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

Optimal. Leaf size=258 \[ \frac{x^{7/2} (A b-a B)}{4 a b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{x^{5/2} (7 a B+A b)}{24 a b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{5 \sqrt{x} (7 a B+A b)}{64 a b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{5 x^{3/2} (7 a B+A b)}{96 a b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{5 (a+b x) (7 a B+A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{64 a^{3/2} b^{9/2} \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

(-5*(A*b + 7*a*B)*Sqrt[x])/(64*a*b^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + ((A*b - a*
B)*x^(7/2))/(4*a*b*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - ((A*b + 7*a*B)*x
^(5/2))/(24*a*b^2*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (5*(A*b + 7*a*B)*
x^(3/2))/(96*a*b^3*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (5*(A*b + 7*a*B)*(
a + b*x)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(64*a^(3/2)*b^(9/2)*Sqrt[a^2 + 2*a*b
*x + b^2*x^2])

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Rubi [A]  time = 0.367501, antiderivative size = 258, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161 \[ \frac{x^{7/2} (A b-a B)}{4 a b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{x^{5/2} (7 a B+A b)}{24 a b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{5 \sqrt{x} (7 a B+A b)}{64 a b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{5 x^{3/2} (7 a B+A b)}{96 a b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{5 (a+b x) (7 a B+A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{64 a^{3/2} b^{9/2} \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(-5*(A*b + 7*a*B)*Sqrt[x])/(64*a*b^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + ((A*b - a*
B)*x^(7/2))/(4*a*b*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - ((A*b + 7*a*B)*x
^(5/2))/(24*a*b^2*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (5*(A*b + 7*a*B)*
x^(3/2))/(96*a*b^3*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (5*(A*b + 7*a*B)*(
a + b*x)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(64*a^(3/2)*b^(9/2)*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(x**(5/2)*(B*x+A)/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Exception raised: RecursionError

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Mathematica [A]  time = 0.242297, size = 169, normalized size = 0.66 \[ \frac{\frac{15 (a+b x)^3 (7 a B+A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{3/2}}+\frac{48 a^2 \sqrt{b} \sqrt{x} (a B-A b)}{a+b x}+\frac{3 \sqrt{b} \sqrt{x} (a+b x)^2 (5 A b-93 a B)}{a}-2 \sqrt{b} \sqrt{x} (a+b x) (59 A b-163 a B)+8 a \sqrt{b} \sqrt{x} (17 A b-25 a B)}{192 b^{9/2} \left ((a+b x)^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(8*a*Sqrt[b]*(17*A*b - 25*a*B)*Sqrt[x] + (48*a^2*Sqrt[b]*(-(A*b) + a*B)*Sqrt[x])
/(a + b*x) - 2*Sqrt[b]*(59*A*b - 163*a*B)*Sqrt[x]*(a + b*x) + (3*Sqrt[b]*(5*A*b
- 93*a*B)*Sqrt[x]*(a + b*x)^2)/a + (15*(A*b + 7*a*B)*(a + b*x)^3*ArcTan[(Sqrt[b]
*Sqrt[x])/Sqrt[a]])/a^(3/2))/(192*b^(9/2)*((a + b*x)^2)^(3/2))

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Maple [B]  time = 0.025, size = 357, normalized size = 1.4 \[{\frac{bx+a}{192\,a{b}^{4}} \left ( 15\,A\sqrt{ab}{x}^{7/2}{b}^{4}-279\,B\sqrt{ab}{x}^{7/2}a{b}^{3}-73\,A\sqrt{ab}{x}^{5/2}a{b}^{3}+15\,A\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{4}{b}^{5}-511\,B\sqrt{ab}{x}^{5/2}{a}^{2}{b}^{2}+105\,B\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{4}a{b}^{4}+60\,A\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{3}a{b}^{4}+420\,B\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{3}{a}^{2}{b}^{3}-55\,A\sqrt{ab}{x}^{3/2}{a}^{2}{b}^{2}+90\,A\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{2}{a}^{2}{b}^{3}-385\,B\sqrt{ab}{x}^{3/2}{a}^{3}b+630\,B\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{2}{a}^{3}{b}^{2}+60\,A\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ) x{a}^{3}{b}^{2}+420\,B\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ) x{a}^{4}b-15\,A\sqrt{ab}\sqrt{x}{a}^{3}b+15\,A\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){a}^{4}b-105\,B\sqrt{ab}\sqrt{x}{a}^{4}+105\,B\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){a}^{5} \right ){\frac{1}{\sqrt{ab}}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(5/2)*(B*x+A)/(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^4-279*B*(a*b)^(1/2)*x^(7/2)*a*b^3-73*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^5-511*B*(a*b)^(1/2)*
x^(5/2)*a^2*b^2+105*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+420*B*arctan(x^(1/2)*b/(a*b)^(1/2))*x^3*a^2*b^3-55*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^3-385*B
*(a*b)^(1/2)*x^(3/2)*a^3*b+630*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+420*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-105*
B*(a*b)^(1/2)*x^(1/2)*a^4+105*B*arctan(x^(1/2)*b/(a*b)^(1/2))*a^5)*(b*x+a)/(a*b)
^(1/2)/b^4/a/((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)*x^(5/2)/(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 = 0.291986, size = 1, normalized size = 0. \[ \left [-\frac{2 \,{\left (105 \, B a^{4} + 15 \, A a^{3} b + 3 \,{\left (93 \, B a b^{3} - 5 \, A b^{4}\right )} x^{3} + 73 \,{\left (7 \, B a^{2} b^{2} + A a b^{3}\right )} x^{2} + 55 \,{\left (7 \, B a^{3} b + A a^{2} b^{2}\right )} x\right )} \sqrt{-a b} \sqrt{x} - 15 \,{\left (7 \, B a^{5} + A a^{4} b +{\left (7 \, B a b^{4} + A b^{5}\right )} x^{4} + 4 \,{\left (7 \, B a^{2} b^{3} + A a b^{4}\right )} x^{3} + 6 \,{\left (7 \, B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{2} + 4 \,{\left (7 \, B a^{4} b + A a^{3} b^{2}\right )} x\right )} \log \left (\frac{2 \, a b \sqrt{x} + \sqrt{-a b}{\left (b x - a\right )}}{b x + a}\right )}{384 \,{\left (a b^{8} x^{4} + 4 \, a^{2} b^{7} x^{3} + 6 \, a^{3} b^{6} x^{2} + 4 \, a^{4} b^{5} x + a^{5} b^{4}\right )} \sqrt{-a b}}, -\frac{{\left (105 \, B a^{4} + 15 \, A a^{3} b + 3 \,{\left (93 \, B a b^{3} - 5 \, A b^{4}\right )} x^{3} + 73 \,{\left (7 \, B a^{2} b^{2} + A a b^{3}\right )} x^{2} + 55 \,{\left (7 \, B a^{3} b + A a^{2} b^{2}\right )} x\right )} \sqrt{a b} \sqrt{x} + 15 \,{\left (7 \, B a^{5} + A a^{4} b +{\left (7 \, B a b^{4} + A b^{5}\right )} x^{4} + 4 \,{\left (7 \, B a^{2} b^{3} + A a b^{4}\right )} x^{3} + 6 \,{\left (7 \, B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{2} + 4 \,{\left (7 \, B a^{4} b + A a^{3} b^{2}\right )} x\right )} \arctan \left (\frac{a}{\sqrt{a b} \sqrt{x}}\right )}{192 \,{\left (a b^{8} x^{4} + 4 \, a^{2} b^{7} x^{3} + 6 \, a^{3} b^{6} x^{2} + 4 \, a^{4} b^{5} x + a^{5} b^{4}\right )} \sqrt{a b}}\right ] \]

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, algorithm="fricas")

[Out]

[-1/384*(2*(105*B*a^4 + 15*A*a^3*b + 3*(93*B*a*b^3 - 5*A*b^4)*x^3 + 73*(7*B*a^2*
b^2 + A*a*b^3)*x^2 + 55*(7*B*a^3*b + A*a^2*b^2)*x)*sqrt(-a*b)*sqrt(x) - 15*(7*B*
a^5 + A*a^4*b + (7*B*a*b^4 + A*b^5)*x^4 + 4*(7*B*a^2*b^3 + A*a*b^4)*x^3 + 6*(7*B
*a^3*b^2 + A*a^2*b^3)*x^2 + 4*(7*B*a^4*b + A*a^3*b^2)*x)*log((2*a*b*sqrt(x) + sq
rt(-a*b)*(b*x - a))/(b*x + a)))/((a*b^8*x^4 + 4*a^2*b^7*x^3 + 6*a^3*b^6*x^2 + 4*
a^4*b^5*x + a^5*b^4)*sqrt(-a*b)), -1/192*((105*B*a^4 + 15*A*a^3*b + 3*(93*B*a*b^
3 - 5*A*b^4)*x^3 + 73*(7*B*a^2*b^2 + A*a*b^3)*x^2 + 55*(7*B*a^3*b + A*a^2*b^2)*x
)*sqrt(a*b)*sqrt(x) + 15*(7*B*a^5 + A*a^4*b + (7*B*a*b^4 + A*b^5)*x^4 + 4*(7*B*a
^2*b^3 + A*a*b^4)*x^3 + 6*(7*B*a^3*b^2 + A*a^2*b^3)*x^2 + 4*(7*B*a^4*b + A*a^3*b
^2)*x)*arctan(a/(sqrt(a*b)*sqrt(x))))/((a*b^8*x^4 + 4*a^2*b^7*x^3 + 6*a^3*b^6*x^
2 + 4*a^4*b^5*x + a^5*b^4)*sqrt(a*b))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{\frac{5}{2}} \left (A + B x\right )}{\left (\left (a + b x\right )^{2}\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x**(5/2)*(A + B*x)/((a + b*x)**2)**(5/2), x)

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GIAC/XCAS [A]  time = 0.278718, size = 198, normalized size = 0.77 \[ \frac{5 \,{\left (7 \, B a + A b\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{64 \, \sqrt{a b} a b^{4}{\rm sign}\left (b x + a\right )} - \frac{279 \, B a b^{3} x^{\frac{7}{2}} - 15 \, A b^{4} x^{\frac{7}{2}} + 511 \, B a^{2} b^{2} x^{\frac{5}{2}} + 73 \, A a b^{3} x^{\frac{5}{2}} + 385 \, B a^{3} b x^{\frac{3}{2}} + 55 \, A a^{2} b^{2} x^{\frac{3}{2}} + 105 \, B a^{4} \sqrt{x} + 15 \, A a^{3} b \sqrt{x}}{192 \,{\left (b x + a\right )}^{4} a b^{4}{\rm sign}\left (b x + a\right )} \]

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, algorithm="giac")

[Out]

5/64*(7*B*a + A*b)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a*b^4*sign(b*x + a)) -
1/192*(279*B*a*b^3*x^(7/2) - 15*A*b^4*x^(7/2) + 511*B*a^2*b^2*x^(5/2) + 73*A*a*
b^3*x^(5/2) + 385*B*a^3*b*x^(3/2) + 55*A*a^2*b^2*x^(3/2) + 105*B*a^4*sqrt(x) + 1
5*A*a^3*b*sqrt(x))/((b*x + a)^4*a*b^4*sign(b*x + a))