3.218 \(\int \frac{(A+B x) \left (b x+c x^2\right )^{5/2}}{x^{5/2}} \, dx\)

Optimal. Leaf size=61 \[ \frac{2 B \left (b x+c x^2\right )^{7/2}}{9 c x^{5/2}}-\frac{2 \left (b x+c x^2\right )^{7/2} (2 b B-9 A c)}{63 c^2 x^{7/2}} \]

[Out]

(-2*(2*b*B - 9*A*c)*(b*x + c*x^2)^(7/2))/(63*c^2*x^(7/2)) + (2*B*(b*x + c*x^2)^(
7/2))/(9*c*x^(5/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.135775, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{2 B \left (b x+c x^2\right )^{7/2}}{9 c x^{5/2}}-\frac{2 \left (b x+c x^2\right )^{7/2} (2 b B-9 A c)}{63 c^2 x^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(2*b*B - 9*A*c)*(b*x + c*x^2)^(7/2))/(63*c^2*x^(7/2)) + (2*B*(b*x + c*x^2)^(
7/2))/(9*c*x^(5/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 8.02996, size = 56, normalized size = 0.92 \[ \frac{2 B \left (b x + c x^{2}\right )^{\frac{7}{2}}}{9 c x^{\frac{5}{2}}} + \frac{4 \left (\frac{9 A c}{2} - B b\right ) \left (b x + c x^{2}\right )^{\frac{7}{2}}}{63 c^{2} x^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*B*(b*x + c*x**2)**(7/2)/(9*c*x**(5/2)) + 4*(9*A*c/2 - B*b)*(b*x + c*x**2)**(7/
2)/(63*c**2*x**(7/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0656756, size = 44, normalized size = 0.72 \[ \frac{2 (b+c x)^3 \sqrt{x (b+c x)} (9 A c-2 b B+7 B c x)}{63 c^2 \sqrt{x}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(b + c*x)^3*Sqrt[x*(b + c*x)]*(-2*b*B + 9*A*c + 7*B*c*x))/(63*c^2*Sqrt[x])

_______________________________________________________________________________________

Maple [A]  time = 0.004, size = 39, normalized size = 0.6 \[{\frac{ \left ( 2\,cx+2\,b \right ) \left ( 7\,Bcx+9\,Ac-2\,Bb \right ) }{63\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}{x}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/63*(c*x+b)*(7*B*c*x+9*A*c-2*B*b)*(c*x^2+b*x)^(5/2)/c^2/x^(5/2)

_______________________________________________________________________________________

Maxima [A]  time = 0.702959, size = 311, normalized size = 5.1 \[ \frac{2 \,{\left (35 \, b^{2} c x^{3} + 35 \, b^{3} x^{2} +{\left (15 \, c^{3} x^{3} + 3 \, b c^{2} x^{2} - 4 \, b^{2} c x + 8 \, b^{3}\right )} x^{2} + 14 \,{\left (3 \, b c^{2} x^{3} + b^{2} c x^{2} - 2 \, b^{3} x\right )} x\right )} \sqrt{c x + b} A}{105 \, c x^{2}} + \frac{2 \,{\left ({\left (35 \, c^{4} x^{4} + 5 \, b c^{3} x^{3} - 6 \, b^{2} c^{2} x^{2} + 8 \, b^{3} c x - 16 \, b^{4}\right )} x^{3} + 6 \,{\left (15 \, b c^{3} x^{4} + 3 \, b^{2} c^{2} x^{3} - 4 \, b^{3} c x^{2} + 8 \, b^{4} x\right )} x^{2} + 21 \,{\left (3 \, b^{2} c^{2} x^{4} + b^{3} c x^{3} - 2 \, b^{4} x^{2}\right )} x\right )} \sqrt{c x + b} B}{315 \, c^{2} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/105*(35*b^2*c*x^3 + 35*b^3*x^2 + (15*c^3*x^3 + 3*b*c^2*x^2 - 4*b^2*c*x + 8*b^3
)*x^2 + 14*(3*b*c^2*x^3 + b^2*c*x^2 - 2*b^3*x)*x)*sqrt(c*x + b)*A/(c*x^2) + 2/31
5*((35*c^4*x^4 + 5*b*c^3*x^3 - 6*b^2*c^2*x^2 + 8*b^3*c*x - 16*b^4)*x^3 + 6*(15*b
*c^3*x^4 + 3*b^2*c^2*x^3 - 4*b^3*c*x^2 + 8*b^4*x)*x^2 + 21*(3*b^2*c^2*x^4 + b^3*
c*x^3 - 2*b^4*x^2)*x)*sqrt(c*x + b)*B/(c^2*x^3)

