3.819 \(\int \frac{x^{5/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=255 \[ \frac{x^{7/2} (A b-a B)}{2 a b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{x^{5/2} (3 A b-7 a B)}{4 a b^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{5 \sqrt{a} (a+b x) (3 A b-7 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 b^{9/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{5 \sqrt{x} (a+b x) (3 A b-7 a B)}{4 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{5 x^{3/2} (a+b x) (3 A b-7 a B)}{12 a b^3 \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

((3*A*b - 7*a*B)*x^(5/2))/(4*a*b^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + ((A*b - a*B)
*x^(7/2))/(2*a*b*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (5*(3*A*b - 7*a*B)*S
qrt[x]*(a + b*x))/(4*b^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (5*(3*A*b - 7*a*B)*x^(
3/2)*(a + b*x))/(12*a*b^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (5*Sqrt[a]*(3*A*b - 7
*a*B)*(a + b*x)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(4*b^(9/2)*Sqrt[a^2 + 2*a*b*x
+ b^2*x^2])

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Rubi [A]  time = 0.349045, antiderivative size = 255, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.194 \[ \frac{x^{7/2} (A b-a B)}{2 a b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{x^{5/2} (3 A b-7 a B)}{4 a b^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{5 \sqrt{a} (a+b x) (3 A b-7 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 b^{9/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{5 \sqrt{x} (a+b x) (3 A b-7 a B)}{4 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{5 x^{3/2} (a+b x) (3 A b-7 a B)}{12 a b^3 \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(x^(5/2)*(A + B*x))/(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

((3*A*b - 7*a*B)*x^(5/2))/(4*a*b^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + ((A*b - a*B)
*x^(7/2))/(2*a*b*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (5*(3*A*b - 7*a*B)*S
qrt[x]*(a + b*x))/(4*b^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (5*(3*A*b - 7*a*B)*x^(
3/2)*(a + b*x))/(12*a*b^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (5*Sqrt[a]*(3*A*b - 7
*a*B)*(a + b*x)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(4*b^(9/2)*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**(5/2)*(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 = 0.150187, size = 129, normalized size = 0.51 \[ \frac{\sqrt{b} \sqrt{x} \left (-105 a^3 B+5 a^2 b (9 A-35 B x)+a b^2 x (75 A-56 B x)+8 b^3 x^2 (3 A+B x)\right )+15 \sqrt{a} (a+b x)^2 (7 a B-3 A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{12 b^{9/2} (a+b x) \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^(5/2)*(A + B*x))/(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

(Sqrt[b]*Sqrt[x]*(-105*a^3*B + a*b^2*x*(75*A - 56*B*x) + 5*a^2*b*(9*A - 35*B*x)
+ 8*b^3*x^2*(3*A + B*x)) + 15*Sqrt[a]*(-3*A*b + 7*a*B)*(a + b*x)^2*ArcTan[(Sqrt[
b]*Sqrt[x])/Sqrt[a]])/(12*b^(9/2)*(a + b*x)*Sqrt[(a + b*x)^2])

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Maple [A]  time = 0.013, size = 247, normalized size = 1. \[{\frac{bx+a}{12\,{b}^{4}} \left ( 8\,B\sqrt{ab}{x}^{7/2}{b}^{3}-56\,B\sqrt{ab}{x}^{5/2}a{b}^{2}+75\,A\sqrt{ab}{x}^{3/2}a{b}^{2}+24\,A\sqrt{ab}{x}^{5/2}{b}^{3}-45\,A\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{2}a{b}^{3}-175\,B\sqrt{ab}{x}^{3/2}{a}^{2}b+105\,B\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{2}{a}^{2}{b}^{2}-90\,A\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ) x{a}^{2}{b}^{2}+210\,B\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ) x{a}^{3}b+45\,A\sqrt{ab}\sqrt{x}{a}^{2}b-45\,A\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){a}^{3}b-105\,B\sqrt{ab}\sqrt{x}{a}^{3}+105\,B\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){a}^{4} \right ){\frac{1}{\sqrt{ab}}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(5/2)*(B*x+A)/(b^2*x^2+2*a*b*x+a^2)^(3/2),x)

[Out]

