3.827 \(\int \frac{x^{9/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=357 \[ \frac{x^{11/2} (A b-a B)}{4 a b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{x^{9/2} (3 A b-11 a B)}{24 a b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{105 \sqrt{a} (a+b x) (3 A b-11 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{64 b^{13/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{105 \sqrt{x} (a+b x) (3 A b-11 a B)}{64 b^6 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{35 x^{3/2} (a+b x) (3 A b-11 a B)}{64 a b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{21 x^{5/2} (3 A b-11 a B)}{64 a b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{3 x^{7/2} (3 A b-11 a B)}{32 a b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

(21*(3*A*b - 11*a*B)*x^(5/2))/(64*a*b^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + ((A*b -
a*B)*x^(11/2))/(4*a*b*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + ((3*A*b - 11
*a*B)*x^(9/2))/(24*a*b^2*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (3*(3*A*b
- 11*a*B)*x^(7/2))/(32*a*b^3*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (105*(3*
A*b - 11*a*B)*Sqrt[x]*(a + b*x))/(64*b^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (35*(3
*A*b - 11*a*B)*x^(3/2)*(a + b*x))/(64*a*b^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (10
5*Sqrt[a]*(3*A*b - 11*a*B)*(a + b*x)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(64*b^(1
3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

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Rubi [A]  time = 0.468065, antiderivative size = 357, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.194 \[ \frac{x^{11/2} (A b-a B)}{4 a b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{x^{9/2} (3 A b-11 a B)}{24 a b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{105 \sqrt{a} (a+b x) (3 A b-11 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{64 b^{13/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{105 \sqrt{x} (a+b x) (3 A b-11 a B)}{64 b^6 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{35 x^{3/2} (a+b x) (3 A b-11 a B)}{64 a b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{21 x^{5/2} (3 A b-11 a B)}{64 a b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{3 x^{7/2} (3 A b-11 a B)}{32 a b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(x^(9/2)*(A + B*x))/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(21*(3*A*b - 11*a*B)*x^(5/2))/(64*a*b^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + ((A*b -
a*B)*x^(11/2))/(4*a*b*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + ((3*A*b - 11
*a*B)*x^(9/2))/(24*a*b^2*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (3*(3*A*b
- 11*a*B)*x^(7/2))/(32*a*b^3*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (105*(3*
A*b - 11*a*B)*Sqrt[x]*(a + b*x))/(64*b^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (35*(3
*A*b - 11*a*B)*x^(3/2)*(a + b*x))/(64*a*b^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (10
5*Sqrt[a]*(3*A*b - 11*a*B)*(a + b*x)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(64*b^(1
3/2)*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**(9/2)*(B*x+A)/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Exception raised: RecursionError

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Mathematica [A]  time = 0.226315, size = 167, normalized size = 0.47 \[ \frac{\sqrt{b} \sqrt{x} \left (-3465 a^5 B+105 a^4 b (9 A-121 B x)+231 a^3 b^2 x (15 A-73 B x)+9 a^2 b^3 x^2 (511 A-1023 B x)+a b^4 x^3 (2511 A-1408 B x)+128 b^5 x^4 (3 A+B x)\right )+315 \sqrt{a} (a+b x)^4 (11 a B-3 A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{192 b^{13/2} (a+b x)^3 \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^(9/2)*(A + B*x))/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(Sqrt[b]*Sqrt[x]*(-3465*a^5*B + a*b^4*x^3*(2511*A - 1408*B*x) + 9*a^2*b^3*x^2*(5
11*A - 1023*B*x) + 105*a^4*b*(9*A - 121*B*x) + 231*a^3*b^2*x*(15*A - 73*B*x) + 1
28*b^5*x^4*(3*A + B*x)) + 315*Sqrt[a]*(-3*A*b + 11*a*B)*(a + b*x)^4*ArcTan[(Sqrt
[b]*Sqrt[x])/Sqrt[a]])/(192*b^(13/2)*(a + b*x)^3*Sqrt[(a + b*x)^2])

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Maple [A]  time = 0.031, size = 407, normalized size = 1.1 \[{\frac{bx+a}{192\,{b}^{6}} \left ( -1408\,B\sqrt{ab}{x}^{9/2}a{b}^{4}+384\,A\sqrt{ab}{x}^{9/2}{b}^{5}-945\,A\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{4}a{b}^{5}-16863\,B\sqrt{ab}{x}^{5/2}{a}^{3}{b}^{2}+3465\,B\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{4}{a}^{2}{b}^{4}-3780\,A\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{3}{a}^{2}{b}^{4}+13860\,B\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{3}{a}^{3}{b}^{3}+3465\,A\sqrt{ab}{x}^{3/2}{a}^{3}{b}^{2}-5670\,A\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{2}{a}^{3}{b}^{3}-12705\,B\sqrt{ab}{x}^{3/2}{a}^{4}b+20790\,B\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{2}{a}^{4}{b}^{2}-3780\,A\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ) x{a}^{4}{b}^{2}+13860\,B\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ) x{a}^{5}b+945\,A\sqrt{ab}\sqrt{x}{a}^{4}b+3465\,B\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){a}^{6}+2511\,A\sqrt{ab}{x}^{7/2}a{b}^{4}-9207\,B\sqrt{ab}{x}^{7/2}{a}^{2}{b}^{3}+128\,B\sqrt{ab}{x}^{11/2}{b}^{5}+4599\,A\sqrt{ab}{x}^{5/2}{a}^{2}{b}^{3}-945\,A\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){a}^{5}b-3465\,B\sqrt{ab}\sqrt{x}{a}^{5} \right ){\frac{1}{\sqrt{ab}}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(9/2)*(B*x+A)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

