3.206 \(\int \frac{(A+B x) \left (b x+c x^2\right )^{3/2}}{\sqrt{x}} \, dx\)

Optimal. Leaf size=96 \[ \frac{4 b \left (b x+c x^2\right )^{5/2} (4 b B-9 A c)}{315 c^3 x^{5/2}}-\frac{2 \left (b x+c x^2\right )^{5/2} (4 b B-9 A c)}{63 c^2 x^{3/2}}+\frac{2 B \left (b x+c x^2\right )^{5/2}}{9 c \sqrt{x}} \]

[Out]

(4*b*(4*b*B - 9*A*c)*(b*x + c*x^2)^(5/2))/(315*c^3*x^(5/2)) - (2*(4*b*B - 9*A*c)
*(b*x + c*x^2)^(5/2))/(63*c^2*x^(3/2)) + (2*B*(b*x + c*x^2)^(5/2))/(9*c*Sqrt[x])

_______________________________________________________________________________________

Rubi [A]  time = 0.195991, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{4 b \left (b x+c x^2\right )^{5/2} (4 b B-9 A c)}{315 c^3 x^{5/2}}-\frac{2 \left (b x+c x^2\right )^{5/2} (4 b B-9 A c)}{63 c^2 x^{3/2}}+\frac{2 B \left (b x+c x^2\right )^{5/2}}{9 c \sqrt{x}} \]

Antiderivative was successfully verified.

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

[Out]

(4*b*(4*b*B - 9*A*c)*(b*x + c*x^2)^(5/2))/(315*c^3*x^(5/2)) - (2*(4*b*B - 9*A*c)
*(b*x + c*x^2)^(5/2))/(63*c^2*x^(3/2)) + (2*B*(b*x + c*x^2)^(5/2))/(9*c*Sqrt[x])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 11.4606, size = 92, normalized size = 0.96 \[ \frac{2 B \left (b x + c x^{2}\right )^{\frac{5}{2}}}{9 c \sqrt{x}} - \frac{4 b \left (9 A c - 4 B b\right ) \left (b x + c x^{2}\right )^{\frac{5}{2}}}{315 c^{3} x^{\frac{5}{2}}} + \frac{2 \left (9 A c - 4 B b\right ) \left (b x + c x^{2}\right )^{\frac{5}{2}}}{63 c^{2} x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+b*x)**(3/2)/x**(1/2),x)

[Out]

2*B*(b*x + c*x**2)**(5/2)/(9*c*sqrt(x)) - 4*b*(9*A*c - 4*B*b)*(b*x + c*x**2)**(5
/2)/(315*c**3*x**(5/2)) + 2*(9*A*c - 4*B*b)*(b*x + c*x**2)**(5/2)/(63*c**2*x**(3
/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0766586, size = 56, normalized size = 0.58 \[ \frac{2 (x (b+c x))^{5/2} \left (-2 b c (9 A+10 B x)+5 c^2 x (9 A+7 B x)+8 b^2 B\right )}{315 c^3 x^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(x*(b + c*x))^(5/2)*(8*b^2*B + 5*c^2*x*(9*A + 7*B*x) - 2*b*c*(9*A + 10*B*x)))
/(315*c^3*x^(5/2))

_______________________________________________________________________________________

Maple [A]  time = 0.006, size = 59, normalized size = 0.6 \[ -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( -35\,B{c}^{2}{x}^{2}-45\,Ax{c}^{2}+20\,Bxbc+18\,Abc-8\,{b}^{2}B \right ) }{315\,{c}^{3}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}{x}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x)^(3/2)/x^(1/2),x)

[Out]

-2/315*(c*x+b)*(-35*B*c^2*x^2-45*A*c^2*x+20*B*b*c*x+18*A*b*c-8*B*b^2)*(c*x^2+b*x
)^(3/2)/c^3/x^(3/2)

