3.5.28 \(\int \frac {x}{\sqrt {a+b x}+\sqrt {a+c x}} \, dx\) [428]

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

[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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Rubi [A]
time = 0.04, antiderivative size = 47, normalized size of antiderivative = 1.00, 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 = {2128, 32} \begin {gather*} \frac {2 (a+b x)^{3/2}}{3 b (b-c)}-\frac {2 (a+c x)^{3/2}}{3 c (b-c)} \end {gather*}

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 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 2128

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]

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*}

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Mathematica [A]
time = 0.49, size = 71, normalized size = 1.51 \begin {gather*} \frac {2 a c \sqrt {a+b x}+2 b c x \sqrt {a+b x}-2 a b \sqrt {a+c x}-2 b c x \sqrt {a+c x}}{3 b^2 c-3 b c^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.01, size = 40, normalized size = 0.85

method	result	size
default	\(\frac {2 \left (b x +a \right )^{\frac {3}{2}}}{3 b \left (b -c \right )}-\frac {2 \left (c x +a \right )^{\frac {3}{2}}}{3 \left (b -c \right ) c}\)	\(40\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.33, size = 50, normalized size = 1.06 \begin {gather*} \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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\sqrt {a + b x} + \sqrt {a + c x}}\, dx \end {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. 107 vs. \(2 (39) = 78\).
time = 5.18, size = 107, normalized size = 2.28 \begin {gather*} -\frac {2 \, {\left ({\left (\frac {{\left (b x + a\right )} b^{2} c {\left | b \right |}}{b^{5} c - b^{4} c^{2}} + \frac {a b^{3} {\left | b \right |} - a b^{2} c {\left | b \right |}}{b^{5} c - b^{4} c^{2}}\right )} \sqrt {a b^{2} + {\left (b x + a\right )} b c - a b c} - \frac {{\left (b x + a\right )}^{\frac {3}{2}}}{b - c}\right )}}{3 \, b} \end {gather*}

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]

-2/3*(((b*x + a)*b^2*c*abs(b)/(b^5*c - b^4*c^2) + (a*b^3*abs(b) - a*b^2*c*abs(b))/(b^5*c - b^4*c^2))*sqrt(a*b^
2 + (b*x + a)*b*c - a*b*c) - (b*x + a)^(3/2)/(b - c))/b

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Mupad [B]
time = 2.91, size = 79, normalized size = 1.68 \begin {gather*} \frac {2\,x\,\sqrt {a+b\,x}}{3\,\left (b-c\right )}-\frac {2\,x\,\sqrt {a+c\,x}}{3\,\left (b-c\right )}+\frac {2\,a\,\sqrt {a+b\,x}}{3\,b\,\left (b-c\right )}-\frac {2\,a\,\sqrt {a+c\,x}}{3\,c\,\left (b-c\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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