Optimal. Leaf size=228 \[ \frac {5 (a+c) (a+b x)^{3/2} (b x+c)^{3/2}}{12 b^3 (a-c)^2}+\frac {\left (4 a c-5 (a+c)^2\right ) (a+b x)^{3/2} \sqrt {b x+c}}{16 b^3 (a-c)^2}-\frac {\left (4 a c-5 (a+c)^2\right ) \sqrt {a+b x} \sqrt {b x+c}}{32 b^3 (a-c)}-\frac {\left (4 a c-5 (a+c)^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {b x+c}}\right )}{32 b^3}-\frac {x (a+b x)^{3/2} (b x+c)^{3/2}}{2 b^2 (a-c)^2}+\frac {b x^4}{2 (a-c)^2}+\frac {x^3 (a+c)}{3 (a-c)^2} \]
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Rubi [A] time = 0.37, antiderivative size = 228, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {6689, 90, 80, 50, 63, 217, 206} \[ -\frac {x (a+b x)^{3/2} (b x+c)^{3/2}}{2 b^2 (a-c)^2}+\frac {5 (a+c) (a+b x)^{3/2} (b x+c)^{3/2}}{12 b^3 (a-c)^2}+\frac {\left (4 a c-5 (a+c)^2\right ) (a+b x)^{3/2} \sqrt {b x+c}}{16 b^3 (a-c)^2}-\frac {\left (4 a c-5 (a+c)^2\right ) \sqrt {a+b x} \sqrt {b x+c}}{32 b^3 (a-c)}-\frac {\left (4 a c-5 (a+c)^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {b x+c}}\right )}{32 b^3}+\frac {b x^4}{2 (a-c)^2}+\frac {x^3 (a+c)}{3 (a-c)^2} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 80
Rule 90
Rule 206
Rule 217
Rule 6689
Rubi steps
\begin {align*} \int \frac {x^2}{\left (\sqrt {a+b x}+\sqrt {c+b x}\right )^2} \, dx &=\frac {\int \left (a \left (1+\frac {c}{a}\right ) x^2+2 b x^3-2 x^2 \sqrt {a+b x} \sqrt {c+b x}\right ) \, dx}{(a-c)^2}\\ &=\frac {(a+c) x^3}{3 (a-c)^2}+\frac {b x^4}{2 (a-c)^2}-\frac {2 \int x^2 \sqrt {a+b x} \sqrt {c+b x} \, dx}{(a-c)^2}\\ &=\frac {(a+c) x^3}{3 (a-c)^2}+\frac {b x^4}{2 (a-c)^2}-\frac {x (a+b x)^{3/2} (c+b x)^{3/2}}{2 b^2 (a-c)^2}-\frac {\int \sqrt {a+b x} \sqrt {c+b x} \left (-a c-\frac {5}{2} b (a+c) x\right ) \, dx}{2 b^2 (a-c)^2}\\ &=\frac {(a+c) x^3}{3 (a-c)^2}+\frac {b x^4}{2 (a-c)^2}+\frac {5 (a+c) (a+b x)^{3/2} (c+b x)^{3/2}}{12 b^3 (a-c)^2}-\frac {x (a+b x)^{3/2} (c+b x)^{3/2}}{2 b^2 (a-c)^2}+\frac {\left (4 a c-5 (a+c)^2\right ) \int \sqrt {a+b x} \sqrt {c+b x} \, dx}{8 b^2 (a-c)^2}\\ &=\frac {(a+c) x^3}{3 (a-c)^2}+\frac {b x^4}{2 (a-c)^2}+\frac {\left (4 a c-5 (a+c)^2\right ) (a+b x)^{3/2} \sqrt {c+b x}}{16 b^3 (a-c)^2}+\frac {5 (a+c) (a+b x)^{3/2} (c+b x)^{3/2}}{12 b^3 (a-c)^2}-\frac {x (a+b x)^{3/2} (c+b x)^{3/2}}{2 b^2 (a-c)^2}+\frac {\left (5 a^2+6 a c+5 c^2\right ) \int \frac {\sqrt {a+b x}}{\sqrt {c+b x}} \, dx}{32 b^2 (a-c)}\\ &=\frac {(a+c) x^3}{3 (a-c)^2}+\frac {b x^4}{2 (a-c)^2}+\frac {\left (5 a^2+6 a c+5 c^2\right ) \sqrt {a+b x} \sqrt {c+b x}}{32 b^3 (a-c)}+\frac {\left (4 a c-5 (a+c)^2\right ) (a+b x)^{3/2} \sqrt {c+b