3.5.15 \(\int \frac {1}{x^2 (\sqrt {a+b x}+\sqrt {c+b x})^3} \, dx\) [415]
Optimal. Leaf size=162 \[ \frac {8 b \sqrt {a+b x}}{(a-c)^3}-\frac {(a+3 c) \sqrt {a+b x}}{(a-c)^3 x}-\frac {8 b \sqrt {c+b x}}{(a-c)^3}+\frac {(3 a+c) \sqrt {c+b x}}{(a-c)^3 x}-\frac {3 b (3 a+c) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{\sqrt {a} (a-c)^3}-\frac {3 b (a+3 c) \tanh ^{-1}\left (\frac {\sqrt {c+b x}}{\sqrt {c}}\right )}{\sqrt {c} (-a+c)^3} \]
[Out]
-3*b*(3*a+c)*arctanh((b*x+a)^(1/2)/a^(1/2))/(a-c)^3/a^(1/2)-3*b*(a+3*c)*arctanh((b*x+c)^(1/2)/c^(1/2))/(-a+c)^
3/c^(1/2)+8*b*(b*x+a)^(1/2)/(a-c)^3-(a+3*c)*(b*x+a)^(1/2)/(a-c)^3/x-8*b*(b*x+c)^(1/2)/(a-c)^3+(3*a+c)*(b*x+c)^
(1/2)/(a-c)^3/x
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Rubi [A]
time = 0.19, antiderivative size = 223, normalized size of antiderivative = 1.38, number of steps
used = 14, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6821, 43, 65,
214, 52} \begin {gather*} \frac {8 b \sqrt {a+b x}}{(a-c)^3}-\frac {8 b \sqrt {b x+c}}{(a-c)^3}-\frac {(a+3 c) \sqrt {a+b x}}{x (a-c)^3}+\frac {(3 a+c) \sqrt {b x+c}}{x (a-c)^3}-\frac {b (a+3 c) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{\sqrt {a} (a-c)^3}-\frac {8 \sqrt {a} b \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{(a-c)^3}+\frac {b (3 a+c) \tanh ^{-1}\left (\frac {\sqrt {b x+c}}{\sqrt {c}}\right )}{\sqrt {c} (a-c)^3}+\frac {8 b \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {b x+c}}{\sqrt {c}}\right )}{(a-c)^3} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[1/(x^2*(Sqrt[a + b*x] + Sqrt[c + b*x])^3),x]
[Out]
(8*b*Sqrt[a + b*x])/(a - c)^3 - ((a + 3*c)*Sqrt[a + b*x])/((a - c)^3*x) - (8*b*Sqrt[c + b*x])/(a - c)^3 + ((3*
a + c)*Sqrt[c + b*x])/((a - c)^3*x) - (8*Sqrt[a]*b*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(a - c)^3 - (b*(a + 3*c)*Ar
cTanh[Sqrt[a + b*x]/Sqrt[a]])/(Sqrt[a]*(a - c)^3) + (8*b*Sqrt[c]*ArcTanh[Sqrt[c + b*x]/Sqrt[c]])/(a - c)^3 + (
b*(3*a + c)*ArcTanh[Sqrt[c + b*x]/Sqrt[c]])/((a - c)^3*Sqrt[c])
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 52
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]
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 6821
Int[(u_.)*((e_.)*Sqrt[(a_.) + (b_.)*(x_)^(n_.)] + (f_.)*Sqrt[(c_.) + (d_.)*(x_)^(n_.)])^(m_), x_Symbol] :> Dis
t[(a*e^2 - c*f^2)^m, Int[ExpandIntegrand[u/(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[b*e^2 - d*f^2, 0]
Rubi steps
