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

Optimal. Leaf size=82 \[ \frac{2 x^{3/2} (A b-a B)}{3 a b (a+b x)^{3/2}}+\frac{2 B \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{b^{5/2}}-\frac{2 B \sqrt{x}}{b^2 \sqrt{a+b x}} \]

[Out]

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

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Rubi [A]  time = 0.0815157, 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 \[ \frac{2 x^{3/2} (A b-a B)}{3 a b (a+b x)^{3/2}}+\frac{2 B \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{b^{5/2}}-\frac{2 B \sqrt{x}}{b^2 \sqrt{a+b x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 9.173, size = 75, normalized size = 0.91 \[ - \frac{2 B \sqrt{x}}{b^{2} \sqrt{a + b x}} + \frac{2 B \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a + b x}} \right )}}{b^{\frac{5}{2}}} + \frac{2 x^{\frac{3}{2}} \left (A b - B a\right )}{3 a b \left (a + b x\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.188397, size = 76, normalized size = 0.93 \[ \frac{2 \sqrt{x} \left (-3 a^2 B-4 a b B x+A b^2 x\right )}{3 a b^2 (a+b x)^{3/2}}+\frac{2 B \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )}{b^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.023, size = 182, normalized size = 2.2 \[{\frac{1}{3\,a} \left ( 3\,B\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ){x}^{2}a{b}^{2}+2\,A\sqrt{x \left ( bx+a \right ) }x{b}^{5/2}+6\,B\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) x{a}^{2}b-8\,Ba\sqrt{x \left ( bx+a \right ) }x{b}^{3/2}+3\,B{a}^{3}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) -6\,B{a}^{2}\sqrt{x \left ( bx+a \right ) }\sqrt{b} \right ) \sqrt{x}{b}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{x \left ( bx+a \right ) }}} \left ( bx+a \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.240182, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (B a b x + B a^{2}\right )} \sqrt{b x + a} \sqrt{x} \log \left (2 \, \sqrt{b x + a} b \sqrt{x} +{\left (2 \, b x + a\right )} \sqrt{b}\right ) - 2 \,{\left (3 \, B a^{2} x +{\left (4 \, B a b - A b^{2}\right )} x^{2}\right )} \sqrt{b}}{3 \,{\left (a b^{3} x + a^{2} b^{2}\right )} \sqrt{b x + a} \sqrt{b} \sqrt{x}}, \frac{2 \,{\left (3 \,{\left (B a b x + B a^{2}\right )} \sqrt{b x + a} \sqrt{x} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right ) -{\left (3 \, B a^{2} x +{\left (4 \, B a b - A b^{2}\right )} x^{2}\right )} \sqrt{-b}\right )}}{3 \,{\left (a b^{3} x + a^{2} b^{2}\right )} \sqrt{b x + a} \sqrt{-b} \sqrt{x}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 77.847, size = 376, normalized size = 4.59 \[ \frac{2 A x^{\frac{3}{2}}}{3 a^{\frac{5}{2}} \sqrt{1 + \frac{b x}{a}} + 3 a^{\frac{3}{2}} b x \sqrt{1 + \frac{b x}{a}}} + B \left (\frac{6 a^{\frac{39}{2}} b^{11} x^{\frac{27}{2}} \sqrt{1 + \frac{b x}{a}} \operatorname{asinh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{3 a^{\frac{39}{2}} b^{\frac{27}{2}} x^{\frac{27}{2}} \sqrt{1 + \frac{b x}{a}} + 3 a^{\frac{37}{2}} b^{\frac{29}{2}} x^{\frac{29}{2}} \sqrt{1 + \frac{b x}{a}}} + \frac{6 a^{\frac{37}{2}} b^{12} x^{\frac{29}{2}} \sqrt{1 + \frac{b x}{a}} \operatorname{asinh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{3 a^{\frac{39}{2}} b^{\frac{27}{2}} x^{\frac{27}{2}} \sqrt{1 + \frac{b x}{a}} + 3 a^{\frac{37}{2}} b^{\frac{29}{2}} x^{\frac{29}{2}} \sqrt{1 + \frac{b x}{a}}} - \frac{6 a^{19} b^{\frac{23}{2}} x^{14}}{3 a^{\frac{39}{2}} b^{\frac{27}{2}} x^{\frac{27}{2}} \sqrt{1 + \frac{b x}{a}} + 3 a^{\frac{37}{2}} b^{\frac{29}{2}} x^{\frac{29}{2}} \sqrt{1 + \frac{b x}{a}}} - \frac{8 a^{18} b^{\frac{25}{2}} x^{15}}{3 a^{\frac{39}{2}} b^{\frac{27}{2}} x^{\frac{27}{2}} \sqrt{1 + \frac{b x}{a}} + 3 a^{\frac{37}{2}} b^{\frac{29}{2}} x^{\frac{29}{2}} \sqrt{1 + \frac{b x}{a}}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*A*x**(3/2)/(3*a**(5/2)*sqrt(1 + b*x/a) + 3*a**(3/2)*b*x*sqrt(1 + b*x/a)) + B*(
6*a**(39/2)*b**11*x**(27/2)*sqrt(1 + b*x/a)*asinh(sqrt(b)*sqrt(x)/sqrt(a))/(3*a*
*(39/2)*b**(27/2)*x**(27/2)*sqrt(1 + b*x/a) + 3*a**(37/2)*b**(29/2)*x**(29/2)*sq
rt(1 + b*x/a)) + 6*a**(37/2)*b**12*x**(29/2)*sqrt(1 + b*x/a)*asinh(sqrt(b)*sqrt(
x)/sqrt(a))/(3*a**(39/2)*b**(27/2)*x**(27/2)*sqrt(1 + b*x/a) + 3*a**(37/2)*b**(2
9/2)*x**(29/2)*sqrt(1 + b*x/a)) - 6*a**19*b**(23/2)*x**14/(3*a**(39/2)*b**(27/2)
*x**(27/2)*sqrt(1 + b*x/a) + 3*a**(37/2)*b**(29/2)*x**(29/2)*sqrt(1 + b*x/a)) -
8*a**18*b**(25/2)*x**15/(3*a**(39/2)*b**(27/2)*x**(27/2)*sqrt(1 + b*x/a) + 3*a**
(37/2)*b**(29/2)*x**(29/2)*sqrt(1 + b*x/a)))

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GIAC/XCAS [A]  time = 0.250153, size = 300, normalized size = 3.66 \[ -\frac{B{\left | b \right |}{\rm ln}\left ({\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2}\right )}{b^{\frac{7}{2}}} - \frac{4 \,{\left (6 \, B a{\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{4} \sqrt{b}{\left | b \right |} + 6 \, B a^{2}{\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac{3}{2}}{\left | b \right |} - 3 \, A{\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{4} b^{\frac{3}{2}}{\left | b \right |} + 4 \, B a^{3} b^{\frac{5}{2}}{\left | b \right |} - A a^{2} b^{\frac{7}{2}}{\left | b \right |}\right )}}{3 \,{\left ({\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )}^{3} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-B*abs(b)*ln((sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2)/b^(7/2) - 4/3*
(6*B*a*(sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^4*sqrt(b)*abs(b) + 6*B*
a^2*(sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2*b^(3/2)*abs(b) - 3*A*(sq
rt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^4*b^(3/2)*abs(b) + 4*B*a^3*b^(5/2
)*abs(b) - A*a^2*b^(7/2)*abs(b))/(((sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a
*b))^2 + a*b)^3*b^3)