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

Optimal. Leaf size=47 \[ \frac{2 (a+b x)^{3/2}}{3 b (b-c)}-\frac{2 (a+c x)^{3/2}}{3 c (b-c)} \]

[Out]

(2*(a + b*x)^(3/2))/(3*b*(b - c)) - (2*(a + c*x)^(3/2))/(3*(b - c)*c)

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Rubi [A]  time = 0.0550301, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {2103, 32} \[ \frac{2 (a+b x)^{3/2}}{3 b (b-c)}-\frac{2 (a+c x)^{3/2}}{3 c (b-c)} \]

Antiderivative was successfully verified.

[In]

Int[x/(Sqrt[a + b*x] + Sqrt[a + c*x]),x]

[Out]

(2*(a + b*x)^(3/2))/(3*b*(b - c)) - (2*(a + c*x)^(3/2))/(3*(b - c)*c)

Rule 2103

Int[(u_)/((e_.)*Sqrt[(a_.) + (b_.)*(x_)] + (f_.)*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[c/(e*(b*c - a*d)
), Int[(u*Sqrt[a + b*x])/x, x], x] - Dist[a/(f*(b*c - a*d)), Int[(u*Sqrt[c + d*x])/x, x], x] /; FreeQ[{a, b, c
, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a*e^2 - c*f^2, 0]

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]

Rubi steps

\begin{align*} \int \frac{x}{\sqrt{a+b x}+\sqrt{a+c x}} \, dx &=\frac{\int \sqrt{a+b x} \, dx}{b-c}-\frac{\int \sqrt{a+c x} \, dx}{b-c}\\ &=\frac{2 (a+b x)^{3/2}}{3 b (b-c)}-\frac{2 (a+c x)^{3/2}}{3 (b-c) c}\\ \end{align*}

Mathematica [A]  time = 0.0760489, size = 39, normalized size = 0.83 \[ \frac{2 \left (\frac{(a+b x)^{3/2}}{b}-\frac{(a+c x)^{3/2}}{c}\right )}{3 (b-c)} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(Sqrt[a + b*x] + Sqrt[a + c*x]),x]

[Out]

(2*((a + b*x)^(3/2)/b - (a + c*x)^(3/2)/c))/(3*(b - c))

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Maple [A]  time = 0.002, size = 40, normalized size = 0.9 \begin{align*}{\frac{2}{3\,b \left ( b-c \right ) } \left ( bx+a \right ) ^{{\frac{3}{2}}}}-{\frac{2}{ \left ( 3\,b-3\,c \right ) c} \left ( cx+a \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(b*x+a)^(3/2)/b/(b-c)-2/3*(c*x+a)^(3/2)/(b-c)/c

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\sqrt{b x + a} + \sqrt{c x + a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x/(sqrt(b*x + a) + sqrt(c*x + a)), x)

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Fricas [A]  time = 1.20339, size = 109, normalized size = 2.32 \begin{align*} \frac{2 \,{\left ({\left (b c x + a c\right )} \sqrt{b x + a} -{\left (b c x + a b\right )} \sqrt{c x + a}\right )}}{3 \,{\left (b^{2} c - b c^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\sqrt{a + b x} + \sqrt{a + c x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x/(sqrt(a + b*x) + sqrt(a + c*x)), x)

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError