3.774 \(\int \frac{x^{9/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx\)

Optimal. Leaf size=213 \[ \frac{63 (A b-11 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{128 \sqrt{a} b^{13/2}}-\frac{63 \sqrt{x} (A b-11 a B)}{128 a b^6}+\frac{21 x^{3/2} (A b-11 a B)}{128 a b^5 (a+b x)}+\frac{21 x^{5/2} (A b-11 a B)}{320 a b^4 (a+b x)^2}+\frac{3 x^{7/2} (A b-11 a B)}{80 a b^3 (a+b x)^3}+\frac{x^{9/2} (A b-11 a B)}{40 a b^2 (a+b x)^4}+\frac{x^{11/2} (A b-a B)}{5 a b (a+b x)^5} \]

[Out]

(-63*(A*b - 11*a*B)*Sqrt[x])/(128*a*b^6) + ((A*b - a*B)*x^(11/2))/(5*a*b*(a + b*
x)^5) + ((A*b - 11*a*B)*x^(9/2))/(40*a*b^2*(a + b*x)^4) + (3*(A*b - 11*a*B)*x^(7
/2))/(80*a*b^3*(a + b*x)^3) + (21*(A*b - 11*a*B)*x^(5/2))/(320*a*b^4*(a + b*x)^2
) + (21*(A*b - 11*a*B)*x^(3/2))/(128*a*b^5*(a + b*x)) + (63*(A*b - 11*a*B)*ArcTa
n[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(128*Sqrt[a]*b^(13/2))

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Rubi [A]  time = 0.247483, antiderivative size = 213, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207 \[ \frac{63 (A b-11 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{128 \sqrt{a} b^{13/2}}-\frac{63 \sqrt{x} (A b-11 a B)}{128 a b^6}+\frac{21 x^{3/2} (A b-11 a B)}{128 a b^5 (a+b x)}+\frac{21 x^{5/2} (A b-11 a B)}{320 a b^4 (a+b x)^2}+\frac{3 x^{7/2} (A b-11 a B)}{80 a b^3 (a+b x)^3}+\frac{x^{9/2} (A b-11 a B)}{40 a b^2 (a+b x)^4}+\frac{x^{11/2} (A b-a B)}{5 a b (a+b x)^5} \]

Antiderivative was successfully verified.

[In]  Int[(x^(9/2)*(A + B*x))/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

(-63*(A*b - 11*a*B)*Sqrt[x])/(128*a*b^6) + ((A*b - a*B)*x^(11/2))/(5*a*b*(a + b*
x)^5) + ((A*b - 11*a*B)*x^(9/2))/(40*a*b^2*(a + b*x)^4) + (3*(A*b - 11*a*B)*x^(7
/2))/(80*a*b^3*(a + b*x)^3) + (21*(A*b - 11*a*B)*x^(5/2))/(320*a*b^4*(a + b*x)^2
) + (21*(A*b - 11*a*B)*x^(3/2))/(128*a*b^5*(a + b*x)) + (63*(A*b - 11*a*B)*ArcTa
n[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(128*Sqrt[a]*b^(13/2))

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Rubi in Sympy [A]  time = 59.177, size = 196, normalized size = 0.92 \[ \frac{x^{\frac{11}{2}} \left (A b - B a\right )}{5 a b \left (a + b x\right )^{5}} + \frac{x^{\frac{9}{2}} \left (A b - 11 B a\right )}{40 a b^{2} \left (a + b x\right )^{4}} + \frac{3 x^{\frac{7}{2}} \left (A b - 11 B a\right )}{80 a b^{3} \left (a + b x\right )^{3}} + \frac{21 x^{\frac{5}{2}} \left (A b - 11 B a\right )}{320 a b^{4} \left (a + b x\right )^{2}} + \frac{21 x^{\frac{3}{2}} \left (A b - 11 B a\right )}{128 a b^{5} \left (a + b x\right )} - \frac{63 \sqrt{x} \left (A b - 11 B a\right )}{128 a b^{6}} + \frac{63 \left (A b - 11 B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{128 \sqrt{a} b^{\frac{13}{2}}} \]

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)**3,x)