\begin {align*} \int \frac {1}{x^2 \left (\sqrt {a+b x}+\sqrt {c+b x}\right )^3} \, dx &=\frac {\int \left (\frac {a \left (1+\frac {3 c}{a}\right ) \sqrt {a+b x}}{x^2}+\frac {4 b \sqrt {a+b x}}{x}-\frac {3 a \left (1+\frac {c}{3 a}\right ) \sqrt {c+b x}}{x^2}-\frac {4 b \sqrt {c+b x}}{x}\right ) \, dx}{(a-c)^3}\\ &=\frac {(4 b) \int \frac {\sqrt {a+b x}}{x} \, dx}{(a-c)^3}-\frac {(4 b) \int \frac {\sqrt {c+b x}}{x} \, dx}{(a-c)^3}-\frac {(3 a+c) \int \frac {\sqrt {c+b x}}{x^2} \, dx}{(a-c)^3}+\frac {(a+3 c) \int \frac {\sqrt {a+b x}}{x^2} \, dx}{(a-c)^3}\\ &=\frac {8 b \sqrt {a+b x}}{(a-c)^3}-\frac {(a+3 c) \sqrt {a+b x}}{(a-c)^3 x}-\frac {8 b \sqrt {c+b x}}{(a-c)^3}+\frac {(3 a+c) \sqrt {c+b x}}{(a-c)^3 x}+\frac {(4 a b) \int \frac {1}{x \sqrt {a+b x}} \, dx}{(a-c)^3}-\frac {(4 b c) \int \frac {1}{x \sqrt {c+b x}} \, dx}{(a-c)^3}-\frac {(b (3 a+c)) \int \frac {1}{x \sqrt {c+b x}} \, dx}{2 (a-c)^3}+\frac {(b (a+3 c)) \int \frac {1}{x \sqrt {a+b x}} \, dx}{2 (a-c)^3}\\ &=\frac {8 b \sqrt {a+b x}}{(a-c)^3}-\frac {(a+3 c) \sqrt {a+b x}}{(a-c)^3 x}-\frac {8 b \sqrt {c+b x}}{(a-c)^3}+\frac {(3 a+c) \sqrt {c+b x}}{(a-c)^3 x}+\frac {(8 a) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x}\right )}{(a-c)^3}-\frac {(8 c) \text {Subst}\left (\int \frac {1}{-\frac {c}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {c+b x}\right )}{(a-c)^3}-\frac {(3 a+c) \text {Subst}\left (\int \frac {1}{-\frac {c}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {c+b x}\right )}{(a-c)^3}+\frac {(a+3 c) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x}\right )}{(a-c)^3}\\ &=\frac {8 b \sqrt {a+b x}}{(a-c)^3}-\frac {(a+3 c) \sqrt {a+b x}}{(a-c)^3 x}-\frac {8 b \sqrt {c+b x}}{(a-c)^3}+\frac {(3 a+c) \sqrt {c+b x}}{(a-c)^3 x}-\frac {8 \sqrt {a} b \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{(a-c)^3}-\frac {b (a+3 c) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{\sqrt {a} (a-c)^3}+\frac {8 b \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c+b x}}{\sqrt {c}}\right )}{(a-c)^3}+\frac {b (3 a+c) \tanh ^{-1}\left (\frac {\sqrt {c+b x}}{\sqrt {c}}\right )}{(a-c)^3 \sqrt {c}}\\ \end {align*}
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Mathematica [A]
time = 10.28, size = 112, normalized size = 0.69 \begin {gather*} \frac {-\frac {(a+3 c-8 b x) \sqrt {a+b x}}{x}+\frac {(3 a+c-8 b x) \sqrt {c+b x}}{x}-\frac {3 b (3 a+c) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{\sqrt {a}}+\frac {3 b (a+3 c) \tanh ^{-1}\left (\frac {\sqrt {c+b x}}{\sqrt {c}}\right )}{\sqrt {c}}}{(a-c)^3} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[1/(x^2*(Sqrt[a + b*x] + Sqrt[c + b*x])^3),x]
[Out]
(-(((a + 3*c - 8*b*x)*Sqrt[a + b*x])/x) + ((3*a + c - 8*b*x)*Sqrt[c + b*x])/x - (3*b*(3*a + c)*ArcTanh[Sqrt[a
