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

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

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Rubi [A]  time = 0.04, antiderivative size = 61, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {2 B \sqrt {x} \sqrt {b x+c x^2}}{3 c}-\frac {2 \sqrt {b x+c x^2} (2 b B-3 A c)}{3 c^2 \sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]

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])

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 36, normalized size = 0.59 \begin {gather*} \frac {2 \sqrt {x (b+c x)} (3 A c-2 b B+B c x)}{3 c^2 \sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.08, size = 38, normalized size = 0.62 \begin {gather*} \frac {2 \sqrt {b x+c x^2} (3 A c-2 b B+B c x)}{3 c^2 \sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.40, size = 32, normalized size = 0.52 \begin {gather*} \frac {2 \, {\left (B c x - 2 \, B b + 3 \, A c\right )} \sqrt {c x^{2} + b x}}{3 \, c^{2} \sqrt {x}} \end {gather*}

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, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.18, size = 53, normalized size = 0.87 \begin {gather*} \frac {2 \, {\left (c x + b\right )}^{\frac {3}{2}} B}{3 \, c^{2}} - \frac {2 \, {\left (B b - A c\right )} \sqrt {c x + b}}{c^{2}} + \frac {2 \, {\left (2 \, B b^{\frac {3}{2}} - 3 \, A \sqrt {b} c\right )}}{3 \, c^{2}} \end {gather*}

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, algorithm="giac")

[Out]

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

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maple [A]  time = 0.05, size = 38, normalized size = 0.62 \begin {gather*} \frac {2 \left (c x +b \right ) \left (B c x +3 A c -2 b B \right ) \sqrt {x}}{3 \sqrt {c \,x^{2}+b x}\, c^{2}} \end {gather*}

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]

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

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maxima [A]  time = 0.62, size = 45, normalized size = 0.74 \begin {gather*} \frac {2 \, \sqrt {c x + b} A}{c} + \frac {2 \, {\left (c^{2} x^{2} - b c x - 2 \, b^{2}\right )} B}{3 \, \sqrt {c x + b} c^{2}} \end {gather*}

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, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\sqrt {x}\,\left (A+B\,x\right )}{\sqrt {c\,x^2+b\,x}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x} \left (A + B x\right )}{\sqrt {x \left (b + c x\right )}}\, dx \end {gather*}

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)*(A + B*x)/sqrt(x*(b + c*x)), x)

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