1/192*(-1408*B*(a*b)^(1/2)*x^(9/2)*a*b^4+384*A*(a*b)^(1/2)*x^(9/2)*b^5-945*A*arc
tan(x^(1/2)*b/(a*b)^(1/2))*x^4*a*b^5-16863*B*(a*b)^(1/2)*x^(5/2)*a^3*b^2+3465*B*
arctan(x^(1/2)*b/(a*b)^(1/2))*x^4*a^2*b^4-3780*A*arctan(x^(1/2)*b/(a*b)^(1/2))*x
^3*a^2*b^4+13860*B*arctan(x^(1/2)*b/(a*b)^(1/2))*x^3*a^3*b^3+3465*A*(a*b)^(1/2)*
x^(3/2)*a^3*b^2-5670*A*arctan(x^(1/2)*b/(a*b)^(1/2))*x^2*a^3*b^3-12705*B*(a*b)^(
1/2)*x^(3/2)*a^4*b+20790*B*arctan(x^(1/2)*b/(a*b)^(1/2))*x^2*a^4*b^2-3780*A*arct
an(x^(1/2)*b/(a*b)^(1/2))*x*a^4*b^2+13860*B*arctan(x^(1/2)*b/(a*b)^(1/2))*x*a^5*
b+945*A*(a*b)^(1/2)*x^(1/2)*a^4*b+3465*B*arctan(x^(1/2)*b/(a*b)^(1/2))*a^6+2511*
A*(a*b)^(1/2)*x^(7/2)*a*b^4-9207*B*(a*b)^(1/2)*x^(7/2)*a^2*b^3+128*B*(a*b)^(1/2)
*x^(11/2)*b^5+4599*A*(a*b)^(1/2)*x^(5/2)*a^2*b^3-945*A*arctan(x^(1/2)*b/(a*b)^(1
/2))*a^5*b-3465*B*(a*b)^(1/2)*x^(1/2)*a^5)*(b*x+a)/(a*b)^(1/2)/b^6/((b*x+a)^2)^(
5/2)

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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^(9/2)/(b^2*x^2 + 2*a*b*x + a^2)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.338997, size = 1, normalized size = 0. \[ \left [-\frac{315 \,{\left (11 \, B a^{5} - 3 \, A a^{4} b +{\left (11 \, B a b^{4} - 3 \, A b^{5}\right )} x^{4} + 4 \,{\left (11 \, B a^{2} b^{3} - 3 \, A a b^{4}\right )} x^{3} + 6 \,{\left (11 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} x^{2} + 4 \,{\left (11 \, B a^{4} b - 3 \, A a^{3} b^{2}\right )} x\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x - 2 \, b \sqrt{x} \sqrt{-\frac{a}{b}} - a}{b x + a}\right ) - 2 \,{\left (128 \, B b^{5} x^{5} - 3465 \, B a^{5} + 945 \, A a^{4} b - 128 \,{\left (11 \, B a b^{4} - 3 \, A b^{5}\right )} x^{4} - 837 \,{\left (11 \, B a^{2} b^{3} - 3 \, A a b^{4}\right )} x^{3} - 1533 \,{\left (11 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} x^{2} - 1155 \,{\left (11 \, B a^{4} b - 3 \, A a^{3} b^{2}\right )} x\right )} \sqrt{x}}{384 \,{\left (b^{10} x^{4} + 4 \, a b^{9} x^{3} + 6 \, a^{2} b^{8} x^{2} + 4 \, a^{3} b^{7} x + a^{4} b^{6}\right )}}, \frac{315 \,{\left (11 \, B a^{5} - 3 \, A a^{4} b +{\left (11 \, B a b^{4} - 3 \, A b^{5}\right )} x^{4} + 4 \,{\left (11 \, B a^{2} b^{3} - 3 \, A a b^{4}\right )} x^{3} + 6 \,{\left (11 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} x^{2} + 4 \,{\left (11 \, B a^{4} b - 3 \, A a^{3} b^{2}\right )} x\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{\sqrt{x}}{\sqrt{\frac{a}{b}}}\right ) +{\left (128 \, B b^{5} x^{5} - 3465 \, B a^{5} + 945 \, A a^{4} b - 128 \,{\left (11 \, B a b^{4} - 3 \, A b^{5}\right )} x^{4} - 837 \,{\left (11 \, B a^{2} b^{3} - 3 \, A a b^{4}\right )} x^{3} - 1533 \,{\left (11 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} x^{2} - 1155 \,{\left (11 \, B a^{4} b - 3 \, A a^{3} b^{2}\right )} x\right )} \sqrt{x}}{192 \,{\left (b^{10} x^{4} + 4 \, a b^{9} x^{3} + 6 \, a^{2} b^{8} x^{2} + 4 \, a^{3} b^{7} x + a^{4} b^{6}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*x^(9/2)/(b^2*x^2 + 2*a*b*x + a^2)^(5/2),x, algorithm="fricas")

