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

Optimal. Leaf size=262 \[ \frac{\sqrt{x} (3 a B+5 A b)}{96 a^2 b^2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\sqrt{x} (3 a B+5 A b)}{24 a b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{x^{3/2} (A b-a B)}{4 a b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(a+b x) (3 a B+5 A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{64 a^{7/2} b^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\sqrt{x} (3 a B+5 A b)}{64 a^3 b^2 \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

((5*A*b + 3*a*B)*Sqrt[x])/(64*a^3*b^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + ((A*b - a
*B)*x^(3/2))/(4*a*b*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - ((5*A*b + 3*a*B
)*Sqrt[x])/(24*a*b^2*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + ((5*A*b + 3*a*
B)*Sqrt[x])/(96*a^2*b^2*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + ((5*A*b + 3*a
*B)*(a + b*x)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(64*a^(7/2)*b^(5/2)*Sqrt[a^2 +
2*a*b*x + b^2*x^2])

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Rubi [A]  time = 0.375458, antiderivative size = 262, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.194 \[ \frac{\sqrt{x} (3 a B+5 A b)}{96 a^2 b^2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\sqrt{x} (3 a B+5 A b)}{24 a b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{x^{3/2} (A b-a B)}{4 a b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(a+b x) (3 a B+5 A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{64 a^{7/2} b^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\sqrt{x} (3 a B+5 A b)}{64 a^3 b^2 \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

((5*A*b + 3*a*B)*Sqrt[x])/(64*a^3*b^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + ((A*b - a
*B)*x^(3/2))/(4*a*b*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - ((5*A*b + 3*a*B
)*Sqrt[x])/(24*a*b^2*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + ((5*A*b + 3*a*
B)*Sqrt[x])/(96*a^2*b^2*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + ((5*A*b + 3*a
*B)*(a + b*x)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(64*a^(7/2)*b^(5/2)*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**(1/2)/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Exception raised: RecursionError

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Mathematica [A]  time = 0.214711, size = 171, normalized size = 0.65 \[ \frac{\frac{3 (a+b x)^3 (3 a B+5 A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{7/2}}+\frac{3 \sqrt{b} \sqrt{x} (a+b x)^2 (3 a B+5 A b)}{a^3}+\frac{2 \sqrt{b} \sqrt{x} (a+b x) (3 a B+5 A b)}{a^2}+\frac{8 \sqrt{b} \sqrt{x} (A b-9 a B)}{a}-\frac{48 \sqrt{b} \sqrt{x} (A b-a B)}{a+b x}}{192 b^{5/2} \left ((a+b x)^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

((8*Sqrt[b]*(A*b - 9*a*B)*Sqrt[x])/a - (48*Sqrt[b]*(A*b - a*B)*Sqrt[x])/(a + b*x
) + (2*Sqrt[b]*(5*A*b + 3*a*B)*Sqrt[x]*(a + b*x))/a^2 + (3*Sqrt[b]*(5*A*b + 3*a*
B)*Sqrt[x]*(a + b*x)^2)/a^3 + (3*(5*A*b + 3*a*B)*(a + b*x)^3*ArcTan[(Sqrt[b]*Sqr
t[x])/Sqrt[a]])/a^(7/2))/(192*b^(5/2)*((a + b*x)^2)^(3/2))

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Maple [A]  time = 0.025, size = 357, normalized size = 1.4 \[{\frac{bx+a}{192\,{a}^{3}{b}^{2}} \left ( 15\,A\sqrt{ab}{x}^{7/2}{b}^{4}+9\,B\sqrt{ab}{x}^{7/2}a{b}^{3}+55\,A\sqrt{ab}{x}^{5/2}a{b}^{3}+15\,A\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{4}{b}^{5}+33\,B\sqrt{ab}{x}^{5/2}{a}^{2}{b}^{2}+9\,B\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{4}a{b}^{4}+60\,A\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{3}a{b}^{4}+36\,B\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{3}{a}^{2}{b}^{3}+73\,A\sqrt{ab}{x}^{3/2}{a}^{2}{b}^{2}+90\,A\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{2}{a}^{2}{b}^{3}-33\,B\sqrt{ab}{x}^{3/2}{a}^{3}b+54\,B\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{2}{a}^{3}{b}^{2}+60\,A\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ) x{a}^{3}{b}^{2}+36\,B\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ) x{a}^{4}b-15\,A\sqrt{ab}\sqrt{x}{a}^{3}b+15\,A\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){a}^{4}b-9\,B\sqrt{ab}\sqrt{x}{a}^{4}+9\,B\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){a}^{5} \right ){\frac{1}{\sqrt{ab}}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*x^(1/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

1/192*(15*A*(a*b)^(1/2)*x^(7/2)*b^4+9*B*(a*b)^(1/2)*x^(7/2)*a*b^3+55*A*(a*b)^(1/
2)*x^(5/2)*a*b^3+15*A*arctan(x^(1/2)*b/(a*b)^(1/2))*x^4*b^5+33*B*(a*b)^(1/2)*x^(
5/2)*a^2*b^2+9*B*arctan(x^(1/2)*b/(a*b)^(1/2))*x^4*a*b^4+60*A*arctan(x^(1/2)*b/(
a*b)^(1/2))*x^3*a*b^4+36*B*arctan(x^(1/2)*b/(a*b)^(1/2))*x^3*a^2*b^3+73*A*(a*b)^
(1/2)*x^(3/2)*a^2*b^2+90*A*arctan(x^(1/2)*b/(a*b)^(1/2))*x^2*a^2*b^3-33*B*(a*b)^
(1/2)*x^(3/2)*a^3*b+54*B*arctan(x^(1/2)*b/(a*b)^(1/2))*x^2*a^3*b^2+60*A*arctan(x
^(1/2)*b/(a*b)^(1/2))*x*a^3*b^2+36*B*arctan(x^(1/2)*b/(a*b)^(1/2))*x*a^4*b-15*A*
(a*b)^(1/2)*x^(1/2)*a^3*b+15*A*arctan(x^(1/2)*b/(a*b)^(1/2))*a^4*b-9*B*(a*b)^(1/
2)*x^(1/2)*a^4+9*B*arctan(x^(1/2)*b/(a*b)^(1/2))*a^5)*(b*x+a)/(a*b)^(1/2)/b^2/a^
3/((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)*sqrt(x)/(b^2*x^2 + 2*a*b*x + a^2)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.294549, size = 1, normalized size = 0. \[ \left [-\frac{2 \,{\left (9 \, B a^{4} + 15 \, A a^{3} b - 3 \,{\left (3 \, B a b^{3} + 5 \, A b^{4}\right )} x^{3} - 11 \,{\left (3 \, B a^{2} b^{2} + 5 \, A a b^{3}\right )} x^{2} +{\left (33 \, B a^{3} b - 73 \, A a^{2} b^{2}\right )} x\right )} \sqrt{-a b} \sqrt{x} - 3 \,{\left (3 \, B a^{5} + 5 \, A a^{4} b +{\left (3 \, B a b^{4} + 5 \, A b^{5}\right )} x^{4} + 4 \,{\left (3 \, B a^{2} b^{3} + 5 \, A a b^{4}\right )} x^{3} + 6 \,{\left (3 \, B a^{3} b^{2} + 5 \, A a^{2} b^{3}\right )} x^{2} + 4 \,{\left (3 \, B a^{4} b + 5 \, A a^{3} b^{2}\right )} x\right )} \log \left (\frac{2 \, a b \sqrt{x} + \sqrt{-a b}{\left (b x - a\right )}}{b x + a}\right )}{384 \,{\left (a^{3} b^{6} x^{4} + 4 \, a^{4} b^{5} x^{3} + 6 \, a^{5} b^{4} x^{2} + 4 \, a^{6} b^{3} x + a^{7} b^{2}\right )} \sqrt{-a b}}, -\frac{{\left (9 \, B a^{4} + 15 \, A a^{3} b - 3 \,{\left (3 \, B a b^{3} + 5 \, A b^{4}\right )} x^{3} - 11 \,{\left (3 \, B a^{2} b^{2} + 5 \, A a b^{3}\right )} x^{2} +{\left (33 \, B a^{3} b - 73 \, A a^{2} b^{2}\right )} x\right )} \sqrt{a b} \sqrt{x} + 3 \,{\left (3 \, B a^{5} + 5 \, A a^{4} b +{\left (3 \, B a b^{4} + 5 \, A b^{5}\right )} x^{4} + 4 \,{\left (3 \, B a^{2} b^{3} + 5 \, A a b^{4}\right )} x^{3} + 6 \,{\left (3 \, B a^{3} b^{2} + 5 \, A a^{2} b^{3}\right )} x^{2} + 4 \,{\left (3 \, B a^{4} b + 5 \, A a^{3} b^{2}\right )} x\right )} \arctan \left (\frac{a}{\sqrt{a b} \sqrt{x}}\right )}{192 \,{\left (a^{3} b^{6} x^{4} + 4 \, a^{4} b^{5} x^{3} + 6 \, a^{5} b^{4} x^{2} + 4 \, a^{6} b^{3} x + a^{7} b^{2}\right )} \sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*sqrt(x)/(b^2*x^2 + 2*a*b*x + a^2)^(5/2),x, algorithm="fricas")

