Optimal. Leaf size=97 \[ \frac {2 \sqrt {a+b x}}{b-c}-\frac {2 \sqrt {a+c x}}{b-c}-\frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{b-c}+\frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+c x}}{\sqrt {a}}\right )}{b-c} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.06, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {6822, 52, 65,
214} \begin {gather*} \frac {2 \sqrt {a+b x}}{b-c}-\frac {2 \sqrt {a+c x}}{b-c}-\frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{b-c}+\frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+c x}}{\sqrt {a}}\right )}{b-c} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 52
Rule 65
Rule 214
Rule 6822
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a+b x}+\sqrt {a+c x}} \, dx &=\frac {\int \left (\frac {\sqrt {a+b x}}{x}-\frac {\sqrt {a+c x}}{x}\right ) \, dx}{b-c}\\ &=\frac {\int \frac {\sqrt {a+b x}}{x} \, dx}{b-c}-\frac {\int \frac {\sqrt {a+c x}}{x} \, dx}{b-c}\\ &=\frac {2 \sqrt {a+b x}}{b-c}-\frac {2 \sqrt {a+c x}}{b-c}+\frac {a \int \frac {1}{x \sqrt {a+b x}} \, dx}{b-c}-\frac {a \int \frac {1}{x \sqrt {a+c x}} \, dx}{b-c}\\ &=\frac {2 \sqrt {a+b x}}{b-c}-\frac {2 \sqrt {a+c x}}{b-c}+\frac {(2 a) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x}\right )}{b (b-c)}-\frac {(2 a) \text {Subst}\left (\int \frac {1}{-\frac {a}{c}+\frac {x^2}{c}} \, dx,x,\sqrt {a+c x}\right )}{(b-c) c}\\ &=\frac {2 \sqrt {a+b x}}{b-c}-\frac {2 \sqrt {a+c x}}{b-c}-\frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{b-c}+\frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+c x}}{\sqrt {a}}\right )}{b-c}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(263\) vs. \(2(97)=194\).
time = 1.40, size = 263, normalized size = 2.71 \begin {gather*} 2 \left (\frac {\sqrt {a+b x}}{b-c}+\frac {\sqrt {a+c x}}{-b+c}+\frac {\sqrt {a} \left (\sqrt {b}-\sqrt {\frac {b}{c}} \sqrt {c}\right ) \tan ^{-1}\left (\frac {\sqrt {c} \left (-\sqrt {a+b x}+\sqrt {\frac {b}{c}} \sqrt {a+c x}\right )}{\sqrt {a} \sqrt {-\left (\sqrt {b}-\sqrt {c}\right )^2}}\right )}{\sqrt {b} \sqrt {-\left (\sqrt {b}-\sqrt {c}\right )^2} \left (\sqrt {b}+\sqrt {c}\right )}+\frac {\sqrt {a} \left (\sqrt {b}+\sqrt {\frac {b}{c}} \sqrt {c}\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \left (-\sqrt {a+b x}+\sqrt {\frac {b}{c}} \sqrt {a+c x}\right )}{\sqrt {a} \left (\sqrt {b}+\sqrt {c}\right )}\right )}{\sqrt {b} (b-c)}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.01, size = 73, normalized size = 0.75
method | result | size |
default | \(\frac {2 \sqrt {b x +a}-2 \sqrt {a}\, \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{b -c}-\frac {2 \sqrt {c x +a}-2 \sqrt {a}\, \arctanh \left (\frac {\sqrt {c x +a}}{\sqrt {a}}\right )}{b -c}\) | \(73\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.35, size = 158, normalized size = 1.63 \begin {gather*} \left [-\frac {\sqrt {a} \log \left (\frac {b x + 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + \sqrt {a} \log \left (\frac {c x - 2 \, \sqrt {c x + a} \sqrt {a} + 2 \, a}{x}\right ) - 2 \, \sqrt {b x + a} + 2 \, \sqrt {c x + a}}{b - c}, \frac {2 \, {\left (\sqrt {-a} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) - \sqrt {-a} \arctan \left (\frac {\sqrt {c x + a} \sqrt {-a}}{a}\right ) + \sqrt {b x + a} - \sqrt {c x + a}\right )}}{b - c}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {a + b x} + \sqrt {a + c x}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1093 vs.
\(2 (81) = 162\).