x}}{16 b^3 (a-c)^2}+\frac {5 (a+c) (a+b x)^{3/2} (c+b x)^{3/2}}{12 b^3 (a-c)^2}-\frac {x (a+b x)^{3/2} (c+b x)^{3/2}}{2 b^2 (a-c)^2}+\frac {\left (5 a^2+6 a c+5 c^2\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+b x}} \, dx}{64 b^2}\\ &=\frac {(a+c) x^3}{3 (a-c)^2}+\frac {b x^4}{2 (a-c)^2}+\frac {\left (5 a^2+6 a c+5 c^2\right ) \sqrt {a+b x} \sqrt {c+b x}}{32 b^3 (a-c)}+\frac {\left (4 a c-5 (a+c)^2\right ) (a+b x)^{3/2} \sqrt {c+b x}}{16 b^3 (a-c)^2}+\frac {5 (a+c) (a+b x)^{3/2} (c+b x)^{3/2}}{12 b^3 (a-c)^2}-\frac {x (a+b x)^{3/2} (c+b x)^{3/2}}{2 b^2 (a-c)^2}+\frac {\left (5 a^2+6 a c+5 c^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-a+c+x^2}} \, dx,x,\sqrt {a+b x}\right )}{32 b^3}\\ &=\frac {(a+c) x^3}{3 (a-c)^2}+\frac {b x^4}{2 (a-c)^2}+\frac {\left (5 a^2+6 a c+5 c^2\right ) \sqrt {a+b x} \sqrt {c+b x}}{32 b^3 (a-c)}+\frac {\left (4 a c-5 (a+c)^2\right ) (a+b x)^{3/2} \sqrt {c+b x}}{16 b^3 (a-c)^2}+\frac {5 (a+c) (a+b x)^{3/2} (c+b x)^{3/2}}{12 b^3 (a-c)^2}-\frac {x (a+b x)^{3/2} (c+b x)^{3/2}}{2 b^2 (a-c)^2}+\frac {\left (5 a^2+6 a c+5 c^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+b x}}\right )}{32 b^3}\\ &=\frac {(a+c) x^3}{3 (a-c)^2}+\frac {b x^4}{2 (a-c)^2}+\frac {\left (5 a^2+6 a c+5 c^2\right ) \sqrt {a+b x} \sqrt {c+b x}}{32 b^3 (a-c)}+\frac {\left (4 a c-5 (a+c)^2\right ) (a+b x)^{3/2} \sqrt {c+b x}}{16 b^3 (a-c)^2}+\frac {5 (a+c) (a+b x)^{3/2} (c+b x)^{3/2}}{12 b^3 (a-c)^2}-\frac {x (a+b x)^{3/2} (c+b x)^{3/2}}{2 b^2 (a-c)^2}+\frac {\left (5 a^2+6 a c+5 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {c+b x}}\right )}{32 b^3}\\ \end {align*}
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Mathematica [A] time = 1.29, size = 361, normalized size = 1.58 \[ \frac {-15 a^3 \sqrt {a+b x} \sqrt {b x+c}+\frac {3 \sqrt {b} (c-a)^3 \left (5 a^2+6 a c+5 c^2\right ) \sqrt {-\frac {b x+c}{a-c}} \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {a+b x}}{\sqrt {b (c-a)}}\right )}{\sqrt {b (c-a)} \sqrt {b x+c}}+a^2 \sqrt {a+b x} \sqrt {b x+c} (10 b x+7 c)-16 b^3 x^3 \left (3 \sqrt {a+b x} \sqrt {b x+c}-2 c\right )-8 b^2 c x^2 \sqrt {a+b x} \sqrt {b x+c}-a \left (8 b^2 x^2 \sqrt {a+b x} \sqrt {b x+c}-7 c^2 \sqrt {a+b x} \sqrt {b x+c}+4 b c x \sqrt {a+b x} \sqrt {b x+c}-32 b^3 x^3\right )-15 c^3 \sqrt {a+b x} \sqrt {b x+c}+10 b c^2 x \sqrt {a+b x} \sqrt {b x+c}+48 b^4 x^4}{96 b^3 (a-c)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 196, normalized size = 0.86 \[ \frac {96 \, b^{4} x^{4} + 64 \, {\left (a b^{3} + b^{3} c\right )} x^{3} - 2 \, {\left (48 \, b^{3} x^{3} + 15 \, a^{3} - 