3.20.1 \(\int \frac {a+b x}{\sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}} \, dx\)

Optimal. Leaf size=39 \[ \frac {2 (a+b x) \sqrt {d+e x}}{e \sqrt {a^2+2 a b x+b^2 x^2}} \]

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {770, 21, 32} \begin {gather*} \frac {2 (a+b x) \sqrt {d+e x}}{e \sqrt {a^2+2 a b x+b^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)/(Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]),x]

[Out]

(2*(a + b*x)*Sqrt[d + e*x])/(e*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 770

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

Rubi steps

\begin {align*} \int \frac {a+b x}{\sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}} \, dx &=\frac {\left (a b+b^2 x\right ) \int \frac {a+b x}{\left (a b+b^2 x\right ) \sqrt {d+e x}} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {\left (a b+b^2 x\right ) \int \frac {1}{\sqrt {d+e x}} \, dx}{b \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {2 (a+b x) \sqrt {d+e x}}{e \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 30, normalized size = 0.77 \begin {gather*} \frac {2 (a+b x) \sqrt {d+e x}}{e \sqrt {(a+b x)^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)/(Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]),x]

[Out]

(2*(a + b*x)*Sqrt[d + e*x])/(e*Sqrt[(a + b*x)^2])

________________________________________________________________________________________

IntegrateAlgebraic [B]  time = 1.82, size = 105, normalized size = 2.69 \begin {gather*} \frac {2 \sqrt {a^2+\frac {2 a b (d+e x)}{e}-\frac {2 a b d}{e}+\frac {b^2 d^2}{e^2}+\frac {b^2 (d+e x)^2}{e^2}-\frac {2 b^2 d (d+e x)}{e^2}}}{\sqrt {b} \left (\sqrt {b} \sqrt {d+e x}-\sqrt {b d-a e}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a + b*x)/(Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]),x]

[Out]

(2*Sqrt[a^2 + (b^2*d^2)/e^2 - (2*a*b*d)/e - (2*b^2*d*(d + e*x))/e^2 + (2*a*b*(d + e*x))/e + (b^2*(d + e*x)^2)/
e^2])/(Sqrt[b]*(-Sqrt[b*d - a*e] + Sqrt[b]*Sqrt[d + e*x]))

________________________________________________________________________________________

fricas [A]  time = 0.41, size = 12, normalized size = 0.31 \begin {gather*} \frac {2 \, \sqrt {e x + d}}{e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/(e*x+d)^(1/2)/((b*x+a)^2)^(1/2),x, algorithm="fricas")

[Out]

2*sqrt(e*x + d)/e

________________________________________________________________________________________

giac [A]  time = 0.16, size = 18, normalized size = 0.46 \begin {gather*} 2 \, \sqrt {x e + d} e^{\left (-1\right )} \mathrm {sgn}\left (b x + a\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/(e*x+d)^(1/2)/((b*x+a)^2)^(1/2),x, algorithm="giac")

[Out]

2*sqrt(x*e + d)*e^(-1)*sgn(b*x + a)

________________________________________________________________________________________

maple [A]  time = 0.05, size = 27, normalized size = 0.69 \begin {gather*} \frac {2 \left (b x +a \right ) \sqrt {e x +d}}{\sqrt {\left (b x +a \right )^{2}}\, e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)/(e*x+d)^(1/2)/((b*x+a)^2)^(1/2),x)

[Out]

2*(b*x+a)*(e*x+d)^(1/2)/e/((b*x+a)^2)^(1/2)

________________________________________________________________________________________

maxima [A]  time = 0.74, size = 12, normalized size = 0.31 \begin {gather*} \frac {2 \, \sqrt {e x + d}}{e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/(e*x+d)^(1/2)/((b*x+a)^2)^(1/2),x, algorithm="maxima")

[Out]

2*sqrt(e*x + d)/e

________________________________________________________________________________________

mupad [B]  time = 2.42, size = 50, normalized size = 1.28 \begin {gather*} \frac {\left (\frac {2\,x}{b}+\frac {2\,d}{b\,e}\right )\,\sqrt {{\left (a+b\,x\right )}^2}}{x\,\sqrt {d+e\,x}+\frac {a\,\sqrt {d+e\,x}}{b}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)/(((a + b*x)^2)^(1/2)*(d + e*x)^(1/2)),x)

[Out]

(((2*x)/b + (2*d)/(b*e))*((a + b*x)^2)^(1/2))/(x*(d + e*x)^(1/2) + (a*(d + e*x)^(1/2))/b)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a + b x}{\sqrt {d + e x} \sqrt {\left (a + b x\right )^{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/(e*x+d)**(1/2)/((b*x+a)**2)**(1/2),x)

[Out]

Integral((a + b*x)/(sqrt(d + e*x)*sqrt((a + b*x)**2)), x)

________________________________________________________________________________________