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

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

[Out]

(21*(11*A*b - 3*a*B))/(64*a^4*b*x^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (A*b -
a*B)/(4*a*b*x^(3/2)*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (11*A*b - 3*a*B
)/(24*a^2*b*x^(3/2)*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (3*(11*A*b - 3*
a*B))/(32*a^3*b*x^(3/2)*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (35*(11*A*b -
3*a*B)*(a + b*x))/(64*a^5*b*x^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (105*(11*A
*b - 3*a*B)*(a + b*x))/(64*a^6*Sqrt[x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (105*Sqr
t[b]*(11*A*b - 3*a*B)*(a + b*x)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(64*a^(13/2)*
Sqrt[a^2 + 2*a*b*x + b^2*x^2])

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Rubi [A]  time = 0.481325, 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 \[ \frac{11 A b-3 a B}{24 a^2 b x^{3/2} (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{A b-a B}{4 a b x^{3/2} (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{105 \sqrt{b} (a+b x) (11 A b-3 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{64 a^{13/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{105 (a+b x) (11 A b-3 a B)}{64 a^6 \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{35 (a+b x) (11 A b-3 a B)}{64 a^5 b x^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{21 (11 A b-3 a B)}{64 a^4 b x^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{3 (11 A b-3 a B)}{32 a^3 b x^{3/2} (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(21*(11*A*b - 3*a*B))/(64*a^4*b*x^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (A*b -
a*B)/(4*a*b*x^(3/2)*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (11*A*b - 3*a*B
)/(24*a^2*b*x^(3/2)*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (3*(11*A*b - 3*
a*B))/(32*a^3*b*x^(3/2)*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (35*(11*A*b -
3*a*B)*(a + b*x))/(64*a^5*b*x^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (105*(11*A
*b - 3*a*B)*(a + b*x))/(64*a^6*Sqrt[x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (105*Sqr
t[b]*(11*A*b - 3*a*B)*(a + b*x)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(64*a^(13/2)*
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((B*x+A)/x**(5/2)/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Exception raised: RecursionError

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Mathematica [A]  time = 0.226968, size = 174, normalized size = 0.49 \[ \frac{\sqrt{a} \left (-128 a^5 (A+3 B x)+a^4 b x (1408 A-2511 B x)+9 a^3 b^2 x^2 (1023 A-511 B x)+231 a^2 b^3 x^3 (73 A-15 B x)+105 a b^4 x^4 (121 A-9 B x)+3465 A b^5 x^5\right )+315 \sqrt{b} x^{3/2} (a+b x)^4 (11 A b-3 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{192 a^{13/2} x^{3/2} (a+b x)^3 \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a]*(3465*A*b^5*x^5 + a^4*b*x*(1408*A - 2511*B*x) + 9*a^3*b^2*x^2*(1023*A -
511*B*x) + 231*a^2*b^3*x^3*(73*A - 15*B*x) + 105*a*b^4*x^4*(121*A - 9*B*x) - 12
8*a^5*(A + 3*B*x)) + 315*Sqrt[b]*(11*A*b - 3*a*B)*x^(3/2)*(a + b*x)^4*ArcTan[(Sq
rt[b]*Sqrt[x])/Sqrt[a]])/(192*a^(13/2)*x^(3/2)*(a + b*x)^3*Sqrt[(a + b*x)^2])