7 \, a^{2} c - 7 \, a c^{2} + 15 \, c^{3} + 8 \, {\left (a b^{2} + b^{2} c\right )} x^{2} - 2 \, {\left (5 \, a^{2} b - 2 \, a b c + 5 \, b c^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {b x + c} - 3 \, {\left (5 \, a^{4} - 4 \, a^{3} c - 2 \, a^{2} c^{2} - 4 \, a c^{3} + 5 \, c^{4}\right )} \log \left (-2 \, b x + 2 \, \sqrt {b x + a} \sqrt {b x + c} - a - c\right )}{192 \, {\left (a^{2} b^{3} - 2 \, a b^{3} c + b^{3} c^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.39, size = 797, normalized size = 3.50 \[ -\frac {1}{96} \, {\left (2 \, {\left (4 \, {\left (b x + a\right )} {\left (\frac {6 \, {\left (a^{5} b^{9} - 5 \, a^{4} b^{9} c + 10 \, a^{3} b^{9} c^{2} - 10 \, a^{2} b^{9} c^{3} + 5 \, a b^{9} c^{4} - b^{9} c^{5}\right )} {\left (b x + a\right )}}{a^{7} b^{12} - 7 \, a^{6} b^{12} c + 21 \, a^{5} b^{12} c^{2} - 35 \, a^{4} b^{12} c^{3} + 35 \, a^{3} b^{12} c^{4} - 21 \, a^{2} b^{12} c^{5} + 7 \, a b^{12} c^{6} - b^{12} c^{7}} - \frac {17 \, a^{6} b^{9} - 86 \, a^{5} b^{9} c + 175 \, a^{4} b^{9} c^{2} - 180 \, a^{3} b^{9} c^{3} + 95 \, a^{2} b^{9} c^{4} - 22 \, a b^{9} c^{5} + b^{9} c^{6}}{a^{7} b^{12} - 7 \, a^{6} b^{12} c + 21 \, a^{5} b^{12} c^{2} - 35 \, a^{4} b^{12} c^{3} + 35 \, a^{3} b^{12} c^{4} - 21 \, a^{2} b^{12} c^{5} + 7 \, a b^{12} c^{6} - b^{12} c^{7}}\right )} + \frac {59 \, a^{7} b^{9} - 301 \, a^{6} b^{9} c + 615 \, a^{5} b^{9} c^{2} - 625 \, a^{4} b^{9} c^{3} + 305 \, a^{3} b^{9} c^{4} - 39 \, a^{2} b^{9} c^{5} - 19 \, a b^{9} c^{6} + 5 \, b^{9} c^{7}}{a^{7} b^{12} - 7 \, a^{6} b^{12} c + 21 \, a^{5} b^{12} c^{2} - 35 \, a^{4} b^{12} c^{3} + 35 \, a^{3} b^{12} c^{4} - 21 \, a^{2} b^{12} c^{5} + 7 \, a b^{12} c^{6} - b^{12} c^{7}}\right )} {\left (b x + a\right )} - \frac {3 \, {\left (5 \, a^{8} b^{9} - 24 \, a^{7} b^{9} c + 44 \, a^{6} b^{9} c^{2} - 40 \, a^{5} b^{9} c^{3} + 30 \, a^{4} b^{9} c^{4} - 40 \, a^{3} b^{9} c^{5} + 44 \, a^{2} b^{9} c^{6} - 24 \, a b^{9} c^{7} + 5 \, b^{9} c^{8}\right )}}{a^{7} b^{12} - 7 \, a^{6} b^{12} c + 21 \, a^{5} b^{12} c^{2} - 35 \, a^{4} b^{12} c^{3} + 35 \, a^{3} b^{12} c^{4} - 21 \, a^{2} b^{12} c^{5} + 7 \, a b^{12} c^{6} - b^{12} c^{7}}\right )} \sqrt {b x + a} \sqrt {b x + c} + \frac {3 \, {\left (b x + a\right )}^{4} - 10 \, {\left (b x + a\right )}^{3} a + 12 \, {\left (b x + a\right )}^{2} a^{2} - 6 \, {\left (b x + a\right )} a^{3} + 2 \, {\left (b x + a\right )}^{3} c - 6 \, {\left (b x + a\right )}^{2} a c + 6 \, {\left (b x + a\right )} a^{2} c}{6 \, {\left (a^{2} b^{3} - 2 \, a b^{3} c + b^{3} c^{2}\right )}} - \frac {{\left (5 \, a^{2} + 6 \, a c + 5 \, c^{2}\right )} \log \left ({\left | -\sqrt {b x + a} + \sqrt {b x + c} \right |}\right )}{32 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 604, normalized size = 2.65 \[ \frac {b \,x^{4}}{2 \left (a -c \right )^{2}}+\frac {a \,x^{3}}{3 \left (a -c \right )^{2}}+\frac {c \,x^{3}}{3 \left (a -c \right )^{2}}-\frac {\sqrt {b x +a}\, \sqrt {b x +c}\, \left (96 \sqrt {b^{2} x^{2}+a b x +b c x +a c}\, b^{3} x^{3} \mathrm {csgn}\relax (b )+16 \sqrt {b^{2} x^{2}+a b x +b c x +a c}\, a \,b^{2} x^{2} \mathrm {csgn}\relax (b )+16 \sqrt {b^{2} x^{2}+a b x +b c x +a c}\, b^{2} c \,x^{2} \mathrm {csgn}\relax (b )-15 a^{4} \ln \left (\frac {\left (2 b x +a +c +2 \sqrt {b^{2} x^{2}+a b x +b c x +a c}\, \mathrm {csgn}\relax (b )\right ) \mathrm {csgn}\relax (b )}{2}\right )+12 a^{3} c \ln \left (\frac {\left (2 b x +a +c +2 \sqrt {b^{2} x^{2}+a b x +b c x +a c}\, \mathrm {csgn}\relax (b )\right ) \mathrm {csgn}\relax (b )}{2}\right )-20 \sqrt {b^{2} x^{2}+a b x +b c x +a c}\, a^{2} b x \,\mathrm {csgn}\relax (b )+6 a^{2} c^{2} \ln \left (\frac {\left (2 b x +a +c +2 \sqrt {b^{2} x^{2}+a b x +b c x +a c}\, \mathrm {csgn}\relax (b )\right ) \mathrm {csgn}\relax (b )}{2}\right )+8 \sqrt {b^{2} x^{2}+a b x +b c x +a c}\, a b c x \,\mathrm {csgn}\relax (b )+12 a \,c^{3} \ln \left (\frac {\left (2 b x +a +c +2 \sqrt {b^{2} x^{2}+a b x +b c x +a c}\, \mathrm {csgn}\relax (b )\right ) \mathrm {csgn}\relax (b )}{2}\right )-20 \sqrt {b^{2} x^{2}+a b x +b c x +a c}\, b \,c^{2} x \,\mathrm {csgn}\relax (b )-15 c^{4} \ln \left (\frac {\left (2 b x +a +c +2 \sqrt {b^{2} x^{2}+a b x +b c x +a c}\, \mathrm {csgn}\relax (b )\right ) \mathrm {csgn}\relax (b )}{2}\right )+30 \sqrt {b^{2} x^{2}+a b x +b c x +a c}\, a^{3} \mathrm {csgn}\relax (b )-14 \sqrt {b^{2} x^{2}+a b x +b c x +a c}\, a^{2} c \,\mathrm {csgn}\relax (b )-14 \sqrt {b^{2} x^{2}+a b x +b c x +a c}\, a \,c^{2} \mathrm {csgn}\relax (b )+30 \sqrt {b^{2} x^{2}+a b x +b c x +a c}\, c^{3} \mathrm {csgn}\relax (b )\right ) \mathrm {csgn}\relax (b )}{192 \left (a -c \right )^{2} \sqrt {b^{2} x^{2}+a b x +b c x +a c}\, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{{\left (\sqrt {b x + a} + \sqrt {b x + c}\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 81.17, size = 1358, normalized size = 5.96 \[ \text {result too large to display} \]
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 \[ \int \frac {x^{2}}{\left (\sqrt {a + b x} + \sqrt {b x + c}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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