3.826 \(\int \frac{x^{11/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=404 \[ \frac{x^{13/2} (A b-a B)}{4 a b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{x^{11/2} (5 A b-13 a B)}{24 a b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{231 a \sqrt{x} (a+b x) (5 A b-13 a B)}{64 b^7 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{77 x^{3/2} (a+b x) (5 A b-13 a B)}{64 b^6 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{231 x^{5/2} (a+b x) (5 A b-13 a B)}{320 a b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{33 x^{7/2} (5 A b-13 a B)}{64 a b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{11 x^{9/2} (5 A b-13 a B)}{96 a b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{231 a^{3/2} (a+b x) (5 A b-13 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{64 b^{15/2} \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

(33*(5*A*b - 13*a*B)*x^(7/2))/(64*a*b^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + ((A*b -
a*B)*x^(13/2))/(4*a*b*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + ((5*A*b - 13
*a*B)*x^(11/2))/(24*a*b^2*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (11*(5*A*
b - 13*a*B)*x^(9/2))/(96*a*b^3*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (231*a
*(5*A*b - 13*a*B)*Sqrt[x]*(a + b*x))/(64*b^7*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (7
7*(5*A*b - 13*a*B)*x^(3/2)*(a + b*x))/(64*b^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (
231*(5*A*b - 13*a*B)*x^(5/2)*(a + b*x))/(320*a*b^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2]
) + (231*a^(3/2)*(5*A*b - 13*a*B)*(a + b*x)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(
64*b^(15/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

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Rubi [A]  time = 0.524595, antiderivative size = 404, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.194 \[ \frac{x^{13/2} (A b-a B)}{4 a b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{x^{11/2} (5 A b-13 a B)}{24 a b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{231 a \sqrt{x} (a+b x) (5 A b-13 a B)}{64 b^7 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{77 x^{3/2} (a+b x) (5 A b-13 a B)}{64 b^6 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{231 x^{5/2} (a+b x) (5 A b-13 a B)}{320 a b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{33 x^{7/2} (5 A b-13 a B)}{64 a b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{11 x^{9/2} (5 A b-13 a B)}{96 a b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{231 a^{3/2} (a+b x) (5 A b-13 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{64 b^{15/2} \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(x^(11/2)*(A + B*x))/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(33*(5*A*b - 13*a*B)*x^(7/2))/(64*a*b^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + ((A*b -
a*B)*x^(13/2))/(4*a*b*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + ((5*A*b - 13
*a*B)*x^(11/2))/(24*a*b^2*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (11*(5*A*
b - 13*a*B)*x^(9/2))/(96*a*b^3*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (231*a
*(5*A*b - 13*a*B)*Sqrt[x]*(a + b*x))/(64*b^7*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (7
7*(5*A*b - 13*a*B)*x^(3/2)*(a + b*x))/(64*b^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (
231*(5*A*b - 13*a*B)*x^(5/2)*(a + b*x))/(320*a*b^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2]
) + (231*a^(3/2)*(5*A*b - 13*a*B)*(a + b*x)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(
64*b^(15/2)*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**(11/2)*(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 = 0.264192, size = 188, normalized size = 0.47 \[ \frac{\sqrt{b} \sqrt{x} \left (45045 a^6 B-1155 a^5 b (15 A-143 B x)+231 a^4 b^2 x (949 B x-275 A)+33 a^3 b^3 x^2 (3627 B x-2555 A)+11 a^2 b^4 x^3 (1664 B x-4185 A)-128 a b^5 x^4 (55 A+13 B x)+128 b^6 x^5 (5 A+3 B x)\right )-3465 a^{3/2} (a+b x)^4 (13 a B-5 A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{960 b^{15/2} (a+b x)^3 \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^(11/2)*(A + B*x))/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(Sqrt[b]*Sqrt[x]*(45045*a^6*B - 1155*a^5*b*(15*A - 143*B*x) + 128*b^6*x^5*(5*A +
3*B*x) - 128*a*b^5*x^4*(55*A + 13*B*x) + 231*a^4*b^2*x*(-275*A + 949*B*x) + 11*
a^2*b^4*x^3*(-4185*A + 1664*B*x) + 33*a^3*b^3*x^2*(-2555*A + 3627*B*x)) - 3465*a
^(3/2)*(-5*A*b + 13*a*B)*(a + b*x)^4*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(960*b^(
15/2)*(a + b*x)^3*Sqrt[(a + b*x)^2])

