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

Optimal. Leaf size=96 \[ \frac{4 b \left (b x+c x^2\right )^{3/2} (4 b B-7 A c)}{105 c^3 x^{3/2}}-\frac{2 \left (b x+c x^2\right )^{3/2} (4 b B-7 A c)}{35 c^2 \sqrt{x}}+\frac{2 B \sqrt{x} \left (b x+c x^2\right )^{3/2}}{7 c} \]

[Out]

(4*b*(4*b*B - 7*A*c)*(b*x + c*x^2)^(3/2))/(105*c^3*x^(3/2)) - (2*(4*b*B - 7*A*c)
*(b*x + c*x^2)^(3/2))/(35*c^2*Sqrt[x]) + (2*B*Sqrt[x]*(b*x + c*x^2)^(3/2))/(7*c)

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Rubi [A]  time = 0.189086, 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 \[ \frac{4 b \left (b x+c x^2\right )^{3/2} (4 b B-7 A c)}{105 c^3 x^{3/2}}-\frac{2 \left (b x+c x^2\right )^{3/2} (4 b B-7 A c)}{35 c^2 \sqrt{x}}+\frac{2 B \sqrt{x} \left (b x+c x^2\right )^{3/2}}{7 c} \]

Antiderivative was successfully verified.

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

[Out]

(4*b*(4*b*B - 7*A*c)*(b*x + c*x^2)^(3/2))/(105*c^3*x^(3/2)) - (2*(4*b*B - 7*A*c)
*(b*x + c*x^2)^(3/2))/(35*c^2*Sqrt[x]) + (2*B*Sqrt[x]*(b*x + c*x^2)^(3/2))/(7*c)

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Rubi in Sympy [A]  time = 11.5315, size = 92, normalized size = 0.96 \[ \frac{2 B \sqrt{x} \left (b x + c x^{2}\right )^{\frac{3}{2}}}{7 c} - \frac{4 b \left (7 A c - 4 B b\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}}}{105 c^{3} x^{\frac{3}{2}}} + \frac{2 \left (7 A c - 4 B b\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}}}{35 c^{2} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*B*sqrt(x)*(b*x + c*x**2)**(3/2)/(7*c) - 4*b*(7*A*c - 4*B*b)*(b*x + c*x**2)**(3
/2)/(105*c**3*x**(3/2)) + 2*(7*A*c - 4*B*b)*(b*x + c*x**2)**(3/2)/(35*c**2*sqrt(
x))

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Mathematica [A]  time = 0.0652849, size = 56, normalized size = 0.58 \[ \frac{2 (x (b+c x))^{3/2} \left (-2 b c (7 A+6 B x)+3 c^2 x (7 A+5 B x)+8 b^2 B\right )}{105 c^3 x^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(x*(b + c*x))^(3/2)*(8*b^2*B + 3*c^2*x*(7*A + 5*B*x) - 2*b*c*(7*A + 6*B*x)))/
(105*c^3*x^(3/2))

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Maple [A]  time = 0.006, size = 59, normalized size = 0.6 \[ -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( -15\,B{c}^{2}{x}^{2}-21\,Ax{c}^{2}+12\,Bxbc+14\,Abc-8\,{b}^{2}B \right ) }{105\,{c}^{3}}\sqrt{c{x}^{2}+bx}{\frac{1}{\sqrt{x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/105*(c*x+b)*(-15*B*c^2*x^2-21*A*c^2*x+12*B*b*c*x+14*A*b*c-8*B*b^2)*(c*x^2+b*x
)^(1/2)/c^3/x^(1/2)

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Maxima [A]  time = 0.699234, size = 101, normalized size = 1.05 \[ \frac{2 \,{\left (3 \, c^{2} x^{2} + b c x - 2 \, b^{2}\right )} \sqrt{c x + b} A}{15 \, c^{2}} + \frac{2 \,{\left (15 \, c^{3} x^{3} + 3 \, b c^{2} x^{2} - 4 \, b^{2} c x + 8 \, b^{3}\right )} \sqrt{c x + b} B}{105 \, c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.274801, size = 144, normalized size = 1.5 \[ \frac{2 \,{\left (15 \, B c^{4} x^{5} + 3 \,{\left (6 \, B b c^{3} + 7 \, A c^{4}\right )} x^{4} -{\left (B b^{2} c^{2} - 28 \, A b c^{3}\right )} x^{3} +{\left (4 \, B b^{3} c - 7 \, A b^{2} c^{2}\right )} x^{2} + 2 \,{\left (4 \, B b^{4} - 7 \, A b^{3} c\right )} x\right )}}{105 \, \sqrt{c x^{2} + b x} c^{3} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/105*(15*B*c^4*x^5 + 3*(6*B*b*c^3 + 7*A*c^4)*x^4 - (B*b^2*c^2 - 28*A*b*c^3)*x^3
+ (4*B*b^3*c - 7*A*b^2*c^2)*x^2 + 2*(4*B*b^4 - 7*A*b^3*c)*x)/(sqrt(c*x^2 + b*x)
*c^3*sqrt(x))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.27323, size = 116, normalized size = 1.21 \[ -\frac{2}{105} \, B{\left (\frac{8 \, b^{\frac{7}{2}}}{c^{3}} - \frac{15 \,{\left (c x + b\right )}^{\frac{7}{2}} - 42 \,{\left (c x + b\right )}^{\frac{5}{2}} b + 35 \,{\left (c x + b\right )}^{\frac{3}{2}} b^{2}}{c^{3}}\right )} + \frac{2}{15} \, A{\left (\frac{2 \, b^{\frac{5}{2}}}{c^{2}} + \frac{3 \,{\left (c x + b\right )}^{\frac{5}{2}} - 5 \,{\left (c x + b\right )}^{\frac{3}{2}} b}{c^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/105*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) + 2/15*A*(2*b^(5/2)/c^2 + (3*(c*x + b)^(5/2) - 5*(c*x + b)^(
3/2)*b)/c^2)