[Out]

x**(11/2)*(A*b - B*a)/(5*a*b*(a + b*x)**5) + x**(9/2)*(A*b - 11*B*a)/(40*a*b**2*
(a + b*x)**4) + 3*x**(7/2)*(A*b - 11*B*a)/(80*a*b**3*(a + b*x)**3) + 21*x**(5/2)
*(A*b - 11*B*a)/(320*a*b**4*(a + b*x)**2) + 21*x**(3/2)*(A*b - 11*B*a)/(128*a*b*
*5*(a + b*x)) - 63*sqrt(x)*(A*b - 11*B*a)/(128*a*b**6) + 63*(A*b - 11*B*a)*atan(
sqrt(b)*sqrt(x)/sqrt(a))/(128*sqrt(a)*b**(13/2))

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Mathematica [A]  time = 0.374489, size = 150, normalized size = 0.7 \[ \frac{\frac{\sqrt{b} \sqrt{x} \left (3465 a^5 B-105 a^4 b (3 A-154 B x)+42 a^3 b^2 x (704 B x-35 A)+6 a^2 b^3 x^2 (4345 B x-448 A)+5 a b^4 x^3 (2123 B x-474 A)+5 b^5 x^4 (256 B x-193 A)\right )}{(a+b x)^5}+\frac{315 (A b-11 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{\sqrt{a}}}{640 b^{13/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^(9/2)*(A + B*x))/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

((Sqrt[b]*Sqrt[x]*(3465*a^5*B - 105*a^4*b*(3*A - 154*B*x) + 5*b^5*x^4*(-193*A +
256*B*x) + 42*a^3*b^2*x*(-35*A + 704*B*x) + 5*a*b^4*x^3*(-474*A + 2123*B*x) + 6*
a^2*b^3*x^2*(-448*A + 4345*B*x)))/(a + b*x)^5 + (315*(A*b - 11*a*B)*ArcTan[(Sqrt
[b]*Sqrt[x])/Sqrt[a]])/Sqrt[a])/(640*b^(13/2))

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Maple [A]  time = 0.03, size = 239, normalized size = 1.1 \[ 2\,{\frac{B\sqrt{x}}{{b}^{6}}}-{\frac{193\,A}{128\,b \left ( bx+a \right ) ^{5}}{x}^{{\frac{9}{2}}}}+{\frac{843\,Ba}{128\,{b}^{2} \left ( bx+a \right ) ^{5}}{x}^{{\frac{9}{2}}}}+{\frac{1327\,{a}^{2}B}{64\,{b}^{3} \left ( bx+a \right ) ^{5}}{x}^{{\frac{7}{2}}}}-{\frac{237\,aA}{64\,{b}^{2} \left ( bx+a \right ) ^{5}}{x}^{{\frac{7}{2}}}}+{\frac{131\,B{a}^{3}}{5\,{b}^{4} \left ( bx+a \right ) ^{5}}{x}^{{\frac{5}{2}}}}-{\frac{21\,A{a}^{2}}{5\,{b}^{3} \left ( bx+a \right ) ^{5}}{x}^{{\frac{5}{2}}}}-{\frac{147\,A{a}^{3}}{64\,{b}^{4} \left ( bx+a \right ) ^{5}}{x}^{{\frac{3}{2}}}}+{\frac{977\,B{a}^{4}}{64\,{b}^{5} \left ( bx+a \right ) ^{5}}{x}^{{\frac{3}{2}}}}+{\frac{437\,B{a}^{5}}{128\,{b}^{6} \left ( bx+a \right ) ^{5}}\sqrt{x}}-{\frac{63\,A{a}^{4}}{128\,{b}^{5} \left ( bx+a \right ) ^{5}}\sqrt{x}}+{\frac{63\,A}{128\,{b}^{5}}\arctan \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{693\,Ba}{128\,{b}^{6}}\arctan \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(9/2)*(B*x+A)/(b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