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Maple [A]  time = 0.037, size = 413, normalized size = 1.2 \[{\frac{bx+a}{192\,{a}^{6}} \left ( -4599\,B\sqrt{ab}{x}^{3}{a}^{3}{b}^{2}+9207\,A\sqrt{ab}{x}^{2}{a}^{3}{b}^{2}+3465\,A\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{3/2}{a}^{4}{b}^{2}-128\,A\sqrt{ab}{a}^{5}+3465\,A\sqrt{ab}{x}^{5}{b}^{5}-384\,B\sqrt{ab}x{a}^{5}-2511\,B\sqrt{ab}{x}^{2}{a}^{4}b-945\,B\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{3/2}{a}^{5}b+1408\,A\sqrt{ab}x{a}^{4}b-945\,B\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{11/2}a{b}^{5}+13860\,A\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{9/2}a{b}^{5}-3780\,B\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{9/2}{a}^{2}{b}^{4}+20790\,A\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{7/2}{a}^{2}{b}^{4}-945\,B\sqrt{ab}{x}^{5}a{b}^{4}+12705\,A\sqrt{ab}{x}^{4}a{b}^{4}+3465\,A\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{11/2}{b}^{6}-3465\,B\sqrt{ab}{x}^{4}{a}^{2}{b}^{3}+16863\,A\sqrt{ab}{x}^{3}{a}^{2}{b}^{3}-5670\,B\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{7/2}{a}^{3}{b}^{3}+13860\,A\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{5/2}{a}^{3}{b}^{3}-3780\,B\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{5/2}{a}^{4}{b}^{2} \right ){\frac{1}{\sqrt{ab}}}{x}^{-{\frac{3}{2}}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.295353, size = 1, normalized size = 0. \[ \left [-\frac{256 \, A a^{5} + 630 \,{\left (3 \, B a b^{4} - 11 \, A b^{5}\right )} x^{5} + 2310 \,{\left (3 \, B a^{2} b^{3} - 11 \, A a b^{4}\right )} x^{4} + 3066 \,{\left (3 \, B a^{3} b^{2} - 11 \, A a^{2} b^{3}\right )} x^{3} + 1674 \,{\left (3 \, B a^{4} b - 11 \, A a^{3} b^{2}\right )} x^{2} + 315 \,{\left ({\left (3 \, B a b^{4} - 11 \, A b^{5}\right )} x^{5} + 4 \,{\left (3 \, B a^{2} b^{3} - 11 \, A a b^{4}\right )} x^{4} + 6 \,{\left (3 \, B a^{3} b^{2} - 11 \, A a^{2} b^{3}\right )} x^{3} + 4 \,{\left (3 \, B a^{4} b - 11 \, A a^{3} b^{2}\right )} x^{2} +{\left (3 \, B a^{5} - 11 \, A a^{4} b\right )} x\right )} \sqrt{x} \sqrt{-\frac{b}{a}} \log \left (\frac{b x + 2 \, a \sqrt{x} \sqrt{-\frac{b}{a}} - a}{b x + a}\right ) + 256 \,{\left (3 \, B a^{5} - 11 \, A a^{4} b\right )} x}{384 \,{\left (a^{6} b^{4} x^{5} + 4 \, a^{7} b^{3} x^{4} + 6 \, a^{8} b^{2} x^{3} + 4 \, a^{9} b x^{2} + a^{10} x\right )} \sqrt{x}}, -\frac{128 \, A a^{5} + 315 \,{\left (3 \, B a b^{4} - 11 \, A b^{5}\right )} x^{5} + 1155 \,{\left (3 \, B a^{2} b^{3} - 11 \, A a b^{4}\right )} x^{4} + 1533 \,{\left (3 \, B a^{3} b^{2} - 11 \, A a^{2} b^{3}\right )} x^{3} + 837 \,{\left (3 \, B a^{4} b - 11 \, A a^{3} b^{2}\right )} x^{2} - 315 \,{\left ({\left (3 \, B a b^{4} - 11 \, A b^{5}\right )} x^{5} + 4 \,{\left (3 \, B a^{2} b^{3} - 11 \, A a b^{4}\right )} x^{4} + 6 \,{\left (3 \, B a^{3} b^{2} - 11 \, A a^{2} b^{3}\right )} x^{3} + 4 \,{\left (3 \, B a^{4} b - 11 \, A a^{3} b^{2}\right )} x^{2} +{\left (3 \, B a^{5} - 11 \, A a^{4} b\right )} x\right )} \sqrt{x} \sqrt{\frac{b}{a}} \arctan \left (\frac{a \sqrt{\frac{b}{a}}}{b \sqrt{x}}\right ) + 128 \,{\left (3 \, B a^{5} - 11 \, A a^{4} b\right )} x}{192 \,{\left (a^{6} b^{4} x^{5} + 4 \, a^{7} b^{3} x^{4} + 6 \, a^{8} b^{2} x^{3} + 4 \, a^{9} b x^{2} + a^{10} x\right )} \sqrt{x}}\right ] \]

