3.195 \(\int x^{5/2} (A+B x) \sqrt{b x+c x^2} \, dx\)

Optimal. Leaf size=170 \[ \frac{32 b^3 \left (b x+c x^2\right )^{3/2} (8 b B-11 A c)}{3465 c^5 x^{3/2}}-\frac{16 b^2 \left (b x+c x^2\right )^{3/2} (8 b B-11 A c)}{1155 c^4 \sqrt{x}}+\frac{4 b \sqrt{x} \left (b x+c x^2\right )^{3/2} (8 b B-11 A c)}{231 c^3}-\frac{2 x^{3/2} \left (b x+c x^2\right )^{3/2} (8 b B-11 A c)}{99 c^2}+\frac{2 B x^{5/2} \left (b x+c x^2\right )^{3/2}}{11 c} \]

[Out]

(32*b^3*(8*b*B - 11*A*c)*(b*x + c*x^2)^(3/2))/(3465*c^5*x^(3/2)) - (16*b^2*(8*b*
B - 11*A*c)*(b*x + c*x^2)^(3/2))/(1155*c^4*Sqrt[x]) + (4*b*(8*b*B - 11*A*c)*Sqrt
[x]*(b*x + c*x^2)^(3/2))/(231*c^3) - (2*(8*b*B - 11*A*c)*x^(3/2)*(b*x + c*x^2)^(
3/2))/(99*c^2) + (2*B*x^(5/2)*(b*x + c*x^2)^(3/2))/(11*c)

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Rubi [A]  time = 0.356978, antiderivative size = 170, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{32 b^3 \left (b x+c x^2\right )^{3/2} (8 b B-11 A c)}{3465 c^5 x^{3/2}}-\frac{16 b^2 \left (b x+c x^2\right )^{3/2} (8 b B-11 A c)}{1155 c^4 \sqrt{x}}+\frac{4 b \sqrt{x} \left (b x+c x^2\right )^{3/2} (8 b B-11 A c)}{231 c^3}-\frac{2 x^{3/2} \left (b x+c x^2\right )^{3/2} (8 b B-11 A c)}{99 c^2}+\frac{2 B x^{5/2} \left (b x+c x^2\right )^{3/2}}{11 c} \]

Antiderivative was successfully verified.

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

[Out]

(32*b^3*(8*b*B - 11*A*c)*(b*x + c*x^2)^(3/2))/(3465*c^5*x^(3/2)) - (16*b^2*(8*b*
B - 11*A*c)*(b*x + c*x^2)^(3/2))/(1155*c^4*Sqrt[x]) + (4*b*(8*b*B - 11*A*c)*Sqrt
[x]*(b*x + c*x^2)^(3/2))/(231*c^3) - (2*(8*b*B - 11*A*c)*x^(3/2)*(b*x + c*x^2)^(
3/2))/(99*c^2) + (2*B*x^(5/2)*(b*x + c*x^2)^(3/2))/(11*c)

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Rubi in Sympy [A]  time = 21.3312, size = 167, normalized size = 0.98 \[ \frac{2 B x^{\frac{5}{2}} \left (b x + c x^{2}\right )^{\frac{3}{2}}}{11 c} - \frac{32 b^{3} \left (11 A c - 8 B b\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}}}{3465 c^{5} x^{\frac{3}{2}}} + \frac{16 b^{2} \left (11 A c - 8 B b\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}}}{1155 c^{4} \sqrt{x}} - \frac{4 b \sqrt{x} \left (11 A c - 8 B b\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}}}{231 c^{3}} + \frac{2 x^{\frac{3}{2}} \left (11 A c - 8 B b\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}}}{99 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(5/2)*(B*x+A)*(c*x**2+b*x)**(1/2),x)

[Out]

2*B*x**(5/2)*(b*x + c*x**2)**(3/2)/(11*c) - 32*b**3*(11*A*c - 8*B*b)*(b*x + c*x*
*2)**(3/2)/(3465*c**5*x**(3/2)) + 16*b**2*(11*A*c - 8*B*b)*(b*x + c*x**2)**(3/2)
/(1155*c**4*sqrt(x)) - 4*b*sqrt(x)*(11*A*c - 8*B*b)*(b*x + c*x**2)**(3/2)/(231*c
**3) + 2*x**(3/2)*(11*A*c - 8*B*b)*(b*x + c*x**2)**(3/2)/(99*c**2)

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Mathematica [A]  time = 0.0983065, size = 94, normalized size = 0.55 \[ \frac{2 (x (b+c x))^{3/2} \left (-16 b^3 c (11 A+12 B x)+24 b^2 c^2 x (11 A+10 B x)-10 b c^3 x^2 (33 A+28 B x)+35 c^4 x^3 (11 A+9 B x)+128 b^4 B\right )}{3465 c^5 x^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(x*(b + c*x))^(3/2)*(128*b^4*B + 35*c^4*x^3*(11*A + 9*B*x) + 24*b^2*c^2*x*(11
*A + 10*B*x) - 16*b^3*c*(11*A + 12*B*x) - 10*b*c^3*x^2*(33*A + 28*B*x)))/(3465*c
^5*x^(3/2))

