3.441 \(\int \frac{x}{(\sqrt{a+b x}+\sqrt{a+c x})^3} \, dx\)

Optimal. Leaf size=157 \[ -\frac{4 a \sqrt{a+b x}}{x (b-c)^3}+\frac{4 a \sqrt{a+c x}}{x (b-c)^3}+\frac{2 (b+3 c) \sqrt{a+b x}}{(b-c)^3}-\frac{2 (3 b+c) \sqrt{a+c x}}{(b-c)^3}-\frac{6 \sqrt{a} (b+c) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{(b-c)^3}+\frac{6 \sqrt{a} (b+c) \tanh ^{-1}\left (\frac{\sqrt{a+c x}}{\sqrt{a}}\right )}{(b-c)^3} \]

[Out]

(2*(b + 3*c)*Sqrt[a + b*x])/(b - c)^3 - (4*a*Sqrt[a + b*x])/((b - c)^3*x) - (2*(3*b + c)*Sqrt[a + c*x])/(b - c
)^3 + (4*a*Sqrt[a + c*x])/((b - c)^3*x) - (6*Sqrt[a]*(b + c)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(b - c)^3 + (6*Sq
rt[a]*(b + c)*ArcTanh[Sqrt[a + c*x]/Sqrt[a]])/(b - c)^3

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Rubi [A]  time = 0.217252, antiderivative size = 223, normalized size of antiderivative = 1.42, number of steps used = 14, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {6690, 47, 63, 208, 50} \[ -\frac{4 a \sqrt{a+b x}}{x (b-c)^3}+\frac{4 a \sqrt{a+c x}}{x (b-c)^3}+\frac{2 (b+3 c) \sqrt{a+b x}}{(b-c)^3}-\frac{2 (3 b+c) \sqrt{a+c x}}{(b-c)^3}-\frac{2 \sqrt{a} (b+3 c) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{(b-c)^3}-\frac{4 \sqrt{a} b \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{(b-c)^3}+\frac{4 \sqrt{a} c \tanh ^{-1}\left (\frac{\sqrt{a+c x}}{\sqrt{a}}\right )}{(b-c)^3}+\frac{2 \sqrt{a} (3 b+c) \tanh ^{-1}\left (\frac{\sqrt{a+c x}}{\sqrt{a}}\right )}{(b-c)^3} \]

Antiderivative was successfully verified.

[In]

Int[x/(Sqrt[a + b*x] + Sqrt[a + c*x])^3,x]

[Out]

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

Rule 6690

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

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rubi steps

\begin{align*} \int \frac{x}{\left (\sqrt{a+b x}+\sqrt{a+c x}\right )^3} \, dx &=\frac{\int \left (\frac{4 a \sqrt{a+b x}}{x^2}+\frac{b \left (1+\frac{3 c}{b}\right ) \sqrt{a+b x}}{x}-\frac{4 a \sqrt{a+c x}}{x^2}-\frac{3 b \left (1+\frac{c}{3 b}\right ) \sqrt{a+c x}}{x}\right ) \, dx}{(b-c)^3}\\ &=\frac{(4 a) \int \frac{\sqrt{a+b x}}{x^2} \, dx}{(b-c)^3}-\frac{(4 a) \int \frac{\sqrt{a+c x}}{x^2} \, dx}{(b-c)^3}-\frac{(3 b+c) \int \frac{\sqrt{a+c x}}{x} \, dx}{(b-c)^3}+\frac{(b+3 c) \int \frac{\sqrt{a+b x}}{x} \, dx}{(b-c)^3}\\ &=\frac{2 (b+3 c) \sqrt{a+b x}}{(b-c)^3}-\frac{4 a \sqrt{a+b x}}{(b-c)^3 x}-\frac{2 (3 b+c) \sqrt{a+c x}}{(b-c)^3}+\frac{4 a \sqrt{a+c x}}{(b-c)^3 x}+\frac{(2 a b) \int \frac{1}{x \sqrt{a+b x}} \, dx}{(b-c)^3}-\frac{(2 a c) \int \frac{1}{x \sqrt{a+c x}} \, dx}{(b-c)^3}-\frac{(a (3 b+c)) \int \frac{1}{x \sqrt{a+c x}} \, dx}{(b-c)^3}+\frac{(a (b+3 c)) \int \frac{1}{x \sqrt{a+b x}} \, dx}{(b-c)^3}\\ &=\frac{2 (b+3 c) \sqrt{a+b x}}{(b-c)^3}-\frac{4 a \sqrt{a+b x}}{(b-c)^3 x}-\frac{2 (3 b+c) \sqrt{a+c x}}{(b-c)^3}+\frac{4 a \sqrt{a+c x}}{(b-c)^3 x}+\frac{(4 a) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x}\right )}{(b-c)^3}-\frac{(4 a) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{c}+\frac{x^2}{c}} \, dx,x,\sqrt{a+c x}\right )}{(b-c)^3}-\frac{(2 a (3 b+c)) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{c}+\frac{x^2}{c}} \, dx,x,\sqrt{a+c x}\right )}{(b-c)^3 c}+\frac{(2 a (b+3 c)) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x}\right )}{b (b-c)^3}\\ &=\frac{2 (b+3 c) \sqrt{a+b x}}{(b-c)^3}-\frac{4 a \sqrt{a+b x}}{(b-c)^3 x}-\frac{2 (3 b+c) \sqrt{a+c x}}{(b-c)^3}+\frac{4 a \sqrt{a+c x}}{(b-c)^3 x}-\frac{4 \sqrt{a} b \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{(b-c)^3}-\frac{2 \sqrt{a} (b+3 c) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{(b-c)^3}+\frac{4 \sqrt{a} c \tanh ^{-1}\left (\frac{\sqrt{a+c x}}{\sqrt{a}}\right )}{(b-c)^3}+\frac{2 \sqrt{a} (3 b+c) \tanh ^{-1}\left (\frac{\sqrt{a+c x}}{\sqrt{a}}\right )}{(b-c)^3}\\ \end{align*}

