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

Optimal. Leaf size=100 \[ \frac{(3 a B+A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 a^{3/2} b^{5/2}}-\frac{\sqrt{x} (3 a B+A b)}{4 a b^2 (a+b x)}+\frac{x^{3/2} (A b-a B)}{2 a b (a+b x)^2} \]

[Out]

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

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Rubi [A]  time = 0.115351, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{(3 a B+A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 a^{3/2} b^{5/2}}-\frac{\sqrt{x} (3 a B+A b)}{4 a b^2 (a+b x)}+\frac{x^{3/2} (A b-a B)}{2 a b (a+b x)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 14.276, size = 85, normalized size = 0.85 \[ \frac{x^{\frac{3}{2}} \left (A b - B a\right )}{2 a b \left (a + b x\right )^{2}} - \frac{\sqrt{x} \left (A b + 3 B a\right )}{4 a b^{2} \left (a + b x\right )} + \frac{\left (A b + 3 B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{4 a^{\frac{3}{2}} b^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.150251, size = 85, normalized size = 0.85 \[ \frac{\frac{(3 a B+A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{3/2}}+\frac{\sqrt{b} \sqrt{x} \left (-3 a^2 B-a b (A+5 B x)+A b^2 x\right )}{a (a+b x)^2}}{4 b^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.018, size = 94, normalized size = 0.9 \[ 2\,{\frac{1}{ \left ( bx+a \right ) ^{2}} \left ( 1/8\,{\frac{ \left ( Ab-5\,Ba \right ){x}^{3/2}}{ab}}-1/8\,{\frac{ \left ( Ab+3\,Ba \right ) \sqrt{x}}{{b}^{2}}} \right ) }+{\frac{A}{4\,ab}\arctan \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{3\,B}{4\,{b}^{2}}\arctan \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.225934, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \,{\left (3 \, B a^{2} + A a b +{\left (5 \, B a b - A b^{2}\right )} x\right )} \sqrt{-a b} \sqrt{x} -{\left (3 \, B a^{3} + A a^{2} b +{\left (3 \, B a b^{2} + A b^{3}\right )} x^{2} + 2 \,{\left (3 \, B a^{2} b + A a b^{2}\right )} x\right )} \log \left (\frac{2 \, a b \sqrt{x} + \sqrt{-a b}{\left (b x - a\right )}}{b x + a}\right )}{8 \,{\left (a b^{4} x^{2} + 2 \, a^{2} b^{3} x + a^{3} b^{2}\right )} \sqrt{-a b}}, -\frac{{\left (3 \, B a^{2} + A a b +{\left (5 \, B a b - A b^{2}\right )} x\right )} \sqrt{a b} \sqrt{x} +{\left (3 \, B a^{3} + A a^{2} b +{\left (3 \, B a b^{2} + A b^{3}\right )} x^{2} + 2 \,{\left (3 \, B a^{2} b + A a b^{2}\right )} x\right )} \arctan \left (\frac{a}{\sqrt{a b} \sqrt{x}}\right )}{4 \,{\left (a b^{4} x^{2} + 2 \, a^{2} b^{3} x + a^{3} b^{2}\right )} \sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 29.4178, size = 643, normalized size = 6.43 \[ - \frac{10 A a^{2} \sqrt{x}}{8 a^{4} b + 16 a^{3} b^{2} x + 8 a^{2} b^{3} x^{2}} - \frac{6 A a x^{\frac{3}{2}}}{8 a^{4} + 16 a^{3} b x + 8 a^{2} b^{2} x^{2}} + \frac{3 A a \sqrt{- \frac{1}{a^{5} b}} \log{\left (- a^{3} \sqrt{- \frac{1}{a^{5} b}} + \sqrt{x} \right )}}{8 b} - \frac{3 A a \sqrt{- \frac{1}{a^{5} b}} \log{\left (a^{3} \sqrt{- \frac{1}{a^{5} b}} + \sqrt{x} \right )}}{8 b} + \frac{2 A \sqrt{x}}{2 a^{2} b + 2 a b^{2} x} - \frac{A \sqrt{- \frac{1}{a^{3} b}} \log{\left (- a^{2} \sqrt{- \frac{1}{a^{3} b}} + \sqrt{x} \right )}}{2 b} + \frac{A \sqrt{- \frac{1}{a^{3} b}} \log{\left (a^{2} \sqrt{- \frac{1}{a^{3} b}} + \sqrt{x} \right )}}{2 b} + \frac{10 B a^{3} \sqrt{x}}{8 a^{4} b^{2} + 16 a^{3} b^{3} x + 8 a^{2} b^{4} x^{2}} + \frac{6 B a^{2} x^{\frac{3}{2}}}{8 a^{4} b + 16 a^{3} b^{2} x + 8 a^{2} b^{3} x^{2}} - \frac{3 B a^{2} \sqrt{- \frac{1}{a^{5} b}} \log{\left (- a^{3} \sqrt{- \frac{1}{a^{5} b}} + \sqrt{x} \right )}}{8 b^{2}} + \frac{3 B a^{2} \sqrt{- \frac{1}{a^{5} b}} \log{\left (a^{3} \sqrt{- \frac{1}{a^{5} b}} + \sqrt{x} \right )}}{8 b^{2}} - \frac{4 B a \sqrt{x}}{2 a^{2} b^{2} + 2 a b^{3} x} + \frac{B a \sqrt{- \frac{1}{a^{3} b}} \log{\left (- a^{2} \sqrt{- \frac{1}{a^{3} b}} + \sqrt{x} \right )}}{b^{2}} - \frac{B a \sqrt{- \frac{1}{a^{3} b}} \log{\left (a^{2} \sqrt{- \frac{1}{a^{3} b}} + \sqrt{x} \right )}}{b^{2}} + \frac{2 B \left (\begin{cases} \frac{\operatorname{atan}{\left (\frac{\sqrt{x}}{\sqrt{\frac{a}{b}}} \right )}}{b \sqrt{\frac{a}{b}}} & \text{for}\: \frac{a}{b} > 0 \\- \frac{\operatorname{acoth}{\left (\frac{\sqrt{x}}{\sqrt{- \frac{a}{b}}} \right )}}{b \sqrt{- \frac{a}{b}}} & \text{for}\: x > - \frac{a}{b} \wedge \frac{a}{b} < 0 \\- \frac{\operatorname{atanh}{\left (\frac{\sqrt{x}}{\sqrt{- \frac{a}{b}}} \right )}}{b \sqrt{- \frac{a}{b}}} & \text{for}\: x < - \frac{a}{b} \wedge \frac{a}{b} < 0 \end{cases}\right )}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-10*A*a**2*sqrt(x)/(8*a**4*b + 16*a**3*b**2*x + 8*a**2*b**3*x**2) - 6*A*a*x**(3/
2)/(8*a**4 + 16*a**3*b*x + 8*a**2*b**2*x**2) + 3*A*a*sqrt(-1/(a**5*b))*log(-a**3
*sqrt(-1/(a**5*b)) + sqrt(x))/(8*b) - 3*A*a*sqrt(-1/(a**5*b))*log(a**3*sqrt(-1/(
a**5*b)) + sqrt(x))/(8*b) + 2*A*sqrt(x)/(2*a**2*b + 2*a*b**2*x) - A*sqrt(-1/(a**
3*b))*log(-a**2*sqrt(-1/(a**3*b)) + sqrt(x))/(2*b) + A*sqrt(-1/(a**3*b))*log(a**
2*sqrt(-1/(a**3*b)) + sqrt(x))/(2*b) + 10*B*a**3*sqrt(x)/(8*a**4*b**2 + 16*a**3*
b**3*x + 8*a**2*b**4*x**2) + 6*B*a**2*x**(3/2)/(8*a**4*b + 16*a**3*b**2*x + 8*a*
*2*b**3*x**2) - 3*B*a**2*sqrt(-1/(a**5*b))*log(-a**3*sqrt(-1/(a**5*b)) + sqrt(x)
)/(8*b**2) + 3*B*a**2*sqrt(-1/(a**5*b))*log(a**3*sqrt(-1/(a**5*b)) + sqrt(x))/(8
*b**2) - 4*B*a*sqrt(x)/(2*a**2*b**2 + 2*a*b**3*x) + B*a*sqrt(-1/(a**3*b))*log(-a
**2*sqrt(-1/(a**3*b)) + sqrt(x))/b**2 - B*a*sqrt(-1/(a**3*b))*log(a**2*sqrt(-1/(
a**3*b)) + sqrt(x))/b**2 + 2*B*Piecewise((atan(sqrt(x)/sqrt(a/b))/(b*sqrt(a/b)),
a/b > 0), (-acoth(sqrt(x)/sqrt(-a/b))/(b*sqrt(-a/b)), (a/b < 0) & (x > -a/b)),
(-atanh(sqrt(x)/sqrt(-a/b))/(b*sqrt(-a/b)), (a/b < 0) & (x < -a/b)))/b**2

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GIAC/XCAS [A]  time = 0.219045, size = 111, normalized size = 1.11 \[ \frac{{\left (3 \, B a + A b\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{4 \, \sqrt{a b} a b^{2}} - \frac{5 \, B a b x^{\frac{3}{2}} - A b^{2} x^{\frac{3}{2}} + 3 \, B a^{2} \sqrt{x} + A a b \sqrt{x}}{4 \,{\left (b x + a\right )}^{2} a b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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