3.487 \(\int \frac{(a+b x)^{3/2} (A+B x)}{x^{13/2}} \, dx\)

Optimal. Leaf size=117 \[ \frac{16 b^2 (a+b x)^{5/2} (6 A b-11 a B)}{3465 a^4 x^{5/2}}-\frac{8 b (a+b x)^{5/2} (6 A b-11 a B)}{693 a^3 x^{7/2}}+\frac{2 (a+b x)^{5/2} (6 A b-11 a B)}{99 a^2 x^{9/2}}-\frac{2 A (a+b x)^{5/2}}{11 a x^{11/2}} \]

[Out]

(-2*A*(a + b*x)^(5/2))/(11*a*x^(11/2)) + (2*(6*A*b - 11*a*B)*(a + b*x)^(5/2))/(9
9*a^2*x^(9/2)) - (8*b*(6*A*b - 11*a*B)*(a + b*x)^(5/2))/(693*a^3*x^(7/2)) + (16*
b^2*(6*A*b - 11*a*B)*(a + b*x)^(5/2))/(3465*a^4*x^(5/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.138476, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{16 b^2 (a+b x)^{5/2} (6 A b-11 a B)}{3465 a^4 x^{5/2}}-\frac{8 b (a+b x)^{5/2} (6 A b-11 a B)}{693 a^3 x^{7/2}}+\frac{2 (a+b x)^{5/2} (6 A b-11 a B)}{99 a^2 x^{9/2}}-\frac{2 A (a+b x)^{5/2}}{11 a x^{11/2}} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^(3/2)*(A + B*x))/x^(13/2),x]

[Out]

(-2*A*(a + b*x)^(5/2))/(11*a*x^(11/2)) + (2*(6*A*b - 11*a*B)*(a + b*x)^(5/2))/(9
9*a^2*x^(9/2)) - (8*b*(6*A*b - 11*a*B)*(a + b*x)^(5/2))/(693*a^3*x^(7/2)) + (16*
b^2*(6*A*b - 11*a*B)*(a + b*x)^(5/2))/(3465*a^4*x^(5/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 10.9886, size = 116, normalized size = 0.99 \[ - \frac{2 A \left (a + b x\right )^{\frac{5}{2}}}{11 a x^{\frac{11}{2}}} + \frac{2 \left (a + b x\right )^{\frac{5}{2}} \left (6 A b - 11 B a\right )}{99 a^{2} x^{\frac{9}{2}}} - \frac{8 b \left (a + b x\right )^{\frac{5}{2}} \left (6 A b - 11 B a\right )}{693 a^{3} x^{\frac{7}{2}}} + \frac{16 b^{2} \left (a + b x\right )^{\frac{5}{2}} \left (6 A b - 11 B a\right )}{3465 a^{4} x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(3/2)*(B*x+A)/x**(13/2),x)

[Out]

-2*A*(a + b*x)**(5/2)/(11*a*x**(11/2)) + 2*(a + b*x)**(5/2)*(6*A*b - 11*B*a)/(99
*a**2*x**(9/2)) - 8*b*(a + b*x)**(5/2)*(6*A*b - 11*B*a)/(693*a**3*x**(7/2)) + 16
*b**2*(a + b*x)**(5/2)*(6*A*b - 11*B*a)/(3465*a**4*x**(5/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.102718, size = 76, normalized size = 0.65 \[ -\frac{2 (a+b x)^{5/2} \left (35 a^3 (9 A+11 B x)-10 a^2 b x (21 A+22 B x)+8 a b^2 x^2 (15 A+11 B x)-48 A b^3 x^3\right )}{3465 a^4 x^{11/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^(3/2)*(A + B*x))/x^(13/2),x]

[Out]

(-2*(a + b*x)^(5/2)*(-48*A*b^3*x^3 + 35*a^3*(9*A + 11*B*x) + 8*a*b^2*x^2*(15*A +
11*B*x) - 10*a^2*b*x*(21*A + 22*B*x)))/(3465*a^4*x^(11/2))

