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

Optimal. Leaf size=120 \[ -\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{5 x^{5/2} (a+b x)}-\frac{2 a A \sqrt{a^2+2 a b x+b^2 x^2}}{7 x^{7/2} (a+b x)}-\frac{2 b B \sqrt{a^2+2 a b x+b^2 x^2}}{3 x^{3/2} (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.155134, antiderivative size = 120, 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 \[ -\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{5 x^{5/2} (a+b x)}-\frac{2 a A \sqrt{a^2+2 a b x+b^2 x^2}}{7 x^{7/2} (a+b x)}-\frac{2 b B \sqrt{a^2+2 a b x+b^2 x^2}}{3 x^{3/2} (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 18.6352, size = 122, normalized size = 1.02 \[ - \frac{A \left (2 a + 2 b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{7 a x^{\frac{7}{2}}} - \frac{\left (\frac{4 A b}{35} - \frac{4 B a}{15}\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{x^{\frac{5}{2}} \left (a + b x\right )} + \frac{2 \left (3 A b - 7 B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{21 a x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A*(2*a + 2*b*x)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(7*a*x**(7/2)) - (4*A*b/35 - 4
*B*a/15)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(x**(5/2)*(a + b*x)) + 2*(3*A*b - 7*B*
a)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(21*a*x**(5/2))

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Mathematica [A]  time = 0.0327355, size = 51, normalized size = 0.42 \[ -\frac{2 \sqrt{(a+b x)^2} (3 a (5 A+7 B x)+7 b x (3 A+5 B x))}{105 x^{7/2} (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.008, size = 44, normalized size = 0.4 \[ -{\frac{70\,Bb{x}^{2}+42\,Abx+42\,aBx+30\,aA}{105\,bx+105\,a}\sqrt{ \left ( bx+a \right ) ^{2}}{x}^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.707051, size = 47, normalized size = 0.39 \[ -\frac{2 \,{\left (5 \, b x^{2} + 3 \, a x\right )} B}{15 \, x^{\frac{7}{2}}} - \frac{2 \,{\left (7 \, b x^{2} + 5 \, a x\right )} A}{35 \, x^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/15*(5*b*x^2 + 3*a*x)*B/x^(7/2) - 2/35*(7*b*x^2 + 5*a*x)*A/x^(9/2)

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Fricas [A]  time = 0.319477, size = 36, normalized size = 0.3 \[ -\frac{2 \,{\left (35 \, B b x^{2} + 15 \, A a + 21 \,{\left (B a + A b\right )} x\right )}}{105 \, x^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/105*(35*B*b*x^2 + 15*A*a + 21*(B*a + A*b)*x)/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)*((b*x+a)**2)**(1/2)/x**(9/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.272274, size = 69, normalized size = 0.57 \[ -\frac{2 \,{\left (35 \, B b x^{2}{\rm sign}\left (b x + a\right ) + 21 \, B a x{\rm sign}\left (b x + a\right ) + 21 \, A b x{\rm sign}\left (b x + a\right ) + 15 \, A a{\rm sign}\left (b x + a\right )\right )}}{105 \, x^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/105*(35*B*b*x^2*sign(b*x + a) + 21*B*a*x*sign(b*x + a) + 21*A*b*x*sign(b*x +
a) + 15*A*a*sign(b*x + a))/x^(7/2)