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

Optimal. Leaf size=219 \[ -\frac{63 (11 A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{128 a^{13/2} \sqrt{b}}-\frac{63 (11 A b-a B)}{128 a^6 b \sqrt{x}}+\frac{21 (11 A b-a B)}{128 a^5 b \sqrt{x} (a+b x)}+\frac{21 (11 A b-a B)}{320 a^4 b \sqrt{x} (a+b x)^2}+\frac{3 (11 A b-a B)}{80 a^3 b \sqrt{x} (a+b x)^3}+\frac{11 A b-a B}{40 a^2 b \sqrt{x} (a+b x)^4}+\frac{A b-a B}{5 a b \sqrt{x} (a+b x)^5} \]

[Out]

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

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Rubi [A]  time = 0.253204, antiderivative size = 219, 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 \[ -\frac{63 (11 A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{128 a^{13/2} \sqrt{b}}-\frac{63 (11 A b-a B)}{128 a^6 b \sqrt{x}}+\frac{21 (11 A b-a B)}{128 a^5 b \sqrt{x} (a+b x)}+\frac{21 (11 A b-a B)}{320 a^4 b \sqrt{x} (a+b x)^2}+\frac{3 (11 A b-a B)}{80 a^3 b \sqrt{x} (a+b x)^3}+\frac{11 A b-a B}{40 a^2 b \sqrt{x} (a+b x)^4}+\frac{A b-a B}{5 a b \sqrt{x} (a+b x)^5} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 67.6926, size = 196, normalized size = 0.89 \[ \frac{A b - B a}{5 a b \sqrt{x} \left (a + b x\right )^{5}} + \frac{11 A b - B a}{40 a^{2} b \sqrt{x} \left (a + b x\right )^{4}} + \frac{3 \left (11 A b - B a\right )}{80 a^{3} b \sqrt{x} \left (a + b x\right )^{3}} + \frac{21 \left (11 A b - B a\right )}{320 a^{4} b \sqrt{x} \left (a + b x\right )^{2}} + \frac{21 \left (11 A b - B a\right )}{128 a^{5} b \sqrt{x} \left (a + b x\right )} - \frac{63 \left (11 A b - B a\right )}{128 a^{6} b \sqrt{x}} - \frac{63 \left (11 A b - B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{128 a^{\frac{13}{2}} \sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.300232, size = 152, normalized size = 0.69 \[ \frac{\frac{\sqrt{a} \left (a^5 (965 B x-1280 A)+5 a^4 b x (474 B x-2123 A)+6 a^3 b^2 x^2 (448 B x-4345 A)+42 a^2 b^3 x^3 (35 B x-704 A)+105 a b^4 x^4 (3 B x-154 A)-3465 A b^5 x^5\right )}{\sqrt{x} (a+b x)^5}+\frac{315 (a B-11 A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{\sqrt{b}}}{640 a^{13/2}} \]

Antiderivative was successfully verified.

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

[Out]

((Sqrt[a]*(-3465*A*b^5*x^5 + 105*a*b^4*x^4*(-154*A + 3*B*x) + 42*a^2*b^3*x^3*(-7
04*A + 35*B*x) + 6*a^3*b^2*x^2*(-4345*A + 448*B*x) + 5*a^4*b*x*(-2123*A + 474*B*
x) + a^5*(-1280*A + 965*B*x)))/(Sqrt[x]*(a + b*x)^5) + (315*(-11*A*b + a*B)*ArcT
an[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/Sqrt[b])/(640*a^(13/2))

