3.817 \(\int \frac{A+B x}{x^{9/2} \sqrt{a^2+2 a b x+b^2 x^2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.313391, 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 \[ \frac{2 (a+b x) (A b-a B)}{5 a^2 x^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 A (a+b x)}{7 a x^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 b^{5/2} (a+b x) (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{9/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 b^2 (a+b x) (A b-a B)}{a^4 \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 b (a+b x) (A b-a B)}{3 a^3 x^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/(x^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]),x]

[Out]

(-2*A*(a + b*x))/(7*a*x^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (2*(A*b - a*B)*(a
+ b*x))/(5*a^2*x^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (2*b*(A*b - a*B)*(a + b
*x))/(3*a^3*x^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (2*b^2*(A*b - a*B)*(a + b*x
))/(a^4*Sqrt[x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (2*b^(5/2)*(A*b - a*B)*(a + b*x
)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(a^(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((B*x+A)/x**(9/2)/((b*x+a)**2)**(1/2),x)

[Out]

Exception raised: RecursionError

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Mathematica [A]  time = 0.134965, size = 127, normalized size = 0.53 \[ -\frac{2 (a+b x) \left (\sqrt{a} \left (3 a^3 (5 A+7 B x)-7 a^2 b x (3 A+5 B x)+35 a b^2 x^2 (A+3 B x)-105 A b^3 x^3\right )-105 b^{5/2} x^{7/2} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )\right )}{105 a^{9/2} x^{7/2} \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/(x^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]),x]

[Out]

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

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Maple [A]  time = 0.011, size = 165, normalized size = 0.7 \[{\frac{2\,bx+2\,a}{105\,{a}^{4}} \left ( 105\,A\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{7/2}{b}^{4}-105\,B\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{7/2}a{b}^{3}+105\,A\sqrt{ab}{x}^{3}{b}^{3}-105\,B\sqrt{ab}{x}^{3}a{b}^{2}-35\,A\sqrt{ab}{x}^{2}a{b}^{2}+35\,B\sqrt{ab}{x}^{2}{a}^{2}b+21\,A\sqrt{ab}x{a}^{2}b-21\,B\sqrt{ab}x{a}^{3}-15\,A{a}^{3}\sqrt{ab} \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}{x}^{-{\frac{7}{2}}}{\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/x^(9/2)/((b*x+a)^2)^(1/2),x)

[Out]

2/105*(b*x+a)*(105*A*arctan(x^(1/2)*b/(a*b)^(1/2))*x^(7/2)*b^4-105*B*arctan(x^(1
/2)*b/(a*b)^(1/2))*x^(7/2)*a*b^3+105*A*(a*b)^(1/2)*x^3*b^3-105*B*(a*b)^(1/2)*x^3
*a*b^2-35*A*(a*b)^(1/2)*x^2*a*b^2+35*B*(a*b)^(1/2)*x^2*a^2*b+21*A*(a*b)^(1/2)*x*
a^2*b-21*B*(a*b)^(1/2)*x*a^3-15*A*a^3*(a*b)^(1/2))/((b*x+a)^2)^(1/2)/a^4/x^(7/2)
/(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^(9/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.285718, size = 1, normalized size = 0. \[ \left [-\frac{105 \,{\left (B a b^{2} - A b^{3}\right )} x^{\frac{7}{2}} \sqrt{-\frac{b}{a}} \log \left (\frac{b x + 2 \, a \sqrt{x} \sqrt{-\frac{b}{a}} - a}{b x + a}\right ) + 30 \, A a^{3} + 210 \,{\left (B a b^{2} - A b^{3}\right )} x^{3} - 70 \,{\left (B a^{2} b - A a b^{2}\right )} x^{2} + 42 \,{\left (B a^{3} - A a^{2} b\right )} x}{105 \, a^{4} x^{\frac{7}{2}}}, \frac{2 \,{\left (105 \,{\left (B a b^{2} - A b^{3}\right )} x^{\frac{7}{2}} \sqrt{\frac{b}{a}} \arctan \left (\frac{a \sqrt{\frac{b}{a}}}{b \sqrt{x}}\right ) - 15 \, A a^{3} - 105 \,{\left (B a b^{2} - A b^{3}\right )} x^{3} + 35 \,{\left (B a^{2} b - A a b^{2}\right )} x^{2} - 21 \,{\left (B a^{3} - A a^{2} b\right )} x\right )}}{105 \, a^{4} x^{\frac{7}{2}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/(sqrt((b*x + a)^2)*x^(9/2)),x, algorithm="fricas")

[Out]

[-1/105*(105*(B*a*b^2 - A*b^3)*x^(7/2)*sqrt(-b/a)*log((b*x + 2*a*sqrt(x)*sqrt(-b
/a) - a)/(b*x + a)) + 30*A*a^3 + 210*(B*a*b^2 - A*b^3)*x^3 - 70*(B*a^2*b - A*a*b
^2)*x^2 + 42*(B*a^3 - A*a^2*b)*x)/(a^4*x^(7/2)), 2/105*(105*(B*a*b^2 - A*b^3)*x^
(7/2)*sqrt(b/a)*arctan(a*sqrt(b/a)/(b*sqrt(x))) - 15*A*a^3 - 105*(B*a*b^2 - A*b^
3)*x^3 + 35*(B*a^2*b - A*a*b^2)*x^2 - 21*(B*a^3 - A*a^2*b)*x)/(a^4*x^(7/2))]

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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**(9/2)/((b*x+a)**2)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.276753, size = 213, normalized size = 0.89 \[ -\frac{2 \,{\left (B a b^{3}{\rm sign}\left (b x + a\right ) - A b^{4}{\rm sign}\left (b x + a\right )\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} a^{4}} - \frac{2 \,{\left (105 \, B a b^{2} x^{3}{\rm sign}\left (b x + a\right ) - 105 \, A b^{3} x^{3}{\rm sign}\left (b x + a\right ) - 35 \, B a^{2} b x^{2}{\rm sign}\left (b x + a\right ) + 35 \, A a b^{2} x^{2}{\rm sign}\left (b x + a\right ) + 21 \, B a^{3} x{\rm sign}\left (b x + a\right ) - 21 \, A a^{2} b x{\rm sign}\left (b x + a\right ) + 15 \, A a^{3}{\rm sign}\left (b x + a\right )\right )}}{105 \, a^{4} x^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/(sqrt((b*x + a)^2)*x^(9/2)),x, algorithm="giac")

[Out]

-2*(B*a*b^3*sign(b*x + a) - A*b^4*sign(b*x + a))*arctan(b*sqrt(x)/sqrt(a*b))/(sq
rt(a*b)*a^4) - 2/105*(105*B*a*b^2*x^3*sign(b*x + a) - 105*A*b^3*x^3*sign(b*x + a
) - 35*B*a^2*b*x^2*sign(b*x + a) + 35*A*a*b^2*x^2*sign(b*x + a) + 21*B*a^3*x*sig
n(b*x + a) - 21*A*a^2*b*x*sign(b*x + a) + 15*A*a^3*sign(b*x + a))/(a^4*x^(7/2))