3.833 \(\int \frac{A+B x}{x^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=311 \[ \frac{A b-a B}{4 a b \sqrt{x} (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{9 A b-a B}{24 a^2 b \sqrt{x} (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{35 (a+b x) (9 A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{64 a^{11/2} \sqrt{b} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{35 (a+b x) (9 A b-a B)}{64 a^5 b \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{35 (9 A b-a B)}{192 a^4 b \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{7 (9 A b-a B)}{96 a^3 b \sqrt{x} (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

(35*(9*A*b - a*B))/(192*a^4*b*Sqrt[x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (A*b - a*
B)/(4*a*b*Sqrt[x]*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (9*A*b - a*B)/(24
*a^2*b*Sqrt[x]*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (7*(9*A*b - a*B))/(9
6*a^3*b*Sqrt[x]*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (35*(9*A*b - a*B)*(a
+ b*x))/(64*a^5*b*Sqrt[x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (35*(9*A*b - a*B)*(a
+ b*x)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(64*a^(11/2)*Sqrt[b]*Sqrt[a^2 + 2*a*b*
x + b^2*x^2])

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Rubi [A]  time = 0.430853, antiderivative size = 311, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.194 \[ \frac{A b-a B}{4 a b \sqrt{x} (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{9 A b-a B}{24 a^2 b \sqrt{x} (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{35 (a+b x) (9 A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{64 a^{11/2} \sqrt{b} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{35 (a+b x) (9 A b-a B)}{64 a^5 b \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{35 (9 A b-a B)}{192 a^4 b \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{7 (9 A b-a B)}{96 a^3 b \sqrt{x} (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/(x^(3/2)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]

[Out]

(35*(9*A*b - a*B))/(192*a^4*b*Sqrt[x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (A*b - a*
B)/(4*a*b*Sqrt[x]*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (9*A*b - a*B)/(24
*a^2*b*Sqrt[x]*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (7*(9*A*b - a*B))/(9
6*a^3*b*Sqrt[x]*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (35*(9*A*b - a*B)*(a
+ b*x))/(64*a^5*b*Sqrt[x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (35*(9*A*b - a*B)*(a
+ b*x)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(64*a^(11/2)*Sqrt[b]*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**(3/2)/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Exception raised: RecursionError

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Mathematica [A]  time = 0.15351, size = 182, normalized size = 0.59 \[ \frac{\frac{48 a^{7/2} \sqrt{x} (a B-A b)}{a+b x}+8 a^{5/2} \sqrt{x} (7 a B-15 A b)+2 a^{3/2} \sqrt{x} (a+b x) (35 a B-123 A b)+3 \sqrt{a} \sqrt{x} (a+b x)^2 (35 a B-187 A b)+\frac{105 (a+b x)^3 (a B-9 A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{\sqrt{b}}-\frac{384 \sqrt{a} A (a+b x)^3}{\sqrt{x}}}{192 a^{11/2} \left ((a+b x)^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/(x^(3/2)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]

[Out]

(8*a^(5/2)*(-15*A*b + 7*a*B)*Sqrt[x] + (48*a^(7/2)*(-(A*b) + a*B)*Sqrt[x])/(a +
b*x) + 2*a^(3/2)*(-123*A*b + 35*a*B)*Sqrt[x]*(a + b*x) + 3*Sqrt[a]*(-187*A*b + 3
5*a*B)*Sqrt[x]*(a + b*x)^2 - (384*Sqrt[a]*A*(a + b*x)^3)/Sqrt[x] + (105*(-9*A*b
+ a*B)*(a + b*x)^3*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/Sqrt[b])/(192*a^(11/2)*((a
+ b*x)^2)^(3/2))

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Maple [A]  time = 0.031, size = 374, normalized size = 1.2 \[ -{\frac{bx+a}{192\,{a}^{5}} \left ( 945\,A\sqrt{ab}{x}^{4}{b}^{4}-105\,B\sqrt{ab}{x}^{4}a{b}^{3}+3465\,A\sqrt{ab}{x}^{3}a{b}^{3}+945\,A\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{9/2}{b}^{5}-385\,B\sqrt{ab}{x}^{3}{a}^{2}{b}^{2}-105\,B\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{9/2}a{b}^{4}+3780\,A\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{7/2}a{b}^{4}-420\,B\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{7/2}{a}^{2}{b}^{3}+4599\,A\sqrt{ab}{x}^{2}{a}^{2}{b}^{2}+5670\,A\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{5/2}{a}^{2}{b}^{3}-511\,B\sqrt{ab}{x}^{2}{a}^{3}b-630\,B\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{5/2}{a}^{3}{b}^{2}+3780\,A\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{3/2}{a}^{3}{b}^{2}-420\,B\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{3/2}{a}^{4}b+2511\,A\sqrt{ab}x{a}^{3}b+945\,A\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ) \sqrt{x}{a}^{4}b-279\,B\sqrt{ab}x{a}^{4}-105\,B\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ) \sqrt{x}{a}^{5}+384\,A\sqrt{ab}{a}^{4} \right ){\frac{1}{\sqrt{ab}}}{\frac{1}{\sqrt{x}}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/x^(3/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

