Optimal. Leaf size=198 \[ -\frac {\sqrt {a+b x} \sqrt {c+d x}}{3 a c x^3}+\frac {5 (b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{12 a^2 c^2 x^2}-\frac {\left (15 b^2 c^2+14 a b c d+15 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{24 a^3 c^3 x}+\frac {(b c+a d) \left (5 b^2 c^2-2 a b c d+5 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{8 a^{7/2} c^{7/2}} \]
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Rubi [A]
time = 0.09, antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {105, 156, 12,
95, 214} \begin {gather*} \frac {5 \sqrt {a+b x} \sqrt {c+d x} (a d+b c)}{12 a^2 c^2 x^2}+\frac {\left (5 a^2 d^2-2 a b c d+5 b^2 c^2\right ) (a d+b c) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{8 a^{7/2} c^{7/2}}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (15 a^2 d^2+14 a b c d+15 b^2 c^2\right )}{24 a^3 c^3 x}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{3 a c x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 95
Rule 105
Rule 156
Rule 214
Rubi steps
\begin {align*} \int \frac {1}{x^4 \sqrt {a+b x} \sqrt {c+d x}} \, dx &=-\frac {\sqrt {a+b x} \sqrt {c+d x}}{3 a c x^3}-\frac {\int \frac {\frac {5}{2} (b c+a d)+2 b d x}{x^3 \sqrt {a+b x} \sqrt {c+d x}} \, dx}{3 a c}\\ &=-\frac {\sqrt {a+b x} \sqrt {c+d x}}{3 a c x^3}+\frac {5 (b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{12 a^2 c^2 x^2}+\frac {\int \frac {\frac {1}{4} \left (15 b^2 c^2+14 a b c d+15 a^2 d^2\right )+\frac {5}{2} b d (b c+a d) x}{x^2 \sqrt {a+b x} \sqrt {c+d x}} \, dx}{6 a^2 c^2}\\ &=-\frac {\sqrt {a+b x} \sqrt {c+d x}}{3 a c x^3}+\frac {5 (b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{12 a^2 c^2 x^2}-\frac {\left (15 b^2 c^2+14 a b c d+15 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{24 a^3 c^3 x}-\frac {\int \frac {3 (b c+a d) \left (5 b^2 c^2-2 a b c d+5 a^2 d^2\right )}{8 x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{6 a^3 c^3}\\ &=-\frac {\sqrt {a+b x} \sqrt {c+d x}}{3 a c x^3}+\frac {5 (b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{12 a^2 c^2 x^2}-\frac {\left (15 b^2 c^2+14 a b c d+15 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{24 a^3 c^3 x}-\frac {\left ((b c+a d) \left (5 b^2 c^2-2 a b c d+5 a^2 d^2\right )\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{16 a^3 c^3}\\ &=-\frac {\sqrt {a+b x} \sqrt {c+d x}}{3 a c x^3}+\frac {5 (b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{12 a^2 c^2 x^2}-\frac {\left (15 b^2 c^2+14 a b c d+15 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{24 a^3 c^3 x}-\frac {\left ((b c+a d) \left (5 b^2 c^2-2 a b c d+5 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{8 a^3 c^3}\\ &=-\frac {\sqrt {a+b x} \sqrt {c+d x}}{3 a c x^3}+\frac {5 (b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{12 a^2 c^2 x^2}-\frac {\left (15 b^2 c^2+14 a b c d+15 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{24 a^3 c^3 x}+\frac {(b c+a d) \left (5 b^2 c^2-2 a b c d+5 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{8 a^{7/2} c^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 0.40, size = 162, normalized size = 0.82 \begin {gather*} -\frac {\sqrt {a+b x} \sqrt {c+d x} \left (15 b^2 c^2 x^2+2 a b c x (-5 c+7 d x)+a^2 \left (8 c^2-10 c d x+15 d^2 x^2\right )\right )}{24 a^3 c^3 x^3}+\frac {\left (5 b^3 c^3+3 a b^2 c^2 d+3 a^2 b c d^2+5 a^3 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )}{8 a^{7/2} c^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(407\) vs.
\(2(166)=332\).
