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

Optimal. Leaf size=84 \[ -\frac{4 b (a+b x)^{3/2} (4 A b-7 a B)}{105 a^3 x^{3/2}}+\frac{2 (a+b x)^{3/2} (4 A b-7 a B)}{35 a^2 x^{5/2}}-\frac{2 A (a+b x)^{3/2}}{7 a x^{7/2}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.101499, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ -\frac{4 b (a+b x)^{3/2} (4 A b-7 a B)}{105 a^3 x^{3/2}}+\frac{2 (a+b x)^{3/2} (4 A b-7 a B)}{35 a^2 x^{5/2}}-\frac{2 A (a+b x)^{3/2}}{7 a x^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 7.74441, size = 82, normalized size = 0.98 \[ - \frac{2 A \left (a + b x\right )^{\frac{3}{2}}}{7 a x^{\frac{7}{2}}} + \frac{2 \left (a + b x\right )^{\frac{3}{2}} \left (4 A b - 7 B a\right )}{35 a^{2} x^{\frac{5}{2}}} - \frac{4 b \left (a + b x\right )^{\frac{3}{2}} \left (4 A b - 7 B a\right )}{105 a^{3} x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*A*(a + b*x)**(3/2)/(7*a*x**(7/2)) + 2*(a + b*x)**(3/2)*(4*A*b - 7*B*a)/(35*a*
*2*x**(5/2)) - 4*b*(a + b*x)**(3/2)*(4*A*b - 7*B*a)/(105*a**3*x**(3/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.064897, size = 57, normalized size = 0.68 \[ -\frac{2 (a+b x)^{3/2} \left (3 a^2 (5 A+7 B x)-2 a b x (6 A+7 B x)+8 A b^2 x^2\right )}{105 a^3 x^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(a + b*x)^(3/2)*(8*A*b^2*x^2 + 3*a^2*(5*A + 7*B*x) - 2*a*b*x*(6*A + 7*B*x)))
/(105*a^3*x^(7/2))

_______________________________________________________________________________________

Maple [A]  time = 0.009, size = 53, normalized size = 0.6 \[ -{\frac{16\,A{b}^{2}{x}^{2}-28\,B{x}^{2}ab-24\,aAbx+42\,{a}^{2}Bx+30\,A{a}^{2}}{105\,{a}^{3}} \left ( bx+a \right ) ^{{\frac{3}{2}}}{x}^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/105*(b*x+a)^(3/2)*(8*A*b^2*x^2-14*B*a*b*x^2-12*A*a*b*x+21*B*a^2*x+15*A*a^2)/x
^(7/2)/a^3

_______________________________________________________________________________________

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)/x^(9/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.228814, size = 103, normalized size = 1.23 \[ -\frac{2 \,{\left (15 \, A a^{3} - 2 \,{\left (7 \, B a b^{2} - 4 \, A b^{3}\right )} x^{3} +{\left (7 \, B a^{2} b - 4 \, A a b^{2}\right )} x^{2} + 3 \,{\left (7 \, B a^{3} + A a^{2} b\right )} x\right )} \sqrt{b x + a}}{105 \, a^{3} x^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/105*(15*A*a^3 - 2*(7*B*a*b^2 - 4*A*b^3)*x^3 + (7*B*a^2*b - 4*A*a*b^2)*x^2 + 3
*(7*B*a^3 + A*a^2*b)*x)*sqrt(b*x + a)/(a^3*x^(7/2))

_______________________________________________________________________________________

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)**(1/2)/x**(9/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.224343, size = 154, normalized size = 1.83 \[ -\frac{{\left (b x + a\right )}^{\frac{3}{2}}{\left ({\left (b x + a\right )}{\left (\frac{2 \,{\left (7 \, B a b^{6} - 4 \, A b^{7}\right )}{\left (b x + a\right )}}{a^{4} b^{12}} - \frac{7 \,{\left (7 \, B a^{2} b^{6} - 4 \, A a b^{7}\right )}}{a^{4} b^{12}}\right )} + \frac{35 \,{\left (B a^{3} b^{6} - A a^{2} b^{7}\right )}}{a^{4} b^{12}}\right )} b}{80640 \,{\left ({\left (b x + a\right )} b - a b\right )}^{\frac{7}{2}}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/80640*(b*x + a)^(3/2)*((b*x + a)*(2*(7*B*a*b^6 - 4*A*b^7)*(b*x + a)/(a^4*b^12
) - 7*(7*B*a^2*b^6 - 4*A*a*b^7)/(a^4*b^12)) + 35*(B*a^3*b^6 - A*a^2*b^7)/(a^4*b^
12))*b/(((b*x + a)*b - a*b)^(7/2)*abs(b))