3.499 \(\int \frac{(a+b x)^{5/2} (A+B x)}{x^{15/2}} \, dx\)

Optimal. Leaf size=117 \[ \frac{16 b^2 (a+b x)^{7/2} (6 A b-13 a B)}{9009 a^4 x^{7/2}}-\frac{8 b (a+b x)^{7/2} (6 A b-13 a B)}{1287 a^3 x^{9/2}}+\frac{2 (a+b x)^{7/2} (6 A b-13 a B)}{143 a^2 x^{11/2}}-\frac{2 A (a+b x)^{7/2}}{13 a x^{13/2}} \]

[Out]

(-2*A*(a + b*x)^(7/2))/(13*a*x^(13/2)) + (2*(6*A*b - 13*a*B)*(a + b*x)^(7/2))/(1
43*a^2*x^(11/2)) - (8*b*(6*A*b - 13*a*B)*(a + b*x)^(7/2))/(1287*a^3*x^(9/2)) + (
16*b^2*(6*A*b - 13*a*B)*(a + b*x)^(7/2))/(9009*a^4*x^(7/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.135993, 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 \[ \frac{16 b^2 (a+b x)^{7/2} (6 A b-13 a B)}{9009 a^4 x^{7/2}}-\frac{8 b (a+b x)^{7/2} (6 A b-13 a B)}{1287 a^3 x^{9/2}}+\frac{2 (a+b x)^{7/2} (6 A b-13 a B)}{143 a^2 x^{11/2}}-\frac{2 A (a+b x)^{7/2}}{13 a x^{13/2}} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^(5/2)*(A + B*x))/x^(15/2),x]

[Out]

(-2*A*(a + b*x)^(7/2))/(13*a*x^(13/2)) + (2*(6*A*b - 13*a*B)*(a + b*x)^(7/2))/(1
43*a^2*x^(11/2)) - (8*b*(6*A*b - 13*a*B)*(a + b*x)^(7/2))/(1287*a^3*x^(9/2)) + (
16*b^2*(6*A*b - 13*a*B)*(a + b*x)^(7/2))/(9009*a^4*x^(7/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 11.0331, size = 116, normalized size = 0.99 \[ - \frac{2 A \left (a + b x\right )^{\frac{7}{2}}}{13 a x^{\frac{13}{2}}} + \frac{2 \left (a + b x\right )^{\frac{7}{2}} \left (6 A b - 13 B a\right )}{143 a^{2} x^{\frac{11}{2}}} - \frac{8 b \left (a + b x\right )^{\frac{7}{2}} \left (6 A b - 13 B a\right )}{1287 a^{3} x^{\frac{9}{2}}} + \frac{16 b^{2} \left (a + b x\right )^{\frac{7}{2}} \left (6 A b - 13 B a\right )}{9009 a^{4} x^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(5/2)*(B*x+A)/x**(15/2),x)

[Out]

-2*A*(a + b*x)**(7/2)/(13*a*x**(13/2)) + 2*(a + b*x)**(7/2)*(6*A*b - 13*B*a)/(14
3*a**2*x**(11/2)) - 8*b*(a + b*x)**(7/2)*(6*A*b - 13*B*a)/(1287*a**3*x**(9/2)) +
16*b**2*(a + b*x)**(7/2)*(6*A*b - 13*B*a)/(9009*a**4*x**(7/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.125341, size = 76, normalized size = 0.65 \[ -\frac{2 (a+b x)^{7/2} \left (63 a^3 (11 A+13 B x)-14 a^2 b x (27 A+26 B x)+8 a b^2 x^2 (21 A+13 B x)-48 A b^3 x^3\right )}{9009 a^4 x^{13/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^(5/2)*(A + B*x))/x^(15/2),x]

[Out]

(-2*(a + b*x)^(7/2)*(-48*A*b^3*x^3 + 63*a^3*(11*A + 13*B*x) + 8*a*b^2*x^2*(21*A
+ 13*B*x) - 14*a^2*b*x*(27*A + 26*B*x)))/(9009*a^4*x^(13/2))