[Out]

[-1/384*(315*(11*B*a^5 - 3*A*a^4*b + (11*B*a*b^4 - 3*A*b^5)*x^4 + 4*(11*B*a^2*b^
3 - 3*A*a*b^4)*x^3 + 6*(11*B*a^3*b^2 - 3*A*a^2*b^3)*x^2 + 4*(11*B*a^4*b - 3*A*a^
3*b^2)*x)*sqrt(-a/b)*log((b*x - 2*b*sqrt(x)*sqrt(-a/b) - a)/(b*x + a)) - 2*(128*
B*b^5*x^5 - 3465*B*a^5 + 945*A*a^4*b - 128*(11*B*a*b^4 - 3*A*b^5)*x^4 - 837*(11*
B*a^2*b^3 - 3*A*a*b^4)*x^3 - 1533*(11*B*a^3*b^2 - 3*A*a^2*b^3)*x^2 - 1155*(11*B*
a^4*b - 3*A*a^3*b^2)*x)*sqrt(x))/(b^10*x^4 + 4*a*b^9*x^3 + 6*a^2*b^8*x^2 + 4*a^3
*b^7*x + a^4*b^6), 1/192*(315*(11*B*a^5 - 3*A*a^4*b + (11*B*a*b^4 - 3*A*b^5)*x^4
+ 4*(11*B*a^2*b^3 - 3*A*a*b^4)*x^3 + 6*(11*B*a^3*b^2 - 3*A*a^2*b^3)*x^2 + 4*(11
*B*a^4*b - 3*A*a^3*b^2)*x)*sqrt(a/b)*arctan(sqrt(x)/sqrt(a/b)) + (128*B*b^5*x^5
- 3465*B*a^5 + 945*A*a^4*b - 128*(11*B*a*b^4 - 3*A*b^5)*x^4 - 837*(11*B*a^2*b^3
- 3*A*a*b^4)*x^3 - 1533*(11*B*a^3*b^2 - 3*A*a^2*b^3)*x^2 - 1155*(11*B*a^4*b - 3*
A*a^3*b^2)*x)*sqrt(x))/(b^10*x^4 + 4*a*b^9*x^3 + 6*a^2*b^8*x^2 + 4*a^3*b^7*x + a
^4*b^6)]

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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**(9/2)*(B*x+A)/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.286569, size = 258, normalized size = 0.72 \[ \frac{105 \,{\left (11 \, B a^{2} - 3 \, A a b\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{64 \, \sqrt{a b} b^{6}{\rm sign}\left (b x + a\right )} - \frac{2295 \, B a^{2} b^{3} x^{\frac{7}{2}} - 975 \, A a b^{4} x^{\frac{7}{2}} + 5855 \, B a^{3} b^{2} x^{\frac{5}{2}} - 2295 \, A a^{2} b^{3} x^{\frac{5}{2}} + 5153 \, B a^{4} b x^{\frac{3}{2}} - 1929 \, A a^{3} b^{2} x^{\frac{3}{2}} + 1545 \, B a^{5} \sqrt{x} - 561 \, A a^{4} b \sqrt{x}}{192 \,{\left (b x + a\right )}^{4} b^{6}{\rm sign}\left (b x + a\right )} + \frac{2 \,{\left (B b^{10} x^{\frac{3}{2}} - 15 \, B a b^{9} \sqrt{x} + 3 \, A b^{10} \sqrt{x}\right )}}{3 \, b^{15}{\rm sign}\left (b x + a\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*x^(9/2)/(b^2*x^2 + 2*a*b*x + a^2)^(5/2),x, algorithm="giac")

[Out]

105/64*(11*B*a^2 - 3*A*a*b)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*b^6*sign(b*x
+ a)) - 1/192*(2295*B*a^2*b^3*x^(7/2) - 975*A*a*b^4*x^(7/2) + 5855*B*a^3*b^2*x^(
5/2) - 2295*A*a^2*b^3*x^(5/2) + 5153*B*a^4*b*x^(3/2) - 1929*A*a^3*b^2*x^(3/2) +
1545*B*a^5*sqrt(x) - 561*A*a^4*b*sqrt(x))/((b*x + a)^4*b^6*sign(b*x + a)) + 2/3*
(B*b^10*x^(3/2) - 15*B*a*b^9*sqrt(x) + 3*A*b^10*sqrt(x))/(b^15*sign(b*x + a))