Optimal. Leaf size=174 \[ \frac {2 (a+b x)^{3/2} (a+c x)^{3/2}}{3 a^2 x^3 (b-c)^2}-\frac {(b+c) \sqrt {a+b x} (a+c x)^{3/2}}{2 a^2 x^2 (b-c)^2}-\frac {(b+c) \sqrt {a+b x} \sqrt {a+c x}}{4 a^2 x (b-c)}+\frac {(b+c) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a+c x}}\right )}{4 a^2}-\frac {2 a}{3 x^3 (b-c)^2}-\frac {b+c}{2 x^2 (b-c)^2} \]
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Rubi [A] time = 0.22, antiderivative size = 174, 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, 96, 94, 93, 208} \[ \frac {2 (a+b x)^{3/2} (a+c x)^{3/2}}{3 a^2 x^3 (b-c)^2}-\frac {(b+c) \sqrt {a+b x} (a+c x)^{3/2}}{2 a^2 x^2 (b-c)^2}-\frac {(b+c) \sqrt {a+b x} \sqrt {a+c x}}{4 a^2 x (b-c)}+\frac {(b+c) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a+c x}}\right )}{4 a^2}-\frac {2 a}{3 x^3 (b-c)^2}-\frac {b+c}{2 x^2 (b-c)^2} \]
Antiderivative was successfully verified.
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Rule 93
Rule 94
Rule 96
Rule 208
Rule 6690
Rubi steps
\begin {align*} \int \frac {1}{x^2 \left (\sqrt {a+b x}+\sqrt {a+c x}\right )^2} \, dx &=\frac {\int \left (\frac {2 a}{x^4}+\frac {b \left (1+\frac {c}{b}\right )}{x^3}-\frac {2 \sqrt {a+b x} \sqrt {a+c x}}{x^4}\right ) \, dx}{(b-c)^2}\\ &=-\frac {2 a}{3 (b-c)^2 x^3}-\frac {b+c}{2 (b-c)^2 x^2}-\frac {2 \int \frac {\sqrt {a+b x} \sqrt {a+c x}}{x^4} \, dx}{(b-c)^2}\\ &=-\frac {2 a}{3 (b-c)^2 x^3}-\frac {b+c}{2 (b-c)^2 x^2}+\frac {2 (a+b x)^{3/2} (a+c x)^{3/2}}{3 a^2 (b-c)^2 x^3}+\frac {(b+c) \int \frac {\sqrt {a+b x} \sqrt {a+c x}}{x^3} \, dx}{a (b-c)^2}\\ &=-\frac {2 a}{3 (b-c)^2 x^3}-\frac {b+c}{2 (b-c)^2 x^2}-\frac {(b+c) \sqrt {a+b x} (a+c x)^{3/2}}{2 a^2 (b-c)^2 x^2}+\frac {2 (a+b x)^{3/2} (a+c x)^{3/2}}{3 a^2 (b-c)^2 x^3}+\frac {(b+c) \int \frac {\sqrt {a+c x}}{x^2 \sqrt {a+b x}} \, dx}{4 a (b-c)}\\ &=-\frac {2 a}{3 (b-c)^2 x^3}-\frac {b+c}{2 (b-c)^2 x^2}-\frac {(b+c) \sqrt {a+b x} \sqrt {a+c x}}{4 a^2 (b-c) x}-\frac {(b+c) \sqrt {a+b x} (a+c x)^{3/2}}{2 a^2 (b-c)^2 x^2}+\frac {2 (a+b x)^{3/2} (a+c x)^{3/2}}{3 a^2 (b-c)^2 x^3}-\frac {(b+c) \int \frac {1}{x \sqrt {a+b x} \sqrt {a+c x}} \, dx}{8 a}\\ &=-\frac {2 a}{3 (b-c)^2 x^3}-\frac {b+c}{2 (b-c)^2 x^2}-\frac {(b+c) \sqrt {a+b x} \sqrt {a+c x}}{4 a^2 (b-c) x}-\frac {(b+c) \sqrt {a+b x} (a+c x)^{3/2}}{2 a^2 (b-c)^2 x^2}+\frac {2 (a+b x)^{3/2} (a+c x)^{3/2}}{3 a^2 (b-c)^2 x^3}-\frac {(b+c) \operatorname {Subst}\left (\int \frac {1}{-a+a x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {a+c x}}\right )}{4 a}\\ &=-\frac {2 a}{3 (b-c)^2 x^3}-\frac {b+c}{2 (b-c)^2 x^2}-\frac {(b+c) \sqrt {a+b x} \sqrt {a+c x}}{4 a^2 (b-c) x}-\frac {(b+c) \sqrt {a+b x} (a+c x)^{3/2}}{2 a^2 (b-c)^2 x^2}+\frac {2 (a+b x)^{3/2} (a+c x)^{3/2}}{3 a^2 (b-c)^2 x^3}+\frac {(b+c) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a+c x}}\right )}{4 a^2}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 153, normalized size = 0.88 \[ \frac {-8 a^3+a^2 \left (8 \sqrt {a+b x} \sqrt {a+c x}-6 b x-6 c x\right )+x^2 \left (-3 b^2+2 b c-3 c^2\right ) \sqrt {a+b x} \sqrt {a+c x}+3 x^3 (b-c)^2 (b+c) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a+c x}}\right )+2 a x (b+c) \sqrt {a+b x} \sqrt {a+c x}}{12 a^2 x^3 (b-c)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 182, normalized size = 1.05 \[ -\frac {12 \, {\left (b^{3} - b^{2} c - b c^{2} + c^{3}\right )} x^{3} \log \left (-\frac {{\left (b + c\right )} x - 2 \, \sqrt {b x + a} \sqrt {c x + a} + 2 \, a}{x}\right ) + {\left (5 \, b^{3} + 3 \, b^{2} c + 3 \, b c^{2} + 5 \, c^{3}\right )} x^{3} + 64 \, a^{3} + 8 \, {\left ({\left (3 \, b^{2} - 2 \, b c + 3 \, c^{2}\right )} x^{2} - 8 \, a^{2} - 2 \, {\left (a b + a c\right )} x\right )} \sqrt {b x + a} \sqrt {c x + a} + 48 \, {\left (a^{2} b + a^{2} c\right )} x}{96 \, {\left (a^{2} b^{2} - 2 \, a^{2} b c + a^{2} c^{2}\right )} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 16.94, size = 802, normalized size = 4.61 \[ -\frac {\sqrt {b c} {\left (b + c\right )} {\left | b \right |} \arctan \left (\frac {a b^{2} + a b c - {\left (\sqrt {b c} \sqrt {b x + a} - \sqrt {a b^{2} + {\left (b x + a\right )} b c - a b c}\right )}^{2}}{2 \, \sqrt {-b c} a b}\right )}{4 \, \sqrt {-b c} a^{2} b} + \frac {3 \, {\left (b^{3} - b^{2} c - b c^{2} + c^{3}\right )} \sqrt {b c} {\left (\sqrt {b c} \sqrt {b x + a} - \sqrt {a b^{2} + {\left (b x + a\right )} b c - a b c}\right )}^{10} {\left | b \right |} - 3 \, {\left (5 \, b^{5} + 22 \, b^{3} c^{2} + 5 \, b c^{4}\right )} \sqrt {b c} {\left (\sqrt {b c} \sqrt {b x + a} - \sqrt {a b^{2} + {\left (b x + a\right )} b c - a b c}\right )}^{8} a {\left | b \right |} + 2 \, {\left (15 \, b^{7} - b^{6} c + 18 \, b^{5} c^{2} + 18 \, b^{4} c^{3} - b^{3} c^{4} + 15 \, b^{2} c^{5}\right )} \sqrt {b c} {\left (\sqrt {b c} \sqrt {b x + a} - \sqrt {a b^{2} + {\left (b x + a\right )} b c - a b c}\right )}^{6} a^{2} {\left | b \right |} - 6 \, {\left (5 \, b^{9} - 6 \, b^{8} c - 5 \, b^{7} c^{2} + 12 \, b^{6} c^{3} - 5 \, b^{5} c^{4} - 6 \, b^{4} c^{5} + 5 \, b^{3} c^{6}\right )} \sqrt {b c} {\left (\sqrt {b c} \sqrt {b x + a} - \sqrt {a b^{2} + {\left (b x + a\right )} b c - a b c}\right )}^{4} a^{3} {\left | b \right |} + 3 \, {\left (5 \, b^{11} - 17 \, b^{10} c + 21 \, b^{9} c^{2} - 9 \, b^{8} c^{3} - 9 \, b^{7} c^{4} + 21 \, b^{6} c^{5} - 17 \, b^{5} c^{6} + 5 \, b^{4} c^{7}\right )} \sqrt {b c} {\left (\sqrt {b c} \sqrt {b x + a} - \sqrt {a b^{2} + {\left (b x + a\right )} b c - a b c}\right )}^{2} a^{4} {\left | b \right |} - {\left (3 \, b^{13} - 20 \, b^{12} c + 60 \, b^{11} c^{2} - 108 \, b^{10} c^{3} + 130 \, b^{9} c^{4} - 108 \, b^{8} c^{5} + 60 \, b^{7} c^{6} - 20 \, b^{6} c^{7} + 3 \, b^{5} c^{8}\right )} \sqrt {b c} a^{5} {\left | b \right |}}{6 \, {\left ({\left (\sqrt {b c} \sqrt {b x + a} - \sqrt {a b^{2} + {\left (b x + a\right )} b c - a b c}\right )}^{4} - 2 \, {\left (b^{2} + b c\right )} {\left (\sqrt {b c} \sqrt {b x + a} - \sqrt {a b^{2} + {\left (b x + a\right )} b c - a b c}\right )}^{2} a + {\left (b^{4} - 2 \, b^{3} c + b^{2} c^{2}\right )} a^{2}\right )}^{3} {\left (b^{2} - 2 \, b c + c^{2}\right )} a} - \frac {3 \, {\left (b x + a\right )} b^{3} + a b^{3} + 3 \, {\left (b x + a\right )} b^{2} c - 3 \, a b^{2} c}{6 \, {\left (b^{2} - 2 \, b c + c^{2}\right )} b^{3} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 457, normalized size = 2.63 \[ -\frac {b}{2 \left (b -c \right )^{2} x^{2}}-\frac {c}{2 \left (b -c \right )^{2} x^{2}}-\frac {2 a}{3 \left (b -c \right )^{2} x^{3}}-\frac {\sqrt {b x +a}\, \sqrt {c x +a}\, \left (-3 b^{3} x^{3} \ln \left (\frac {\left (b x +c x +2 a +2 \sqrt {b c \,x^{2}+a b x +a c x +a^{2}}\, \mathrm {csgn}\relax (a )\right ) a}{x}\right )+3 b^{2} c \,x^{3} \ln \left (\frac {\left (b x +c x +2 a +2 \sqrt {b c \,x^{2}+a b x +a c x +a^{2}}\, \mathrm {csgn}\relax (a )\right ) a}{x}\right )+3 b \,c^{2} x^{3} \ln \left (\frac {\left (b x +c x +2 a +2 \sqrt {b c \,x^{2}+a b x +a c x +a^{2}}\, \mathrm {csgn}\relax (a )\right ) a}{x}\right )-3 c^{3} x^{3} \ln \left (\frac {\left (b x +c x +2 a +2 \sqrt {b c \,x^{2}+a b x +a c x +a^{2}}\, \mathrm {csgn}\relax (a )\right ) a}{x}\right )+6 \sqrt {b c \,x^{2}+a b x +a c x +a^{2}}\, b^{2} x^{2} \mathrm {csgn}\relax (a )-4 \sqrt {b c \,x^{2}+a b x +a c x +a^{2}}\, b c \,x^{2} \mathrm {csgn}\relax (a )+6 \sqrt {b c \,x^{2}+a b x +a c x +a^{2}}\, c^{2} x^{2} \mathrm {csgn}\relax (a )-4 \sqrt {b c \,x^{2}+a b x +a c x +a^{2}}\, a b x \,\mathrm {csgn}\relax (a )-4 \sqrt {b c \,x^{2}+a b x +a c x +a^{2}}\, a c x \,\mathrm {csgn}\relax (a )-16 \sqrt {b c \,x^{2}+a b x +a c x +a^{2}}\, a^{2} \mathrm {csgn}\relax (a )\right ) \mathrm {csgn}\relax (a )}{24 \left (b -c \right )^{2} \sqrt {b c \,x^{2}+a b x +a c x +a^{2}}\, a^{2} x^{3}} \]
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 \[ \int \frac {1}{x^{2} {\left (\sqrt {b x + a} + \sqrt {c x + a}\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 18.74, size = 1290, normalized size = 7.41 \[ \text {result too large to display} \]
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 \[ \int \frac {1}{x^{2} \left (\sqrt {a + b x} + \sqrt {a + c x}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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