Mathematica [A]  time = 0.812005, size = 192, normalized size = 1.22 \[ \frac{2 \left (-(3 b+c) \sqrt{a+c x}+(b+3 c) \sqrt{a+b x}+\sqrt{a} (3 b+c) \tanh ^{-1}\left (\frac{\sqrt{a+c x}}{\sqrt{a}}\right )-\sqrt{a} (b+3 c) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )-\frac{2 a \left (b x \sqrt{\frac{b x}{a}+1} \tanh ^{-1}\left (\sqrt{\frac{b x}{a}+1}\right )+a+b x\right )}{x \sqrt{a+b x}}+\frac{2 a \left (c x \sqrt{\frac{c x}{a}+1} \tanh ^{-1}\left (\sqrt{\frac{c x}{a}+1}\right )+a+c x\right )}{x \sqrt{a+c x}}\right )}{(b-c)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(Sqrt[a + b*x] + Sqrt[a + c*x])^3,x]

[Out]

(2*((b + 3*c)*Sqrt[a + b*x] - (3*b + c)*Sqrt[a + c*x] - Sqrt[a]*(b + 3*c)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]] - (2*
a*(a + b*x + b*x*Sqrt[1 + (b*x)/a]*ArcTanh[Sqrt[1 + (b*x)/a]]))/(x*Sqrt[a + b*x]) + Sqrt[a]*(3*b + c)*ArcTanh[
Sqrt[a + c*x]/Sqrt[a]] + (2*a*(a + c*x + c*x*Sqrt[1 + (c*x)/a]*ArcTanh[Sqrt[1 + (c*x)/a]]))/(x*Sqrt[a + c*x]))
)/(b - c)^3

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Maple [A]  time = 0.003, size = 237, normalized size = 1.5 \begin{align*}{\frac{b}{ \left ( b-c \right ) ^{3}} \left ( 2\,\sqrt{bx+a}-2\,\sqrt{a}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) \right ) }+8\,{\frac{ab}{ \left ( b-c \right ) ^{3}} \left ( -1/2\,{\frac{\sqrt{bx+a}}{bx}}-1/2\,{\frac{1}{\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) }-8\,{\frac{ac}{ \left ( b-c \right ) ^{3}} \left ( -1/2\,{\frac{\sqrt{cx+a}}{cx}}-1/2\,{\frac{1}{\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{cx+a}}{\sqrt{a}}} \right ) } \right ) }+3\,{\frac{c}{ \left ( b-c \right ) ^{3}} \left ( 2\,\sqrt{bx+a}-2\,\sqrt{a}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) \right ) }-3\,{\frac{b}{ \left ( b-c \right ) ^{3}} \left ( 2\,\sqrt{cx+a}-2\,\sqrt{a}{\it Artanh} \left ({\frac{\sqrt{cx+a}}{\sqrt{a}}} \right ) \right ) }-{\frac{c}{ \left ( b-c \right ) ^{3}} \left ( 2\,\sqrt{cx+a}-2\,\sqrt{a}{\it Artanh} \left ({\frac{\sqrt{cx+a}}{\sqrt{a}}} \right ) \right ) } \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))^3,x)

