3.511 \(\int \frac{A+B x}{x^{11/2} \sqrt{a+b x}} \, dx\)

Optimal. Leaf size=150 \[ -\frac{32 b^3 \sqrt{a+b x} (8 A b-9 a B)}{315 a^5 \sqrt{x}}+\frac{16 b^2 \sqrt{a+b x} (8 A b-9 a B)}{315 a^4 x^{3/2}}-\frac{4 b \sqrt{a+b x} (8 A b-9 a B)}{105 a^3 x^{5/2}}+\frac{2 \sqrt{a+b x} (8 A b-9 a B)}{63 a^2 x^{7/2}}-\frac{2 A \sqrt{a+b x}}{9 a x^{9/2}} \]

[Out]

(-2*A*Sqrt[a + b*x])/(9*a*x^(9/2)) + (2*(8*A*b - 9*a*B)*Sqrt[a + b*x])/(63*a^2*x
^(7/2)) - (4*b*(8*A*b - 9*a*B)*Sqrt[a + b*x])/(105*a^3*x^(5/2)) + (16*b^2*(8*A*b
- 9*a*B)*Sqrt[a + b*x])/(315*a^4*x^(3/2)) - (32*b^3*(8*A*b - 9*a*B)*Sqrt[a + b*
x])/(315*a^5*Sqrt[x])

_______________________________________________________________________________________

Rubi [A]  time = 0.178147, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ -\frac{32 b^3 \sqrt{a+b x} (8 A b-9 a B)}{315 a^5 \sqrt{x}}+\frac{16 b^2 \sqrt{a+b x} (8 A b-9 a B)}{315 a^4 x^{3/2}}-\frac{4 b \sqrt{a+b x} (8 A b-9 a B)}{105 a^3 x^{5/2}}+\frac{2 \sqrt{a+b x} (8 A b-9 a B)}{63 a^2 x^{7/2}}-\frac{2 A \sqrt{a+b x}}{9 a x^{9/2}} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/(x^(11/2)*Sqrt[a + b*x]),x]

[Out]

(-2*A*Sqrt[a + b*x])/(9*a*x^(9/2)) + (2*(8*A*b - 9*a*B)*Sqrt[a + b*x])/(63*a^2*x
^(7/2)) - (4*b*(8*A*b - 9*a*B)*Sqrt[a + b*x])/(105*a^3*x^(5/2)) + (16*b^2*(8*A*b
- 9*a*B)*Sqrt[a + b*x])/(315*a^4*x^(3/2)) - (32*b^3*(8*A*b - 9*a*B)*Sqrt[a + b*
x])/(315*a^5*Sqrt[x])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 15.0654, size = 150, normalized size = 1. \[ - \frac{2 A \sqrt{a + b x}}{9 a x^{\frac{9}{2}}} + \frac{2 \sqrt{a + b x} \left (8 A b - 9 B a\right )}{63 a^{2} x^{\frac{7}{2}}} - \frac{4 b \sqrt{a + b x} \left (8 A b - 9 B a\right )}{105 a^{3} x^{\frac{5}{2}}} + \frac{16 b^{2} \sqrt{a + b x} \left (8 A b - 9 B a\right )}{315 a^{4} x^{\frac{3}{2}}} - \frac{32 b^{3} \sqrt{a + b x} \left (8 A b - 9 B a\right )}{315 a^{5} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/x**(11/2)/(b*x+a)**(1/2),x)

[Out]

-2*A*sqrt(a + b*x)/(9*a*x**(9/2)) + 2*sqrt(a + b*x)*(8*A*b - 9*B*a)/(63*a**2*x**
(7/2)) - 4*b*sqrt(a + b*x)*(8*A*b - 9*B*a)/(105*a**3*x**(5/2)) + 16*b**2*sqrt(a
+ b*x)*(8*A*b - 9*B*a)/(315*a**4*x**(3/2)) - 32*b**3*sqrt(a + b*x)*(8*A*b - 9*B*
a)/(315*a**5*sqrt(x))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0920892, size = 95, normalized size = 0.63 \[ -\frac{2 \sqrt{a+b x} \left (5 a^4 (7 A+9 B x)-2 a^3 b x (20 A+27 B x)+24 a^2 b^2 x^2 (2 A+3 B x)-16 a b^3 x^3 (4 A+9 B x)+128 A b^4 x^4\right )}{315 a^5 x^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/(x^(11/2)*Sqrt[a + b*x]),x]

[Out]