_______________________________________________________________________________________

Maxima [A]  time = 0.711907, size = 246, normalized size = 2.56 \[ \frac{2 \,{\left ({\left (15 \, c^{3} x^{3} + 3 \, b c^{2} x^{2} - 4 \, b^{2} c x + 8 \, b^{3}\right )} x^{2} + 7 \,{\left (3 \, b c^{2} x^{3} + b^{2} c x^{2} - 2 \, b^{3} x\right )} x\right )} \sqrt{c x + b} A}{105 \, c^{2} x^{2}} + \frac{2 \,{\left ({\left (35 \, c^{4} x^{4} + 5 \, b c^{3} x^{3} - 6 \, b^{2} c^{2} x^{2} + 8 \, b^{3} c x - 16 \, b^{4}\right )} x^{3} + 3 \,{\left (15 \, b c^{3} x^{4} + 3 \, b^{2} c^{2} x^{3} - 4 \, b^{3} c x^{2} + 8 \, b^{4} x\right )} x^{2}\right )} \sqrt{c x + b} B}{315 \, c^{3} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(3/2)*(B*x + A)/sqrt(x),x, algorithm="maxima")

[Out]

2/105*((15*c^3*x^3 + 3*b*c^2*x^2 - 4*b^2*c*x + 8*b^3)*x^2 + 7*(3*b*c^2*x^3 + b^2
*c*x^2 - 2*b^3*x)*x)*sqrt(c*x + b)*A/(c^2*x^2) + 2/315*((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 + 3*(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)*sqrt(c*x + b)*B/(c^3*x^3)

_______________________________________________________________________________________

Fricas [A]  time = 0.285546, size = 176, normalized size = 1.83 \[ \frac{2 \,{\left (35 \, B c^{5} x^{6} + 5 \,{\left (17 \, B b c^{4} + 9 \, A c^{5}\right )} x^{5} +{\left (53 \, B b^{2} c^{3} + 117 \, A b c^{4}\right )} x^{4} -{\left (B b^{3} c^{2} - 81 \, A b^{2} c^{3}\right )} x^{3} +{\left (4 \, B b^{4} c - 9 \, A b^{3} c^{2}\right )} x^{2} + 2 \,{\left (4 \, B b^{5} - 9 \, A b^{4} c\right )} x\right )}}{315 \, \sqrt{c x^{2} + b x} c^{3} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/315*(35*B*c^5*x^6 + 5*(17*B*b*c^4 + 9*A*c^5)*x^5 + (53*B*b^2*c^3 + 117*A*b*c^4
)*x^4 - (B*b^3*c^2 - 81*A*b^2*c^3)*x^3 + (4*B*b^4*c - 9*A*b^3*c^2)*x^2 + 2*(4*B*
b^5 - 9*A*b^4*c)*x)/(sqrt(c*x^2 + b*x)*c^3*sqrt(x))

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (x \left (b + c x\right )\right )^{\frac{3}{2}} \left (A + B x\right )}{\sqrt{x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+b*x)**(3/2)/x**(1/2),x)

[Out]

Integral((x*(b + c*x))**(3/2)*(A + B*x)/sqrt(x), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.284332, size = 269, normalized size = 2.8 \[ \frac{2}{315} \, B c{\left (\frac{16 \, b^{\frac{9}{2}}}{c^{4}} + \frac{35 \,{\left (c x + b\right )}^{\frac{9}{2}} - 135 \,{\left (c x + b\right )}^{\frac{7}{2}} b + 189 \,{\left (c x + b\right )}^{\frac{5}{2}} b^{2} - 105 \,{\left (c x + b\right )}^{\frac{3}{2}} b^{3}}{c^{4}}\right )} - \frac{2}{105} \, B b{\left (\frac{8 \, b^{\frac{7}{2}}}{c^{3}} - \frac{15 \,{\left (c x + b\right )}^{\frac{7}{2}} - 42 \,{\left (c x + b\right )}^{\frac{5}{2}} b + 35 \,{\left (c x + b\right )}^{\frac{3}{2}} b^{2}}{c^{3}}\right )} - \frac{2}{105} \, A c{\left (\frac{8 \, b^{\frac{7}{2}}}{c^{3}} - \frac{15 \,{\left (c x + b\right )}^{\frac{7}{2}} - 42 \,{\left (c x + b\right )}^{\frac{5}{2}} b + 35 \,{\left (c x + b\right )}^{\frac{3}{2}} b^{2}}{c^{3}}\right )} + \frac{2}{15} \, A b{\left (\frac{2 \, b^{\frac{5}{2}}}{c^{2}} + \frac{3 \,{\left (c x + b\right )}^{\frac{5}{2}} - 5 \,{\left (c x + b\right )}^{\frac{3}{2}} b}{c^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/315*B*c*(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) - 2/105*B*b*(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*(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*A*b*(2*b^(5/2)/c^2 + (3*(c*x + b)^(5/2) - 5*(c*x + b)^(3/
2)*b)/c^2)