[Out]

1/(b-c)^3*b*(2*(b*x+a)^(1/2)-2*a^(1/2)*arctanh((b*x+a)^(1/2)/a^(1/2)))+8/(b-c)^3*a*b*(-1/2*(b*x+a)^(1/2)/b/x-1
/2*arctanh((b*x+a)^(1/2)/a^(1/2))/a^(1/2))-8/(b-c)^3*a*c*(-1/2*(c*x+a)^(1/2)/c/x-1/2/a^(1/2)*arctanh((c*x+a)^(
1/2)/a^(1/2)))+3/(b-c)^3*c*(2*(b*x+a)^(1/2)-2*a^(1/2)*arctanh((b*x+a)^(1/2)/a^(1/2)))-3/(b-c)^3*b*(2*(c*x+a)^(
1/2)-2*a^(1/2)*arctanh((c*x+a)^(1/2)/a^(1/2)))-1/(b-c)^3*c*(2*(c*x+a)^(1/2)-2*a^(1/2)*arctanh((c*x+a)^(1/2)/a^
(1/2)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{{\left (\sqrt{b x + a} + \sqrt{c x + a}\right )}^{3}}\,{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))^3,x, algorithm="maxima")

[Out]

integrate(x/(sqrt(b*x + a) + sqrt(c*x + a))^3, x)

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Fricas [A]  time = 1.39099, size = 641, normalized size = 4.08 \begin{align*} \left [-\frac{3 \, \sqrt{a}{\left (b + c\right )} x \log \left (\frac{b x + 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) + 3 \, \sqrt{a}{\left (b + c\right )} x \log \left (\frac{c x - 2 \, \sqrt{c x + a} \sqrt{a} + 2 \, a}{x}\right ) - 2 \,{\left ({\left (b + 3 \, c\right )} x - 2 \, a\right )} \sqrt{b x + a} + 2 \,{\left ({\left (3 \, b + c\right )} x - 2 \, a\right )} \sqrt{c x + a}}{{\left (b^{3} - 3 \, b^{2} c + 3 \, b c^{2} - c^{3}\right )} x}, \frac{2 \,{\left (3 \, \sqrt{-a}{\left (b + c\right )} x \arctan \left (\frac{\sqrt{b x + a} \sqrt{-a}}{a}\right ) - 3 \, \sqrt{-a}{\left (b + c\right )} x \arctan \left (\frac{\sqrt{c x + a} \sqrt{-a}}{a}\right ) +{\left ({\left (b + 3 \, c\right )} x - 2 \, a\right )} \sqrt{b x + a} -{\left ({\left (3 \, b + c\right )} x - 2 \, a\right )} \sqrt{c x + a}\right )}}{{\left (b^{3} - 3 \, b^{2} c + 3 \, b c^{2} - c^{3}\right )} x}\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))^3,x, algorithm="fricas")

[Out]

[-(3*sqrt(a)*(b + c)*x*log((b*x + 2*sqrt(b*x + a)*sqrt(a) + 2*a)/x) + 3*sqrt(a)*(b + c)*x*log((c*x - 2*sqrt(c*
x + a)*sqrt(a) + 2*a)/x) - 2*((b + 3*c)*x - 2*a)*sqrt(b*x + a) + 2*((3*b + c)*x - 2*a)*sqrt(c*x + a))/((b^3 -
3*b^2*c + 3*b*c^2 - c^3)*x), 2*(3*sqrt(-a)*(b + c)*x*arctan(sqrt(b*x + a)*sqrt(-a)/a) - 3*sqrt(-a)*(b + c)*x*a
rctan(sqrt(c*x + a)*sqrt(-a)/a) + ((b + 3*c)*x - 2*a)*sqrt(b*x + a) - ((3*b + c)*x - 2*a)*sqrt(c*x + a))/((b^3
- 3*b^2*c + 3*b*c^2 - c^3)*x)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\left (\sqrt{a + b x} + \sqrt{a + c x}\right )^{3}}\, 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))**3,x)

[Out]

Integral(x/(sqrt(a + b*x) + sqrt(a + c*x))**3, x)

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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))^3,x, algorithm="giac")

[Out]

Timed out