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Maple [A]  time = 0.034, size = 239, normalized size = 1.1 \[ -2\,{\frac{A}{{a}^{6}\sqrt{x}}}-{\frac{437\,A{b}^{5}}{128\,{a}^{6} \left ( bx+a \right ) ^{5}}{x}^{{\frac{9}{2}}}}+{\frac{63\,{b}^{4}B}{128\,{a}^{5} \left ( bx+a \right ) ^{5}}{x}^{{\frac{9}{2}}}}-{\frac{977\,A{b}^{4}}{64\,{a}^{5} \left ( bx+a \right ) ^{5}}{x}^{{\frac{7}{2}}}}+{\frac{147\,B{b}^{3}}{64\,{a}^{4} \left ( bx+a \right ) ^{5}}{x}^{{\frac{7}{2}}}}-{\frac{131\,A{b}^{3}}{5\,{a}^{4} \left ( bx+a \right ) ^{5}}{x}^{{\frac{5}{2}}}}+{\frac{21\,{b}^{2}B}{5\,{a}^{3} \left ( bx+a \right ) ^{5}}{x}^{{\frac{5}{2}}}}-{\frac{1327\,{b}^{2}A}{64\,{a}^{3} \left ( bx+a \right ) ^{5}}{x}^{{\frac{3}{2}}}}+{\frac{237\,Bb}{64\,{a}^{2} \left ( bx+a \right ) ^{5}}{x}^{{\frac{3}{2}}}}-{\frac{843\,Ab}{128\,{a}^{2} \left ( bx+a \right ) ^{5}}\sqrt{x}}+{\frac{193\,B}{128\,a \left ( bx+a \right ) ^{5}}\sqrt{x}}-{\frac{693\,Ab}{128\,{a}^{6}}\arctan \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{63\,B}{128\,{a}^{5}}\arctan \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*A/a^6/x^(1/2)-437/128/a^6/(b*x+a)^5*x^(9/2)*A*b^5+63/128/a^5/(b*x+a)^5*x^(9/2
)*B*b^4-977/64/a^5/(b*x+a)^5*A*x^(7/2)*b^4+147/64/a^4/(b*x+a)^5*B*x^(7/2)*b^3-13
1/5/a^4/(b*x+a)^5*x^(5/2)*A*b^3+21/5/a^3/(b*x+a)^5*x^(5/2)*B*b^2-1327/64/a^3/(b*
x+a)^5*x^(3/2)*A*b^2+237/64/a^2/(b*x+a)^5*x^(3/2)*B*b-843/128/a^2/(b*x+a)^5*x^(1
/2)*A*b+193/128/a/(b*x+a)^5*x^(1/2)*B-693/128/a^6/(a*b)^(1/2)*arctan(x^(1/2)*b/(
a*b)^(1/2))*A*b+63/128/a^5/(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)/((b^2*x^2 + 2*a*b*x + a^2)^3*x^(3/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.330148, size = 1, normalized size = 0. \[ \left [-\frac{315 \,{\left (B a^{6} - 11 \, A a^{5} b +{\left (B a b^{5} - 11 \, A b^{6}\right )} x^{5} + 5 \,{\left (B a^{2} b^{4} - 11 \, A a b^{5}\right )} x^{4} + 10 \,{\left (B a^{3} b^{3} - 11 \, A a^{2} b^{4}\right )} x^{3} + 10 \,{\left (B a^{4} b^{2} - 11 \, A a^{3} b^{3}\right )} x^{2} + 5 \,{\left (B a^{5} b - 11 \, A a^{4} b^{2}\right )} x\right )} \sqrt{x} \log \left (-\frac{2 \, a b \sqrt{x} - \sqrt{-a b}{\left (b x - a\right )}}{b x + a}\right ) + 2 \,{\left (1280 \, A a^{5} - 315 \,{\left (B a b^{4} - 11 \, A b^{5}\right )} x^{5} - 1470 \,{\left (B a^{2} b^{3} - 11 \, A a b^{4}\right )} x^{4} - 2688 \,{\left (B a^{3} b^{2} - 11 \, A a^{2} b^{3}\right )} x^{3} - 2370 \,{\left (B a^{4} b - 11 \, A a^{3} b^{2}\right )} x^{2} - 965 \,{\left (B a^{5} - 11 \, A a^{4} b\right )} x\right )} \sqrt{-a b}}{1280 \,{\left (a^{6} b^{5} x^{5} + 5 \, a^{7} b^{4} x^{4} + 10 \, a^{8} b^{3} x^{3} + 10 \, a^{9} b^{2} x^{2} + 5 \, a^{10} b x + a^{11}\right )} \sqrt{-a b} \sqrt{x}}, -\frac{315 \,{\left (B a^{6} - 11 \, A a^{5} b +{\left (B a b^{5} - 11 \, A b^{6}\right )} x^{5} + 5 \,{\left (B a^{2} b^{4} - 11 \, A a b^{5}\right )} x^{4} + 10 \,{\left (B a^{3} b^{3} - 11 \, A a^{2} b^{4}\right )} x^{3} + 