3.820 \(\int \frac{x^{3/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=206 \[ \frac{x^{5/2} (A b-a B)}{2 a b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{x^{3/2} (A b-5 a B)}{4 a b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{3 (a+b x) (A b-5 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 \sqrt{a} b^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 \sqrt{x} (a+b x) (A b-5 a B)}{4 a b^3 \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

((A*b - 5*a*B)*x^(3/2))/(4*a*b^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + ((A*b - a*B)*x
^(5/2))/(2*a*b*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (3*(A*b - 5*a*B)*Sqrt[
x]*(a + b*x))/(4*a*b^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (3*(A*b - 5*a*B)*(a + b*
x)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(4*Sqrt[a]*b^(7/2)*Sqrt[a^2 + 2*a*b*x + b^
2*x^2])

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Rubi [A]  time = 0.293969, antiderivative size = 206, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.194 \[ \frac{x^{5/2} (A b-a B)}{2 a b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{x^{3/2} (A b-5 a B)}{4 a b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{3 (a+b x) (A b-5 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 \sqrt{a} b^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 \sqrt{x} (a+b x) (A b-5 a B)}{4 a b^3 \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

[Out]

Exception raised: RecursionError

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Mathematica [A]  time = 0.138123, size = 115, normalized size = 0.56 \[ \frac{\sqrt{a} \sqrt{b} \sqrt{x} \left (15 a^2 B+a (25 b B x-3 A b)+b^2 x (8 B x-5 A)\right )+3 (a+b x)^2 (A b-5 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 \sqrt{a} b^{7/2} (a+b x) \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a]*Sqrt[b]*Sqrt[x]*(15*a^2*B + b^2*x*(-5*A + 8*B*x) + a*(-3*A*b + 25*b*B*x
)) + 3*(A*b - 5*a*B)*(a + b*x)^2*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(4*Sqrt[a]*b
^(7/2)*(a + b*x)*Sqrt[(a + b*x)^2])

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Maple [A]  time = 0.013, size = 208, normalized size = 1. \[ -{\frac{bx+a}{4\,{b}^{3}} \left ( 5\,A\sqrt{ab}{x}^{3/2}{b}^{2}-3\,A\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{2}{b}^{3}-25\,B\sqrt{ab}{x}^{3/2}ab-8\,B\sqrt{ab}{x}^{5/2}{b}^{2}+15\,B\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{2}a{b}^{2}-6\,A\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ) xa{b}^{2}+30\,B\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ) x{a}^{2}b+3\,A\sqrt{ab}\sqrt{x}ab-3\,A\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){a}^{2}b-15\,B\sqrt{ab}\sqrt{x}{a}^{2}+15\,B\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){a}^{3} \right ){\frac{1}{\sqrt{ab}}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*(5*A*(a*b)^(1/2)*x^(3/2)*b^2-3*A*arctan(x^(1/2)*b/(a*b)^(1/2))*x^2*b^3-25*B
*(a*b)^(1/2)*x^(3/2)*a*b-8*B*(a*b)^(1/2)*x^(5/2)*b^2+15*B*arctan(x^(1/2)*b/(a*b)
^(1/2))*x^2*a*b^2-6*A*arctan(x^(1/2)*b/(a*b)^(1/2))*x*a*b^2+30*B*arctan(x^(1/2)*
b/(a*b)^(1/2))*x*a^2*b+3*A*(a*b)^(1/2)*x^(1/2)*a*b-3*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)/(a*b)^(1/2)/b^3/((b*x+a)^2)^(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)*x^(3/2)/(b^2*x^2 + 2*a*b*x + a^2)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.288673, size = 1, normalized size = 0. \[ \left [\frac{2 \,{\left (8 \, B b^{2} x^{2} + 15 \, B a^{2} - 3 \, A a b + 5 \,{\left (5 \, B a b - A b^{2}\right )} x\right )} \sqrt{-a b} \sqrt{x} - 3 \,{\left (5 \, B a^{3} - A a^{2} b +{\left (5 \, B a b^{2} - A b^{3}\right )} x^{2} + 2 \,{\left (5 \, B a^{2} b - A a b^{2}\right )} x\right )} \log \left (\frac{2 \, a b \sqrt{x} + \sqrt{-a b}{\left (b x - a\right )}}{b x + a}\right )}{8 \,{\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )} \sqrt{-a b}}, \frac{{\left (8 \, B b^{2} x^{2} + 15 \, B a^{2} - 3 \, A a b + 5 \,{\left (5 \, B a b - A b^{2}\right )} x\right )} \sqrt{a b} \sqrt{x} + 3 \,{\left (5 \, B a^{3} - A a^{2} b +{\left (5 \, B a b^{2} - A b^{3}\right )} x^{2} + 2 \,{\left (5 \, B a^{2} b - A a b^{2}\right )} x\right )} \arctan \left (\frac{a}{\sqrt{a b} \sqrt{x}}\right )}{4 \,{\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )} \sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/8*(2*(8*B*b^2*x^2 + 15*B*a^2 - 3*A*a*b + 5*(5*B*a*b - A*b^2)*x)*sqrt(-a*b)*sq
rt(x) - 3*(5*B*a^3 - A*a^2*b + (5*B*a*b^2 - A*b^3)*x^2 + 2*(5*B*a^2*b - A*a*b^2)
*x)*log((2*a*b*sqrt(x) + sqrt(-a*b)*(b*x - a))/(b*x + a)))/((b^5*x^2 + 2*a*b^4*x
+ a^2*b^3)*sqrt(-a*b)), 1/4*((8*B*b^2*x^2 + 15*B*a^2 - 3*A*a*b + 5*(5*B*a*b - A
*b^2)*x)*sqrt(a*b)*sqrt(x) + 3*(5*B*a^3 - A*a^2*b + (5*B*a*b^2 - A*b^3)*x^2 + 2*
(5*B*a^2*b - A*a*b^2)*x)*arctan(a/(sqrt(a*b)*sqrt(x))))/((b^5*x^2 + 2*a*b^4*x +
a^2*b^3)*sqrt(a*b))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{\frac{3}{2}} \left (A + B x\right )}{\left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x**(3/2)*(A + B*x)/((a + b*x)**2)**(3/2), x)

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GIAC/XCAS [A]  time = 0.278548, size = 150, normalized size = 0.73 \[ \frac{2 \, B \sqrt{x}}{b^{3}{\rm sign}\left (b x + a\right )} - \frac{3 \,{\left (5 \, B a - A b\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{4 \, \sqrt{a b} b^{3}{\rm sign}\left (b x + a\right )} + \frac{9 \, B a b x^{\frac{3}{2}} - 5 \, A b^{2} x^{\frac{3}{2}} + 7 \, B a^{2} \sqrt{x} - 3 \, A a b \sqrt{x}}{4 \,{\left (b x + a\right )}^{2} b^{3}{\rm sign}\left (b x + a\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*B*sqrt(x)/(b^3*sign(b*x + a)) - 3/4*(5*B*a - A*b)*arctan(b*sqrt(x)/sqrt(a*b))/
(sqrt(a*b)*b^3*sign(b*x + a)) + 1/4*(9*B*a*b*x^(3/2) - 5*A*b^2*x^(3/2) + 7*B*a^2
*sqrt(x) - 3*A*a*b*sqrt(x))/((b*x + a)^2*b^3*sign(b*x + a))