3.5.42 \(\int \frac {1}{(\sqrt {a+b x}+\sqrt {a+c x})^3} \, dx\) [442]

Optimal. Leaf size=164 \[ -\frac {2 a \sqrt {a+b x}}{(b-c)^3 x^2}-\frac {(2 b+3 c) \sqrt {a+b x}}{(b-c)^3 x}+\frac {2 a \sqrt {a+c x}}{(b-c)^3 x^2}+\frac {(3 b+2 c) \sqrt {a+c x}}{(b-c)^3 x}-\frac {3 b c \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{\sqrt {a} (b-c)^3}+\frac {3 b c \tanh ^{-1}\left (\frac {\sqrt {a+c x}}{\sqrt {a}}\right )}{\sqrt {a} (b-c)^3} \]

[Out]

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

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Rubi [A]
time = 0.13, antiderivative size = 275, normalized size of antiderivative = 1.68, number of steps used = 16, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {6822, 43, 44, 65, 214} \begin {gather*} \frac {b^2 \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{\sqrt {a} (b-c)^3}-\frac {c^2 \tanh ^{-1}\left (\frac {\sqrt {a+c x}}{\sqrt {a}}\right )}{\sqrt {a} (b-c)^3}-\frac {2 a \sqrt {a+b x}}{x^2 (b-c)^3}+\frac {2 a \sqrt {a+c x}}{x^2 (b-c)^3}-\frac {b \sqrt {a+b x}}{x (b-c)^3}-\frac {(b+3 c) \sqrt {a+b x}}{x (b-c)^3}+\frac {c \sqrt {a+c x}}{x (b-c)^3}+\frac {(3 b+c) \sqrt {a+c x}}{x (b-c)^3}-\frac {b (b+3 c) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{\sqrt {a} (b-c)^3}+\frac {c (3 b+c) \tanh ^{-1}\left (\frac {\sqrt {a+c x}}{\sqrt {a}}\right )}{\sqrt {a} (b-c)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 43

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, n
}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && GtQ[n, 0]

Rule 44

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

Rule 65

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 214

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

Rule 6822

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]

Rubi steps

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

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Mathematica [A]
time = 10.16, size = 146, normalized size = 0.89 \begin {gather*} \frac {\sqrt {a} \left (-2 a \sqrt {a+b x}-2 b x \sqrt {a+b x}-3 c x \sqrt {a+b x}+2 a \sqrt {a+c x}+3 b x \sqrt {a+c x}+2 c x \sqrt {a+c x}\right )-3 b c x^2 \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )+3 b c x^2 \tanh ^{-1}\left (\frac {\sqrt {a+c x}}{\sqrt {a}}\right )}{\sqrt {a} (b-c)^3 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(299\) vs. \(2(144)=288\).
time = 0.02, size = 300, normalized size = 1.83

method	result	size
default	\(\frac {2 b^{2} \left (-\frac {\sqrt {b x +a}}{2 x b}-\frac {\arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{2 \sqrt {a}}\right )}{\left (b -c \right )^{3}}+\frac {8 a \,b^{2} \left (\frac {-\frac {\left (b x +a \right )^{\frac {3}{2}}}{8 a}-\frac {\sqrt {b x +a}}{8}}{x^{2} b^{2}}+\frac {\arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{8 a^{\frac {3}{2}}}\right )}{\left (b -c \right )^{3}}-\frac {8 a \,c^{2} \left (\frac {-\frac {\left (c x +a \right )^{\frac {3}{2}}}{8 a}-\frac {\sqrt {c x +a}}{8}}{x^{2} c^{2}}+\frac {\arctanh \left (\frac {\sqrt {c x +a}}{\sqrt {a}}\right )}{8 a^{\frac {3}{2}}}\right )}{\left (b -c \right )^{3}}+\frac {6 c b \left (-\frac {\sqrt {b x +a}}{2 x b}-\frac {\arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{2 \sqrt {a}}\right )}{\left (b -c \right )^{3}}-\frac {6 b c \left (-\frac {\sqrt {c x +a}}{2 x c}-\frac {\arctanh \left (\frac {\sqrt {c x +a}}{\sqrt {a}}\right )}{2 \sqrt {a}}\right )}{\left (b -c \right )^{3}}-\frac {2 c^{2} \left (-\frac {\sqrt {c x +a}}{2 x c}-\frac {\arctanh \left (\frac {\sqrt {c x +a}}{\sqrt {a}}\right )}{2 \sqrt {a}}\right )}{\left (b -c \right )^{3}}\)	\(300\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((b*x+a)^(1/2)+(c*x+a)^(1/2))^3,x,method=_RETURNVERBOSE)

[Out]

2/(b-c)^3*b^2*(-1/2*(b*x+a)^(1/2)/x/b-1/2*arctanh((b*x+a)^(1/2)/a^(1/2))/a^(1/2))+8/(b-c)^3*a*b^2*((-1/8/a*(b*
x+a)^(3/2)-1/8*(b*x+a)^(1/2))/x^2/b^2+1/8/a^(3/2)*arctanh((b*x+a)^(1/2)/a^(1/2)))-8/(b-c)^3*a*c^2*((-1/8/a*(c*
x+a)^(3/2)-1/8*(c*x+a)^(1/2))/x^2/c^2+1/8/a^(3/2)*arctanh((c*x+a)^(1/2)/a^(1/2)))+6/(b-c)^3*c*b*(-1/2*(b*x+a)^
(1/2)/x/b-1/2*arctanh((b*x+a)^(1/2)/a^(1/2))/a^(1/2))-6/(b-c)^3*b*c*(-1/2*(c*x+a)^(1/2)/x/c-1/2/a^(1/2)*arctan
h((c*x+a)^(1/2)/a^(1/2)))-2/(b-c)^3*c^2*(-1/2*(c*x+a)^(1/2)/x/c-1/2/a^(1/2)*arctanh((c*x+a)^(1/2)/a^(1/2)))

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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(1/((b*x+a)^(1/2)+(c*x+a)^(1/2))^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.36, size = 297, normalized size = 1.81 \begin {gather*} \left [-\frac {3 \, \sqrt {a} b c x^{2} \log \left (\frac {b x + 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 3 \, \sqrt {a} b c x^{2} \log \left (\frac {c x - 2 \, \sqrt {c x + a} \sqrt {a} + 2 \, a}{x}\right ) + 2 \, {\left (2 \, a^{2} + {\left (2 \, a b + 3 \, a c\right )} x\right )} \sqrt {b x + a} - 2 \, {\left (2 \, a^{2} + {\left (3 \, a b + 2 \, a c\right )} x\right )} \sqrt {c x + a}}{2 \, {\left (a b^{3} - 3 \, a b^{2} c + 3 \, a b c^{2} - a c^{3}\right )} x^{2}}, \frac {3 \, \sqrt {-a} b c x^{2} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) - 3 \, \sqrt {-a} b c x^{2} \arctan \left (\frac {\sqrt {c x + a} \sqrt {-a}}{a}\right ) - {\left (2 \, a^{2} + {\left (2 \, a b + 3 \, a c\right )} x\right )} \sqrt {b x + a} + {\left (2 \, a^{2} + {\left (3 \, a b + 2 \, a c\right )} x\right )} \sqrt {c x + a}}{{\left (a b^{3} - 3 \, a b^{2} c + 3 \, a b c^{2} - a c^{3}\right )} x^{2}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((b*x+a)^(1/2)+(c*x+a)^(1/2))^3,x, algorithm="fricas")

[Out]

[-1/2*(3*sqrt(a)*b*c*x^2*log((b*x + 2*sqrt(b*x + a)*sqrt(a) + 2*a)/x) + 3*sqrt(a)*b*c*x^2*log((c*x - 2*sqrt(c*
x + a)*sqrt(a) + 2*a)/x) + 2*(2*a^2 + (2*a*b + 3*a*c)*x)*sqrt(b*x + a) - 2*(2*a^2 + (3*a*b + 2*a*c)*x)*sqrt(c*
x + a))/((a*b^3 - 3*a*b^2*c + 3*a*b*c^2 - a*c^3)*x^2), (3*sqrt(-a)*b*c*x^2*arctan(sqrt(b*x + a)*sqrt(-a)/a) -
3*sqrt(-a)*b*c*x^2*arctan(sqrt(c*x + a)*sqrt(-a)/a) - (2*a^2 + (2*a*b + 3*a*c)*x)*sqrt(b*x + a) + (2*a^2 + (3*
a*b + 2*a*c)*x)*sqrt(c*x + a))/((a*b^3 - 3*a*b^2*c + 3*a*b*c^2 - a*c^3)*x^2)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (\sqrt {a + b x} + \sqrt {a + c x}\right )^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((b*x+a)**(1/2)+(c*x+a)**(1/2))**3,x)