10 \,{\left (B a^{4} b^{2} - 11 \, A a^{3} b^{3}\right )} x^{2} + 5 \,{\left (B a^{5} b - 11 \, A a^{4} b^{2}\right )} x\right )} \sqrt{x} \arctan \left (\frac{a}{\sqrt{a b} \sqrt{x}}\right ) +{\left (1280 \, A a^{5} - 315 \,{\left (B a b^{4} - 11 \, A b^{5}\right )} x^{5} - 1470 \,{\left (B a^{2} b^{3} - 11 \, A a b^{4}\right )} x^{4} - 2688 \,{\left (B a^{3} b^{2} - 11 \, A a^{2} b^{3}\right )} x^{3} - 2370 \,{\left (B a^{4} b - 11 \, A a^{3} b^{2}\right )} x^{2} - 965 \,{\left (B a^{5} - 11 \, A a^{4} b\right )} x\right )} \sqrt{a b}}{640 \,{\left (a^{6} b^{5} x^{5} + 5 \, a^{7} b^{4} x^{4} + 10 \, a^{8} b^{3} x^{3} + 10 \, a^{9} b^{2} x^{2} + 5 \, a^{10} b x + a^{11}\right )} \sqrt{a b} \sqrt{x}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/1280*(315*(B*a^6 - 11*A*a^5*b + (B*a*b^5 - 11*A*b^6)*x^5 + 5*(B*a^2*b^4 - 11
*A*a*b^5)*x^4 + 10*(B*a^3*b^3 - 11*A*a^2*b^4)*x^3 + 10*(B*a^4*b^2 - 11*A*a^3*b^3
)*x^2 + 5*(B*a^5*b - 11*A*a^4*b^2)*x)*sqrt(x)*log(-(2*a*b*sqrt(x) - sqrt(-a*b)*(
b*x - a))/(b*x + a)) + 2*(1280*A*a^5 - 315*(B*a*b^4 - 11*A*b^5)*x^5 - 1470*(B*a^
2*b^3 - 11*A*a*b^4)*x^4 - 2688*(B*a^3*b^2 - 11*A*a^2*b^3)*x^3 - 2370*(B*a^4*b -
11*A*a^3*b^2)*x^2 - 965*(B*a^5 - 11*A*a^4*b)*x)*sqrt(-a*b))/((a^6*b^5*x^5 + 5*a^
7*b^4*x^4 + 10*a^8*b^3*x^3 + 10*a^9*b^2*x^2 + 5*a^10*b*x + a^11)*sqrt(-a*b)*sqrt
(x)), -1/640*(315*(B*a^6 - 11*A*a^5*b + (B*a*b^5 - 11*A*b^6)*x^5 + 5*(B*a^2*b^4
- 11*A*a*b^5)*x^4 + 10*(B*a^3*b^3 - 11*A*a^2*b^4)*x^3 + 10*(B*a^4*b^2 - 11*A*a^3
*b^3)*x^2 + 5*(B*a^5*b - 11*A*a^4*b^2)*x)*sqrt(x)*arctan(a/(sqrt(a*b)*sqrt(x)))
+ (1280*A*a^5 - 315*(B*a*b^4 - 11*A*b^5)*x^5 - 1470*(B*a^2*b^3 - 11*A*a*b^4)*x^4
- 2688*(B*a^3*b^2 - 11*A*a^2*b^3)*x^3 - 2370*(B*a^4*b - 11*A*a^3*b^2)*x^2 - 965
*(B*a^5 - 11*A*a^4*b)*x)*sqrt(a*b))/((a^6*b^5*x^5 + 5*a^7*b^4*x^4 + 10*a^8*b^3*x
^3 + 10*a^9*b^2*x^2 + 5*a^10*b*x + a^11)*sqrt(a*b)*sqrt(x))]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.279268, size = 213, normalized size = 0.97 \[ \frac{63 \,{\left (B a - 11 \, A b\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{128 \, \sqrt{a b} a^{6}} - \frac{2 \, A}{a^{6} \sqrt{x}} + \frac{315 \, B a b^{4} x^{\frac{9}{2}} - 2185 \, A b^{5} x^{\frac{9}{2}} + 1470 \, B a^{2} b^{3} x^{\frac{7}{2}} - 9770 \, A a b^{4} x^{\frac{7}{2}} + 2688 \, B a^{3} b^{2} x^{\frac{5}{2}} - 16768 \, A a^{2} b^{3} x^{\frac{5}{2}} + 2370 \, B a^{4} b x^{\frac{3}{2}} - 13270 \, A a^{3} b^{2} x^{\frac{3}{2}} + 965 \, B a^{5} \sqrt{x} - 4215 \, A a^{4} b \sqrt{x}}{640 \,{\left (b x + a\right )}^{5} a^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

63/128*(B*a - 11*A*b)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^6) - 2*A/(a^6*sqr
t(x)) + 1/640*(315*B*a*b^4*x^(9/2) - 2185*A*b^5*x^(9/2) + 1470*B*a^2*b^3*x^(7/2)
- 9770*A*a*b^4*x^(7/2) + 2688*B*a^3*b^2*x^(5/2) - 16768*A*a^2*b^3*x^(5/2) + 237
0*B*a^4*b*x^(3/2) - 13270*A*a^3*b^2*x^(3/2) + 965*B*a^5*sqrt(x) - 4215*A*a^4*b*s
qrt(x))/((b*x + a)^5*a^6)