time = 0.08, size = 408, normalized size = 2.06
method | result | size |
default | \(\frac {\left (15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{3} d^{3} x^{3}+9 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{2} b c \,d^{2} x^{3}+9 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a \,b^{2} c^{2} d \,x^{3}+15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) b^{3} c^{3} x^{3}-30 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} d^{2} x^{2}-28 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a b c d \,x^{2}-30 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, b^{2} c^{2} x^{2}+20 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} c d x +20 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a b \,c^{2} x -16 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} c^{2} \sqrt {a c}\right ) \sqrt {d x +c}\, \sqrt {b x +a}}{48 a^{3} c^{3} \sqrt {a c}\, x^{3} \sqrt {\left (d x +c \right ) \left (b x +a \right )}}\) | \(408\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.80, size = 436, normalized size = 2.20 \begin {gather*} \left [\frac {3 \, {\left (5 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + 5 \, a^{3} d^{3}\right )} \sqrt {a c} x^{3} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 4 \, {\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) - 4 \, {\left (8 \, a^{3} c^{3} + {\left (15 \, a b^{2} c^{3} + 14 \, a^{2} b c^{2} d + 15 \, a^{3} c d^{2}\right )} x^{2} - 10 \, {\left (a^{2} b c^{3} + a^{3} c^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{96 \, a^{4} c^{4} x^{3}}, -\frac {3 \, {\left (5 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + 5 \, a^{3} d^{3}\right )} \sqrt {-a c} x^{3} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {-a c} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b c d x^{2} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) + 2 \, {\left (8 \, a^{3} c^{3} + {\left (15 \, a b^{2} c^{3} + 14 \, a^{2} b c^{2} d + 15 \, a^{3} c d^{2}\right )} x^{2} - 10 \, {\left (a^{2} b c^{3} + a^{3} c^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{48 \, a^{4} c^{4} x^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{4} \sqrt {a + b x} \sqrt {c + d x}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1932 vs.
\(2 (166) = 332\).
time = 6.07, size = 1932, normalized size = 9.76 \begin {gather*} \frac {\sqrt {b d} b^{8} d^{3} {\left (\frac {3 \, {\left (5 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + 5 \, a^{3} d^{3}\right )} \arctan \left (-\frac {b^{2} c + a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt {-a b c d} b}\right )}{\sqrt {-a b c d} a^{3} b^{7} c^{3} d^{3}} - \frac {2 \, {\left (15 \, b^{13} c^{8} - 76 \, a b^{12} c^{7} d + 156 \, a^{2} b^{11} c^{6} d^{2} - 180 \, a^{3} b^{10} c^{5} d^{3} + 170 \, a^{4} b^{9} c^{4} d^{4} - 180 \, a^{5} b^{8} c^{3} d^{5} + 156 \, a^{6} b^{7} c^{2} d^{6} - 76 \, a^{7} b^{6} c d^{7} + 15 \, a^{8} b^{5} d^{8} - 75 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{11} c^{7} + 135 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b^{10} c^{6} d + 45 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{2} b^{9} c^{5} d^{2} - 105 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{3} b^{8} c^{4} d^{3} - 105 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{4} b^{7} c^{3} d^{4} + 45 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{5} b^{6} c^{2} d^{5} + 135 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{6} b^{5} c d^{6} - 75 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{7} b^{4} d^{7} + 150 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} b^{9} c^{6} + 60 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a b^{8} c^{5} d + 42 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a^{2} b^{7} c^{4} d^{2} - 504 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a^{3} b^{6} c^{3} d^{3} + 42 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a^{4} b^{5} c^{2} d^{4} + 60 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a^{5} b^{4} c d^{5} + 150 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a^{6} b^{3} d^{6} - 150 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{6} b^{7} c^{5} - 230 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{6} a b^{6} c^{4} d - 324 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{6} a^{2} b^{5} c^{3} d^{2} - 324 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{6} a^{3} b^{4} c^{2} d^{3} - 230 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{6} a^{4} b^{3} c d^{4} - 150 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{6} a^{5} b^{2} d^{5} + 75 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{8} b^{5} c^{4} + 120 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{8} a b^{4} c^{3} d + 90 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{8} a^{2} b^{3} c^{2} d^{2} + 120 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{8} a^{3} b^{2} c d^{3} + 75 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{8} a^{4} b d^{4} - 15 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{10} b^{3} c^{3} - 9 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{10} a b^{2} c^{2} d - 9 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{10} a^{2} b c d^{2} - 15 