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 2766 vs. \(2 (144) = 288\).
time = 55.11, size = 2766, normalized size = 16.87 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((b*x+a)^(1/2)+(c*x+a)^(1/2))^3,x, algorithm="giac")

[Out]

3*b*c*arctan(sqrt(b*x + a)/sqrt(-a))/((b^3 - 3*b^2*c + 3*b*c^2 - c^3)*sqrt(-a)) - 2*(3*(sqrt(b*c)*sqrt(b*x + a
) - sqrt(a*b^2 + (b*x + a)*b*c - a*b*c))*a^3*b^7*c*abs(b) - 7*(sqrt(b*c)*sqrt(b*x + a) - sqrt(a*b^2 + (b*x + a
)*b*c - a*b*c))*a^3*b^6*c^2*abs(b) + 3*(sqrt(b*c)*sqrt(b*x + a) - sqrt(a*b^2 + (b*x + a)*b*c - a*b*c))*a^3*b^5
*c^3*abs(b) + 3*(sqrt(b*c)*sqrt(b*x + a) - sqrt(a*b^2 + (b*x + a)*b*c - a*b*c))*a^3*b^4*c^4*abs(b) - 2*(sqrt(b
*c)*sqrt(b*x + a) - sqrt(a*b^2 + (b*x + a)*b*c - a*b*c))*a^3*b^3*c^5*abs(b) - 3*(sqrt(b*c)*sqrt(b*x + a) - sqr
t(a*b^2 + (b*x + a)*b*c - a*b*c))^3*a^2*b^5*c*abs(b) - 3*(sqrt(b*c)*sqrt(b*x + a) - sqrt(a*b^2 + (b*x + a)*b*c
- a*b*c))^3*a^2*b^3*c^3*abs(b) + 6*(sqrt(b*c)*sqrt(b*x + a) - sqrt(a*b^2 + (b*x + a)*b*c - a*b*c))^3*a^2*b^2*
c^4*abs(b) - 3*(sqrt(b*c)*sqrt(b*x + a) - sqrt(a*b^2 + (b*x + a)*b*c - a*b*c))^5*a*b^3*c*abs(b) - 3*(sqrt(b*c)
*sqrt(b*x + a) - sqrt(a*b^2 + (b*x + a)*b*c - a*b*c))^5*a*b^2*c^2*abs(b) - 6*(sqrt(b*c)*sqrt(b*x + a) - sqrt(a
*b^2 + (b*x + a)*b*c - a*b*c))^5*a*b*c^3*abs(b) + 3*(sqrt(b*c)*sqrt(b*x + a) - sqrt(a*b^2 + (b*x + a)*b*c - a*
b*c))^7*b*c*abs(b) + 2*(sqrt(b*c)*sqrt(b*x + a) - sqrt(a*b^2 + (b*x + a)*b*c - a*b*c))^7*c^2*abs(b))/((a^2*b^4
- 2*a^2*b^3*c + a^2*b^2*c^2 - 2*(sqrt(b*c)*sqrt(b*x + a) - sqrt(a*b^2 + (b*x + a)*b*c - a*b*c))^2*a*b^2 - 2*(
sqrt(b*c)*sqrt(b*x + a) - sqrt(a*b^2 + (b*x + a)*b*c - a*b*c))^2*a*b*c + (sqrt(b*c)*sqrt(b*x + a) - sqrt(a*b^2
+ (b*x + a)*b*c - a*b*c))^4)^2*(b^3 - 3*b^2*c + 3*b*c^2 - c^3)) - 3*(2*(a*b^3*c^2 - a*b^2*c^3)*(a*b^4 - 3*a*b
^3*c + 3*a*b^2*c^2 - a*b*c^3)^2*sqrt(-a)*abs(b)*sgn(2*b^3 - 6*b^2*c + 6*b*c^2 - 2*c^3) + 2*(a*b^4 - 3*a*b^3*c
+ 3*a*b^2*c^2 - a*b*c^3)^2*(a*b^3*c - a*b^2*c^2)*sqrt(-a*b*c)*abs(b) + (a^2*b^7*c - 5*a^2*b^6*c^2 + 10*a^2*b^5
*c^3 - 10*a^2*b^4*c^4 + 5*a^2*b^3*c^5 - a^2*b^2*c^6)*sqrt(-a*b*c)*abs(a*b^4 - 3*a*b^3*c + 3*a*b^2*c^2 - a*b*c^
3)*abs(b)*sgn(2*b^3 - 6*b^2*c + 6*b*c^2 - 2*c^3) + (a^2*b^8*c - 5*a^2*b^7*c^2 + 10*a^2*b^6*c^3 - 10*a^2*b^5*c^
4 + 5*a^2*b^4*c^5 - a^2*b^3*c^6)*sqrt(-a)*abs(a*b^4 - 3*a*b^3*c + 3*a*b^2*c^2 - a*b*c^3)*abs(b) + (a^3*b^11*c^
