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

Optimal. Leaf size=133 \[ -\frac{16 b^2 \left (b x+c x^2\right )^{5/2} (6 b B-11 A c)}{3465 c^4 x^{5/2}}+\frac{8 b \left (b x+c x^2\right )^{5/2} (6 b B-11 A c)}{693 c^3 x^{3/2}}-\frac{2 \left (b x+c x^2\right )^{5/2} (6 b B-11 A c)}{99 c^2 \sqrt{x}}+\frac{2 B \sqrt{x} \left (b x+c x^2\right )^{5/2}}{11 c} \]

[Out]

(-16*b^2*(6*b*B - 11*A*c)*(b*x + c*x^2)^(5/2))/(3465*c^4*x^(5/2)) + (8*b*(6*b*B
- 11*A*c)*(b*x + c*x^2)^(5/2))/(693*c^3*x^(3/2)) - (2*(6*b*B - 11*A*c)*(b*x + c*
x^2)^(5/2))/(99*c^2*Sqrt[x]) + (2*B*Sqrt[x]*(b*x + c*x^2)^(5/2))/(11*c)

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Rubi [A]  time = 0.262634, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{16 b^2 \left (b x+c x^2\right )^{5/2} (6 b B-11 A c)}{3465 c^4 x^{5/2}}+\frac{8 b \left (b x+c x^2\right )^{5/2} (6 b B-11 A c)}{693 c^3 x^{3/2}}-\frac{2 \left (b x+c x^2\right )^{5/2} (6 b B-11 A c)}{99 c^2 \sqrt{x}}+\frac{2 B \sqrt{x} \left (b x+c x^2\right )^{5/2}}{11 c} \]

Antiderivative was successfully verified.

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

[Out]

(-16*b^2*(6*b*B - 11*A*c)*(b*x + c*x^2)^(5/2))/(3465*c^4*x^(5/2)) + (8*b*(6*b*B
- 11*A*c)*(b*x + c*x^2)^(5/2))/(693*c^3*x^(3/2)) - (2*(6*b*B - 11*A*c)*(b*x + c*
x^2)^(5/2))/(99*c^2*Sqrt[x]) + (2*B*Sqrt[x]*(b*x + c*x^2)^(5/2))/(11*c)

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Rubi in Sympy [A]  time = 16.0151, size = 129, normalized size = 0.97 \[ \frac{2 B \sqrt{x} \left (b x + c x^{2}\right )^{\frac{5}{2}}}{11 c} + \frac{16 b^{2} \left (11 A c - 6 B b\right ) \left (b x + c x^{2}\right )^{\frac{5}{2}}}{3465 c^{4} x^{\frac{5}{2}}} - \frac{8 b \left (11 A c - 6 B b\right ) \left (b x + c x^{2}\right )^{\frac{5}{2}}}{693 c^{3} x^{\frac{3}{2}}} + \frac{2 \left (11 A c - 6 B b\right ) \left (b x + c x^{2}\right )^{\frac{5}{2}}}{99 c^{2} \sqrt{x}} \]

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

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Mathematica [A]  time = 0.0934002, size = 75, normalized size = 0.56 \[ \frac{2 (x (b+c x))^{5/2} \left (8 b^2 c (11 A+15 B x)-10 b c^2 x (22 A+21 B x)+35 c^3 x^2 (11 A+9 B x)-48 b^3 B\right )}{3465 c^4 x^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 83, normalized size = 0.6 \[{\frac{ \left ( 2\,cx+2\,b \right ) \left ( 315\,B{c}^{3}{x}^{3}+385\,A{c}^{3}{x}^{2}-210\,Bb{c}^{2}{x}^{2}-220\,Ab{c}^{2}x+120\,B{b}^{2}cx+88\,A{b}^{2}c-48\,B{b}^{3} \right ) }{3465\,{c}^{4}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}{x}^{-{\frac{3}{2}}}} \]

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]

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

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Maxima [A]  time = 0.708581, size = 309, normalized size = 2.32 \[ \frac{2 \,{\left ({\left (35 \, c^{4} x^{4} + 5 \, b c^{3} x^{3} - 6 \, b^{2} c^{2} x^{2} + 8 \, b^{3} c x - 16 \, b^{4}\right )} x^{3} + 3 \,{\left (15 \, b c^{3} x^{4} + 3 \, b^{2} c^{2} x^{3} - 4 \, b^{3} c x^{2} + 8 \, b^{4} x\right )} x^{2}\right )} \sqrt{c x + b} A}{315 \, c^{3} x^{3}} + \frac{2 \,{\left ({\left (315 \, c^{5} x^{5} + 35 \, b c^{4} x^{4} - 40 \, b^{2} c^{3} x^{3} + 48 \, b^{3} c^{2} x^{2} - 64 \, b^{4} c x + 128 \, b^{5}\right )} x^{4} + 11 \,{\left (35 \, b c^{4} x^{5} + 5 \, b^{2} c^{3} x^{4} - 6 \, b^{3} c^{2} x^{3} + 8 \, b^{4} c x^{2} - 16 \, b^{5} x\right )} x^{3}\right )} \sqrt{c x + b} B}{3465 \, c^{4} x^{4}} \]

