3.472 \(\int \frac{\sqrt{a+b x} (A+B x)}{x^{5/2}} \, dx\)

Optimal. Leaf size=69 \[ -\frac{2 A (a+b x)^{3/2}}{3 a x^{3/2}}-\frac{2 B \sqrt{a+b x}}{\sqrt{x}}+2 \sqrt{b} B \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0718976, antiderivative size = 69, 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 \[ -\frac{2 A (a+b x)^{3/2}}{3 a x^{3/2}}-\frac{2 B \sqrt{a+b x}}{\sqrt{x}}+2 \sqrt{b} B \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[a + b*x]*(A + B*x))/x^(5/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 7.37916, size = 65, normalized size = 0.94 \[ - \frac{2 A \left (a + b x\right )^{\frac{3}{2}}}{3 a x^{\frac{3}{2}}} + 2 B \sqrt{b} \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a + b x}} \right )} - \frac{2 B \sqrt{a + b x}}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(b*x+a)**(1/2)/x**(5/2),x)

[Out]

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

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Mathematica [A]  time = 0.0983743, size = 67, normalized size = 0.97 \[ 2 \sqrt{b} B \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )-\frac{2 \sqrt{a+b x} (a (A+3 B x)+A b x)}{3 a x^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[a + b*x]*(A + B*x))/x^(5/2),x]

[Out]

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

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Maple [A]  time = 0.019, size = 103, normalized size = 1.5 \[ -{\frac{1}{3\,a}\sqrt{bx+a} \left ( -3\,B\sqrt{b}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) a{x}^{2}+2\,Axb\sqrt{x \left ( bx+a \right ) }+6\,Bxa\sqrt{x \left ( bx+a \right ) }+2\,Aa\sqrt{x \left ( bx+a \right ) } \right ){x}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{x \left ( bx+a \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(b*x+a)^(1/2)/x^(5/2),x)

[Out]

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

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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)/x^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.245866, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, B a \sqrt{b} x^{2} \log \left (2 \, b x + 2 \, \sqrt{b x + a} \sqrt{b} \sqrt{x} + a\right ) - 2 \,{\left (A a +{\left (3 \, B a + A b\right )} x\right )} \sqrt{b x + a} \sqrt{x}}{3 \, a x^{2}}, \frac{2 \,{\left (3 \, B a \sqrt{-b} x^{2} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-b} \sqrt{x}}\right ) -{\left (A a +{\left (3 \, B a + A b\right )} x\right )} \sqrt{b x + a} \sqrt{x}\right )}}{3 \, a x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*sqrt(b*x + a)/x^(5/2),x, algorithm="fricas")

[Out]

[1/3*(3*B*a*sqrt(b)*x^2*log(2*b*x + 2*sqrt(b*x + a)*sqrt(b)*sqrt(x) + a) - 2*(A*
a + (3*B*a + A*b)*x)*sqrt(b*x + a)*sqrt(x))/(a*x^2), 2/3*(3*B*a*sqrt(-b)*x^2*arc
tan(sqrt(b*x + a)/(sqrt(-b)*sqrt(x))) - (A*a + (3*B*a + A*b)*x)*sqrt(b*x + a)*sq
rt(x))/(a*x^2)]

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Sympy [A]  time = 70.4865, size = 114, normalized size = 1.65 \[ A \left (- \frac{2 \sqrt{b} \sqrt{\frac{a}{b x} + 1}}{3 x} - \frac{2 b^{\frac{3}{2}} \sqrt{\frac{a}{b x} + 1}}{3 a}\right ) + B \left (- \frac{2 \sqrt{a}}{\sqrt{x} \sqrt{1 + \frac{b x}{a}}} + 2 \sqrt{b} \operatorname{asinh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )} - \frac{2 b \sqrt{x}}{\sqrt{a} \sqrt{1 + \frac{b x}{a}}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(b*x+a)**(1/2)/x**(5/2),x)

[Out]

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

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*sqrt(b*x + a)/x^(5/2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError