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

Optimal. Leaf size=238 \[ \frac{2 x^{5/2} (a+b x) (A b-a B)}{5 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 a^2 \sqrt{x} (a+b x) (A b-a B)}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 a x^{3/2} (a+b x) (A b-a B)}{3 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 B x^{7/2} (a+b x)}{7 b \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 a^{5/2} (a+b x) (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{9/2} \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.310522, antiderivative size = 238, 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 \[ \frac{2 x^{5/2} (a+b x) (A b-a B)}{5 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 a^2 \sqrt{x} (a+b x) (A b-a B)}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 a x^{3/2} (a+b x) (A b-a B)}{3 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 B x^{7/2} (a+b x)}{7 b \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 a^{5/2} (a+b x) (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{9/2} \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

[Out]

Exception raised: RecursionError

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Mathematica [A]  time = 0.139106, size = 120, normalized size = 0.5 \[ \frac{2 (a+b x) \left (105 a^{5/2} (a B-A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )+\sqrt{b} \sqrt{x} \left (-105 a^3 B+35 a^2 b (3 A+B x)-7 a b^2 x (5 A+3 B x)+3 b^3 x^2 (7 A+5 B x)\right )\right )}{105 b^{9/2} \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(a + b*x)*(Sqrt[b]*Sqrt[x]*(-105*a^3*B + 35*a^2*b*(3*A + B*x) - 7*a*b^2*x*(5*
A + 3*B*x) + 3*b^3*x^2*(7*A + 5*B*x)) + 105*a^(5/2)*(-(A*b) + a*B)*ArcTan[(Sqrt[
b]*Sqrt[x])/Sqrt[a]]))/(105*b^(9/2)*Sqrt[(a + b*x)^2])

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Maple [A]  time = 0.01, size = 163, normalized size = 0.7 \[{\frac{2\,bx+2\,a}{105\,{b}^{4}} \left ( 15\,B\sqrt{ab}{x}^{7/2}{b}^{3}+21\,A\sqrt{ab}{x}^{5/2}{b}^{3}-21\,B\sqrt{ab}{x}^{5/2}a{b}^{2}-35\,A\sqrt{ab}{x}^{3/2}a{b}^{2}+35\,B\sqrt{ab}{x}^{3/2}{a}^{2}b+105\,A\sqrt{ab}\sqrt{x}{a}^{2}b-105\,A\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){a}^{3}b-105\,B\sqrt{ab}\sqrt{x}{a}^{3}+105\,B\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){a}^{4} \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}{\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/105*(b*x+a)*(15*B*(a*b)^(1/2)*x^(7/2)*b^3+21*A*(a*b)^(1/2)*x^(5/2)*b^3-21*B*(a
*b)^(1/2)*x^(5/2)*a*b^2-35*A*(a*b)^(1/2)*x^(3/2)*a*b^2+35*B*(a*b)^(1/2)*x^(3/2)*
a^2*b+105*A*(a*b)^(1/2)*x^(1/2)*a^2*b-105*A*arctan(x^(1/2)*b/(a*b)^(1/2))*a^3*b-
105*B*(a*b)^(1/2)*x^(1/2)*a^3+105*B*arctan(x^(1/2)*b/(a*b)^(1/2))*a^4)/((b*x+a)^
2)^(1/2)/b^4/(a*b)^(1/2)

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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)/sqrt((b*x + a)^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.282254, size = 1, normalized size = 0. \[ \left [-\frac{105 \,{\left (B a^{3} - A a^{2} b\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x - 2 \, b \sqrt{x} \sqrt{-\frac{a}{b}} - a}{b x + a}\right ) - 2 \,{\left (15 \, B b^{3} x^{3} - 105 \, B a^{3} + 105 \, A a^{2} b - 21 \,{\left (B a b^{2} - A b^{3}\right )} x^{2} + 35 \,{\left (B a^{2} b - A a b^{2}\right )} x\right )} \sqrt{x}}{105 \, b^{4}}, \frac{2 \,{\left (105 \,{\left (B a^{3} - A a^{2} b\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{\sqrt{x}}{\sqrt{\frac{a}{b}}}\right ) +{\left (15 \, B b^{3} x^{3} - 105 \, B a^{3} + 105 \, A a^{2} b - 21 \,{\left (B a b^{2} - A b^{3}\right )} x^{2} + 35 \,{\left (B a^{2} b - A a b^{2}\right )} x\right )} \sqrt{x}\right )}}{105 \, b^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/105*(105*(B*a^3 - A*a^2*b)*sqrt(-a/b)*log((b*x - 2*b*sqrt(x)*sqrt(-a/b) - a)
/(b*x + a)) - 2*(15*B*b^3*x^3 - 105*B*a^3 + 105*A*a^2*b - 21*(B*a*b^2 - A*b^3)*x
^2 + 35*(B*a^2*b - A*a*b^2)*x)*sqrt(x))/b^4, 2/105*(105*(B*a^3 - A*a^2*b)*sqrt(a
/b)*arctan(sqrt(x)/sqrt(a/b)) + (15*B*b^3*x^3 - 105*B*a^3 + 105*A*a^2*b - 21*(B*
a*b^2 - A*b^3)*x^2 + 35*(B*a^2*b - A*a*b^2)*x)*sqrt(x))/b^4]

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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(x**(5/2)*(B*x+A)/((b*x+a)**2)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.277153, size = 228, normalized size = 0.96 \[ \frac{2 \,{\left (B a^{4}{\rm sign}\left (b x + a\right ) - A a^{3} b{\rm sign}\left (b x + a\right )\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} b^{4}} + \frac{2 \,{\left (15 \, B b^{6} x^{\frac{7}{2}}{\rm sign}\left (b x + a\right ) - 21 \, B a b^{5} x^{\frac{5}{2}}{\rm sign}\left (b x + a\right ) + 21 \, A b^{6} x^{\frac{5}{2}}{\rm sign}\left (b x + a\right ) + 35 \, B a^{2} b^{4} x^{\frac{3}{2}}{\rm sign}\left (b x + a\right ) - 35 \, A a b^{5} x^{\frac{3}{2}}{\rm sign}\left (b x + a\right ) - 105 \, B a^{3} b^{3} \sqrt{x}{\rm sign}\left (b x + a\right ) + 105 \, A a^{2} b^{4} \sqrt{x}{\rm sign}\left (b x + a\right )\right )}}{105 \, b^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(B*a^4*sign(b*x + a) - A*a^3*b*sign(b*x + a))*arctan(b*sqrt(x)/sqrt(a*b))/(sqr
t(a*b)*b^4) + 2/105*(15*B*b^6*x^(7/2)*sign(b*x + a) - 21*B*a*b^5*x^(5/2)*sign(b*
x + a) + 21*A*b^6*x^(5/2)*sign(b*x + a) + 35*B*a^2*b^4*x^(3/2)*sign(b*x + a) - 3
5*A*a*b^5*x^(3/2)*sign(b*x + a) - 105*B*a^3*b^3*sqrt(x)*sign(b*x + a) + 105*A*a^
2*b^4*sqrt(x)*sign(b*x + a))/b^7