3.816 \(\int \frac{A+B x}{x^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}} \, dx\)

Optimal. Leaf size=190 \[ \frac{2 (a+b x) (A b-a B)}{3 a^2 x^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 A (a+b x)}{5 a x^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 b^{3/2} (a+b x) (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 b (a+b x) (A b-a B)}{a^3 \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.268269, antiderivative size = 190, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161 \[ \frac{2 (a+b x) (A b-a B)}{3 a^2 x^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 A (a+b x)}{5 a x^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 b^{3/2} (a+b x) (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 b (a+b x) (A b-a B)}{a^3 \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/(x^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]),x]

[Out]

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

[Out]

Exception raised: RecursionError

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Mathematica [A]  time = 0.101276, size = 107, normalized size = 0.56 \[ -\frac{2 (a+b x) \left (\sqrt{a} \left (a^2 (3 A+5 B x)-5 a b x (A+3 B x)+15 A b^2 x^2\right )+15 b^{3/2} x^{5/2} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )\right )}{15 a^{7/2} x^{5/2} \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/(x^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]),x]

[Out]

(-2*(a + b*x)*(Sqrt[a]*(15*A*b^2*x^2 - 5*a*b*x*(A + 3*B*x) + a^2*(3*A + 5*B*x))
+ 15*b^(3/2)*(A*b - a*B)*x^(5/2)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]]))/(15*a^(7/2)
*x^(5/2)*Sqrt[(a + b*x)^2])

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Maple [A]  time = 0.011, size = 131, normalized size = 0.7 \[ -{\frac{2\,bx+2\,a}{15\,{a}^{3}} \left ( 15\,A\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{5/2}{b}^{3}-15\,B\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{5/2}a{b}^{2}+15\,A\sqrt{ab}{x}^{2}{b}^{2}-15\,B\sqrt{ab}{x}^{2}ab-5\,A\sqrt{ab}xab+5\,B\sqrt{ab}x{a}^{2}+3\,A{a}^{2}\sqrt{ab} \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}{x}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/x^(7/2)/((b*x+a)^2)^(1/2),x)

[Out]

-2/15*(b*x+a)*(15*A*arctan(x^(1/2)*b/(a*b)^(1/2))*x^(5/2)*b^3-15*B*arctan(x^(1/2
)*b/(a*b)^(1/2))*x^(5/2)*a*b^2+15*A*(a*b)^(1/2)*x^2*b^2-15*B*(a*b)^(1/2)*x^2*a*b
-5*A*(a*b)^(1/2)*x*a*b+5*B*(a*b)^(1/2)*x*a^2+3*A*a^2*(a*b)^(1/2))/((b*x+a)^2)^(1
/2)/a^3/x^(5/2)/(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)/(sqrt((b*x + a)^2)*x^(7/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.293318, size = 1, normalized size = 0.01 \[ \left [-\frac{15 \,{\left (B a b - A b^{2}\right )} x^{\frac{5}{2}} \sqrt{-\frac{b}{a}} \log \left (\frac{b x - 2 \, a \sqrt{x} \sqrt{-\frac{b}{a}} - a}{b x + a}\right ) + 6 \, A a^{2} - 30 \,{\left (B a b - A b^{2}\right )} x^{2} + 10 \,{\left (B a^{2} - A a b\right )} x}{15 \, a^{3} x^{\frac{5}{2}}}, -\frac{2 \,{\left (15 \,{\left (B a b - A b^{2}\right )} x^{\frac{5}{2}} \sqrt{\frac{b}{a}} \arctan \left (\frac{a \sqrt{\frac{b}{a}}}{b \sqrt{x}}\right ) + 3 \, A a^{2} - 15 \,{\left (B a b - A b^{2}\right )} x^{2} + 5 \,{\left (B a^{2} - A a b\right )} x\right )}}{15 \, a^{3} x^{\frac{5}{2}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/(sqrt((b*x + a)^2)*x^(7/2)),x, algorithm="fricas")

[Out]

[-1/15*(15*(B*a*b - A*b^2)*x^(5/2)*sqrt(-b/a)*log((b*x - 2*a*sqrt(x)*sqrt(-b/a)
- a)/(b*x + a)) + 6*A*a^2 - 30*(B*a*b - A*b^2)*x^2 + 10*(B*a^2 - A*a*b)*x)/(a^3*
x^(5/2)), -2/15*(15*(B*a*b - A*b^2)*x^(5/2)*sqrt(b/a)*arctan(a*sqrt(b/a)/(b*sqrt
(x))) + 3*A*a^2 - 15*(B*a*b - A*b^2)*x^2 + 5*(B*a^2 - A*a*b)*x)/(a^3*x^(5/2))]

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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**(7/2)/((b*x+a)**2)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.281713, size = 165, normalized size = 0.87 \[ \frac{2 \,{\left (B a b^{2}{\rm sign}\left (b x + a\right ) - A b^{3}{\rm sign}\left (b x + a\right )\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} a^{3}} + \frac{2 \,{\left (15 \, B a b x^{2}{\rm sign}\left (b x + a\right ) - 15 \, A b^{2} x^{2}{\rm sign}\left (b x + a\right ) - 5 \, B a^{2} x{\rm sign}\left (b x + a\right ) + 5 \, A a b x{\rm sign}\left (b x + a\right ) - 3 \, A a^{2}{\rm sign}\left (b x + a\right )\right )}}{15 \, a^{3} x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/(sqrt((b*x + a)^2)*x^(7/2)),x, algorithm="giac")

[Out]

2*(B*a*b^2*sign(b*x + a) - A*b^3*sign(b*x + a))*arctan(b*sqrt(x)/sqrt(a*b))/(sqr
t(a*b)*a^3) + 2/15*(15*B*a*b*x^2*sign(b*x + a) - 15*A*b^2*x^2*sign(b*x + a) - 5*
B*a^2*x*sign(b*x + a) + 5*A*a*b*x*sign(b*x + a) - 3*A*a^2*sign(b*x + a))/(a^3*x^
(5/2))