+ b*x]/Sqrt[a]])/Sqrt[a] + (3*b*(a + 3*c)*ArcTanh[Sqrt[c + b*x]/Sqrt[c]])/Sqrt[c])/(a - c)^3
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Maple [A]
time = 0.02, size = 252, normalized size = 1.56
| | |
| method |
result |
size |
| | |
| default |
\(\frac {2 a b \left (-\frac {\sqrt {b x +a}}{2 x b}-\frac {\arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{2 \sqrt {a}}\right )}{\left (a -c \right )^{3}}+\frac {6 c b \left (-\frac {\sqrt {b x +a}}{2 x b}-\frac {\arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{2 \sqrt {a}}\right )}{\left (a -c \right )^{3}}-\frac {6 a b \left (-\frac {\sqrt {b x +c}}{2 x b}-\frac {\arctanh \left (\frac {\sqrt {b x +c}}{\sqrt {c}}\right )}{2 \sqrt {c}}\right )}{\left (a -c \right )^{3}}-\frac {2 c b \left (-\frac {\sqrt {b x +c}}{2 x b}-\frac {\arctanh \left (\frac {\sqrt {b x +c}}{\sqrt {c}}\right )}{2 \sqrt {c}}\right )}{\left (a -c \right )^{3}}+\frac {4 b \left (2 \sqrt {b x +a}-2 \sqrt {a}\, \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )\right )}{\left (a -c \right )^{3}}-\frac {4 b \left (2 \sqrt {b x +c}-2 \sqrt {c}\, \arctanh \left (\frac {\sqrt {b x +c}}{\sqrt {c}}\right )\right )}{\left (a -c \right )^{3}}\) |
\(252\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/x^2/((b*x+a)^(1/2)+(b*x+c)^(1/2))^3,x,method=_RETURNVERBOSE)
[Out]
2/(a-c)^3*a*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/(a-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/(a-c)^3*a*b*(-1/2*(b*x+c)^(1/2)/x/b-1/2/c^(1/2)*arctan
h((b*x+c)^(1/2)/c^(1/2)))-2/(a-c)^3*c*b*(-1/2*(b*x+c)^(1/2)/x/b-1/2/c^(1/2)*arctanh((b*x+c)^(1/2)/c^(1/2)))+4/
(a-c)^3*b*(2*(b*x+a)^(1/2)-2*a^(1/2)*arctanh((b*x+a)^(1/2)/a^(1/2)))-4/(a-c)^3*b*(2*(b*x+c)^(1/2)-2*c^(1/2)*ar
ctanh((b*x+c)^(1/2)/c^(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/x^2/((b*x+a)^(1/2)+(b*x+c)^(1/2))^3,x, algorithm="maxima")
[Out]
integrate(1/(x^2*(sqrt(b*x + a) + sqrt(b*x + c))^3), x)
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Fricas [A]
time = 0.38, size = 675, normalized size = 4.17 \begin {gather*} \left [-\frac {3 \, {\left (3 \, a b c + b c^{2}\right )} \sqrt {a} x \log \left (\frac {b x + 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 3 \, {\left (a^{2} b + 3 \, a b c\right )} \sqrt {c} x \log \left (\frac {b x - 2 \, \sqrt {b x + c} \sqrt {c} + 2 \, c}{x}\right ) - 2 \, {\left (8 \, a b c x - a^{2} c - 3 \, a c^{2}\right )} \sqrt {b x + a} + 2 \, {\left (8 \, a b c x - 3 \, a^{2} c - a c^{2}\right )} \sqrt {b x + c}}{2 \, {\left (a^{4} c - 3 \, a^{3} c^{2} + 3 \, a^{2} c^{3} - a c^{4}\right )} x}, -\frac {6 \, {\left (a^{2} b + 3 \, a b c\right )} \sqrt {-c} x \arctan \left (\frac {\sqrt {b x + c} \sqrt {-c}}{c}\right ) + 3 \, {\left (3 \, a b c + b c^{2}\right )} \sqrt {a} x \log \left (\frac {b x + 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) - 2 \, {\left (8 \, a b c x - a^{2} c - 3 \, a c^{2}\right )} \sqrt {b x + a} + 2 \, {\left (8 \, a b c x - 3 \, a^{2} c - a c^{2}\right )} \sqrt {b x + c}}{2 \, {\left (a^{4} c - 3 \, a^{3} c^{2} + 3 \, a^{2} c^{3} - a c^{4}\right )} x}, \frac {6 \, {\left (3 \, a b c + b c^{2}\right )} \sqrt {-a} x \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) - 3 \, {\left (a^{2} b + 3 \, a b c\right )} \sqrt {c} x \log \left (\frac {b x - 2 \, \sqrt {b x + c} \sqrt {c} + 2 \, c}{x}\right ) + 2 \, {\left (8 \, a b c x - a^{2} c - 3 \, a c^{2}\right )} \sqrt {b x + a} - 2 \, {\left (8 \, a b c x - 3 \, a^{2} c - a c^{2}\right )} \sqrt {b x + c}}{2 \, {\left (a^{4} c - 3 \, a^{3} c^{2} + 3 \, a^{2} c^{3} - a c^{4}\right )} x}, \frac {3 \, {\left (3 \, a b c + b c^{2}\right )} \sqrt {-a} x \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) - 3 \, {\left (a^{2} b + 3 \, a b c\right )} \sqrt {-c} x \arctan \left (\frac {\sqrt {b x + c} \sqrt {-c}}{c}\right ) + {\left (8 \, a b c x - a^{2} c - 3 \, a c^{2}\right )} \sqrt {b x + a} - {\left (8 \, a b c x - 3 \, a^{2} c - a c^{2}\right )} \sqrt {b x + c}}{{\left (a^{4} c - 3 \, a^{3} c^{2} + 3 \, a^{2} c^{3} - a c^{4}\right )} x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x^2/((b*x+a)^(1/2)+(b*x+c)^(1/2))^3,x, algorithm="fricas")
[Out]
[-1/2*(3*(3*a*b*c + b*c^2)*sqrt(a)*x*log((b*x + 2*sqrt(b*x + a)*sqrt(a) + 2*a)/x) + 3*(a^2*b + 3*a*b*c)*sqrt(c
)*x*log((b*x - 2*sqrt(b*x + c)*sqrt(c) + 2*c)/x) - 2*(8*a*b*c*x - a^2*c - 3*a*c^2)*sqrt(b*x + a) + 2*(8*a*b*c*
x - 3*a^2*c - a*c^2)*sqrt(b*x + c))/((a^4*c - 3*a^3*c^2 + 3*a^2*c^3 - a*c^4)*x), -1/2*(6*(a^2*b + 3*a*b*c)*sqr
t(-c)*x*arctan(sqrt(b*x + c)*sqrt(-c)/c) + 3*(3*a*b*c + b*c^2)*sqrt(a)*x*log((b*x + 2*sqrt(b*x + a)*sqrt(a) +
2*a)/x) - 2*(8*a*b*c*x - a^2*c - 3*a*c^2)*sqrt(b*x + a) + 2*(8*a*b*c*x - 3*a^2*c - a*c^2)*sqrt(b*x + c))/((a^4
*c - 3*a^3*c^2 + 3*a^2*c^3 - a*c^4)*x), 1/2*(6*(3*a*b*c + b*c^2)*sqrt(-a)*x*arctan(sqrt(b*x + a)*sqrt(-a)/a) -
3*(a^2*b + 3*a*b*c)*sqrt(c)*x*log((b*x - 2*sqrt(b*x + c)*sqrt(c) + 2*c)/x) + 2*(8*a*b*c*x - a^2*c - 3*a*c^2)*
sqrt(b*x + a) - 2*(8*a*b*c*x - 3*a^2*c - a*c^2)*sqrt(b*x + c))/((a^4*c - 3*a^3*c^2 + 3*a^2*c^3 - a*c^4)*x), (3