2*B/b^6*x^(1/2)-193/128/b/(b*x+a)^5*x^(9/2)*A+843/128/b^2/(b*x+a)^5*x^(9/2)*B*a+
1327/64/b^3/(b*x+a)^5*x^(7/2)*B*a^2-237/64/b^2/(b*x+a)^5*x^(7/2)*A*a+131/5/b^4/(
b*x+a)^5*x^(5/2)*B*a^3-21/5/b^3/(b*x+a)^5*x^(5/2)*A*a^2-147/64/b^4/(b*x+a)^5*A*x
^(3/2)*a^3+977/64/b^5/(b*x+a)^5*B*x^(3/2)*a^4+437/128/b^6/(b*x+a)^5*x^(1/2)*B*a^
5-63/128/b^5/(b*x+a)^5*x^(1/2)*A*a^4+63/128/b^5/(a*b)^(1/2)*arctan(x^(1/2)*b/(a*
b)^(1/2))*A-693/128/b^6/(a*b)^(1/2)*arctan(x^(1/2)*b/(a*b)^(1/2))*B*a

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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)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.320663, size = 1, normalized size = 0. \[ \left [\frac{2 \,{\left (1280 \, B b^{5} x^{5} + 3465 \, B a^{5} - 315 \, A a^{4} b + 965 \,{\left (11 \, B a b^{4} - A b^{5}\right )} x^{4} + 2370 \,{\left (11 \, B a^{2} b^{3} - A a b^{4}\right )} x^{3} + 2688 \,{\left (11 \, B a^{3} b^{2} - A a^{2} b^{3}\right )} x^{2} + 1470 \,{\left (11 \, B a^{4} b - A a^{3} b^{2}\right )} x\right )} \sqrt{-a b} \sqrt{x} - 315 \,{\left (11 \, B a^{6} - A a^{5} b +{\left (11 \, B a b^{5} - A b^{6}\right )} x^{5} + 5 \,{\left (11 \, B a^{2} b^{4} - A a b^{5}\right )} x^{4} + 10 \,{\left (11 \, B a^{3} b^{3} - A a^{2} b^{4}\right )} x^{3} + 10 \,{\left (11 \, B a^{4} b^{2} - A a^{3} b^{3}\right )} x^{2} + 5 \,{\left (11 \, B a^{5} b - A a^{4} b^{2}\right )} x\right )} \log \left (\frac{2 \, a b \sqrt{x} + \sqrt{-a b}{\left (b x - a\right )}}{b x + a}\right )}{1280 \,{\left (b^{11} x^{5} + 5 \, a b^{10} x^{4} + 10 \, a^{2} b^{9} x^{3} + 10 \, a^{3} b^{8} x^{2} + 5 \, a^{4} b^{7} x + a^{5} b^{6}\right )} \sqrt{-a b}}, \frac{{\left (1280 \, B b^{5} x^{5} + 3465 \, B a^{5} - 315 \, A a^{4} b + 965 \,{\left (11 \, B a b^{4} - A b^{5}\right )} x^{4} + 2370 \,{\left (11 \, B a^{2} b^{3} - A a b^{4}\right )} x^{3} + 2688 \,{\left (11 \, B a^{3} b^{2} - A a^{2} b^{3}\right )} x^{2} + 1470 \,{\left (11 \, B a^{4} b - A a^{3} b^{2}\right )} x\right )} \sqrt{a b} \sqrt{x} + 315 \,{\left (11 \, B a^{6} - A a^{5} b +{\left (11 \, B a b^{5} - A b^{6}\right )} x^{5} + 5 \,{\left (11 \, B a^{2} b^{4} - A a b^{5}\right )} x^{4} + 10 \,{\left (11 \, B a^{3} b^{3} - A a^{2} b^{4}\right )} x^{3} + 10 \,{\left (11 \, B a^{4} b^{2} - A a^{3} b^{3}\right )} x^{2} + 5 \,{\left (11 \, B a^{5} b - A a^{4} b^{2}\right )} x\right )} \arctan \left (\frac{a}{\sqrt{a b} \sqrt{x}}\right )}{640 \,{\left (b^{11} x^{5} + 5 \, a b^{10} x^{4} + 10 \, a^{2} b^{9} x^{3} + 10 \, a^{3} b^{8} x^{2} + 5 \, a^{4} b^{7} x + a^{5} b^{6}\right )} \sqrt{a b}}\right ] \]

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)^3,x, algorithm="fricas")

[Out]