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{10} a^{3} d^{3}\right )}}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2} - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{2} c - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b d + {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4}\right )}^{3} a^{3} b^{6} c^{3} d^{3}}\right )}}{24 \, {\left | b \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 90.68, size = 1518, normalized size = 7.67 \begin {gather*} \frac {\left (\sqrt {a+b\,x}-\sqrt {a}\right )\,\left (\frac {7\,b\,d^2}{64\,a^2\,c^2}-\frac {d\,{\left (a\,d+b\,c\right )}^2}{4\,a^3\,c^3}+\frac {d\,\left (3\,a^2\,d^2+8\,a\,b\,c\,d+3\,b^2\,c^2\right )}{32\,a^3\,c^3}\right )}{\sqrt {c+d\,x}-\sqrt {c}}-\frac {\frac {b^6}{192\,a^2\,c^2\,d^3}+\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2\,\left (\frac {5\,a^2\,b^4\,d^2}{64}+\frac {5\,b^6\,c^2}{64}\right )}{a^3\,c^3\,d^3\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^2}+\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^4\,\left (\frac {13\,a^4\,b^2\,d^4}{64}+\frac {55\,a^3\,b^3\,c\,d^3}{32}+\frac {57\,a^2\,b^4\,c^2\,d^2}{32}+\frac {55\,a\,b^5\,c^3\,d}{32}+\frac {13\,b^6\,c^4}{64}\right )}{a^4\,c^4\,d^3\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^4}-\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^5\,\left (-\frac {17\,a^5\,b\,d^5}{64}+\frac {39\,a^4\,b^2\,c\,d^4}{32}+\frac {25\,a^3\,b^3\,c^2\,d^3}{8}+\frac {25\,a^2\,b^4\,c^3\,d^2}{8}+\frac {39\,a\,b^5\,c^4\,d}{32}-\frac {17\,b^6\,c^5}{64}\right )}{a^{9/2}\,c^{9/2}\,d^3\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^5}-\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^7\,\left (-\frac {27\,a^5\,d^5}{64}+\frac {15\,a^4\,b\,c\,d^4}{64}+\frac {97\,a^3\,b^2\,c^2\,d^3}{64}+\frac {97\,a^2\,b^3\,c^3\,d^2}{64}+\frac {15\,a\,b^4\,c^4\,d}{64}-\frac {27\,b^5\,c^5}{64}\right )}{a^{9/2}\,c^{9/2}\,d^2\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^7}+\frac {\left (\frac {c\,b^6}{64}+\frac {a\,d\,b^5}{64}\right )\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}{a^{5/2}\,c^{5/2}\,d^3\,\left (\sqrt {c+d\,x}-\sqrt {c}\right )}-\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^3\,\left (\frac {73\,a^3\,b^3\,d^3}{192}+\frac {9\,a^2\,b^4\,c\,d^2}{16}+\frac {9\,a\,b^5\,c^2\,d}{16}+\frac {73\,b^6\,c^3}{192}\right )}{a^{7/2}\,c^{7/2}\,d^3\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^3}-\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^6\,\left (\frac {37\,a^6\,d^6}{192}+\frac {11\,a^5\,b\,c\,d^5}{32}-\frac {71\,a^4\,b^2\,c^2\,d^4}{32}-\frac {49\,a^3\,b^3\,c^3\,d^3}{16}-\frac {71\,a^2\,b^4\,c^4\,d^2}{32}+\frac {11\,a\,b^5\,c^5\,d}{32}+\frac {37\,b^6\,c^6}{192}\right )}{a^5\,c^5\,d^3\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^6}+\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^8\,\left (-\frac {15\,a^4\,d^4}{64}+\frac {11\,a^3\,b\,c\,d^3}{32}+\frac {15\,a^2\,b^2\,c^2\,d^2}{64}+\frac {11\,a\,b^3\,c^3\,d}{32}-\frac {15\,b^4\,c^4}{64}\right )}{a^4\,c^4\,d\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^8}}{\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^9}{{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^9}+\frac {b^3\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^3}{d^3\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^3}+\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^7\,\left (3\,a^2\,d^2+9\,a\,b\,c\,d+3\,b^2\,c^2\right )}{a\,c\,d^2\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^7}-\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^6\,\left (a^3\,d^3+9\,a^2\,b\,c\,d^2+9\,a\,b^2\,c^2\,d+b^3\,c^3\right )}{a^{3/2}\,c^{3/2}\,d^3\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^6}-\frac {\left (3\,a\,d+3\,b\,c\right )\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^8}{\sqrt {a}\,\sqrt {c}\,d\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^8}-\frac {\left (3\,c\,b^3+3\,a\,d\,b^2\right )\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^4}{\sqrt {a}\,\sqrt {c}\,d^3\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^4}+\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^5\,\left (3\,a^2\,b\,d^2+9\,a\,b^2\,c\,d+3\,b^3\,c^2\right )}{a\,c\,d^3\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^5}}-\frac {\ln \left (\frac {\left (\sqrt {c}\,\sqrt {a+b\,x}-\sqrt {a}\,\sqrt {c+d\,x}\right )\,\left (b\,\sqrt {c}-\frac {\sqrt {a}\,d\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}{\sqrt {c+d\,x}-\sqrt {c}}\right )}{\sqrt {c+d\,x}-\sqrt {c}}\right )\,\left (5\,\sqrt {a}\,b^3\,c^{7/2}+5\,a^{7/2}\,\sqrt {c}\,d^3+3\,a^{3/2}\,b^2\,c^{5/2}\,d+3\,a^{5/2}\,b\,c^{3/2}\,d^2\right )}{16\,a^4\,c^4}+\frac {\ln \left (\frac {\sqrt {a+b\,x}-\sqrt {a}}{\sqrt {c+d\,x}-\sqrt {c}}\right )\,\left (5\,\sqrt {a}\,b^3\,c^{7/2}+5\,a^{7/2}\,\sqrt {c}\,d^3+3\,a^{3/2}\,b^2\,c^{5/2}\,d+3\,a^{5/2}\,b\,c^{3/2}\,d^2\right )}{16\,a^4\,c^4}-\frac {d^3\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^3}{192\,a^2\,c^2\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^3}-\frac {d^2\,\left (a\,d+b\,c\right )\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}{32\,a^{5/2}\,c^{5/2}\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
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