3.773 \(\int \frac{x^{11/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx\)

Optimal. Leaf size=240 \[ -\frac{231 \sqrt{a} (3 A b-13 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{128 b^{15/2}}+\frac{231 \sqrt{x} (3 A b-13 a B)}{128 b^7}-\frac{77 x^{3/2} (3 A b-13 a B)}{128 a b^6}+\frac{231 x^{5/2} (3 A b-13 a B)}{640 a b^5 (a+b x)}+\frac{33 x^{7/2} (3 A b-13 a B)}{320 a b^4 (a+b x)^2}+\frac{11 x^{9/2} (3 A b-13 a B)}{240 a b^3 (a+b x)^3}+\frac{x^{11/2} (3 A b-13 a B)}{40 a b^2 (a+b x)^4}+\frac{x^{13/2} (A b-a B)}{5 a b (a+b x)^5} \]

[Out]

(231*(3*A*b - 13*a*B)*Sqrt[x])/(128*b^7) - (77*(3*A*b - 13*a*B)*x^(3/2))/(128*a*
b^6) + ((A*b - a*B)*x^(13/2))/(5*a*b*(a + b*x)^5) + ((3*A*b - 13*a*B)*x^(11/2))/
(40*a*b^2*(a + b*x)^4) + (11*(3*A*b - 13*a*B)*x^(9/2))/(240*a*b^3*(a + b*x)^3) +
(33*(3*A*b - 13*a*B)*x^(7/2))/(320*a*b^4*(a + b*x)^2) + (231*(3*A*b - 13*a*B)*x
^(5/2))/(640*a*b^5*(a + b*x)) - (231*Sqrt[a]*(3*A*b - 13*a*B)*ArcTan[(Sqrt[b]*Sq
rt[x])/Sqrt[a]])/(128*b^(15/2))

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Rubi [A]  time = 0.284723, antiderivative size = 240, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207 \[ -\frac{231 \sqrt{a} (3 A b-13 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{128 b^{15/2}}+\frac{231 \sqrt{x} (3 A b-13 a B)}{128 b^7}-\frac{77 x^{3/2} (3 A b-13 a B)}{128 a b^6}+\frac{231 x^{5/2} (3 A b-13 a B)}{640 a b^5 (a+b x)}+\frac{33 x^{7/2} (3 A b-13 a B)}{320 a b^4 (a+b x)^2}+\frac{11 x^{9/2} (3 A b-13 a B)}{240 a b^3 (a+b x)^3}+\frac{x^{11/2} (3 A b-13 a B)}{40 a b^2 (a+b x)^4}+\frac{x^{13/2} (A b-a B)}{5 a b (a+b x)^5} \]

Antiderivative was successfully verified.

[In]  Int[(x^(11/2)*(A + B*x))/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

(231*(3*A*b - 13*a*B)*Sqrt[x])/(128*b^7) - (77*(3*A*b - 13*a*B)*x^(3/2))/(128*a*
b^6) + ((A*b - a*B)*x^(13/2))/(5*a*b*(a + b*x)^5) + ((3*A*b - 13*a*B)*x^(11/2))/
(40*a*b^2*(a + b*x)^4) + (11*(3*A*b - 13*a*B)*x^(9/2))/(240*a*b^3*(a + b*x)^3) +
(33*(3*A*b - 13*a*B)*x^(7/2))/(320*a*b^4*(a + b*x)^2) + (231*(3*A*b - 13*a*B)*x
^(5/2))/(640*a*b^5*(a + b*x)) - (231*Sqrt[a]*(3*A*b - 13*a*B)*ArcTan[(Sqrt[b]*Sq
rt[x])/Sqrt[a]])/(128*b^(15/2))

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Rubi in Sympy [A]  time = 66.1666, size = 228, normalized size = 0.95 \[ - \frac{231 \sqrt{a} \left (3 A b - 13 B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{128 b^{\frac{15}{2}}} + \frac{231 \sqrt{x} \left (3 A b - 13 B a\right )}{128 b^{7}} + \frac{x^{\frac{13}{2}} \left (A b - B a\right )}{5 a b \left (a + b x\right )^{5}} + \frac{x^{\frac{11}{2}} \left (3 A b - 13 B a\right )}{40 a b^{2} \left (a + b x\right )^{4}} + \frac{11 x^{\frac{9}{2}} \left (3 A b - 13 B a\right )}{240 a b^{3} \left (a + b x\right )^{3}} + \frac{33 x^{\frac{7}{2}} \left (3 A b - 13 B a\right )}{320 a b^{4} \left (a + b x\right )^{2}} + \frac{231 x^{\frac{5}{2}} \left (3 A b - 13 B a\right )}{640 a b^{5} \left (a + b x\right )} - \frac{77 x^{\frac{3}{2}} \left (3 A b - 13 B a\right )}{128 a b^{6}} \]

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)**3,x)

[Out]