time = 4.71, size = 1093, normalized size = 11.27 \begin {gather*} -\frac {2 \, \sqrt {a b^{2} + {\left (b x + a\right )} b c - a b c} {\left | b \right |}}{b^{3} - b^{2} c} + \frac {2 \, a \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} {\left (b - c\right )}} + \frac {2 \, \sqrt {b x + a}}{b - c} - \frac {2 \, {\left (2 \, {\left (a b^{3} c - a b^{2} c^{2}\right )} {\left (a b^{2} - a b c\right )}^{2} \sqrt {-a} {\left | b \right |} \mathrm {sgn}\left (b - c\right ) + 2 \, {\left (a b^{3} - a b^{2} c\right )} {\left (a b^{2} - a b c\right )}^{2} \sqrt {-a b c} {\left | b \right |} + {\left (a^{2} b^{5} - 3 \, a^{2} b^{4} c + 3 \, a^{2} b^{3} c^{2} - a^{2} b^{2} c^{3}\right )} \sqrt {-a b c} {\left | a b^{2} - a b c \right |} {\left | b \right |} \mathrm {sgn}\left (b - c\right ) + {\left (a^{2} b^{6} - 3 \, a^{2} b^{5} c + 3 \, a^{2} b^{4} c^{2} - a^{2} b^{3} c^{3}\right )} \sqrt {-a} {\left | a b^{2} - a b c \right |} {\left | b \right |} + {\left (a^{3} b^{7} c - 2 \, a^{3} b^{6} c^{2} + 2 \, a^{3} b^{4} c^{4} - a^{3} b^{3} c^{5}\right )} \sqrt {-a} {\left | b \right |} \mathrm {sgn}\left (b - c\right ) + {\left (a^{3} b^{7} - 2 \, a^{3} b^{6} c + 2 \, a^{3} b^{4} c^{3} - a^{3} b^{3} c^{4}\right )} \sqrt {-a b c} {\left | b \right |}\right )} \arctan \left (-\frac {\sqrt {b c} \sqrt {b x + a} - \sqrt {a b^{2} + {\left (b x + a\right )} b c - a b c}}{\sqrt {-\frac {a b^{3} - a b c^{2} + \sqrt {{\left (a b^{3} - a b c^{2}\right )}^{2} - {\left (a^{2} b^{5} - 3 \, a^{2} b^{4} c + 3 \, a^{2} b^{3} c^{2} - a^{2} b^{2} c^{3}\right )} {\left (b - c\right )}}}{b - c}}}\right )}{{\left (b^{8} - 5 \, b^{7} c + 10 \, b^{6} c^{2} - 10 \, b^{5} c^{3} + 5 \, b^{4} c^{4} - b^{3} c^{5}\right )} a^{2} {\left | a b^{2} - a b c \right |}} + \frac {2 \, {\left (2 \, {\left (a b^{3} c - a b^{2} c^{2}\right )} {\left (a b^{2} - a b c\right )}^{2} \sqrt {-a} {\left | b \right |} \mathrm {sgn}\left (b - c\right ) + 2 \, {\left (a b^{3} - a b^{2} c\right )} {\left (a b^{2} - a b c\right )}^{2} \sqrt {-a b c} {\left | b \right |} + {\left (a^{2} b^{5} - 3 \, a^{2} b^{4} c + 3 \, a^{2} b^{3} c^{2} - a^{2} b^{2} c^{3}\right )} \sqrt {-a b c} {\left | a b^{2} - a b c \right |} {\left | b \right |} \mathrm {sgn}\left (b - c\right ) + {\left (a^{2} b^{6} - 3 \, a^{2} b^{5} c + 3 \, a^{2} b^{4} c^{2} - a^{2} b^{3} c^{3}\right )} \sqrt {-a} {\left | a b^{2} - a b c \right |} {\left | b \right |} + {\left (a^{3} b^{7} c - 2 \, a^{3} b^{6} c^{2} + 2 \, a^{3} b^{4} c^{4} - a^{3} b^{3} c^{5}\right )} \sqrt {-a} {\left | b \right |} \mathrm {sgn}\left (b - c\right ) + {\left (a^{3} b^{7} - 2 \, a^{3} b^{6} c + 2 \, a^{3} b^{4} c^{3} - a^{3} b^{3} c^{4}\right )} \sqrt {-a b c} {\left | b \right |}\right )} \arctan \left (-\frac {\sqrt {b c} \sqrt {b x + a} - \sqrt {a b^{2} + {\left (b x + a\right )} b c - a b c}}{\sqrt {-\frac {a b^{3} - a b c^{2} - \sqrt {{\left (a b^{3} - a b c^{2}\right )}^{2} - {\left (a^{2} b^{5} - 3 \, a^{2} b^{4} c + 3 \, a^{2} b^{3} c^{2} - a^{2} b^{2} c^{3}\right )} {\left (b - c\right )}}}{b - c}}}\right )}{{\left (b^{8} - 5 \, b^{7} c + 10 \, b^{6} c^{2} - 10 \, b^{5} c^{3} + 5 \, b^{4} c^{4} - b^{3} c^{5}\right )} a^{2} {\left | a b^{2} - a b c \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 4.33, size = 213, normalized size = 2.20 \begin {gather*} -\frac {2\,\sqrt {a}\,c\,\left (\frac {2\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}{\sqrt {a+c\,x}-\sqrt {a}}+\frac {\ln \left (\frac {\sqrt {a+b\,x}-\sqrt {a}}{\sqrt {a+c\,x}-\sqrt {a}}\right )\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}{{\left (\sqrt {a+c\,x}-\sqrt {a}\right )}^2}\right )-2\,\sqrt {a}\,b\,\left (\ln \left (\frac {\sqrt {a+b\,x}-\sqrt {a}}{\sqrt {a+c\,x}-\sqrt {a}}\right )-\frac {2\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}{\sqrt {a+c\,x}-\sqrt {a}}+4\right )}{\left (b-c\right )\,\left (b-\frac {c\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}{{\left (\sqrt {a+c\,x}-\sqrt {a}\right )}^2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________