*(3*a*b*c + b*c^2)*sqrt(-a)*x*arctan(sqrt(b*x + a)*sqrt(-a)/a) - 3*(a^2*b + 3*a*b*c)*sqrt(-c)*x*arctan(sqrt(b*
x + c)*sqrt(-c)/c) + (8*a*b*c*x - a^2*c - 3*a*c^2)*sqrt(b*x + a) - (8*a*b*c*x - 3*a^2*c - a*c^2)*sqrt(b*x + c)
)/((a^4*c - 3*a^3*c^2 + 3*a^2*c^3 - a*c^4)*x)]
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{2} \left (\sqrt {a + b x} + \sqrt {b x + c}\right )^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x**2/((b*x+a)**(1/2)+(b*x+c)**(1/2))**3,x)
[Out]
Integral(1/(x**2*(sqrt(a + b*x) + sqrt(b*x + c))**3), x)
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 2594 vs.
\(2 (142) = 284\).
time = 24.44, size = 2594, normalized size = 16.01 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x^2/((b*x+a)^(1/2)+(b*x+c)^(1/2))^3,x, algorithm="giac")
[Out]
8*sqrt(b*x + a)*b/(a^3 - 3*a^2*c + 3*a*c^2 - c^3) - 8*sqrt(b*x + c)*b/(a^3 - 3*a^2*c + 3*a*c^2 - c^3) + 3*(3*a
*b + b*c)*arctan(sqrt(b*x + a)/sqrt(-a))/((a^3 - 3*a^2*c + 3*a*c^2 - c^3)*sqrt(-a)) - 3*(2*(a^2*c^2 + 3*a*c^3
+ (a*c^2 + 3*c^3)*sqrt(a*c))*(a^3 - 3*a^2*c + 3*a*c^2 - c^3)^2*b*sgn(-2*a^3 + 6*a^2*c - 6*a*c^2 + 2*c^3) - 2*(
a^2*c^2 + 3*a*c^3 - (a^2*c + 3*a*c^2)*sqrt(a*c))*(a^3 - 3*a^2*c + 3*a*c^2 - c^3)^2*b + (a^5*c^2 - a^4*c^3 - 6*
a^3*c^4 + 14*a^2*c^5 - 11*a*c^6 + 3*c^7 + (a^5*c - a^4*c^2 - 6*a^3*c^3 + 14*a^2*c^4 - 11*a*c^5 + 3*c^6)*sqrt(a
*c))*b*abs(-a^3 + 3*a^2*c - 3*a*c^2 + c^3)*sgn(-2*a^3 + 6*a^2*c - 6*a*c^2 + 2*c^3) - (a^6*c - a^5*c^2 - 6*a^4*
c^3 + 14*a^3*c^4 - 11*a^2*c^5 + 3*a*c^6 + (a^5*c - a^4*c^2 - 6*a^3*c^3 + 14*a^2*c^4 - 11*a*c^5 + 3*c^6)*sqrt(a
*c))*b*abs(-a^3 + 3*a^2*c - 3*a*c^2 + c^3) - (a^9*c - 2*a^8*c^2 - 6*a^7*c^3 + 22*a^6*c^4 - 20*a^5*c^5 - 6*a^4*
c^6 + 22*a^3*c^7 - 14*a^2*c^8 + 3*a*c^9 + (a^8*c - 2*a^7*c^2 - 6*a^6*c^3 + 22*a^5*c^4 - 20*a^4*c^5 - 6*a^3*c^6
+ 22*a^2*c^7 - 14*a*c^8 + 3*c^9)*sqrt(a*c))*b*sgn(-2*a^3 + 6*a^2*c - 6*a*c^2 + 2*c^3) + (a^9*c - 2*a^8*c^2 -
6*a^7*c^3 + 22*a^6*c^4 - 20*a^5*c^5 - 6*a^4*c^6 + 22*a^3*c^7 - 14*a^2*c^8 + 3*a*c^9 + (a^9 - 2*a^8*c - 6*a^7*c
^2 + 22*a^6*c^3 - 20*a^5*c^4 - 6*a^4*c^5 + 22*a^3*c^6 - 14*a^2*c^7 + 3*a*c^8)*sqrt(a*c))*b)*arctan(-(sqrt(b*x
+ a) - sqrt(b*x + c))/sqrt(-(a^4 - 2*a^3*c + 2*a*c^3 - c^4 + sqrt((a^4 - 2*a^3*c + 2*a*c^3 - c^4)^2 - (a^5 - 5
*a^4*c + 10*a^3*c^2 - 10*a^2*c^3 + 5*a*c^4 - c^5)*(a^3 - 3*a^2*c + 3*a*c^2 - c^3)))/(a^3 - 3*a^2*c + 3*a*c^2 -
c^3)))/((sqrt(-a)*a^8*c - a^8*sqrt(-c)*c - 8*sqrt(-a)*a^7*c^2 + 8*a^7*sqrt(-c)*c^2 + 28*sqrt(-a)*a^6*c^3 - 28
*a^6*sqrt(-c)*c^3 - 56*sqrt(-a)*a^5*c^4 + 56*a^5*sqrt(-c)*c^4 + 70*sqrt(-a)*a^4*c^5 - 70*a^4*sqrt(-c)*c^5 - 56
