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

Optimal. Leaf size=61 \[ \frac{2 B \left (b x+c x^2\right )^{3/2}}{5 c \sqrt{x}}-\frac{2 \left (b x+c x^2\right )^{3/2} (2 b B-5 A c)}{15 c^2 x^{3/2}} \]

[Out]

(-2*(2*b*B - 5*A*c)*(b*x + c*x^2)^(3/2))/(15*c^2*x^(3/2)) + (2*B*(b*x + c*x^2)^(3/2))/(5*c*Sqrt[x])

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Rubi [A]  time = 0.042698, 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, Rules used = {794, 648} \[ \frac{2 B \left (b x+c x^2\right )^{3/2}}{5 c \sqrt{x}}-\frac{2 \left (b x+c x^2\right )^{3/2} (2 b B-5 A c)}{15 c^2 x^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(2*b*B - 5*A*c)*(b*x + c*x^2)^(3/2))/(15*c^2*x^(3/2)) + (2*B*(b*x + c*x^2)^(3/2))/(5*c*Sqrt[x])

Rule 794

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[(g*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 2)), x] + Dist[(m*(g*(c*d - b*e) + c*e*f) + e*(p + 1)
*(2*c*f - b*g))/(c*e*(m + 2*p + 2)), Int[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g
, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[m + 2*p + 2, 0] && (NeQ[m, 2] || Eq
Q[d, 0])

Rule 648

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)
*(a + b*x + c*x^2)^(p + 1))/(c*(p + 1)), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c
*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + p, 0]

Rubi steps

\begin{align*} \int \frac{(A+B x) \sqrt{b x+c x^2}}{\sqrt{x}} \, dx &=\frac{2 B \left (b x+c x^2\right )^{3/2}}{5 c \sqrt{x}}+\frac{\left (2 \left (\frac{1}{2} (b B-A c)+\frac{3}{2} (-b B+2 A c)\right )\right ) \int \frac{\sqrt{b x+c x^2}}{\sqrt{x}} \, dx}{5 c}\\ &=-\frac{2 (2 b B-5 A c) \left (b x+c x^2\right )^{3/2}}{15 c^2 x^{3/2}}+\frac{2 B \left (b x+c x^2\right )^{3/2}}{5 c \sqrt{x}}\\ \end{align*}

Mathematica [A]  time = 0.026309, size = 37, normalized size = 0.61 \[ \frac{2 (x (b+c x))^{3/2} (5 A c-2 b B+3 B c x)}{15 c^2 x^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.003, size = 39, normalized size = 0.6 \begin{align*}{\frac{ \left ( 2\,cx+2\,b \right ) \left ( 3\,Bcx+5\,Ac-2\,bB \right ) }{15\,{c}^{2}}\sqrt{c{x}^{2}+bx}{\frac{1}{\sqrt{x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(c*x+b)*(3*B*c*x+5*A*c-2*B*b)*(c*x^2+b*x)^(1/2)/c^2/x^(1/2)

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Maxima [A]  time = 1.17729, size = 61, normalized size = 1. \begin{align*} \frac{2 \,{\left (c x + b\right )}^{\frac{3}{2}} A}{3 \, c} + \frac{2 \,{\left (3 \, c^{2} x^{2} + b c x - 2 \, b^{2}\right )} \sqrt{c x + b} B}{15 \, c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^(1/2)/x^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.55229, size = 127, normalized size = 2.08 \begin{align*} \frac{2 \,{\left (3 \, B c^{2} x^{2} - 2 \, B b^{2} + 5 \, A b c +{\left (B b c + 5 \, A c^{2}\right )} x\right )} \sqrt{c x^{2} + b x}}{15 \, c^{2} \sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^(1/2)/x^(1/2),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.14927, size = 81, normalized size = 1.33 \begin{align*} \frac{2}{15} \, 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 )} + \frac{2}{3} \, A{\left (\frac{{\left (c x + b\right )}^{\frac{3}{2}}}{c} - \frac{b^{\frac{3}{2}}}{c}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^(1/2)/x^(1/2),x, algorithm="giac")

[Out]

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