3.634 \(\int \frac{x}{a+b \sqrt{c+d x}} \, dx\)

Optimal. Leaf size=90 \[ \frac{2 \left (a^2-b^2 c\right ) \sqrt{c+d x}}{b^3 d^2}-\frac{2 a \left (a^2-b^2 c\right ) \log \left (a+b \sqrt{c+d x}\right )}{b^4 d^2}-\frac{a x}{b^2 d}+\frac{2 (c+d x)^{3/2}}{3 b d^2} \]

[Out]

-((a*x)/(b^2*d)) + (2*(a^2 - b^2*c)*Sqrt[c + d*x])/(b^3*d^2) + (2*(c + d*x)^(3/2))/(3*b*d^2) - (2*a*(a^2 - b^2
*c)*Log[a + b*Sqrt[c + d*x]])/(b^4*d^2)

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Rubi [A]  time = 0.0803427, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {371, 1398, 772} \[ \frac{2 \left (a^2-b^2 c\right ) \sqrt{c+d x}}{b^3 d^2}-\frac{2 a \left (a^2-b^2 c\right ) \log \left (a+b \sqrt{c+d x}\right )}{b^4 d^2}-\frac{a x}{b^2 d}+\frac{2 (c+d x)^{3/2}}{3 b d^2} \]

Antiderivative was successfully verified.

[In]

Int[x/(a + b*Sqrt[c + d*x]),x]

[Out]

-((a*x)/(b^2*d)) + (2*(a^2 - b^2*c)*Sqrt[c + d*x])/(b^3*d^2) + (2*(c + d*x)^(3/2))/(3*b*d^2) - (2*a*(a^2 - b^2
*c)*Log[a + b*Sqrt[c + d*x]])/(b^4*d^2)

Rule 371

Int[((a_) + (b_.)*(v_)^(n_))^(p_.)*(x_)^(m_.), x_Symbol] :> With[{c = Coefficient[v, x, 0], d = Coefficient[v,
x, 1]}, Dist[1/d^(m + 1), Subst[Int[SimplifyIntegrand[(x - c)^m*(a + b*x^n)^p, x], x], x, v], x] /; NeQ[c, 0]
] /; FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && IntegerQ[m]

Rule 1398

Int[((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> With[{g = Denominator[n]}, D
ist[g, Subst[Int[x^(g - 1)*(d + e*x^(g*n))^q*(a + c*x^(2*g*n))^p, x], x, x^(1/g)], x]] /; FreeQ[{a, c, d, e, p
, q}, x] && EqQ[n2, 2*n] && FractionQ[n]

Rule 772

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegr
and[(d + e*x)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \frac{x}{a+b \sqrt{c+d x}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{-c+x}{a+b \sqrt{x}} \, dx,x,c+d x\right )}{d^2}\\ &=\frac{2 \operatorname{Subst}\left (\int \frac{x \left (-c+x^2\right )}{a+b x} \, dx,x,\sqrt{c+d x}\right )}{d^2}\\ &=\frac{2 \operatorname{Subst}\left (\int \left (\frac{a^2-b^2 c}{b^3}-\frac{a x}{b^2}+\frac{x^2}{b}+\frac{-a^3+a b^2 c}{b^3 (a+b x)}\right ) \, dx,x,\sqrt{c+d x}\right )}{d^2}\\ &=-\frac{a x}{b^2 d}+\frac{2 \left (a^2-b^2 c\right ) \sqrt{c+d x}}{b^3 d^2}+\frac{2 (c+d x)^{3/2}}{3 b d^2}-\frac{2 a \left (a^2-b^2 c\right ) \log \left (a+b \sqrt{c+d x}\right )}{b^4 d^2}\\ \end{align*}

Mathematica [A]  time = 0.0651472, size = 82, normalized size = 0.91 \[ \frac{b \left (6 a^2 \sqrt{c+d x}-3 a b d x+2 b^2 (d x-2 c) \sqrt{c+d x}\right )-6 \left (a^3-a b^2 c\right ) \log \left (a+b \sqrt{c+d x}\right )}{3 b^4 d^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(a + b*Sqrt[c + d*x]),x]

[Out]

(b*(-3*a*b*d*x + 6*a^2*Sqrt[c + d*x] + 2*b^2*(-2*c + d*x)*Sqrt[c + d*x]) - 6*(a^3 - a*b^2*c)*Log[a + b*Sqrt[c
+ d*x]])/(3*b^4*d^2)