*sqrt(-a)*a^3*c^6 + 56*a^3*sqrt(-c)*c^6 + 28*sqrt(-a)*a^2*c^7 - 28*a^2*sqrt(-c)*c^7 - 8*sqrt(-a)*a*c^8 + 8*a*s
qrt(-c)*c^8 + sqrt(-a)*c^9 - sqrt(-c)*c^9)*abs(-a^3 + 3*a^2*c - 3*a*c^2 + c^3)) - 3*(2*(a^2*c^2 + 3*a*c^3 + (a
*c^2 + 3*c^3)*sqrt(a*c))*(a^3 - 3*a^2*c + 3*a*c^2 - c^3)^2*b*sgn(-2*a^3 + 6*a^2*c - 6*a*c^2 + 2*c^3) + 2*(a^2*
c^2 + 3*a*c^3 + (a^2*c + 3*a*c^2)*sqrt(a*c))*(a^3 - 3*a^2*c + 3*a*c^2 - c^3)^2*b - (a^5*c^2 - a^4*c^3 - 6*a^3*
c^4 + 14*a^2*c^5 - 11*a*c^6 + 3*c^7 + (a^5*c - a^4*c^2 - 6*a^3*c^3 + 14*a^2*c^4 - 11*a*c^5 + 3*c^6)*sqrt(a*c))
*b*abs(-a^3 + 3*a^2*c - 3*a*c^2 + c^3)*sgn(-2*a^3 + 6*a^2*c - 6*a*c^2 + 2*c^3) - (a^6*c - a^5*c^2 - 6*a^4*c^3
+ 14*a^3*c^4 - 11*a^2*c^5 + 3*a*c^6 + (a^5*c - a^4*c^2 - 6*a^3*c^3 + 14*a^2*c^4 - 11*a*c^5 + 3*c^6)*sqrt(a*c))
*b*abs(-a^3 + 3*a^2*c - 3*a*c^2 + c^3) - (a^9*c - 2*a^8*c^2 - 6*a^7*c^3 + 22*a^6*c^4 - 20*a^5*c^5 - 6*a^4*c^6
+ 22*a^3*c^7 - 14*a^2*c^8 + 3*a*c^9 + (a^8*c - 2*a^7*c^2 - 6*a^6*c^3 + 22*a^5*c^4 - 20*a^4*c^5 - 6*a^3*c^6 + 2
2*a^2*c^7 - 14*a*c^8 + 3*c^9)*sqrt(a*c))*b*sgn(-2*a^3 + 6*a^2*c - 6*a*c^2 + 2*c^3) - (a^9*c - 2*a^8*c^2 - 6*a^
7*c^3 + 22*a^6*c^4 - 20*a^5*c^5 - 6*a^4*c^6 + 22*a^3*c^7 - 14*a^2*c^8 + 3*a*c^9 - (a^9 - 2*a^8*c - 6*a^7*c^2 +
22*a^6*c^3 - 20*a^5*c^4 - 6*a^4*c^5 + 22*a^3*c^6 - 14*a^2*c^7 + 3*a*c^8)*sqrt(a*c))*b)*arctan(-(sqrt(b*x + a)
- sqrt(b*x + c))/sqrt(-(a^4 - 2*a^3*c + 2*a*c^3 - c^4 - sqrt((a^4 - 2*a^3*c + 2*a*c^3 - c^4)^2 - (a^5 - 5*a^4
*c + 10*a^3*c^2 - 10*a^2*c^3 + 5*a*c^4 - c^5)*(a^3 - 3*a^2*c + 3*a*c^2 - c^3)))/(a^3 - 3*a^2*c + 3*a*c^2 - c^3
)))/((sqrt(-a)*a^8*c - a^8*sqrt(-c)*c - 8*sqrt(-a)*a^7*c^2 + 8*a^7*sqrt(-c)*c^2 + 28*sqrt(-a)*a^6*c^3 - 28*a^6
*sqrt(-c)*c^3 - 56*sqrt(-a)*a^5*c^4 + 56*a^5*sqrt(-c)*c^4 + 70*sqrt(-a)*a^4*c^5 - 70*a^4*sqrt(-c)*c^5 - 56*sqr
t(-a)*a^3*c^6 + 56*a^3*sqrt(-c)*c^6 + 28*sqrt(-a)*a^2*c^7 - 28*a^2*sqrt(-c)*c^7 - 8*sqrt(-a)*a*c^8 + 8*a*sqrt(
-c)*c^8 + sqrt(-a)*c^9 - sqrt(-c)*c^9)*abs(-a^3 + 3*a^2*c - 3*a*c^2 + c^3)) - 2*(3*a*b*(sqrt(b*x + a) - sqrt(b
*x + c))^3 + b*c*(sqrt(b*x + a) - sqrt(b*x + c))^3 - 3*a^2*b*(sqrt(b*x + a) - sqrt(b*x + c)) + 2*a*b*c*(sqrt(b
*x + a) - sqrt(b*x + c)) + b*c^2*(sqrt(b*x + a) - sqrt(b*x + c)))/(((sqrt(b*x + a) - sqrt(b*x + c))^4 - 2*a*(s
qrt(b*x + a) - sqrt(b*x + c))^2 - 2*c*(sqrt(b*x + a) - sqrt(b*x + c))^2 + a^2 - 2*a*c + c^2)*(a^3 - 3*a^2*c +
3*a*c^2 - c^3)) - (sqrt(b*x + a)*a*b + 3*sqrt(b*x + a)*b*c)/((a^3 - 3*a^2*c + 3*a*c^2 - c^3)*b*x)
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Mupad [B]
time = 33.22, size = 2500, normalized size = 15.43 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(x^2*((a + b*x)^(1/2) + (c + b*x)^(1/2))^3),x)