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Maple [A]  time = 0.032, size = 443, normalized size = 1.1 \[{\frac{bx+a}{960\,{b}^{7}} \left ( 384\,B\sqrt{ab}{x}^{13/2}{b}^{6}+640\,A\sqrt{ab}{x}^{11/2}{b}^{6}-1664\,B\sqrt{ab}{x}^{11/2}a{b}^{5}-7040\,A\sqrt{ab}{x}^{9/2}a{b}^{5}+18304\,B\sqrt{ab}{x}^{9/2}{a}^{2}{b}^{4}-46035\,A\sqrt{ab}{x}^{7/2}{a}^{2}{b}^{4}+119691\,B\sqrt{ab}{x}^{7/2}{a}^{3}{b}^{3}-84315\,A\sqrt{ab}{x}^{5/2}{a}^{3}{b}^{3}+17325\,A\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{4}{a}^{2}{b}^{5}+219219\,B\sqrt{ab}{x}^{5/2}{a}^{4}{b}^{2}-45045\,B\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{4}{a}^{3}{b}^{4}+69300\,A\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{3}{a}^{3}{b}^{4}-180180\,B\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{3}{a}^{4}{b}^{3}-63525\,A\sqrt{ab}{x}^{3/2}{a}^{4}{b}^{2}+103950\,A\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{2}{a}^{4}{b}^{3}+165165\,B\sqrt{ab}{x}^{3/2}{a}^{5}b-270270\,B\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{2}{a}^{5}{b}^{2}+69300\,A\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ) x{a}^{5}{b}^{2}-180180\,B\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ) x{a}^{6}b-17325\,A\sqrt{ab}\sqrt{x}{a}^{5}b+17325\,A\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){a}^{6}b+45045\,B\sqrt{ab}\sqrt{x}{a}^{6}-45045\,B\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){a}^{7} \right ){\frac{1}{\sqrt{ab}}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(11/2)*(B*x+A)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

