Optimal. Leaf size=165 \[ -\frac {2 (a+b x)^{3/2} (b x+c)^{3/2}}{3 b^2 (a-c)^2}+\frac {(a+c) (a+b x)^{3/2} \sqrt {b x+c}}{2 b^2 (a-c)^2}-\frac {(a+c) \sqrt {a+b x} \sqrt {b x+c}}{4 b^2 (a-c)}-\frac {(a+c) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {b x+c}}\right )}{4 b^2}+\frac {2 b x^3}{3 (a-c)^2}+\frac {x^2 (a+c)}{2 (a-c)^2} \]
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Rubi [A] time = 0.21, antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {6689, 80, 50, 63, 217, 206} \[ -\frac {2 (a+b x)^{3/2} (b x+c)^{3/2}}{3 b^2 (a-c)^2}+\frac {(a+c) (a+b x)^{3/2} \sqrt {b x+c}}{2 b^2 (a-c)^2}-\frac {(a+c) \sqrt {a+b x} \sqrt {b x+c}}{4 b^2 (a-c)}-\frac {(a+c) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {b x+c}}\right )}{4 b^2}+\frac {2 b x^3}{3 (a-c)^2}+\frac {x^2 (a+c)}{2 (a-c)^2} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 80
Rule 206
Rule 217
Rule 6689
Rubi steps
\begin {align*} \int \frac {x}{\left (\sqrt {a+b x}+\sqrt {c+b x}\right )^2} \, dx &=\frac {\int \left (a \left (1+\frac {c}{a}\right ) x+2 b x^2-2 x \sqrt {a+b x} \sqrt {c+b x}\right ) \, dx}{(a-c)^2}\\ &=\frac {(a+c) x^2}{2 (a-c)^2}+\frac {2 b x^3}{3 (a-c)^2}-\frac {2 \int x \sqrt {a+b x} \sqrt {c+b x} \, dx}{(a-c)^2}\\ &=\frac {(a+c) x^2}{2 (a-c)^2}+\frac {2 b x^3}{3 (a-c)^2}-\frac {2 (a+b x)^{3/2} (c+b x)^{3/2}}{3 b^2 (a-c)^2}+\frac {(a+c) \int \sqrt {a+b x} \sqrt {c+b x} \, dx}{b (a-c)^2}\\ &=\frac {(a+c) x^2}{2 (a-c)^2}+\frac {2 b x^3}{3 (a-c)^2}+\frac {(a+c) (a+b x)^{3/2} \sqrt {c+b x}}{2 b^2 (a-c)^2}-\frac {2 (a+b x)^{3/2} (c+b x)^{3/2}}{3 b^2 (a-c)^2}-\frac {(a+c) \int \frac {\sqrt {a+b x}}{\sqrt {c+b x}} \, dx}{4 b (a-c)}\\ &=\frac {(a+c) x^2}{2 (a-c)^2}+\frac {2 b x^3}{3 (a-c)^2}-\frac {(a+c) \sqrt {a+b x} \sqrt {c+b x}}{4 b^2 (a-c)}+\frac {(a+c) (a+b x)^{3/2} \sqrt {c+b x}}{2 b^2 (a-c)^2}-\frac {2 (a+b x)^{3/2} (c+b x)^{3/2}}{3 b^2 (a-c)^2}-\frac {(a+c) \int \frac {1}{\sqrt {a+b x} \sqrt {c+b x}} \, dx}{8 b}\\ &=\frac {(a+c) x^2}{2 (a-c)^2}+\frac {2 b x^3}{3 (a-c)^2}-\frac {(a+c) \sqrt {a+b x} \sqrt {c+b x}}{4 b^2 (a-c)}+\frac {(a+c) (a+b x)^{3/2} \sqrt {c+b x}}{2 b^2 (a-c)^2}-\frac {2 (a+b x)^{3/2} (c+b x)^{3/2}}{3 b^2 (a-c)^2}-\frac {(a+c) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-a+c+x^2}} \, dx,x,\sqrt {a+b x}\right )}{4 b^2}\\ &=\frac {(a+c) x^2}{2 (a-c)^2}+\frac {2 b x^3}{3 (a-c)^2}-\frac {(a+c) \sqrt {a+b x} \sqrt {c+b x}}{4 b^2 (a-c)}+\frac {(a+c) (a+b x)^{3/2} \sqrt {c+b x}}{2 b^2 (a-c)^2}-\frac {2 (a+b x)^{3/2} (c+b x)^{3/2}}{3 b^2 (a-c)^2}-\frac {(a+c) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+b x}}\right )}{4 b^2}\\ &=\frac {(a+c) x^2}{2 (a-c)^2}+\frac {2 b x^3}{3 (a-c)^2}-\frac {(a+c) \sqrt {a+b x} \sqrt {c+b x}}{4 b^2 (a-c)}+\frac {(a+c) (a+b x)^{3/2} \sqrt {c+b x}}{2 b^2 (a-c)^2}-\frac {2 (a+b x)^{3/2} (c+b x)^{3/2}}{3 b^2 (a-c)^2}-\frac {(a+c) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {c+b x}}\right )}{4 b^2}\\ \end {align*}