-1/192*(945*A*(a*b)^(1/2)*x^4*b^4-105*B*(a*b)^(1/2)*x^4*a*b^3+3465*A*(a*b)^(1/2)
*x^3*a*b^3+945*A*arctan(x^(1/2)*b/(a*b)^(1/2))*x^(9/2)*b^5-385*B*(a*b)^(1/2)*x^3
*a^2*b^2-105*B*arctan(x^(1/2)*b/(a*b)^(1/2))*x^(9/2)*a*b^4+3780*A*arctan(x^(1/2)
*b/(a*b)^(1/2))*x^(7/2)*a*b^4-420*B*arctan(x^(1/2)*b/(a*b)^(1/2))*x^(7/2)*a^2*b^
3+4599*A*(a*b)^(1/2)*x^2*a^2*b^2+5670*A*arctan(x^(1/2)*b/(a*b)^(1/2))*x^(5/2)*a^
2*b^3-511*B*(a*b)^(1/2)*x^2*a^3*b-630*B*arctan(x^(1/2)*b/(a*b)^(1/2))*x^(5/2)*a^
3*b^2+3780*A*arctan(x^(1/2)*b/(a*b)^(1/2))*x^(3/2)*a^3*b^2-420*B*arctan(x^(1/2)*
b/(a*b)^(1/2))*x^(3/2)*a^4*b+2511*A*(a*b)^(1/2)*x*a^3*b+945*A*arctan(x^(1/2)*b/(
a*b)^(1/2))*x^(1/2)*a^4*b-279*B*(a*b)^(1/2)*x*a^4-105*B*arctan(x^(1/2)*b/(a*b)^(
1/2))*x^(1/2)*a^5+384*A*(a*b)^(1/2)*a^4)*(b*x+a)/(a*b)^(1/2)/x^(1/2)/a^5/((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^(3/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.293971, size = 1, normalized size = 0. \[ \left [-\frac{105 \,{\left (B a^{5} - 9 \, A a^{4} b +{\left (B a b^{4} - 9 \, A b^{5}\right )} x^{4} + 4 \,{\left (B a^{2} b^{3} - 9 \, A a b^{4}\right )} x^{3} + 6 \,{\left (B a^{3} b^{2} - 9 \, A a^{2} b^{3}\right )} x^{2} + 4 \,{\left (B a^{4} b - 9 \, A a^{3} b^{2}\right )} x\right )} \sqrt{x} \log \left (-\frac{2 \, a b \sqrt{x} - \sqrt{-a b}{\left (b x - a\right )}}{b x + a}\right ) + 2 \,{\left (384 \, A a^{4} - 105 \,{\left (B a b^{3} - 9 \, A b^{4}\right )} x^{4} - 385 \,{\left (B a^{2} b^{2} - 9 \, A a b^{3}\right )} x^{3} - 511 \,{\left (B a^{3} b - 9 \, A a^{2} b^{2}\right )} x^{2} - 279 \,{\left (B a^{4} - 9 \, A a^{3} b\right )} x\right )} \sqrt{-a b}}{384 \,{\left (a^{5} b^{4} x^{4} + 4 \, a^{6} b^{3} x^{3} + 6 \, a^{7} b^{2} x^{2} + 4 \, a^{8} b x + a^{9}\right )} \sqrt{-a b} \sqrt{x}}, -\frac{105 \,{\left (B a^{5} - 9 \, A a^{4} b +{\left (B a b^{4} - 9 \, A b^{5}\right )} x^{4} + 4 \,{\left (B a^{2} b^{3} - 9 \, A a b^{4}\right )} x^{3} + 6 \,{\left (B a^{3} b^{2} - 9 \, A a^{2} b^{3}\right )} x^{2} + 4 \,{\left (B a^{4} b - 9 \, A a^{3} b^{2}\right )} x\right )} \sqrt{x} \arctan \left (\frac{a}{\sqrt{a b} \sqrt{x}}\right ) +{\left (384 \, A a^{4} - 105 \,{\left (B a b^{3} - 9 \, A b^{4}\right )} x^{4} - 385 \,{\left (B a^{2} b^{2} - 9 \, A a b^{3}\right )} x^{3} - 511 \,{\left (B a^{3} b - 9 \, A a^{2} b^{2}\right )} x^{2} - 279 \,{\left (B a^{4} - 9 \, A a^{3} b\right )} x\right )} \sqrt{a b}}{192 \,{\left (a^{5} b^{4} x^{4} + 4 \, a^{6} b^{3} x^{3} + 6 \, a^{7} b^{2} x^{2} + 4 \, a^{8} b x + a^{9}\right )} \sqrt{a b} \sqrt{x}}\right ] \]

