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

Optimal. Leaf size=190 \[ \frac{2 x^{3/2} (a+b x) (A b-a B)}{3 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 a \sqrt{x} (a+b x) (A b-a B)}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 B x^{5/2} (a+b x)}{5 b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 a^{3/2} (a+b x) (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

(-2*a*(A*b - a*B)*Sqrt[x]*(a + b*x))/(b^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (2*(A
*b - a*B)*x^(3/2)*(a + b*x))/(3*b^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (2*B*x^(5/2
)*(a + b*x))/(5*b*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (2*a^(3/2)*(A*b - a*B)*(a + b
*x)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(b^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

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Rubi [A]  time = 0.258799, antiderivative size = 190, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161 \[ \frac{2 x^{3/2} (a+b x) (A b-a B)}{3 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 a \sqrt{x} (a+b x) (A b-a B)}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 B x^{5/2} (a+b x)}{5 b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 a^{3/2} (a+b x) (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*a*(A*b - a*B)*Sqrt[x]*(a + b*x))/(b^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (2*(A
*b - a*B)*x^(3/2)*(a + b*x))/(3*b^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (2*B*x^(5/2
)*(a + b*x))/(5*b*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (2*a^(3/2)*(A*b - a*B)*(a + b
*x)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(b^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

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Rubi in Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RecursionError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: RecursionError

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Mathematica [A]  time = 0.113646, size = 100, normalized size = 0.53 \[ \frac{2 (a+b x) \left (\sqrt{b} \sqrt{x} \left (15 a^2 B-5 a b (3 A+B x)+b^2 x (5 A+3 B x)\right )-15 a^{3/2} (a B-A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )\right )}{15 b^{7/2} \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(a + b*x)*(Sqrt[b]*Sqrt[x]*(15*a^2*B - 5*a*b*(3*A + B*x) + b^2*x*(5*A + 3*B*x
)) - 15*a^(3/2)*(-(A*b) + a*B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]]))/(15*b^(7/2)*S
qrt[(a + b*x)^2])

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Maple [A]  time = 0.01, size = 129, normalized size = 0.7 \[{\frac{2\,bx+2\,a}{15\,{b}^{3}} \left ( 3\,B\sqrt{ab}{x}^{5/2}{b}^{2}+5\,A\sqrt{ab}{x}^{3/2}{b}^{2}-5\,B\sqrt{ab}{x}^{3/2}ab-15\,A\sqrt{ab}\sqrt{x}ab+15\,A\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){a}^{2}b+15\,B\sqrt{ab}\sqrt{x}{a}^{2}-15\,B\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){a}^{3} \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}{\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*(b*x+a)*(3*B*(a*b)^(1/2)*x^(5/2)*b^2+5*A*(a*b)^(1/2)*x^(3/2)*b^2-5*B*(a*b)^
(1/2)*x^(3/2)*a*b-15*A*(a*b)^(1/2)*x^(1/2)*a*b+15*A*arctan(x^(1/2)*b/(a*b)^(1/2)
)*a^2*b+15*B*(a*b)^(1/2)*x^(1/2)*a^2-15*B*arctan(x^(1/2)*b/(a*b)^(1/2))*a^3)/((b
*x+a)^2)^(1/2)/b^3/(a*b)^(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)*x^(3/2)/sqrt((b*x + a)^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.290425, size = 1, normalized size = 0.01 \[ \left [-\frac{15 \,{\left (B a^{2} - A a b\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x + 2 \, b \sqrt{x} \sqrt{-\frac{a}{b}} - a}{b x + a}\right ) - 2 \,{\left (3 \, B b^{2} x^{2} + 15 \, B a^{2} - 15 \, A a b - 5 \,{\left (B a b - A b^{2}\right )} x\right )} \sqrt{x}}{15 \, b^{3}}, -\frac{2 \,{\left (15 \,{\left (B a^{2} - A a b\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{\sqrt{x}}{\sqrt{\frac{a}{b}}}\right ) -{\left (3 \, B b^{2} x^{2} + 15 \, B a^{2} - 15 \, A a b - 5 \,{\left (B a b - A b^{2}\right )} x\right )} \sqrt{x}\right )}}{15 \, b^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/15*(15*(B*a^2 - A*a*b)*sqrt(-a/b)*log((b*x + 2*b*sqrt(x)*sqrt(-a/b) - a)/(b*
x + a)) - 2*(3*B*b^2*x^2 + 15*B*a^2 - 15*A*a*b - 5*(B*a*b - A*b^2)*x)*sqrt(x))/b
^3, -2/15*(15*(B*a^2 - A*a*b)*sqrt(a/b)*arctan(sqrt(x)/sqrt(a/b)) - (3*B*b^2*x^2
+ 15*B*a^2 - 15*A*a*b - 5*(B*a*b - A*b^2)*x)*sqrt(x))/b^3]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.275529, size = 180, normalized size = 0.95 \[ -\frac{2 \,{\left (B a^{3}{\rm sign}\left (b x + a\right ) - A a^{2} b{\rm sign}\left (b x + a\right )\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} b^{3}} + \frac{2 \,{\left (3 \, B b^{4} x^{\frac{5}{2}}{\rm sign}\left (b x + a\right ) - 5 \, B a b^{3} x^{\frac{3}{2}}{\rm sign}\left (b x + a\right ) + 5 \, A b^{4} x^{\frac{3}{2}}{\rm sign}\left (b x + a\right ) + 15 \, B a^{2} b^{2} \sqrt{x}{\rm sign}\left (b x + a\right ) - 15 \, A a b^{3} \sqrt{x}{\rm sign}\left (b x + a\right )\right )}}{15 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(B*a^3*sign(b*x + a) - A*a^2*b*sign(b*x + a))*arctan(b*sqrt(x)/sqrt(a*b))/(sq
rt(a*b)*b^3) + 2/15*(3*B*b^4*x^(5/2)*sign(b*x + a) - 5*B*a*b^3*x^(3/2)*sign(b*x
+ a) + 5*A*b^4*x^(3/2)*sign(b*x + a) + 15*B*a^2*b^2*sqrt(x)*sign(b*x + a) - 15*A
*a*b^3*sqrt(x)*sign(b*x + a))/b^5