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Maple [A]  time = 0.004, size = 116, normalized size = 1.3 \begin{align*}{\frac{2}{3\,b{d}^{2}} \left ( dx+c \right ) ^{{\frac{3}{2}}}}-{\frac{ax}{{b}^{2}d}}-{\frac{ac}{{b}^{2}{d}^{2}}}-2\,{\frac{c\sqrt{dx+c}}{b{d}^{2}}}+2\,{\frac{\sqrt{dx+c}{a}^{2}}{{b}^{3}{d}^{2}}}+2\,{\frac{a\ln \left ( a+b\sqrt{dx+c} \right ) c}{{b}^{2}{d}^{2}}}-2\,{\frac{{a}^{3}\ln \left ( a+b\sqrt{dx+c} \right ) }{{b}^{4}{d}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(d*x+c)^(3/2)/b/d^2-a*x/b^2/d-1/d^2/b^2*a*c-2/d^2/b*c*(d*x+c)^(1/2)+2/d^2/b^3*(d*x+c)^(1/2)*a^2+2/d^2*a/b^
2*ln(a+b*(d*x+c)^(1/2))*c-2/d^2*a^3/b^4*ln(a+b*(d*x+c)^(1/2))

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Maxima [A]  time = 1.11058, size = 109, normalized size = 1.21 \begin{align*} \frac{\frac{2 \,{\left (d x + c\right )}^{\frac{3}{2}} b^{2} - 3 \,{\left (d x + c\right )} a b - 6 \,{\left (b^{2} c - a^{2}\right )} \sqrt{d x + c}}{b^{3}} + \frac{6 \,{\left (a b^{2} c - a^{3}\right )} \log \left (\sqrt{d x + c} b + a\right )}{b^{4}}}{3 \, d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*((2*(d*x + c)^(3/2)*b^2 - 3*(d*x + c)*a*b - 6*(b^2*c - a^2)*sqrt(d*x + c))/b^3 + 6*(a*b^2*c - a^3)*log(sqr
t(d*x + c)*b + a)/b^4)/d^2

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Fricas [A]  time = 1.64355, size = 166, normalized size = 1.84 \begin{align*} -\frac{3 \, a b^{2} d x - 6 \,{\left (a b^{2} c - a^{3}\right )} \log \left (\sqrt{d x + c} b + a\right ) - 2 \,{\left (b^{3} d x - 2 \, b^{3} c + 3 \, a^{2} b\right )} \sqrt{d x + c}}{3 \, b^{4} d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(3*a*b^2*d*x - 6*(a*b^2*c - a^3)*log(sqrt(d*x + c)*b + a) - 2*(b^3*d*x - 2*b^3*c + 3*a^2*b)*sqrt(d*x + c)
)/(b^4*d^2)

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Sympy [A]  time = 3.07034, size = 109, normalized size = 1.21 \begin{align*} \begin{cases} \frac{2 \left (- \frac{a \left (c + d x\right )}{2 b^{2} d} - \frac{a \left (a^{2} - b^{2} c\right ) \left (\begin{cases} \frac{\sqrt{c + d x}}{a} & \text{for}\: b = 0 \\\frac{\log{\left (a + b \sqrt{c + d x} \right )}}{b} & \text{otherwise} \end{cases}\right )}{b^{3} d} + \frac{\left (c + d x\right )^{\frac{3}{2}}}{3 b d} + \frac{\left (a^{2} - b^{2} c\right ) \sqrt{c + d x}}{b^{3} d}\right )}{d} & \text{for}\: d \neq 0 \\\frac{x^{2}}{2 \left (a + b \sqrt{c}\right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((2*(-a*(c + d*x)/(2*b**2*d) - a*(a**2 - b**2*c)*Piecewise((sqrt(c + d*x)/a, Eq(b, 0)), (log(a + b*sq
rt(c + d*x))/b, True))/(b**3*d) + (c + d*x)**(3/2)/(3*b*d) + (a**2 - b**2*c)*sqrt(c + d*x)/(b**3*d))/d, Ne(d,
0)), (x**2/(2*(a + b*sqrt(c))), True))

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Giac [A]  time = 1.19539, size = 177, normalized size = 1.97 \begin{align*} \frac{\frac{6 \,{\left (a b^{2} c - a^{3}\right )} \log \left ({\left | \sqrt{d x + c} b + a \right |}\right )}{b^{4} d} - \frac{6 \,{\left (a b^{2} c \log \left ({\left | a \right |}\right ) - a^{3} \log \left ({\left | a \right |}\right )\right )}}{b^{4} d} + \frac{2 \,{\left (d x + c\right )}^{\frac{3}{2}} b^{2} d^{2} - 6 \, \sqrt{d x + c} b^{2} c d^{2} - 3 \,{\left (d x + c\right )} a b d^{2} + 6 \, \sqrt{d x + c} a^{2} d^{2}}{b^{3} d^{3}}}{3 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(6*(a*b^2*c - a^3)*log(abs(sqrt(d*x + c)*b + a))/(b^4*d) - 6*(a*b^2*c*log(abs(a)) - a^3*log(abs(a)))/(b^4*
d) + (2*(d*x + c)^(3/2)*b^2*d^2 - 6*sqrt(d*x + c)*b^2*c*d^2 - 3*(d*x + c)*a*b*d^2 + 6*sqrt(d*x + c)*a^2*d^2)/(
b^3*d^3))/d