(-2*Sqrt[a + b*x]*(128*A*b^4*x^4 + 24*a^2*b^2*x^2*(2*A + 3*B*x) - 16*a*b^3*x^3*(
4*A + 9*B*x) + 5*a^4*(7*A + 9*B*x) - 2*a^3*b*x*(20*A + 27*B*x)))/(315*a^5*x^(9/2
))

_______________________________________________________________________________________

Maple [A]  time = 0.009, size = 101, normalized size = 0.7 \[ -{\frac{256\,A{b}^{4}{x}^{4}-288\,Ba{b}^{3}{x}^{4}-128\,Aa{b}^{3}{x}^{3}+144\,B{a}^{2}{b}^{2}{x}^{3}+96\,A{a}^{2}{b}^{2}{x}^{2}-108\,B{a}^{3}b{x}^{2}-80\,A{a}^{3}bx+90\,B{a}^{4}x+70\,A{a}^{4}}{315\,{a}^{5}}\sqrt{bx+a}{x}^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/x^(11/2)/(b*x+a)^(1/2),x)

[Out]

-2/315*(b*x+a)^(1/2)*(128*A*b^4*x^4-144*B*a*b^3*x^4-64*A*a*b^3*x^3+72*B*a^2*b^2*
x^3+48*A*a^2*b^2*x^2-54*B*a^3*b*x^2-40*A*a^3*b*x+45*B*a^4*x+35*A*a^4)/x^(9/2)/a^
5

_______________________________________________________________________________________

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^(11/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.223064, size = 138, normalized size = 0.92 \[ -\frac{2 \,{\left (35 \, A a^{4} - 16 \,{\left (9 \, B a b^{3} - 8 \, A b^{4}\right )} x^{4} + 8 \,{\left (9 \, B a^{2} b^{2} - 8 \, A a b^{3}\right )} x^{3} - 6 \,{\left (9 \, B a^{3} b - 8 \, A a^{2} b^{2}\right )} x^{2} + 5 \,{\left (9 \, B a^{4} - 8 \, A a^{3} b\right )} x\right )} \sqrt{b x + a}}{315 \, a^{5} x^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/(sqrt(b*x + a)*x^(11/2)),x, algorithm="fricas")

[Out]

-2/315*(35*A*a^4 - 16*(9*B*a*b^3 - 8*A*b^4)*x^4 + 8*(9*B*a^2*b^2 - 8*A*a*b^3)*x^
3 - 6*(9*B*a^3*b - 8*A*a^2*b^2)*x^2 + 5*(9*B*a^4 - 8*A*a^3*b)*x)*sqrt(b*x + a)/(
a^5*x^(9/2))

_______________________________________________________________________________________

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)/x**(11/2)/(b*x+a)**(1/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.223583, size = 248, normalized size = 1.65 \[ -\frac{{\left ({\left (2 \,{\left (b x + a\right )}{\left (4 \,{\left (b x + a\right )}{\left (\frac{2 \,{\left (9 \, B a b^{8} - 8 \, A b^{9}\right )}{\left (b x + a\right )}}{a^{5} b^{15}} - \frac{9 \,{\left (9 \, B a^{2} b^{8} - 8 \, A a b^{9}\right )}}{a^{5} b^{15}}\right )} + \frac{63 \,{\left (9 \, B a^{3} b^{8} - 8 \, A a^{2} b^{9}\right )}}{a^{5} b^{15}}\right )} - \frac{105 \,{\left (9 \, B a^{4} b^{8} - 8 \, A a^{3} b^{9}\right )}}{a^{5} b^{15}}\right )}{\left (b x + a\right )} + \frac{315 \,{\left (B a^{5} b^{8} - A a^{4} b^{9}\right )}}{a^{5} b^{15}}\right )} \sqrt{b x + a} b}{322560 \,{\left ({\left (b x + a\right )} b - a b\right )}^{\frac{9}{2}}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/(sqrt(b*x + a)*x^(11/2)),x, algorithm="giac")

[Out]

-1/322560*((2*(b*x + a)*(4*(b*x + a)*(2*(9*B*a*b^8 - 8*A*b^9)*(b*x + a)/(a^5*b^1
5) - 9*(9*B*a^2*b^8 - 8*A*a*b^9)/(a^5*b^15)) + 63*(9*B*a^3*b^8 - 8*A*a^2*b^9)/(a
^5*b^15)) - 105*(9*B*a^4*b^8 - 8*A*a^3*b^9)/(a^5*b^15))*(b*x + a) + 315*(B*a^5*b
^8 - A*a^4*b^9)/(a^5*b^15))*sqrt(b*x + a)*b/(((b*x + a)*b - a*b)^(9/2)*abs(b))