-231*sqrt(a)*(3*A*b - 13*B*a)*atan(sqrt(b)*sqrt(x)/sqrt(a))/(128*b**(15/2)) + 23
1*sqrt(x)*(3*A*b - 13*B*a)/(128*b**7) + x**(13/2)*(A*b - B*a)/(5*a*b*(a + b*x)**
5) + x**(11/2)*(3*A*b - 13*B*a)/(40*a*b**2*(a + b*x)**4) + 11*x**(9/2)*(3*A*b -
13*B*a)/(240*a*b**3*(a + b*x)**3) + 33*x**(7/2)*(3*A*b - 13*B*a)/(320*a*b**4*(a
+ b*x)**2) + 231*x**(5/2)*(3*A*b - 13*B*a)/(640*a*b**5*(a + b*x)) - 77*x**(3/2)*
(3*A*b - 13*B*a)/(128*a*b**6)

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Mathematica [A]  time = 0.370937, size = 169, normalized size = 0.7 \[ \frac{\frac{\sqrt{b} \sqrt{x} \left (-45045 a^6 B+1155 a^5 b (9 A-182 B x)+462 a^4 b^2 x (105 A-832 B x)+66 a^3 b^3 x^2 (1344 A-5135 B x)+55 a^2 b^4 x^3 (1422 A-2509 B x)+5 a b^5 x^4 (6369 A-3328 B x)+1280 b^6 x^5 (3 A+B x)\right )}{(a+b x)^5}+3465 \sqrt{a} (13 a B-3 A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{1920 b^{15/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^(11/2)*(A + B*x))/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

((Sqrt[b]*Sqrt[x]*(-45045*a^6*B + 66*a^3*b^3*x^2*(1344*A - 5135*B*x) + 5*a*b^5*x
^4*(6369*A - 3328*B*x) + 55*a^2*b^4*x^3*(1422*A - 2509*B*x) + 462*a^4*b^2*x*(105
*A - 832*B*x) + 1155*a^5*b*(9*A - 182*B*x) + 1280*b^6*x^5*(3*A + B*x)))/(a + b*x
)^5 + 3465*Sqrt[a]*(-3*A*b + 13*a*B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(1920*b^
(15/2))

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Maple [A]  time = 0.033, size = 266, normalized size = 1.1 \[{\frac{2\,B}{3\,{b}^{6}}{x}^{{\frac{3}{2}}}}+2\,{\frac{A\sqrt{x}}{{b}^{6}}}-12\,{\frac{aB\sqrt{x}}{{b}^{7}}}+{\frac{843\,aA}{128\,{b}^{2} \left ( bx+a \right ) ^{5}}{x}^{{\frac{9}{2}}}}-{\frac{2373\,{a}^{2}B}{128\,{b}^{3} \left ( bx+a \right ) ^{5}}{x}^{{\frac{9}{2}}}}+{\frac{1327\,A{a}^{2}}{64\,{b}^{3} \left ( bx+a \right ) ^{5}}{x}^{{\frac{7}{2}}}}-{\frac{12131\,B{a}^{3}}{192\,{b}^{4} \left ( bx+a \right ) ^{5}}{x}^{{\frac{7}{2}}}}+{\frac{131\,A{a}^{3}}{5\,{b}^{4} \left ( bx+a \right ) ^{5}}{x}^{{\frac{5}{2}}}}-{\frac{1253\,B{a}^{4}}{15\,{b}^{5} \left ( bx+a \right ) ^{5}}{x}^{{\frac{5}{2}}}}+{\frac{977\,A{a}^{4}}{64\,{b}^{5} \left ( bx+a \right ) ^{5}}{x}^{{\frac{3}{2}}}}-{\frac{9629\,B{a}^{5}}{192\,{b}^{6} \left ( bx+a \right ) ^{5}}{x}^{{\frac{3}{2}}}}+{\frac{437\,A{a}^{5}}{128\,{b}^{6} \left ( bx+a \right ) ^{5}}\sqrt{x}}-{\frac{1467\,B{a}^{6}}{128\,{b}^{7} \left ( bx+a \right ) ^{5}}\sqrt{x}}-{\frac{693\,aA}{128\,{b}^{6}}\arctan \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{3003\,{a}^{2}B}{128\,{b}^{7}}\arctan \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

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)^3,x)

[Out]

2/3/b^6*B*x^(3/2)+2/b^6*A*x^(1/2)-12/b^7*B*x^(1/2)*a+843/128*a/b^2/(b*x+a)^5*x^(
9/2)*A-2373/128*a^2/b^3/(b*x+a)^5*x^(9/2)*B+1327/64*a^2/b^3/(b*x+a)^5*x^(7/2)*A-
12131/192*a^3/b^4/(b*x+a)^5*x^(7/2)*B+131/5*a^3/b^4/(b*x+a)^5*x^(5/2)*A-1253/15*
a^4/b^5/(b*x+a)^5*x^(5/2)*B+977/64*a^4/b^5/(b*x+a)^5*A*x^(3/2)-9629/192*a^5/b^6/
(b*x+a)^5*B*x^(3/2)+437/128*a^5/b^6/(b*x+a)^5*x^(1/2)*A-1467/128*a^6/b^7/(b*x+a)
^5*x^(1/2)*B-693/128*a/b^6/(a*b)^(1/2)*arctan(x^(1/2)*b/(a*b)^(1/2))*A+3003/128*
a^2/b^7/(a*b)^(1/2)*arctan(x^(1/2)*b/(a*b)^(1/2))*B