2 - 6*a^3*b^10*c^3 + 14*a^3*b^9*c^4 - 14*a^3*b^8*c^5 + 14*a^3*b^6*c^7 - 14*a^3*b^5*c^8 + 6*a^3*b^4*c^9 - a^3*b
^3*c^10)*sqrt(-a)*abs(b)*sgn(2*b^3 - 6*b^2*c + 6*b*c^2 - 2*c^3) + (a^3*b^11*c - 6*a^3*b^10*c^2 + 14*a^3*b^9*c^
3 - 14*a^3*b^8*c^4 + 14*a^3*b^6*c^6 - 14*a^3*b^5*c^7 + 6*a^3*b^4*c^8 - a^3*b^3*c^9)*sqrt(-a*b*c)*abs(b))*arcta
n(-(sqrt(b*c)*sqrt(b*x + a) - sqrt(a*b^2 + (b*x + a)*b*c - a*b*c))/sqrt(-(a*b^5 - 2*a*b^4*c + 2*a*b^2*c^3 - a*
b*c^4 + sqrt((a*b^5 - 2*a*b^4*c + 2*a*b^2*c^3 - a*b*c^4)^2 - (a^2*b^7 - 5*a^2*b^6*c + 10*a^2*b^5*c^2 - 10*a^2*
b^4*c^3 + 5*a^2*b^3*c^4 - a^2*b^2*c^5)*(b^3 - 3*b^2*c + 3*b*c^2 - c^3)))/(b^3 - 3*b^2*c + 3*b*c^2 - c^3)))/((b
^11 - 9*b^10*c + 36*b^9*c^2 - 84*b^8*c^3 + 126*b^7*c^4 - 126*b^6*c^5 + 84*b^5*c^6 - 36*b^4*c^7 + 9*b^3*c^8 - b
^2*c^9)*a^3*abs(a*b^4 - 3*a*b^3*c + 3*a*b^2*c^2 - a*b*c^3)) + 3*(2*(a*b^3*c^2 - a*b^2*c^3)*(a*b^4 - 3*a*b^3*c
+ 3*a*b^2*c^2 - a*b*c^3)^2*sqrt(-a)*abs(b)*sgn(2*b^3 - 6*b^2*c + 6*b*c^2 - 2*c^3) + 2*(a*b^4 - 3*a*b^3*c + 3*a
*b^2*c^2 - a*b*c^3)^2*(a*b^3*c - a*b^2*c^2)*sqrt(-a*b*c)*abs(b) + (a^2*b^7*c - 5*a^2*b^6*c^2 + 10*a^2*b^5*c^3
- 10*a^2*b^4*c^4 + 5*a^2*b^3*c^5 - a^2*b^2*c^6)*sqrt(-a*b*c)*abs(a*b^4 - 3*a*b^3*c + 3*a*b^2*c^2 - a*b*c^3)*ab
s(b)*sgn(2*b^3 - 6*b^2*c + 6*b*c^2 - 2*c^3) + (a^2*b^8*c - 5*a^2*b^7*c^2 + 10*a^2*b^6*c^3 - 10*a^2*b^5*c^4 + 5
*a^2*b^4*c^5 - a^2*b^3*c^6)*sqrt(-a)*abs(a*b^4 - 3*a*b^3*c + 3*a*b^2*c^2 - a*b*c^3)*abs(b) + (a^3*b^11*c^2 - 6
*a^3*b^10*c^3 + 14*a^3*b^9*c^4 - 14*a^3*b^8*c^5 + 14*a^3*b^6*c^7 - 14*a^3*b^5*c^8 + 6*a^3*b^4*c^9 - a^3*b^3*c^
10)*sqrt(-a)*abs(b)*sgn(2*b^3 - 6*b^2*c + 6*b*c^2 - 2*c^3) + (a^3*b^11*c - 6*a^3*b^10*c^2 + 14*a^3*b^9*c^3 - 1
4*a^3*b^8*c^4 + 14*a^3*b^6*c^6 - 14*a^3*b^5*c^7 + 6*a^3*b^4*c^8 - a^3*b^3*c^9)*sqrt(-a*b*c)*abs(b))*arctan(-(s
qrt(b*c)*sqrt(b*x + a) - sqrt(a*b^2 + (b*x + a)*b*c - a*b*c))/sqrt(-(a*b^5 - 2*a*b^4*c + 2*a*b^2*c^3 - a*b*c^4
- sqrt((a*b^5 - 2*a*b^4*c + 2*a*b^2*c^3 - a*b*c^4)^2 - (a^2*b^7 - 5*a^2*b^6*c + 10*a^2*b^5*c^2 - 10*a^2*b^4*c
^3 + 5*a^2*b^3*c^4 - a^2*b^2*c^5)*(b^3 - 3*b^2*c + 3*b*c^2 - c^3)))/(b^3 - 3*b^2*c + 3*b*c^2 - c^3)))/((b^11 -
9*b^10*c + 36*b^9*c^2 - 84*b^8*c^3 + 126*b^7*c^4 - 126*b^6*c^5 + 84*b^5*c^6 - 36*b^4*c^7 + 9*b^3*c^8 - b^2*c^
9)*a^3*abs(a*b^4 - 3*a*b^3*c + 3*a*b^2*c^2 - a*b*c^3)) - (2*(b*x + a)^(3/2)*b^2 + 3*(b*x + a)^(3/2)*b*c - 3*sq
rt(b*x + a)*a*b*c)/((b^3 - 3*b^2*c + 3*b*c^2 - c^3)*b^2*x^2)