_______________________________________________________________________________________

Fricas [A]  time = 0.291716, size = 177, normalized size = 2.9 \[ \frac{2 \,{\left (7 \, B c^{5} x^{6} +{\left (26 \, B b c^{4} + 9 \, A c^{5}\right )} x^{5} + 2 \,{\left (17 \, B b^{2} c^{3} + 18 \, A b c^{4}\right )} x^{4} + 2 \,{\left (8 \, B b^{3} c^{2} + 27 \, A b^{2} c^{3}\right )} x^{3} -{\left (B b^{4} c - 36 \, A b^{3} c^{2}\right )} x^{2} -{\left (2 \, B b^{5} - 9 \, A b^{4} c\right )} x\right )}}{63 \, \sqrt{c x^{2} + b x} c^{2} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/63*(7*B*c^5*x^6 + (26*B*b*c^4 + 9*A*c^5)*x^5 + 2*(17*B*b^2*c^3 + 18*A*b*c^4)*x
^4 + 2*(8*B*b^3*c^2 + 27*A*b^2*c^3)*x^3 - (B*b^4*c - 36*A*b^3*c^2)*x^2 - (2*B*b^
5 - 9*A*b^4*c)*x)/(sqrt(c*x^2 + b*x)*c^2*sqrt(x))

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (x \left (b + c x\right )\right )^{\frac{5}{2}} \left (A + B x\right )}{x^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+b*x)**(5/2)/x**(5/2),x)

[Out]

Integral((x*(b + c*x))**(5/2)*(A + B*x)/x**(5/2), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.286013, size = 365, normalized size = 5.98 \[ \frac{2}{315} \, B c^{2}{\left (\frac{16 \, b^{\frac{9}{2}}}{c^{4}} + \frac{35 \,{\left (c x + b\right )}^{\frac{9}{2}} - 135 \,{\left (c x + b\right )}^{\frac{7}{2}} b + 189 \,{\left (c x + b\right )}^{\frac{5}{2}} b^{2} - 105 \,{\left (c x + b\right )}^{\frac{3}{2}} b^{3}}{c^{4}}\right )} - \frac{4}{105} \, B b c{\left (\frac{8 \, b^{\frac{7}{2}}}{c^{3}} - \frac{15 \,{\left (c x + b\right )}^{\frac{7}{2}} - 42 \,{\left (c x + b\right )}^{\frac{5}{2}} b + 35 \,{\left (c x + b\right )}^{\frac{3}{2}} b^{2}}{c^{3}}\right )} - \frac{2}{105} \, A c^{2}{\left (\frac{8 \, b^{\frac{7}{2}}}{c^{3}} - \frac{15 \,{\left (c x + b\right )}^{\frac{7}{2}} - 42 \,{\left (c x + b\right )}^{\frac{5}{2}} b + 35 \,{\left (c x + b\right )}^{\frac{3}{2}} b^{2}}{c^{3}}\right )} + \frac{2}{15} \, B b^{2}{\left (\frac{2 \, b^{\frac{5}{2}}}{c^{2}} + \frac{3 \,{\left (c x + b\right )}^{\frac{5}{2}} - 5 \,{\left (c x + b\right )}^{\frac{3}{2}} b}{c^{2}}\right )} + \frac{4}{15} \, A b c{\left (\frac{2 \, b^{\frac{5}{2}}}{c^{2}} + \frac{3 \,{\left (c x + b\right )}^{\frac{5}{2}} - 5 \,{\left (c x + b\right )}^{\frac{3}{2}} b}{c^{2}}\right )} + \frac{2}{3} \, A b^{2}{\left (\frac{{\left (c x + b\right )}^{\frac{3}{2}}}{c} - \frac{b^{\frac{3}{2}}}{c}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/315*B*c^2*(16*b^(9/2)/c^4 + (35*(c*x + b)^(9/2) - 135*(c*x + b)^(7/2)*b + 189*
(c*x + b)^(5/2)*b^2 - 105*(c*x + b)^(3/2)*b^3)/c^4) - 4/105*B*b*c*(8*b^(7/2)/c^3
- (15*(c*x + b)^(7/2) - 42*(c*x + b)^(5/2)*b + 35*(c*x + b)^(3/2)*b^2)/c^3) - 2
/105*A*c^2*(8*b^(7/2)/c^3 - (15*(c*x + b)^(7/2) - 42*(c*x + b)^(5/2)*b + 35*(c*x
+ b)^(3/2)*b^2)/c^3) + 2/15*B*b^2*(2*b^(5/2)/c^2 + (3*(c*x + b)^(5/2) - 5*(c*x
+ b)^(3/2)*b)/c^2) + 4/15*A*b*c*(2*b^(5/2)/c^2 + (3*(c*x + b)^(5/2) - 5*(c*x + b
)^(3/2)*b)/c^2) + 2/3*A*b^2*((c*x + b)^(3/2)/c - b^(3/2)/c)