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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)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.339072, size = 1, normalized size = 0. \[ \text{result too large to display} \]

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)^3,x, algorithm="fricas")

[Out]

[-1/3840*(3465*(13*B*a^6 - 3*A*a^5*b + (13*B*a*b^5 - 3*A*b^6)*x^5 + 5*(13*B*a^2*
b^4 - 3*A*a*b^5)*x^4 + 10*(13*B*a^3*b^3 - 3*A*a^2*b^4)*x^3 + 10*(13*B*a^4*b^2 -
3*A*a^3*b^3)*x^2 + 5*(13*B*a^5*b - 3*A*a^4*b^2)*x)*sqrt(-a/b)*log((b*x - 2*b*sqr
t(x)*sqrt(-a/b) - a)/(b*x + a)) - 2*(1280*B*b^6*x^6 - 45045*B*a^6 + 10395*A*a^5*
b - 1280*(13*B*a*b^5 - 3*A*b^6)*x^5 - 10615*(13*B*a^2*b^4 - 3*A*a*b^5)*x^4 - 260
70*(13*B*a^3*b^3 - 3*A*a^2*b^4)*x^3 - 29568*(13*B*a^4*b^2 - 3*A*a^3*b^3)*x^2 - 1
6170*(13*B*a^5*b - 3*A*a^4*b^2)*x)*sqrt(x))/(b^12*x^5 + 5*a*b^11*x^4 + 10*a^2*b^
10*x^3 + 10*a^3*b^9*x^2 + 5*a^4*b^8*x + a^5*b^7), 1/1920*(3465*(13*B*a^6 - 3*A*a
^5*b + (13*B*a*b^5 - 3*A*b^6)*x^5 + 5*(13*B*a^2*b^4 - 3*A*a*b^5)*x^4 + 10*(13*B*
a^3*b^3 - 3*A*a^2*b^4)*x^3 + 10*(13*B*a^4*b^2 - 3*A*a^3*b^3)*x^2 + 5*(13*B*a^5*b
- 3*A*a^4*b^2)*x)*sqrt(a/b)*arctan(sqrt(x)/sqrt(a/b)) + (1280*B*b^6*x^6 - 45045
*B*a^6 + 10395*A*a^5*b - 1280*(13*B*a*b^5 - 3*A*b^6)*x^5 - 10615*(13*B*a^2*b^4 -
3*A*a*b^5)*x^4 - 26070*(13*B*a^3*b^3 - 3*A*a^2*b^4)*x^3 - 29568*(13*B*a^4*b^2 -
3*A*a^3*b^3)*x^2 - 16170*(13*B*a^5*b - 3*A*a^4*b^2)*x)*sqrt(x))/(b^12*x^5 + 5*a
*b^11*x^4 + 10*a^2*b^10*x^3 + 10*a^3*b^9*x^2 + 5*a^4*b^8*x + a^5*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)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.275834, size = 258, normalized size = 1.08 \[ \frac{231 \,{\left (13 \, B a^{2} - 3 \, A a b\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{128 \, \sqrt{a b} b^{7}} - \frac{35595 \, B a^{2} b^{4} x^{\frac{9}{2}} - 12645 \, A a b^{5} x^{\frac{9}{2}} + 121310 \, B a^{3} b^{3} x^{\frac{7}{2}} - 39810 \, A a^{2} b^{4} x^{\frac{7}{2}} + 160384 \, B a^{4} b^{2} x^{\frac{5}{2}} - 50304 \, A a^{3} b^{3} x^{\frac{5}{2}} + 96290 \, B a^{5} b x^{\frac{3}{2}} - 29310 \, A a^{4} b^{2} x^{\frac{3}{2}} + 22005 \, B a^{6} \sqrt{x} - 6555 \, A a^{5} b \sqrt{x}}{1920 \,{\left (b x + a\right )}^{5} b^{7}} + \frac{2 \,{\left (B b^{12} x^{\frac{3}{2}} - 18 \, B a b^{11} \sqrt{x} + 3 \, A b^{12} \sqrt{x}\right )}}{3 \, b^{18}} \]

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)^3,x, algorithm="giac")

[Out]

231/128*(13*B*a^2 - 3*A*a*b)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*b^7) - 1/192
0*(35595*B*a^2*b^4*x^(9/2) - 12645*A*a*b^5*x^(9/2) + 121310*B*a^3*b^3*x^(7/2) -
39810*A*a^2*b^4*x^(7/2) + 160384*B*a^4*b^2*x^(5/2) - 50304*A*a^3*b^3*x^(5/2) + 9
6290*B*a^5*b*x^(3/2) - 29310*A*a^4*b^2*x^(3/2) + 22005*B*a^6*sqrt(x) - 6555*A*a^
5*b*sqrt(x))/((b*x + a)^5*b^7) + 2/3*(B*b^12*x^(3/2) - 18*B*a*b^11*sqrt(x) + 3*A
*b^12*sqrt(x))/b^18