_______________________________________________________________________________________

Maple [A]  time = 0.007, size = 77, normalized size = 0.7 \[ -{\frac{-96\,A{b}^{3}{x}^{3}+208\,B{x}^{3}a{b}^{2}+336\,aA{b}^{2}{x}^{2}-728\,B{x}^{2}{a}^{2}b-756\,{a}^{2}Abx+1638\,{a}^{3}Bx+1386\,A{a}^{3}}{9009\,{a}^{4}} \left ( bx+a \right ) ^{{\frac{7}{2}}}{x}^{-{\frac{13}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(5/2)*(B*x+A)/x^(15/2),x)

[Out]

-2/9009*(b*x+a)^(7/2)*(-48*A*b^3*x^3+104*B*a*b^2*x^3+168*A*a*b^2*x^2-364*B*a^2*b
*x^2-378*A*a^2*b*x+819*B*a^3*x+693*A*a^3)/x^(13/2)/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)^(5/2)/x^(15/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.26613, size = 201, normalized size = 1.72 \[ -\frac{2 \,{\left (693 \, A a^{6} + 8 \,{\left (13 \, B a b^{5} - 6 \, A b^{6}\right )} x^{6} - 4 \,{\left (13 \, B a^{2} b^{4} - 6 \, A a b^{5}\right )} x^{5} + 3 \,{\left (13 \, B a^{3} b^{3} - 6 \, A a^{2} b^{4}\right )} x^{4} +{\left (1469 \, B a^{4} b^{2} + 15 \, A a^{3} b^{3}\right )} x^{3} + 7 \,{\left (299 \, B a^{5} b + 159 \, A a^{4} b^{2}\right )} x^{2} + 63 \,{\left (13 \, B a^{6} + 27 \, A a^{5} b\right )} x\right )} \sqrt{b x + a}}{9009 \, a^{4} x^{\frac{13}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^(5/2)/x^(15/2),x, algorithm="fricas")

[Out]

-2/9009*(693*A*a^6 + 8*(13*B*a*b^5 - 6*A*b^6)*x^6 - 4*(13*B*a^2*b^4 - 6*A*a*b^5)
*x^5 + 3*(13*B*a^3*b^3 - 6*A*a^2*b^4)*x^4 + (1469*B*a^4*b^2 + 15*A*a^3*b^3)*x^3
+ 7*(299*B*a^5*b + 159*A*a^4*b^2)*x^2 + 63*(13*B*a^6 + 27*A*a^5*b)*x)*sqrt(b*x +
a)/(a^4*x^(13/2))

_______________________________________________________________________________________

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)**(5/2)*(B*x+A)/x**(15/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.240523, size = 211, normalized size = 1.8 \[ \frac{{\left ({\left (b x + a\right )}{\left (4 \,{\left (b x + a\right )}{\left (\frac{2 \,{\left (13 \, B a^{3} b^{12} - 6 \, A a^{2} b^{13}\right )}{\left (b x + a\right )}}{a^{7} b^{21}} - \frac{13 \,{\left (13 \, B a^{4} b^{12} - 6 \, A a^{3} b^{13}\right )}}{a^{7} b^{21}}\right )} + \frac{143 \,{\left (13 \, B a^{5} b^{12} - 6 \, A a^{4} b^{13}\right )}}{a^{7} b^{21}}\right )} - \frac{1287 \,{\left (B a^{6} b^{12} - A a^{5} b^{13}\right )}}{a^{7} b^{21}}\right )}{\left (b x + a\right )}^{\frac{7}{2}} b}{6642155520 \,{\left ({\left (b x + a\right )} b - a b\right )}^{\frac{13}{2}}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^(5/2)/x^(15/2),x, algorithm="giac")

[Out]

1/6642155520*((b*x + a)*(4*(b*x + a)*(2*(13*B*a^3*b^12 - 6*A*a^2*b^13)*(b*x + a)
/(a^7*b^21) - 13*(13*B*a^4*b^12 - 6*A*a^3*b^13)/(a^7*b^21)) + 143*(13*B*a^5*b^12
- 6*A*a^4*b^13)/(a^7*b^21)) - 1287*(B*a^6*b^12 - A*a^5*b^13)/(a^7*b^21))*(b*x +
a)^(7/2)*b/(((b*x + a)*b - a*b)^(13/2)*abs(b))