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

Optimal. Leaf size=118 \[ -\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{3 x^{3/2} (a+b x)}-\frac{2 a A \sqrt{a^2+2 a b x+b^2 x^2}}{5 x^{5/2} (a+b x)}-\frac{2 b B \sqrt{a^2+2 a b x+b^2 x^2}}{\sqrt{x} (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.153308, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065 \[ -\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{3 x^{3/2} (a+b x)}-\frac{2 a A \sqrt{a^2+2 a b x+b^2 x^2}}{5 x^{5/2} (a+b x)}-\frac{2 b B \sqrt{a^2+2 a b x+b^2 x^2}}{\sqrt{x} (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 18.6949, size = 121, normalized size = 1.03 \[ - \frac{A \left (2 a + 2 b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{5 a x^{\frac{5}{2}}} - \frac{\left (\frac{4 A b}{15} - \frac{4 B a}{3}\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{x^{\frac{3}{2}} \left (a + b x\right )} + \frac{2 \left (A b - 5 B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{5 a x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A*(2*a + 2*b*x)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(5*a*x**(5/2)) - (4*A*b/15 - 4
*B*a/3)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(x**(3/2)*(a + b*x)) + 2*(A*b - 5*B*a)*
sqrt(a**2 + 2*a*b*x + b**2*x**2)/(5*a*x**(3/2))

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Mathematica [A]  time = 0.0335073, size = 48, normalized size = 0.41 \[ -\frac{2 \sqrt{(a+b x)^2} (a (3 A+5 B x)+5 b x (A+3 B x))}{15 x^{5/2} (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.006, size = 44, normalized size = 0.4 \[ -{\frac{30\,Bb{x}^{2}+10\,Abx+10\,aBx+6\,aA}{15\,bx+15\,a}\sqrt{ \left ( bx+a \right ) ^{2}}{x}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/15*(15*B*b*x^2+5*A*b*x+5*B*a*x+3*A*a)*((b*x+a)^2)^(1/2)/x^(5/2)/(b*x+a)

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Maxima [A]  time = 0.709248, size = 46, normalized size = 0.39 \[ -\frac{2 \,{\left (3 \, b x^{2} + a x\right )} B}{3 \, x^{\frac{5}{2}}} - \frac{2 \,{\left (5 \, b x^{2} + 3 \, a x\right )} A}{15 \, x^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.319956, size = 36, normalized size = 0.31 \[ -\frac{2 \,{\left (15 \, B b x^{2} + 3 \, A a + 5 \,{\left (B a + A b\right )} x\right )}}{15 \, x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/15*(15*B*b*x^2 + 3*A*a + 5*(B*a + A*b)*x)/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)*((b*x+a)**2)**(1/2)/x**(7/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.273476, size = 69, normalized size = 0.58 \[ -\frac{2 \,{\left (15 \, B b x^{2}{\rm sign}\left (b x + a\right ) + 5 \, B a x{\rm sign}\left (b x + a\right ) + 5 \, A b x{\rm sign}\left (b x + a\right ) + 3 \, A a{\rm sign}\left (b x + a\right )\right )}}{15 \, x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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