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Mupad [B]
time = 5.74, size = 287, normalized size = 1.75 \begin {gather*} \frac {c^2\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}{4\,\sqrt {a}\,{\left (b-c\right )}^3\,{\left (\sqrt {a+c\,x}-\sqrt {a}\right )}^2}-\frac {\left (\frac {\sqrt {a}\,b^2}{4\,\left (a\,b^3-3\,a\,b^2\,c+3\,a\,b\,c^2-a\,c^3\right )}-\frac {\sqrt {a}\,\left (b^2+c\,b\right )\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}{\left (\sqrt {a+c\,x}-\sqrt {a}\right )\,\left (a\,b^3-3\,a\,b^2\,c+3\,a\,b\,c^2-a\,c^3\right )}\right )\,{\left (\sqrt {a+c\,x}-\sqrt {a}\right )}^2}{{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}+\frac {3\,b\,c\,\ln \left (\frac {\sqrt {a+b\,x}-\sqrt {a}}{\sqrt {a+c\,x}-\sqrt {a}}\right )}{\sqrt {a}\,\left (b^3-3\,b^2\,c+3\,b\,c^2-c^3\right )}-\frac {c\,\left (b+c\right )\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}{\sqrt {a}\,{\left (b-c\right )}^3\,\left (\sqrt {a+c\,x}-\sqrt {a}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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