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]

2/315*((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)*A/(c^3*
x^3) + 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 - 6
4*b^4*c*x + 128*b^5)*x^4 + 11*(35*b*c^4*x^5 + 5*b^2*c^3*x^4 - 6*b^3*c^2*x^3 + 8*
b^4*c*x^2 - 16*b^5*x)*x^3)*sqrt(c*x + b)*B/(c^4*x^4)

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Fricas [A]  time = 0.286727, size = 211, normalized size = 1.59 \[ \frac{2 \,{\left (315 \, B c^{6} x^{7} + 35 \,{\left (21 \, B b c^{5} + 11 \, A c^{6}\right )} x^{6} + 5 \,{\left (87 \, B b^{2} c^{4} + 187 \, A b c^{5}\right )} x^{5} -{\left (3 \, B b^{3} c^{3} - 583 \, A b^{2} c^{4}\right )} x^{4} +{\left (6 \, B b^{4} c^{2} - 11 \, A b^{3} c^{3}\right )} x^{3} - 4 \,{\left (6 \, B b^{5} c - 11 \, A b^{4} c^{2}\right )} x^{2} - 8 \,{\left (6 \, B b^{6} - 11 \, A b^{5} c\right )} x\right )}}{3465 \, \sqrt{c x^{2} + b x} c^{4} \sqrt{x}} \]

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]

2/3465*(315*B*c^6*x^7 + 35*(21*B*b*c^5 + 11*A*c^6)*x^6 + 5*(87*B*b^2*c^4 + 187*A
*b*c^5)*x^5 - (3*B*b^3*c^3 - 583*A*b^2*c^4)*x^4 + (6*B*b^4*c^2 - 11*A*b^3*c^3)*x
^3 - 4*(6*B*b^5*c - 11*A*b^4*c^2)*x^2 - 8*(6*B*b^6 - 11*A*b^5*c)*x)/(sqrt(c*x^2
+ b*x)*c^4*sqrt(x))

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

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

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GIAC/XCAS [A]  time = 0.280953, size = 333, normalized size = 2.5 \[ -\frac{2}{3465} \, B c{\left (\frac{128 \, b^{\frac{11}{2}}}{c^{5}} - \frac{315 \,{\left (c x + b\right )}^{\frac{11}{2}} - 1540 \,{\left (c x + b\right )}^{\frac{9}{2}} b + 2970 \,{\left (c x + b\right )}^{\frac{7}{2}} b^{2} - 2772 \,{\left (c x + b\right )}^{\frac{5}{2}} b^{3} + 1155 \,{\left (c x + b\right )}^{\frac{3}{2}} b^{4}}{c^{5}}\right )} + \frac{2}{315} \, B b{\left (\frac{16 \, b^{\frac{9}{2}}}{c^{4}} + \frac{35 \,{\left (c x + b\right )}^{\frac{9}{2}} - 135 \,{\left (c x + b\right )}^{\frac{7}{2}} b + 189 \,{\left (c x + b\right )}^{\frac{5}{2}} b^{2} - 105 \,{\left (c x + b\right )}^{\frac{3}{2}} b^{3}}{c^{4}}\right )} + \frac{2}{315} \, A c{\left (\frac{16 \, b^{\frac{9}{2}}}{c^{4}} + \frac{35 \,{\left (c x + b\right )}^{\frac{9}{2}} - 135 \,{\left (c x + b\right )}^{\frac{7}{2}} b + 189 \,{\left (c x + b\right )}^{\frac{5}{2}} b^{2} - 105 \,{\left (c x + b\right )}^{\frac{3}{2}} b^{3}}{c^{4}}\right )} - \frac{2}{105} \, A b{\left (\frac{8 \, b^{\frac{7}{2}}}{c^{3}} - \frac{15 \,{\left (c x + b\right )}^{\frac{7}{2}} - 42 \,{\left (c x + b\right )}^{\frac{5}{2}} b + 35 \,{\left (c x + b\right )}^{\frac{3}{2}} b^{2}}{c^{3}}\right )} \]

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]

-2/3465*B*c*(128*b^(11/2)/c^5 - (315*(c*x + b)^(11/2) - 1540*(c*x + b)^(9/2)*b +
2970*(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*B*b*(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) + 2/315*A*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*A*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)