a*b)*sqrt(b*x + a)*B*abs(b)/b^4 + 1/2*(3*B*a*sqrt(b)*abs(b) -
2*A*b^(3/2)*abs(b))*ln((sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2)/b^4
+ 4*(B*a^2*sqrt(b)*abs(b) - A*a*b^(3/2)*abs(b))/(((sqrt(b*x + a)*sqrt(b) - sqrt
((b*x + a)*b - a*b))^2 + a*b)*b^3)