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Mathematica [A] time = 0.88, size = 229, normalized size = 1.39 \[ \frac {3 a^2 \sqrt {a+b x} \sqrt {b x+c}-2 a \left (b x \sqrt {a+b x} \sqrt {b x+c}+c \sqrt {a+b x} \sqrt {b x+c}-3 b^2 x^2\right )+(4 b x+3 c) \left (-2 b x \sqrt {a+b x} \sqrt {b x+c}+c \sqrt {a+b x} \sqrt {b x+c}+2 b^2 x^2\right )}{12 b^2 (a-c)^2}-\frac {(a+c) \sqrt {b (c-a)} \sqrt {-\frac {b x+c}{a-c}} \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {a+b x}}{\sqrt {b (c-a)}}\right )}{4 b^{5/2} \sqrt {b x+c}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 149, normalized size = 0.90 \[ \frac {16 \, b^{3} x^{3} + 12 \, {\left (a b^{2} + b^{2} c\right )} x^{2} - 2 \, {\left (8 \, b^{2} x^{2} - 3 \, a^{2} + 2 \, a c - 3 \, c^{2} + 2 \, {\left (a b + b c\right )} x\right )} \sqrt {b x + a} \sqrt {b x + c} + 3 \, {\left (a^{3} - a^{2} c - a c^{2} + c^{3}\right )} \log \left (-2 \, b x + 2 \, \sqrt {b x + a} \sqrt {b x + c} - a - c\right )}{24 \, {\left (a^{2} b^{2} - 2 \, a b^{2} c + b^{2} c^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.27, size = 445, normalized size = 2.70 \[ -\frac {{\left (2 \, {\left (b x + a\right )} {\left (\frac {4 \, {\left (a^{3} b^{2} - 3 \, a^{2} b^{2} c + 3 \, a b^{2} c^{2} - b^{2} c^{3}\right )} {\left (b x + a\right )}}{a^{5} b^{3} - 5 \, a^{4} b^{3} c + 10 \, a^{3} b^{3} c^{2} - 10 \, a^{2} b^{3} c^{3} + 5 \, a b^{3} c^{4} - b^{3} c^{5}} - \frac {7 \, a^{4} b^{2} - 22 \, a^{3} b^{2} c + 24 \, a^{2} b^{2} c^{2} - 10 \, a b^{2} c^{3} + b^{2} c^{4}}{a^{5} b^{3} - 5 \, a^{4} b^{3} c + 10 \, a^{3} b^{3} c^{2} - 10 \, a^{2} b^{3} c^{3} + 5 \, a b^{3} c^{4} - b^{3} c^{5}}\right )} + \frac {3 \, {\left (a^{5} b^{2} - 3 \, a^{4} b^{2} c + 2 \, a^{3} b^{2} c^{2} + 2 \, a^{2} b^{2} c^{3} - 3 \, a b^{2} c^{4} + b^{2} c^{5}\right )}}{a^{5} b^{3} - 5 \, a^{4} b^{3} c + 10 \, a^{3} b^{3} c^{2} - 10 \, a^{2} b^{3} c^{3} + 5 \, a b^{3} c^{4} - b^{3} c^{5}}\right )} \sqrt {b x + a} \sqrt {b x + c} - \frac {3 \, {\left (a + c\right )} \log \left ({\left | -\sqrt {b x + a} + \sqrt {b x + c} \right |}\right )}{b} - \frac {2 \, {\left (4 \, {\left (b x + a\right )}^{3} - 9 \, {\left (b x + a\right )}^{2} a + 6 \, {\left (b x + a\right )} a^{2} + 3 \, {\left (b x + a\right )}^{2} c - 6 \, {\left (b x + a\right )} a c\right )}}{a^{2} b - 2 \, a b c + b c^{2}}}{12 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 431, normalized size = 2.61 \[ \frac {2 b \,x^{3}}{3 \left (a -c \right )^{2}}+\frac {a \,x^{2}}{2 \left (a -c \right )^{2}}+\frac {c \,x^{2}}{2 \left (a -c \right )^{2}}-\frac {\sqrt {b x +a}\, \sqrt {b x +c}\, \left (16 \sqrt {b^{2} x^{2}+a b x +b c x +a c}\, b^{2} x^{2} \mathrm {csgn}\relax (b )+3 a^{3} \ln \left (\frac {\left (2 b x +a +c +2 \sqrt {b^{2} x^{2}+a b x +b c x +a c}\, \mathrm {csgn}\relax (b )\right ) \mathrm {csgn}\relax (b )}{2}\right )-3 a^{2} c \ln \left (\frac {\left (2 b x +a +c +2 \sqrt {b^{2} x^{2}+a b x +b c x +a c}\, \mathrm {csgn}\relax (b )\right ) \mathrm {csgn}\relax (b )}{2}\right )+4 \sqrt {b^{2} x^{2}+a b x +b c x +a c}\, a b x \,\mathrm {csgn}\relax (b )-3 a \,c^{2} \ln \left (\frac {\left (2 b x +a +c +2 \sqrt {b^{2} x^{2}+a b x +b c x +a c}\, \mathrm {csgn}\relax (b )\right ) \mathrm {csgn}\relax (b )}{2}\right )+4 \sqrt {b^{2} x^{2}+a b x +b c x +a c}\, b c x \,\mathrm {csgn}\relax (b )+3 c^{3} \ln \left (\frac {\left (2 b x +a +c +2 \sqrt {b^{2} x^{2}+a b x +b c x +a c}\, \mathrm {csgn}\relax (b )\right ) \mathrm {csgn}\relax (b )}{2}\right )-6 \sqrt {b^{2} x^{2}+a b x +b c x +a c}\, a^{2} \mathrm {csgn}\relax (b )+4 \sqrt {b^{2} x^{2}+a b x +b c x +a c}\, a c \,\mathrm {csgn}\relax (b )-6 \sqrt {b^{2} x^{2}+a b x +b c x +a c}\, c^{2} \mathrm {csgn}\relax (b )\right ) \mathrm {csgn}\relax (b )}{24 \left (a -c \right )^{2} \sqrt {b^{2} x^{2}+a b x +b c x +a c}\, b^{2}} \]
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 \[ \int \frac {x}{{\left (\sqrt {b x + a} + \sqrt {b x + c}\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 37.52, size = 1012, normalized size = 6.13 \[ \frac {\frac {\left (\sqrt {a+b\,x}-\sqrt {a}\right )\,\left (\frac {a}{2}+\frac {c}{2}\right )}{b^2\,\left (\sqrt {c+b\,x}-\sqrt {c}\right )}+\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^{11}\,\left (\frac {a}{2}+\frac {c}{2}\right )}{b^2\,{\left (\sqrt {c+b\,x}-\sqrt {c}\right )}^{11}}-\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^3\,\left (\frac {17\,a^3}{6}+\frac {101\,a^2\,c}{2}+\frac {101\,a\,c^2}{2}+\frac {17\,c^3}{6}\right )}{{\left (\sqrt {c+b\,x}-\sqrt {c}\right )}^3\,\left (a^2\,b^2-2\,a\,b^2\,c+b^2\,c^2\right )}-\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^9\,\left (\frac {17\,a^3}{6}+\frac {101\,a^2\,c}{2}+\frac {101\,a\,c^2}{2}+\frac {17\,c^3}{6}\right )}{{\left (\sqrt {c+b\,x}-\sqrt {c}\right )}^9\,\left (a^2\,b^2-2\,a\,b^2\,c+b^2\,c^2\right )}-\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^5\,\left (19\,a^3+269\,a^2\,c+269\,a\,c^2+19\,c^3\right )}{{\left (\sqrt {c+b\,x}-\sqrt {c}\right )}^5\,\left (a^2\,b^2-2\,a\,b^2\,c+b^2\,c^2\right )}-\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^7\,\left (19\,a^3+269\,a^2\,c+269\,a\,c^2+19\,c^3\right )}{{\left (\sqrt {c+b\,x}-\sqrt {c}\right )}^7\,\left (a^2\,b^2-2\,a\,b^2\,c+b^2\,c^2\right )}+\frac {16\,a^{3/2}\,c^{3/2}\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}{{\left (\sqrt {c+b\,x}-\sqrt {c}\right )}^2\,\left (a^2\,b^2-2\,a\,b^2\,c+b^2\,c^2\right )}+\frac {16\,a^{3/2}\,c^{3/2}\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^{10}}{{\left (\sqrt {c+b\,x}-\sqrt {c}\right )}^{10}\,\left (a^2\,b^2-2\,a\,b^2\,c+b^2\,c^2\right )}+\frac {\sqrt {a}\,\sqrt {c}\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^4\,\left (64\,a^2+192\,a\,c+64\,c^2\right )}{{\left (\sqrt {c+b\,x}-\sqrt {c}\right )}^4\,\left (a^2\,b^2-2\,a\,b^2\,c+b^2\,c^2\right )}+\frac {\sqrt {a}\,\sqrt {c}\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^8\,\left (64\,a^2+192\,a\,c+64\,c^2\right )}{{\left (\sqrt {c+b\,x}-\sqrt {c}\right )}^8\,\left (a^2\,b^2-2\,a\,b^2\,c+b^2\,c^2\right )}+\frac {\sqrt {a}\,\sqrt {c}\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^6\,\left (128\,a^2+\frac {1312\,a\,c}{3}+128\,c^2\right )}{{\left (\sqrt {c+b\,x}-\sqrt {c}\right )}^6\,\left (a^2\,b^2-2\,a\,b^2\,c+b^2\,c^2\right )}}{\frac {15\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^4}{{\left (\sqrt {c+b\,x}-\sqrt {c}\right )}^4}-\frac {6\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}{{\left (\sqrt {c+b\,x}-\sqrt {c}\right )}^2}-\frac {20\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^6}{{\left (\sqrt {c+b\,x}-\sqrt {c}\right )}^6}+\frac {15\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^8}{{\left (\sqrt {c+b\,x}-\sqrt {c}\right )}^8}-\frac {6\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^{10}}{{\left (\sqrt {c+b\,x}-\sqrt {c}\right )}^{10}}+\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^{12}}{{\left (\sqrt {c+b\,x}-\sqrt {c}\right )}^{12}}+1}-\frac {\mathrm {atanh}\left (\frac {\sqrt {a+b\,x}-\sqrt {a}}{\sqrt {c+b\,x}-\sqrt {c}}\right )\,\left (a+c\right )}{2\,b^2}+\frac {x^2\,\left (a+c\right )}{2\,{\left (a-c\right )}^2}+\frac {2\,b\,x^3}{3\,{\left (a-c\right )}^2} \]
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 \[ \int \frac {x}{\left (\sqrt {a + b x} + \sqrt {b x + c}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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