1/12*(8*B*(a*b)^(1/2)*x^(7/2)*b^3-56*B*(a*b)^(1/2)*x^(5/2)*a*b^2+75*A*(a*b)^(1/2
)*x^(3/2)*a*b^2+24*A*(a*b)^(1/2)*x^(5/2)*b^3-45*A*arctan(x^(1/2)*b/(a*b)^(1/2))*
x^2*a*b^3-175*B*(a*b)^(1/2)*x^(3/2)*a^2*b+105*B*arctan(x^(1/2)*b/(a*b)^(1/2))*x^
2*a^2*b^2-90*A*arctan(x^(1/2)*b/(a*b)^(1/2))*x*a^2*b^2+210*B*arctan(x^(1/2)*b/(a
*b)^(1/2))*x*a^3*b+45*A*(a*b)^(1/2)*x^(1/2)*a^2*b-45*A*arctan(x^(1/2)*b/(a*b)^(1
/2))*a^3*b-105*B*(a*b)^(1/2)*x^(1/2)*a^3+105*B*arctan(x^(1/2)*b/(a*b)^(1/2))*a^4
)*(b*x+a)/(a*b)^(1/2)/b^4/((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^(5/2)/(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 = 0.295604, size = 1, normalized size = 0. \[ \left [-\frac{15 \,{\left (7 \, B a^{3} - 3 \, A a^{2} b +{\left (7 \, B a b^{2} - 3 \, A b^{3}\right )} x^{2} + 2 \,{\left (7 \, B a^{2} b - 3 \, A a b^{2}\right )} x\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x - 2 \, b \sqrt{x} \sqrt{-\frac{a}{b}} - a}{b x + a}\right ) - 2 \,{\left (8 \, B b^{3} x^{3} - 105 \, B a^{3} + 45 \, A a^{2} b - 8 \,{\left (7 \, B a b^{2} - 3 \, A b^{3}\right )} x^{2} - 25 \,{\left (7 \, B a^{2} b - 3 \, A a b^{2}\right )} x\right )} \sqrt{x}}{24 \,{\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}}, \frac{15 \,{\left (7 \, B a^{3} - 3 \, A a^{2} b +{\left (7 \, B a b^{2} - 3 \, A b^{3}\right )} x^{2} + 2 \,{\left (7 \, B a^{2} b - 3 \, A a b^{2}\right )} x\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{\sqrt{x}}{\sqrt{\frac{a}{b}}}\right ) +{\left (8 \, B b^{3} x^{3} - 105 \, B a^{3} + 45 \, A a^{2} b - 8 \,{\left (7 \, B a b^{2} - 3 \, A b^{3}\right )} x^{2} - 25 \,{\left (7 \, B a^{2} b - 3 \, A a b^{2}\right )} x\right )} \sqrt{x}}{12 \,{\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*x^(5/2)/(b^2*x^2 + 2*a*b*x + a^2)^(3/2),x, algorithm="fricas")

[Out]

[-1/24*(15*(7*B*a^3 - 3*A*a^2*b + (7*B*a*b^2 - 3*A*b^3)*x^2 + 2*(7*B*a^2*b - 3*A
*a*b^2)*x)*sqrt(-a/b)*log((b*x - 2*b*sqrt(x)*sqrt(-a/b) - a)/(b*x + a)) - 2*(8*B
*b^3*x^3 - 105*B*a^3 + 45*A*a^2*b - 8*(7*B*a*b^2 - 3*A*b^3)*x^2 - 25*(7*B*a^2*b
- 3*A*a*b^2)*x)*sqrt(x))/(b^6*x^2 + 2*a*b^5*x + a^2*b^4), 1/12*(15*(7*B*a^3 - 3*
A*a^2*b + (7*B*a*b^2 - 3*A*b^3)*x^2 + 2*(7*B*a^2*b - 3*A*a*b^2)*x)*sqrt(a/b)*arc
tan(sqrt(x)/sqrt(a/b)) + (8*B*b^3*x^3 - 105*B*a^3 + 45*A*a^2*b - 8*(7*B*a*b^2 -
3*A*b^3)*x^2 - 25*(7*B*a^2*b - 3*A*a*b^2)*x)*sqrt(x))/(b^6*x^2 + 2*a*b^5*x + a^2
*b^4)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{\frac{5}{2}} \left (A + B x\right )}{\left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(5/2)*(B*x+A)/(b**2*x**2+2*a*b*x+a**2)**(3/2),x)

[Out]

Integral(x**(5/2)*(A + B*x)/((a + b*x)**2)**(3/2), x)

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GIAC/XCAS [A]  time = 0.279441, size = 193, normalized size = 0.76 \[ \frac{5 \,{\left (7 \, B a^{2} - 3 \, A a b\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{4 \, \sqrt{a b} b^{4}{\rm sign}\left (b x + a\right )} - \frac{13 \, B a^{2} b x^{\frac{3}{2}} - 9 \, A a b^{2} x^{\frac{3}{2}} + 11 \, B a^{3} \sqrt{x} - 7 \, A a^{2} b \sqrt{x}}{4 \,{\left (b x + a\right )}^{2} b^{4}{\rm sign}\left (b x + a\right )} + \frac{2 \,{\left (B b^{6} x^{\frac{3}{2}} - 9 \, B a b^{5} \sqrt{x} + 3 \, A b^{6} \sqrt{x}\right )}}{3 \, b^{9}{\rm sign}\left (b x + a\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*x^(5/2)/(b^2*x^2 + 2*a*b*x + a^2)^(3/2),x, algorithm="giac")

[Out]

5/4*(7*B*a^2 - 3*A*a*b)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*b^4*sign(b*x + a)
) - 1/4*(13*B*a^2*b*x^(3/2) - 9*A*a*b^2*x^(3/2) + 11*B*a^3*sqrt(x) - 7*A*a^2*b*s
qrt(x))/((b*x + a)^2*b^4*sign(b*x + a)) + 2/3*(B*b^6*x^(3/2) - 9*B*a*b^5*sqrt(x)
+ 3*A*b^6*sqrt(x))/(b^9*sign(b*x + a))