Optimal. Leaf size=142 \[ \frac {a^2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {b} \sqrt {a+c x}}\right )}{2 b^{3/2} c^{3/2}}+\frac {2 a x}{(b-c)^2}-\frac {a \sqrt {a+b x} \sqrt {a+c x}}{2 b c (b-c)}-\frac {(a+b x)^{3/2} \sqrt {a+c x}}{b (b-c)^2}+\frac {x^2 (b+c)}{2 (b-c)^2} \]
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Rubi [A] time = 0.23, antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6690, 50, 63, 217, 206} \[ \frac {a^2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {b} \sqrt {a+c x}}\right )}{2 b^{3/2} c^{3/2}}+\frac {2 a x}{(b-c)^2}-\frac {a \sqrt {a+b x} \sqrt {a+c x}}{2 b c (b-c)}-\frac {(a+b x)^{3/2} \sqrt {a+c x}}{b (b-c)^2}+\frac {x^2 (b+c)}{2 (b-c)^2} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 206
Rule 217
Rule 6690
Rubi steps
\begin {align*} \int \frac {x^2}{\left (\sqrt {a+b x}+\sqrt {a+c x}\right )^2} \, dx &=\frac {\int \left (2 a+b \left (1+\frac {c}{b}\right ) x-2 \sqrt {a+b x} \sqrt {a+c x}\right ) \, dx}{(b-c)^2}\\ &=\frac {2 a x}{(b-c)^2}+\frac {(b+c) x^2}{2 (b-c)^2}-\frac {2 \int \sqrt {a+b x} \sqrt {a+c x} \, dx}{(b-c)^2}\\ &=\frac {2 a x}{(b-c)^2}+\frac {(b+c) x^2}{2 (b-c)^2}-\frac {(a+b x)^{3/2} \sqrt {a+c x}}{b (b-c)^2}-\frac {a \int \frac {\sqrt {a+b x}}{\sqrt {a+c x}} \, dx}{2 b (b-c)}\\ &=\frac {2 a x}{(b-c)^2}+\frac {(b+c) x^2}{2 (b-c)^2}-\frac {a \sqrt {a+b x} \sqrt {a+c x}}{2 b (b-c) c}-\frac {(a+b x)^{3/2} \sqrt {a+c x}}{b (b-c)^2}+\frac {a^2 \int \frac {1}{\sqrt {a+b x} \sqrt {a+c x}} \, dx}{4 b c}\\ &=\frac {2 a x}{(b-c)^2}+\frac {(b+c) x^2}{2 (b-c)^2}-\frac {a \sqrt {a+b x} \sqrt {a+c x}}{2 b (b-c) c}-\frac {(a+b x)^{3/2} \sqrt {a+c x}}{b (b-c)^2}+\frac {a^2 \operatorname {Subst}\left (\int \frac {1}{\sqrt {a-\frac {a c}{b}+\frac {c x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{2 b^2 c}\\ &=\frac {2 a x}{(b-c)^2}+\frac {(b+c) x^2}{2 (b-c)^2}-\frac {a \sqrt {a+b x} \sqrt {a+c x}}{2 b (b-c) c}-\frac {(a+b x)^{3/2} \sqrt {a+c x}}{b (b-c)^2}+\frac {a^2 \operatorname {Subst}\left (\int \frac {1}{1-\frac {c x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {a+c x}}\right )}{2 b^2 c}\\ &=\frac {2 a x}{(b-c)^2}+\frac {(b+c) x^2}{2 (b-c)^2}-\frac {a \sqrt {a+b x} \sqrt {a+c x}}{2 b (b-c) c}-\frac {(a+b x)^{3/2} \sqrt {a+c x}}{b (b-c)^2}+\frac {a^2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {b} \sqrt {a+c x}}\right )}{2 b^{3/2} c^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.56, size = 177, normalized size = 1.25 \[ \frac {b \sqrt {c} \left (b c x \left (-2 \sqrt {a+b x} \sqrt {a+c x}+b x+c x\right )-a \left (b \sqrt {a+b x} \sqrt {a+c x}+c \sqrt {a+b x} \sqrt {a+c x}-4 b c x\right )\right )+\frac {(a (b-c))^{5/2} \sqrt {\frac {b (a+c x)}{a (b-c)}} \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a (b-c)}}\right )}{\sqrt {a+c x}}}{2 b^2 c^{3/2} (b-c)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 372, normalized size = 2.62 \[ \left [\frac {8 \, a b^{2} c^{2} x + 2 \, {\left (b^{3} c^{2} + b^{2} c^{3}\right )} x^{2} + {\left (a^{2} b^{2} - 2 \, a^{2} b c + a^{2} c^{2}\right )} \sqrt {b c} \log \left (a b^{2} + 2 \, a b c + a c^{2} + 2 \, {\left (2 \, b c + \sqrt {b c} {\left (b + c\right )}\right )} \sqrt {b x + a} \sqrt {c x + a} + 2 \, {\left (b^{2} c + b c^{2}\right )} x + 2 \, {\left (2 \, b c x + a b + a c\right )} \sqrt {b c}\right ) - 2 \, {\left (2 \, b^{2} c^{2} x + a b^{2} c + a b c^{2}\right )} \sqrt {b x + a} \sqrt {c x + a}}{4 \, {\left (b^{4} c^{2} - 2 \, b^{3} c^{3} + b^{2} c^{4}\right )}}, \frac {4 \, a