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Maple [A]  time = 0.009, size = 107, normalized size = 0.6 \[ -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( -315\,B{x}^{4}{c}^{4}-385\,A{c}^{4}{x}^{3}+280\,Bb{c}^{3}{x}^{3}+330\,Ab{c}^{3}{x}^{2}-240\,B{b}^{2}{c}^{2}{x}^{2}-264\,A{b}^{2}{c}^{2}x+192\,B{b}^{3}cx+176\,A{b}^{3}c-128\,{b}^{4}B \right ) }{3465\,{c}^{5}}\sqrt{c{x}^{2}+bx}{\frac{1}{\sqrt{x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(5/2)*(B*x+A)*(c*x^2+b*x)^(1/2),x)

[Out]

-2/3465*(c*x+b)*(-315*B*c^4*x^4-385*A*c^4*x^3+280*B*b*c^3*x^3+330*A*b*c^3*x^2-24
0*B*b^2*c^2*x^2-264*A*b^2*c^2*x+192*B*b^3*c*x+176*A*b^3*c-128*B*b^4)*(c*x^2+b*x)
^(1/2)/c^5/x^(1/2)

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Maxima [A]  time = 0.703693, size = 162, normalized size = 0.95 \[ \frac{2 \,{\left (35 \, c^{4} x^{4} + 5 \, b c^{3} x^{3} - 6 \, b^{2} c^{2} x^{2} + 8 \, b^{3} c x - 16 \, b^{4}\right )} \sqrt{c x + b} A}{315 \, c^{4}} + \frac{2 \,{\left (315 \, c^{5} x^{5} + 35 \, b c^{4} x^{4} - 40 \, b^{2} c^{3} x^{3} + 48 \, b^{3} c^{2} x^{2} - 64 \, b^{4} c x + 128 \, b^{5}\right )} \sqrt{c x + b} B}{3465 \, c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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)*sqrt(c*x +
b)*A/c^4 + 2/3465*(315*c^5*x^5 + 35*b*c^4*x^4 - 40*b^2*c^3*x^3 + 48*b^3*c^2*x^2
- 64*b^4*c*x + 128*b^5)*sqrt(c*x + b)*B/c^5

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Fricas [A]  time = 0.271406, size = 209, normalized size = 1.23 \[ \frac{2 \,{\left (315 \, B c^{6} x^{7} + 35 \,{\left (10 \, B b c^{5} + 11 \, A c^{6}\right )} x^{6} - 5 \,{\left (B b^{2} c^{4} - 88 \, A b c^{5}\right )} x^{5} +{\left (8 \, B b^{3} c^{3} - 11 \, A b^{2} c^{4}\right )} x^{4} - 2 \,{\left (8 \, B b^{4} c^{2} - 11 \, A b^{3} c^{3}\right )} x^{3} + 8 \,{\left (8 \, B b^{5} c - 11 \, A b^{4} c^{2}\right )} x^{2} + 16 \,{\left (8 \, B b^{6} - 11 \, A b^{5} c\right )} x\right )}}{3465 \, \sqrt{c x^{2} + b x} c^{5} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3465*(315*B*c^6*x^7 + 35*(10*B*b*c^5 + 11*A*c^6)*x^6 - 5*(B*b^2*c^4 - 88*A*b*c
^5)*x^5 + (8*B*b^3*c^3 - 11*A*b^2*c^4)*x^4 - 2*(8*B*b^4*c^2 - 11*A*b^3*c^3)*x^3
+ 8*(8*B*b^5*c - 11*A*b^4*c^2)*x^2 + 16*(8*B*b^6 - 11*A*b^5*c)*x)/(sqrt(c*x^2 +
b*x)*c^5*sqrt(x))

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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(x**(5/2)*(B*x+A)*(c*x**2+b*x)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.275694, size = 181, normalized size = 1.06 \[ -\frac{2}{3465} \, B{\left (\frac{128 \, b^{\frac{11}{2}}}{c^{5}} - \frac{315 \,{\left (c x + b\right )}^{\frac{11}{2}} - 1540 \,{\left (c x + b\right )}^{\frac{9}{2}} b + 2970 \,{\left (c x + b\right )}^{\frac{7}{2}} b^{2} - 2772 \,{\left (c x + b\right )}^{\frac{5}{2}} b^{3} + 1155 \,{\left (c x + b\right )}^{\frac{3}{2}} b^{4}}{c^{5}}\right )} + \frac{2}{315} \, A{\left (\frac{16 \, b^{\frac{9}{2}}}{c^{4}} + \frac{35 \,{\left (c x + b\right )}^{\frac{9}{2}} - 135 \,{\left (c x + b\right )}^{\frac{7}{2}} b + 189 \,{\left (c x + b\right )}^{\frac{5}{2}} b^{2} - 105 \,{\left (c x + b\right )}^{\frac{3}{2}} b^{3}}{c^{4}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3465*B*(128*b^(11/2)/c^5 - (315*(c*x + b)^(11/2) - 1540*(c*x + b)^(9/2)*b + 2
970*(c*x + b)^(7/2)*b^2 - 2772*(c*x + b)^(5/2)*b^3 + 1155*(c*x + b)^(3/2)*b^4)/c
^5) + 2/315*A*(16*b^(9/2)/c^4 + (35*(c*x + b)^(9/2) - 135*(c*x + b)^(7/2)*b + 18
9*(c*x + b)^(5/2)*b^2 - 105*(c*x + b)^(3/2)*b^3)/c^4)