[Out]

[-1/384*(2*(9*B*a^4 + 15*A*a^3*b - 3*(3*B*a*b^3 + 5*A*b^4)*x^3 - 11*(3*B*a^2*b^2
+ 5*A*a*b^3)*x^2 + (33*B*a^3*b - 73*A*a^2*b^2)*x)*sqrt(-a*b)*sqrt(x) - 3*(3*B*a
^5 + 5*A*a^4*b + (3*B*a*b^4 + 5*A*b^5)*x^4 + 4*(3*B*a^2*b^3 + 5*A*a*b^4)*x^3 + 6
*(3*B*a^3*b^2 + 5*A*a^2*b^3)*x^2 + 4*(3*B*a^4*b + 5*A*a^3*b^2)*x)*log((2*a*b*sqr
t(x) + sqrt(-a*b)*(b*x - a))/(b*x + a)))/((a^3*b^6*x^4 + 4*a^4*b^5*x^3 + 6*a^5*b
^4*x^2 + 4*a^6*b^3*x + a^7*b^2)*sqrt(-a*b)), -1/192*((9*B*a^4 + 15*A*a^3*b - 3*(
3*B*a*b^3 + 5*A*b^4)*x^3 - 11*(3*B*a^2*b^2 + 5*A*a*b^3)*x^2 + (33*B*a^3*b - 73*A
*a^2*b^2)*x)*sqrt(a*b)*sqrt(x) + 3*(3*B*a^5 + 5*A*a^4*b + (3*B*a*b^4 + 5*A*b^5)*
x^4 + 4*(3*B*a^2*b^3 + 5*A*a*b^4)*x^3 + 6*(3*B*a^3*b^2 + 5*A*a^2*b^3)*x^2 + 4*(3
*B*a^4*b + 5*A*a^3*b^2)*x)*arctan(a/(sqrt(a*b)*sqrt(x))))/((a^3*b^6*x^4 + 4*a^4*
b^5*x^3 + 6*a^5*b^4*x^2 + 4*a^6*b^3*x + a^7*b^2)*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((B*x+A)*x**(1/2)/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.285915, size = 200, normalized size = 0.76 \[ \frac{{\left (3 \, B a + 5 \, A b\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{64 \, \sqrt{a b} a^{3} b^{2}{\rm sign}\left (b x + a\right )} + \frac{9 \, B a b^{3} x^{\frac{7}{2}} + 15 \, A b^{4} x^{\frac{7}{2}} + 33 \, B a^{2} b^{2} x^{\frac{5}{2}} + 55 \, A a b^{3} x^{\frac{5}{2}} - 33 \, B a^{3} b x^{\frac{3}{2}} + 73 \, A a^{2} b^{2} x^{\frac{3}{2}} - 9 \, B a^{4} \sqrt{x} - 15 \, A a^{3} b \sqrt{x}}{192 \,{\left (b x + a\right )}^{4} a^{3} b^{2}{\rm sign}\left (b x + a\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*sqrt(x)/(b^2*x^2 + 2*a*b*x + a^2)^(5/2),x, algorithm="giac")

[Out]

1/64*(3*B*a + 5*A*b)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^3*b^2*sign(b*x + a
)) + 1/192*(9*B*a*b^3*x^(7/2) + 15*A*b^4*x^(7/2) + 33*B*a^2*b^2*x^(5/2) + 55*A*a
*b^3*x^(5/2) - 33*B*a^3*b*x^(3/2) + 73*A*a^2*b^2*x^(3/2) - 9*B*a^4*sqrt(x) - 15*
A*a^3*b*sqrt(x))/((b*x + a)^4*a^3*b^2*sign(b*x + a))