[Out]
(b*atan(((b*((a^(1/2)*c^(3/2) - 2*a*c + a^(3/2)*c^(1/2))*(2*a*c + a^(1/2)*c^(3/2) + a^(3/2)*c^(1/2)))^(1/2)*((
9*a^6*b*c^(7/2) - 9*a^(7/2)*b*c^6 - 24*a^5*b*c^(9/2) + 24*a^(9/2)*b*c^5 + 18*a^4*b*c^(11/2) - 18*a^(11/2)*b*c^
4 - 3*a^2*b*c^(15/2) + 3*a^(15/2)*b*c^2)/(a^3*c^9 - 6*a^4*c^8 + 15*a^5*c^7 - 20*a^6*c^6 + 15*a^7*c^5 - 6*a^8*c
^4 + a^9*c^3) + (((a + b*x)^(1/2) - a^(1/2))*(6*a^(3/2)*b*c^8 - 6*a^8*b*c^(3/2) + 36*a^6*b*c^(7/2) - 36*a^(7/2
)*b*c^6 - 48*a^5*b*c^(9/2) + 48*a^(9/2)*b*c^5 + 18*a^4*b*c^(11/2) - 18*a^(11/2)*b*c^4))/(2*((c + b*x)^(1/2) -
c^(1/2))*(a^3*c^9 - 6*a^4*c^8 + 15*a^5*c^7 - 20*a^6*c^6 + 15*a^7*c^5 - 6*a^8*c^4 + a^9*c^3)) - (3*b*((a^(5/2)*
c^(19/2) - 5*a^(7/2)*c^(17/2) + 9*a^(9/2)*c^(15/2) - 5*a^(11/2)*c^(13/2) - 5*a^(13/2)*c^(11/2) + 9*a^(15/2)*c^
(9/2) - 5*a^(17/2)*c^(7/2) + a^(19/2)*c^(5/2))/(a^3*c^9 - 6*a^4*c^8 + 15*a^5*c^7 - 20*a^6*c^6 + 15*a^7*c^5 - 6
*a^8*c^4 + a^9*c^3) - (((a + b*x)^(1/2) - a^(1/2))*(4*a^2*c^10 - 28*a^3*c^9 + 88*a^4*c^8 - 164*a^5*c^7 + 200*a
^6*c^6 - 164*a^7*c^5 + 88*a^8*c^4 - 28*a^9*c^3 + 4*a^10*c^2))/(2*((c + b*x)^(1/2) - c^(1/2))*(a^3*c^9 - 6*a^4*
c^8 + 15*a^5*c^7 - 20*a^6*c^6 + 15*a^7*c^5 - 6*a^8*c^4 + a^9*c^3)))*((a^(1/2)*c^(3/2) - 2*a*c + a^(3/2)*c^(1/2
))*(2*a*c + a^(1/2)*c^(3/2) + a^(3/2)*c^(1/2)))^(1/2)*(a*c^(7/2) + a^(7/2)*c - 3*a^3*c^(3/2) - 3*a^(3/2)*c^3 +
2*a^2*c^(5/2) + 2*a^(5/2)*c^2))/(2*(a^2*c^7 - 5*a^3*c^6 + 10*a^4*c^5 - 10*a^5*c^4 + 5*a^6*c^3 - a^7*c^2)))*(a
*c^(7/2) + a^(7/2)*c - 3*a^3*c^(3/2) - 3*a^(3/2)*c^3 + 2*a^2*c^(5/2) + 2*a^(5/2)*c^2)*3i)/(2*(a^2*c^7 - 5*a^3*
c^6 + 10*a^4*c^5 - 10*a^5*c^4 + 5*a^6*c^3 - a^7*c^2)) + (b*((a^(1/2)*c^(3/2) - 2*a*c + a^(3/2)*c^(1/2))*(2*a*c
+ a^(1/2)*c^(3/2) + a^(3/2)*c^(1/2)))^(1/2)*((9*a^6*b*c^(7/2) - 9*a^(7/2)*b*c^6 - 24*a^5*b*c^(9/2) + 24*a^(9/
2)*b*c^5 + 18*a^4*b*c^(11/2) - 18*a^(11/2)*b*c^4 - 3*a^2*b*c^(15/2) + 3*a^(15/2)*b*c^2)/(a^3*c^9 - 6*a^4*c^8 +
15*a^5*c^7 - 20*a^6*c^6 + 15*a^7*c^5 - 6*a^8*c^4 + a^9*c^3) + (((a + b*x)^(1/2) - a^(1/2))*(6*a^(3/2)*b*c^8 -
6*a^8*b*c^(3/2) + 36*a^6*b*c^(7/2) - 36*a^(7/2)*b*c^6 - 48*a^5*b*c^(9/2) + 48*a^(9/2)*b*c^5 + 18*a^4*b*c^(11/
2) - 18*a^(11/2)*b*c^4))/(2*((c + b*x)^(1/2) - c^(1/2))*(a^3*c^9 - 6*a^4*c^8 + 15*a^5*c^7 - 20*a^6*c^6 + 15*a^
7*c^5 - 6*a^8*c^4 + a^9*c^3)) + (3*b*((a^(5/2)*c^(19/2) - 5*a^(7/2)*c^(17/2) + 9*a^(9/2)*c^(15/2) - 5*a^(11/2)
*c^(13/2) - 5*a^(13/2)*c^(11/2) + 9*a^(15/2)*c^(9/2) - 5*a^(17/2)*c^(7/2) + a^(19/2)*c^(5/2))/(a^3*c^9 - 6*a^4
*c^8 + 15*a^5*c^7 - 20*a^6*c^6 + 15*a^7*c^5 - 6*a^8*c^4 + a^9*c^3) - (((a + b*x)^(1/2) - a^(1/2))*(4*a^2*c^10