[1/1280*(2*(1280*B*b^5*x^5 + 3465*B*a^5 - 315*A*a^4*b + 965*(11*B*a*b^4 - A*b^5)
*x^4 + 2370*(11*B*a^2*b^3 - A*a*b^4)*x^3 + 2688*(11*B*a^3*b^2 - A*a^2*b^3)*x^2 +
1470*(11*B*a^4*b - A*a^3*b^2)*x)*sqrt(-a*b)*sqrt(x) - 315*(11*B*a^6 - A*a^5*b +
(11*B*a*b^5 - A*b^6)*x^5 + 5*(11*B*a^2*b^4 - A*a*b^5)*x^4 + 10*(11*B*a^3*b^3 -
A*a^2*b^4)*x^3 + 10*(11*B*a^4*b^2 - A*a^3*b^3)*x^2 + 5*(11*B*a^5*b - A*a^4*b^2)*
x)*log((2*a*b*sqrt(x) + sqrt(-a*b)*(b*x - a))/(b*x + a)))/((b^11*x^5 + 5*a*b^10*
x^4 + 10*a^2*b^9*x^3 + 10*a^3*b^8*x^2 + 5*a^4*b^7*x + a^5*b^6)*sqrt(-a*b)), 1/64
0*((1280*B*b^5*x^5 + 3465*B*a^5 - 315*A*a^4*b + 965*(11*B*a*b^4 - A*b^5)*x^4 + 2
370*(11*B*a^2*b^3 - A*a*b^4)*x^3 + 2688*(11*B*a^3*b^2 - A*a^2*b^3)*x^2 + 1470*(1
1*B*a^4*b - A*a^3*b^2)*x)*sqrt(a*b)*sqrt(x) + 315*(11*B*a^6 - A*a^5*b + (11*B*a*
b^5 - A*b^6)*x^5 + 5*(11*B*a^2*b^4 - A*a*b^5)*x^4 + 10*(11*B*a^3*b^3 - A*a^2*b^4
)*x^3 + 10*(11*B*a^4*b^2 - A*a^3*b^3)*x^2 + 5*(11*B*a^5*b - A*a^4*b^2)*x)*arctan
(a/(sqrt(a*b)*sqrt(x))))/((b^11*x^5 + 5*a*b^10*x^4 + 10*a^2*b^9*x^3 + 10*a^3*b^8
*x^2 + 5*a^4*b^7*x + a^5*b^6)*sqrt(a*b))]

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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)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.27374, size = 215, normalized size = 1.01 \[ \frac{2 \, B \sqrt{x}}{b^{6}} - \frac{63 \,{\left (11 \, B a - A b\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{128 \, \sqrt{a b} b^{6}} + \frac{4215 \, B a b^{4} x^{\frac{9}{2}} - 965 \, A b^{5} x^{\frac{9}{2}} + 13270 \, B a^{2} b^{3} x^{\frac{7}{2}} - 2370 \, A a b^{4} x^{\frac{7}{2}} + 16768 \, B a^{3} b^{2} x^{\frac{5}{2}} - 2688 \, A a^{2} b^{3} x^{\frac{5}{2}} + 9770 \, B a^{4} b x^{\frac{3}{2}} - 1470 \, A a^{3} b^{2} x^{\frac{3}{2}} + 2185 \, B a^{5} \sqrt{x} - 315 \, A a^{4} b \sqrt{x}}{640 \,{\left (b x + a\right )}^{5} b^{6}} \]

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)^3,x, algorithm="giac")

[Out]

2*B*sqrt(x)/b^6 - 63/128*(11*B*a - A*b)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*b
^6) + 1/640*(4215*B*a*b^4*x^(9/2) - 965*A*b^5*x^(9/2) + 13270*B*a^2*b^3*x^(7/2)
- 2370*A*a*b^4*x^(7/2) + 16768*B*a^3*b^2*x^(5/2) - 2688*A*a^2*b^3*x^(5/2) + 9770
*B*a^4*b*x^(3/2) - 1470*A*a^3*b^2*x^(3/2) + 2185*B*a^5*sqrt(x) - 315*A*a^4*b*sqr
t(x))/((b*x + a)^5*b^6)