1/960*(384*B*(a*b)^(1/2)*x^(13/2)*b^6+640*A*(a*b)^(1/2)*x^(11/2)*b^6-1664*B*(a*b
)^(1/2)*x^(11/2)*a*b^5-7040*A*(a*b)^(1/2)*x^(9/2)*a*b^5+18304*B*(a*b)^(1/2)*x^(9
/2)*a^2*b^4-46035*A*(a*b)^(1/2)*x^(7/2)*a^2*b^4+119691*B*(a*b)^(1/2)*x^(7/2)*a^3
*b^3-84315*A*(a*b)^(1/2)*x^(5/2)*a^3*b^3+17325*A*arctan(x^(1/2)*b/(a*b)^(1/2))*x
^4*a^2*b^5+219219*B*(a*b)^(1/2)*x^(5/2)*a^4*b^2-45045*B*arctan(x^(1/2)*b/(a*b)^(
1/2))*x^4*a^3*b^4+69300*A*arctan(x^(1/2)*b/(a*b)^(1/2))*x^3*a^3*b^4-180180*B*arc
tan(x^(1/2)*b/(a*b)^(1/2))*x^3*a^4*b^3-63525*A*(a*b)^(1/2)*x^(3/2)*a^4*b^2+10395
0*A*arctan(x^(1/2)*b/(a*b)^(1/2))*x^2*a^4*b^3+165165*B*(a*b)^(1/2)*x^(3/2)*a^5*b
-270270*B*arctan(x^(1/2)*b/(a*b)^(1/2))*x^2*a^5*b^2+69300*A*arctan(x^(1/2)*b/(a*
b)^(1/2))*x*a^5*b^2-180180*B*arctan(x^(1/2)*b/(a*b)^(1/2))*x*a^6*b-17325*A*(a*b)
^(1/2)*x^(1/2)*a^5*b+17325*A*arctan(x^(1/2)*b/(a*b)^(1/2))*a^6*b+45045*B*(a*b)^(
1/2)*x^(1/2)*a^6-45045*B*arctan(x^(1/2)*b/(a*b)^(1/2))*a^7)*(b*x+a)/(a*b)^(1/2)/
b^7/((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^(11/2)/(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 = 0.343411, size = 1, normalized size = 0. \[ \left [-\frac{3465 \,{\left (13 \, B a^{6} - 5 \, A a^{5} b +{\left (13 \, B a^{2} b^{4} - 5 \, A a b^{5}\right )} x^{4} + 4 \,{\left (13 \, B a^{3} b^{3} - 5 \, A a^{2} b^{4}\right )} x^{3} + 6 \,{\left (13 \, B a^{4} b^{2} - 5 \, A a^{3} b^{3}\right )} x^{2} + 4 \,{\left (13 \, B a^{5} b - 5 \, A a^{4} b^{2}\right )} x\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x + 2 \, b \sqrt{x} \sqrt{-\frac{a}{b}} - a}{b x + a}\right ) - 2 \,{\left (384 \, B b^{6} x^{6} + 45045 \, B a^{6} - 17325 \, A a^{5} b - 128 \,{\left (13 \, B a b^{5} - 5 \, A b^{6}\right )} x^{5} + 1408 \,{\left (13 \, B a^{2} b^{4} - 5 \, A a b^{5}\right )} x^{4} + 9207 \,{\left (13 \, B a^{3} b^{3} - 5 \, A a^{2} b^{4}\right )} x^{3} + 16863 \,{\left (13 \, B a^{4} b^{2} - 5 \, A a^{3} b^{3}\right )} x^{2} + 12705 \,{\left (13 \, B a^{5} b - 5 \, A a^{4} b^{2}\right )} x\right )} \sqrt{x}}{1920 \,{\left (b^{11} x^{4} + 4 \, a b^{10} x^{3} + 6 \, a^{2} b^{9} x^{2} + 4 \, a^{3} b^{8} x + a^{4} b^{7}\right )}}, -\frac{3465 \,{\left (13 \, B a^{6} - 5 \, A a^{5} b +{\left (13 \, B a^{2} b^{4} - 5 \, A a b^{5}\right )} x^{4} + 4 \,{\left (13 \, B a^{3} b^{3} - 5 \, A a^{2} b^{4}\right )} x^{3} + 6 \,{\left (13 \, B a^{4} b^{2} - 5 \, A a^{3} b^{3}\right )} x^{2} + 4 \,{\left (13 \, B a^{5} b - 5 \, A a^{4} b^{2}\right )} x\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{\sqrt{x}}{\sqrt{\frac{a}{b}}}\right ) -{\left (384 \, B b^{6} x^{6} + 45045 \, B a^{6} - 17325 \, A a^{5} b - 128 \,{\left (13 \, B a b^{5} - 5 \, A b^{6}\right )} x^{5} + 1408 \,{\left (13 \, B a^{2} b^{4} - 5 \, A a b^{5}\right )} x^{4} + 9207 \,{\left (13 \, B a^{3} b^{3} - 5 \, A a^{2} b^{4}\right )} x^{3} + 16863 \,{\left (13 \, B a^{4} b^{2} - 5 \, A a^{3} b^{3}\right )} x^{2} + 12705 \,{\left (13 \, B a^{5} b - 5 \, A a^{4} b^{2}\right )} x\right )} \sqrt{x}}{960 \,{\left (b^{11} x^{4} + 4 \, a b^{10} x^{3} + 6 \, a^{2} b^{9} x^{2} + 4 \, a^{3} b^{8} x + a^{4} b^{7}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*x^(11/2)/(b^2*x^2 + 2*a*b*x + a^2)^(5/2),x, algorithm="fricas")

[Out]