- 28*a^3*c^9 + 88*a^4*c^8 - 164*a^5*c^7 + 200*a^6*c^6 - 164*a^7*c^5 + 88*a^8*c^4 - 28*a^9*c^3 + 4*a^10*c^2))/(
2*((c + b*x)^(1/2) - c^(1/2))*(a^3*c^9 - 6*a^4*c^8 + 15*a^5*c^7 - 20*a^6*c^6 + 15*a^7*c^5 - 6*a^8*c^4 + a^9*c^
3)))*((a^(1/2)*c^(3/2) - 2*a*c + a^(3/2)*c^(1/2))*(2*a*c + a^(1/2)*c^(3/2) + a^(3/2)*c^(1/2)))^(1/2)*(a*c^(7/2
) + a^(7/2)*c - 3*a^3*c^(3/2) - 3*a^(3/2)*c^3 + 2*a^2*c^(5/2) + 2*a^(5/2)*c^2))/(2*(a^2*c^7 - 5*a^3*c^6 + 10*a
^4*c^5 - 10*a^5*c^4 + 5*a^6*c^3 - a^7*c^2)))*(a*c^(7/2) + a^(7/2)*c - 3*a^3*c^(3/2) - 3*a^(3/2)*c^3 + 2*a^2*c^
(5/2) + 2*a^(5/2)*c^2)*3i)/(2*(a^2*c^7 - 5*a^3*c^6 + 10*a^4*c^5 - 10*a^5*c^4 + 5*a^6*c^3 - a^7*c^2)))/(((9*a^(
3/2)*b^2*c^(11/2))/2 - 18*a^(5/2)*b^2*c^(9/2) + 27*a^(7/2)*b^2*c^(7/2) - 18*a^(9/2)*b^2*c^(5/2) + (9*a^(11/2)*
b^2*c^(3/2))/2)/(a^3*c^9 - 6*a^4*c^8 + 15*a^5*c^7 - 20*a^6*c^6 + 15*a^7*c^5 - 6*a^8*c^4 + a^9*c^3) - (((a + b*
x)^(1/2) - a^(1/2))*(72*a^3*b^2*c^4 - 72*a^2*b^2*c^5 + 72*a^4*b^2*c^3 - 72*a^5*b^2*c^2 + 27*a^(3/2)*b^2*c^(11/
2) + 36*a^(5/2)*b^2*c^(9/2) - 126*a^(7/2)*b^2*c^(7/2) + 36*a^(9/2)*b^2*c^(5/2) + 27*a^(11/2)*b^2*c^(3/2)))/(((
c + b*x)^(1/2) - c^(1/2))*(a^3*c^9 - 6*a^4*c^8 + 15*a^5*c^7 - 20*a^6*c^6 + 15*a^7*c^5 - 6*a^8*c^4 + a^9*c^3))
- (3*b*((a^(1/2)*c^(3/2) - 2*a*c + a^(3/2)*c^(1/2))*(2*a*c + a^(1/2)*c^(3/2) + a^(3/2)*c^(1/2)))^(1/2)*((9*a^6
*b*c^(7/2) - 9*a^(7/2)*b*c^6 - 24*a^5*b*c^(9/2) + 24*a^(9/2)*b*c^5 + 18*a^4*b*c^(11/2) - 18*a^(11/2)*b*c^4 - 3
*a^2*b*c^(15/2) + 3*a^(15/2)*b*c^2)/(a^3*c^9 - 6*a^4*c^8 + 15*a^5*c^7 - 20*a^6*c^6 + 15*a^7*c^5 - 6*a^8*c^4 +
a^9*c^3) + (((a + b*x)^(1/2) - a^(1/2))*(6*a^(3/2)*b*c^8 - 6*a^8*b*c^(3/2) + 36*a^6*b*c^(7/2) - 36*a^(7/2)*b*c
^6 - 48*a^5*b*c^(9/2) + 48*a^(9/2)*b*c^5 + 18*a^4*b*c^(11/2) - 18*a^(11/2)*b*c^4))/(2*((c + b*x)^(1/2) - c^(1/
2))*(a^3*c^9 - 6*a^4*c^8 + 15*a^5*c^7 - 20*a^6*c^6 + 15*a^7*c^5 - 6*a^8*c^4 + a^9*c^3)) - (3*b*((a^(5/2)*c^(19
/2) - 5*a^(7/2)*c^(17/2) + 9*a^(9/2)*c^(15/2) - 5*a^(11/2)*c^(13/2) - 5*a^(13/2)*c^(11/2) + 9*a^(15/2)*c^(9/2)
- 5*a^(17/2)*c^(7/2) + a^(19/2)*c^(5/2))/(a^3*c^9 - 6*a^4*c^8 + 15*a^5*c^7 - 20*a^6*c^6 + 15*a^7*c^5 - 6*a^8*
c^4 + a^9*c^3) - (((a + b*x)^(1/2) - a^(1/2))*(4*a^2*c^10 - 28*a^3*c^9 + 88*a^4*c^8 - 164*a^5*c^7 + 200*a^6*c^
6 - 164*a^7*c^5 + 88*a^8*c^4 - 28*a^9*c^3 + 4*a^10*c^2))/(2*((c + b*x)^(1/2) - c^(1/2))*(a^3*c^9 - 6*a^4*c^8 +
15*a^5*c^7 - 20*a^6*c^6 + 15*a^7*c^5 - 6*a^8*c^4 + a^9*c^3)))*((a^(1/2)*c^(3/2) - 2*a*c + a^(3/2)*c^(1/2))*(2
*a*c + a^(1/2)*c^(3/2) + a^(3/2)*c^(1/2)))^(1/2...
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