_______________________________________________________________________________________

Maple [A]  time = 0.007, size = 77, normalized size = 0.7 \[ -{\frac{-96\,A{b}^{3}{x}^{3}+176\,B{x}^{3}a{b}^{2}+240\,aA{b}^{2}{x}^{2}-440\,B{x}^{2}{a}^{2}b-420\,{a}^{2}Abx+770\,{a}^{3}Bx+630\,A{a}^{3}}{3465\,{a}^{4}} \left ( bx+a \right ) ^{{\frac{5}{2}}}{x}^{-{\frac{11}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(3/2)*(B*x+A)/x^(13/2),x)

[Out]

-2/3465*(b*x+a)^(5/2)*(-48*A*b^3*x^3+88*B*a*b^2*x^3+120*A*a*b^2*x^2-220*B*a^2*b*
x^2-210*A*a^2*b*x+385*B*a^3*x+315*A*a^3)/x^(11/2)/a^4

_______________________________________________________________________________________

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)*(b*x + a)^(3/2)/x^(13/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.234155, size = 170, normalized size = 1.45 \[ -\frac{2 \,{\left (315 \, A a^{5} + 8 \,{\left (11 \, B a b^{4} - 6 \, A b^{5}\right )} x^{5} - 4 \,{\left (11 \, B a^{2} b^{3} - 6 \, A a b^{4}\right )} x^{4} + 3 \,{\left (11 \, B a^{3} b^{2} - 6 \, A a^{2} b^{3}\right )} x^{3} + 5 \,{\left (110 \, B a^{4} b + 3 \, A a^{3} b^{2}\right )} x^{2} + 35 \,{\left (11 \, B a^{5} + 12 \, A a^{4} b\right )} x\right )} \sqrt{b x + a}}{3465 \, a^{4} x^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^(3/2)/x^(13/2),x, algorithm="fricas")

[Out]

-2/3465*(315*A*a^5 + 8*(11*B*a*b^4 - 6*A*b^5)*x^5 - 4*(11*B*a^2*b^3 - 6*A*a*b^4)
*x^4 + 3*(11*B*a^3*b^2 - 6*A*a^2*b^3)*x^3 + 5*(110*B*a^4*b + 3*A*a^3*b^2)*x^2 +
35*(11*B*a^5 + 12*A*a^4*b)*x)*sqrt(b*x + a)/(a^4*x^(11/2))

_______________________________________________________________________________________

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)**(3/2)*(B*x+A)/x**(13/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.238642, size = 208, normalized size = 1.78 \[ \frac{{\left ({\left (b x + a\right )}{\left (4 \,{\left (b x + a\right )}{\left (\frac{2 \,{\left (11 \, B a^{2} b^{10} - 6 \, A a b^{11}\right )}{\left (b x + a\right )}}{a^{6} b^{18}} - \frac{11 \,{\left (11 \, B a^{3} b^{10} - 6 \, A a^{2} b^{11}\right )}}{a^{6} b^{18}}\right )} + \frac{99 \,{\left (11 \, B a^{4} b^{10} - 6 \, A a^{3} b^{11}\right )}}{a^{6} b^{18}}\right )} - \frac{693 \,{\left (B a^{5} b^{10} - A a^{4} b^{11}\right )}}{a^{6} b^{18}}\right )}{\left (b x + a\right )}^{\frac{5}{2}} b}{14192640 \,{\left ({\left (b x + a\right )} b - a b\right )}^{\frac{11}{2}}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^(3/2)/x^(13/2),x, algorithm="giac")

[Out]

1/14192640*((b*x + a)*(4*(b*x + a)*(2*(11*B*a^2*b^10 - 6*A*a*b^11)*(b*x + a)/(a^
6*b^18) - 11*(11*B*a^3*b^10 - 6*A*a^2*b^11)/(a^6*b^18)) + 99*(11*B*a^4*b^10 - 6*
A*a^3*b^11)/(a^6*b^18)) - 693*(B*a^5*b^10 - A*a^4*b^11)/(a^6*b^18))*(b*x + a)^(5
/2)*b/(((b*x + a)*b - a*b)^(11/2)*abs(b))