[-1/1920*(3465*(13*B*a^6 - 5*A*a^5*b + (13*B*a^2*b^4 - 5*A*a*b^5)*x^4 + 4*(13*B*
a^3*b^3 - 5*A*a^2*b^4)*x^3 + 6*(13*B*a^4*b^2 - 5*A*a^3*b^3)*x^2 + 4*(13*B*a^5*b
- 5*A*a^4*b^2)*x)*sqrt(-a/b)*log((b*x + 2*b*sqrt(x)*sqrt(-a/b) - a)/(b*x + a)) -
2*(384*B*b^6*x^6 + 45045*B*a^6 - 17325*A*a^5*b - 128*(13*B*a*b^5 - 5*A*b^6)*x^5
+ 1408*(13*B*a^2*b^4 - 5*A*a*b^5)*x^4 + 9207*(13*B*a^3*b^3 - 5*A*a^2*b^4)*x^3 +
16863*(13*B*a^4*b^2 - 5*A*a^3*b^3)*x^2 + 12705*(13*B*a^5*b - 5*A*a^4*b^2)*x)*sq
rt(x))/(b^11*x^4 + 4*a*b^10*x^3 + 6*a^2*b^9*x^2 + 4*a^3*b^8*x + a^4*b^7), -1/960
*(3465*(13*B*a^6 - 5*A*a^5*b + (13*B*a^2*b^4 - 5*A*a*b^5)*x^4 + 4*(13*B*a^3*b^3
- 5*A*a^2*b^4)*x^3 + 6*(13*B*a^4*b^2 - 5*A*a^3*b^3)*x^2 + 4*(13*B*a^5*b - 5*A*a^
4*b^2)*x)*sqrt(a/b)*arctan(sqrt(x)/sqrt(a/b)) - (384*B*b^6*x^6 + 45045*B*a^6 - 1
7325*A*a^5*b - 128*(13*B*a*b^5 - 5*A*b^6)*x^5 + 1408*(13*B*a^2*b^4 - 5*A*a*b^5)*
x^4 + 9207*(13*B*a^3*b^3 - 5*A*a^2*b^4)*x^3 + 16863*(13*B*a^4*b^2 - 5*A*a^3*b^3)
*x^2 + 12705*(13*B*a^5*b - 5*A*a^4*b^2)*x)*sqrt(x))/(b^11*x^4 + 4*a*b^10*x^3 + 6
*a^2*b^9*x^2 + 4*a^3*b^8*x + a^4*b^7)]

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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**(11/2)*(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 = 0.280248, size = 294, normalized size = 0.73 \[ -\frac{231 \,{\left (13 \, B a^{3} - 5 \, A a^{2} b\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{64 \, \sqrt{a b} b^{7}{\rm sign}\left (b x + a\right )} + \frac{4431 \, B a^{3} b^{3} x^{\frac{7}{2}} - 2295 \, A a^{2} b^{4} x^{\frac{7}{2}} + 11767 \, B a^{4} b^{2} x^{\frac{5}{2}} - 5855 \, A a^{3} b^{3} x^{\frac{5}{2}} + 10633 \, B a^{5} b x^{\frac{3}{2}} - 5153 \, A a^{4} b^{2} x^{\frac{3}{2}} + 3249 \, B a^{6} \sqrt{x} - 1545 \, A a^{5} b \sqrt{x}}{192 \,{\left (b x + a\right )}^{4} b^{7}{\rm sign}\left (b x + a\right )} + \frac{2 \,{\left (3 \, B b^{20} x^{\frac{5}{2}} - 25 \, B a b^{19} x^{\frac{3}{2}} + 5 \, A b^{20} x^{\frac{3}{2}} + 225 \, B a^{2} b^{18} \sqrt{x} - 75 \, A a b^{19} \sqrt{x}\right )}}{15 \, b^{25}{\rm sign}\left (b x + a\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*x^(11/2)/(b^2*x^2 + 2*a*b*x + a^2)^(5/2),x, algorithm="giac")

[Out]

-231/64*(13*B*a^3 - 5*A*a^2*b)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*b^7*sign(b
*x + a)) + 1/192*(4431*B*a^3*b^3*x^(7/2) - 2295*A*a^2*b^4*x^(7/2) + 11767*B*a^4*
b^2*x^(5/2) - 5855*A*a^3*b^3*x^(5/2) + 10633*B*a^5*b*x^(3/2) - 5153*A*a^4*b^2*x^
(3/2) + 3249*B*a^6*sqrt(x) - 1545*A*a^5*b*sqrt(x))/((b*x + a)^4*b^7*sign(b*x + a
)) + 2/15*(3*B*b^20*x^(5/2) - 25*B*a*b^19*x^(3/2) + 5*A*b^20*x^(3/2) + 225*B*a^2
*b^18*sqrt(x) - 75*A*a*b^19*sqrt(x))/(b^25*sign(b*x + a))