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)^(5/2)*x^(5/2)),x, algorithm="fricas")

[Out]

[-1/384*(256*A*a^5 + 630*(3*B*a*b^4 - 11*A*b^5)*x^5 + 2310*(3*B*a^2*b^3 - 11*A*a
*b^4)*x^4 + 3066*(3*B*a^3*b^2 - 11*A*a^2*b^3)*x^3 + 1674*(3*B*a^4*b - 11*A*a^3*b
^2)*x^2 + 315*((3*B*a*b^4 - 11*A*b^5)*x^5 + 4*(3*B*a^2*b^3 - 11*A*a*b^4)*x^4 + 6
*(3*B*a^3*b^2 - 11*A*a^2*b^3)*x^3 + 4*(3*B*a^4*b - 11*A*a^3*b^2)*x^2 + (3*B*a^5
- 11*A*a^4*b)*x)*sqrt(x)*sqrt(-b/a)*log((b*x + 2*a*sqrt(x)*sqrt(-b/a) - a)/(b*x
+ a)) + 256*(3*B*a^5 - 11*A*a^4*b)*x)/((a^6*b^4*x^5 + 4*a^7*b^3*x^4 + 6*a^8*b^2*
x^3 + 4*a^9*b*x^2 + a^10*x)*sqrt(x)), -1/192*(128*A*a^5 + 315*(3*B*a*b^4 - 11*A*
b^5)*x^5 + 1155*(3*B*a^2*b^3 - 11*A*a*b^4)*x^4 + 1533*(3*B*a^3*b^2 - 11*A*a^2*b^
3)*x^3 + 837*(3*B*a^4*b - 11*A*a^3*b^2)*x^2 - 315*((3*B*a*b^4 - 11*A*b^5)*x^5 +
4*(3*B*a^2*b^3 - 11*A*a*b^4)*x^4 + 6*(3*B*a^3*b^2 - 11*A*a^2*b^3)*x^3 + 4*(3*B*a
^4*b - 11*A*a^3*b^2)*x^2 + (3*B*a^5 - 11*A*a^4*b)*x)*sqrt(x)*sqrt(b/a)*arctan(a*
sqrt(b/a)/(b*sqrt(x))) + 128*(3*B*a^5 - 11*A*a^4*b)*x)/((a^6*b^4*x^5 + 4*a^7*b^3
*x^4 + 6*a^8*b^2*x^3 + 4*a^9*b*x^2 + a^10*x)*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**(5/2)/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.282491, size = 243, normalized size = 0.68 \[ -\frac{105 \,{\left (3 \, B a b - 11 \, A b^{2}\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{64 \, \sqrt{a b} a^{6}{\rm sign}\left (b x + a\right )} - \frac{2 \,{\left (3 \, B a x - 15 \, A b x + A a\right )}}{3 \, a^{6} x^{\frac{3}{2}}{\rm sign}\left (b x + a\right )} - \frac{561 \, B a b^{4} x^{\frac{7}{2}} - 1545 \, A b^{5} x^{\frac{7}{2}} + 1929 \, B a^{2} b^{3} x^{\frac{5}{2}} - 5153 \, A a b^{4} x^{\frac{5}{2}} + 2295 \, B a^{3} b^{2} x^{\frac{3}{2}} - 5855 \, A a^{2} b^{3} x^{\frac{3}{2}} + 975 \, B a^{4} b \sqrt{x} - 2295 \, A a^{3} b^{2} \sqrt{x}}{192 \,{\left (b x + a\right )}^{4} a^{6}{\rm sign}\left (b x + a\right )} \]

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)^(5/2)*x^(5/2)),x, algorithm="giac")

[Out]

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