b^{2} c^{2} x + {\left (b^{3} c^{2} + b^{2} c^{3}\right )} x^{2} - {\left (a^{2} b^{2} - 2 \, a^{2} b c + a^{2} c^{2}\right )} \sqrt {-b c} \arctan \left (\frac {\sqrt {-b c} \sqrt {b x + a} \sqrt {c x + a} - \sqrt {-b c} a}{b c x}\right ) - {\left (2 \, b^{2} c^{2} x + a b^{2} c + a b c^{2}\right )} \sqrt {b x + a} \sqrt {c x + a}}{2 \, {\left (b^{4} c^{2} - 2 \, b^{3} c^{3} + b^{2} c^{4}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 3.28, size = 272, normalized size = 1.92 \[ -\frac {1}{2} \, \sqrt {a b^{2} + {\left (b x + a\right )} b c - a b c} \sqrt {b x + a} {\left (\frac {2 \, {\left (b^{4} c^{2} {\left | b \right |} - b^{3} c^{3} {\left | b \right |}\right )} {\left (b x + a\right )}}{b^{9} c^{2} - 3 \, b^{8} c^{3} + 3 \, b^{7} c^{4} - b^{6} c^{5}} + \frac {a b^{5} c {\left | b \right |} - 2 \, a b^{4} c^{2} {\left | b \right |} + a b^{3} c^{3} {\left | b \right |}}{b^{9} c^{2} - 3 \, b^{8} c^{3} + 3 \, b^{7} c^{4} - b^{6} c^{5}}\right )} - \frac {a^{2} {\left | b \right |} \log \left ({\left | -\sqrt {b c} \sqrt {b x + a} + \sqrt {a b^{2} + {\left (b x + a\right )} b c - a b c} \right |}\right )}{2 \, \sqrt {b c} b^{2} c} + \frac {{\left (b x + a\right )}^{2} b + 2 \, {\left (b x + a\right )} a b + {\left (b x + a\right )}^{2} c - 2 \, {\left (b x + a\right )} a c}{2 \, {\left (b^{4} - 2 \, b^{3} c + b^{2} c^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 385, normalized size = 2.71 \[ \frac {\sqrt {\left (b x +a \right ) \left (c x +a \right )}\, a^{2} b \ln \left (\frac {b c x +\frac {1}{2} a b +\frac {1}{2} a c}{\sqrt {b c}}+\sqrt {b c \,x^{2}+a^{2}+\left (a b +a c \right ) x}\right )}{4 \left (b -c \right )^{2} \sqrt {c x +a}\, \sqrt {b x +a}\, \sqrt {b c}\, c}+\frac {\sqrt {\left (b x +a \right ) \left (c x +a \right )}\, a^{2} c \ln \left (\frac {b c x +\frac {1}{2} a b +\frac {1}{2} a c}{\sqrt {b c}}+\sqrt {b c \,x^{2}+a^{2}+\left (a b +a c \right ) x}\right )}{4 \left (b -c \right )^{2} \sqrt {c x +a}\, \sqrt {b x +a}\, \sqrt {b c}\, b}-\frac {\sqrt {\left (b x +a \right ) \left (c x +a \right )}\, a^{2} \ln \left (\frac {b c x +\frac {1}{2} a b +\frac {1}{2} a c}{\sqrt {b c}}+\sqrt {b c \,x^{2}+a^{2}+\left (a b +a c \right ) x}\right )}{2 \left (b -c \right )^{2} \sqrt {c x +a}\, \sqrt {b x +a}\, \sqrt {b c}}+\frac {b \,x^{2}}{2 \left (b -c \right )^{2}}+\frac {c \,x^{2}}{2 \left (b -c \right )^{2}}+\frac {2 a x}{\left (b -c \right )^{2}}-\frac {\sqrt {c x +a}\, \sqrt {b x +a}\, a}{2 \left (b -c \right )^{2} b}+\frac {\sqrt {c x +a}\, \sqrt {b x +a}\, a}{2 \left (b -c \right )^{2} c}-\frac {\sqrt {b x +a}\, \left (c x +a \right )^{\frac {3}{2}}}{\left (b -c \right )^{2} c} \]
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 \[ \int \frac {x^{2}}{{\left (\sqrt {b x + a} + \sqrt {c x + a}\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.25, size = 129, normalized size = 0.91 \[ \frac {2\,a\,x}{{\left (b-c\right )}^2}+\frac {x^2\,\left (b+c\right )}{2\,{\left (b-c\right )}^2}-\frac {2\,\left (\frac {x}{2}+\frac {a\,b+a\,c}{4\,b\,c}\right )\,\sqrt {a+b\,x}\,\sqrt {a+c\,x}}{{\left (b-c\right )}^2}+\frac {\ln \left (a\,b+a\,c+2\,b\,c\,x+2\,\sqrt {b}\,\sqrt {c}\,\sqrt {a+b\,x}\,\sqrt {a+c\,x}\right )\,{\left (a\,b-a\,c\right )}^2}{4\,b^{3/2}\,c^{3/2}\,{\left (b-c\right )}^2} \]
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 \[ \int \frac {x^{2}}{\left (\sqrt {a + b x} + \sqrt {a + c x}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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