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)^(5/2)*x^(3/2)),x, algorithm="fricas")

[Out]

[-1/384*(105*(B*a^5 - 9*A*a^4*b + (B*a*b^4 - 9*A*b^5)*x^4 + 4*(B*a^2*b^3 - 9*A*a
*b^4)*x^3 + 6*(B*a^3*b^2 - 9*A*a^2*b^3)*x^2 + 4*(B*a^4*b - 9*A*a^3*b^2)*x)*sqrt(
x)*log(-(2*a*b*sqrt(x) - sqrt(-a*b)*(b*x - a))/(b*x + a)) + 2*(384*A*a^4 - 105*(
B*a*b^3 - 9*A*b^4)*x^4 - 385*(B*a^2*b^2 - 9*A*a*b^3)*x^3 - 511*(B*a^3*b - 9*A*a^
2*b^2)*x^2 - 279*(B*a^4 - 9*A*a^3*b)*x)*sqrt(-a*b))/((a^5*b^4*x^4 + 4*a^6*b^3*x^
3 + 6*a^7*b^2*x^2 + 4*a^8*b*x + a^9)*sqrt(-a*b)*sqrt(x)), -1/192*(105*(B*a^5 - 9
*A*a^4*b + (B*a*b^4 - 9*A*b^5)*x^4 + 4*(B*a^2*b^3 - 9*A*a*b^4)*x^3 + 6*(B*a^3*b^
2 - 9*A*a^2*b^3)*x^2 + 4*(B*a^4*b - 9*A*a^3*b^2)*x)*sqrt(x)*arctan(a/(sqrt(a*b)*
sqrt(x))) + (384*A*a^4 - 105*(B*a*b^3 - 9*A*b^4)*x^4 - 385*(B*a^2*b^2 - 9*A*a*b^
3)*x^3 - 511*(B*a^3*b - 9*A*a^2*b^2)*x^2 - 279*(B*a^4 - 9*A*a^3*b)*x)*sqrt(a*b))
/((a^5*b^4*x^4 + 4*a^6*b^3*x^3 + 6*a^7*b^2*x^2 + 4*a^8*b*x + a^9)*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)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.282487, size = 213, normalized size = 0.68 \[ \frac{35 \,{\left (B a - 9 \, A b\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{64 \, \sqrt{a b} a^{5}{\rm sign}\left (b x + a\right )} - \frac{2 \, A}{a^{5} \sqrt{x}{\rm sign}\left (b x + a\right )} + \frac{105 \, B a b^{3} x^{\frac{7}{2}} - 561 \, A b^{4} x^{\frac{7}{2}} + 385 \, B a^{2} b^{2} x^{\frac{5}{2}} - 1929 \, A a b^{3} x^{\frac{5}{2}} + 511 \, B a^{3} b x^{\frac{3}{2}} - 2295 \, A a^{2} b^{2} x^{\frac{3}{2}} + 279 \, B a^{4} \sqrt{x} - 975 \, A a^{3} b \sqrt{x}}{192 \,{\left (b x + a\right )}^{4} a^{5}{\rm sign}\left (b x + a\right )} \]

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^(3/2)),x, algorithm="giac")

[Out]

35/64*(B*a - 9*A*b)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^5*sign(b*x + a)) -
2*A/(a^5*sqrt(x)*sign(b*x + a)) + 1/192*(105*B*a*b^3*x^(7/2) - 561*A*b^4*x^(7/2)
+ 385*B*a^2*b^2*x^(5/2) - 1929*A*a*b^3*x^(5/2) + 511*B*a^3*b*x^(3/2) - 2295*A*a
^2*b^2*x^(3/2) + 279*B*a^4*sqrt(x) - 975*A*a^3*b*sqrt(x))/((b*x + a)^4*a^5*sign(
b*x + a))