Optimal. Leaf size=95 \[ \frac {2 (a+b x)^{5/2}}{5 b^2 (b-c)}-\frac {2 a (a+b x)^{3/2}}{3 b^2 (b-c)}-\frac {2 (a+c x)^{5/2}}{5 c^2 (b-c)}+\frac {2 a (a+c x)^{3/2}}{3 c^2 (b-c)} \]
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Rubi [A] time = 0.10, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2103, 43} \[ \frac {2 (a+b x)^{5/2}}{5 b^2 (b-c)}-\frac {2 a (a+b x)^{3/2}}{3 b^2 (b-c)}-\frac {2 (a+c x)^{5/2}}{5 c^2 (b-c)}+\frac {2 a (a+c x)^{3/2}}{3 c^2 (b-c)} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2103
Rubi steps
\begin {align*} \int \frac {x^2}{\sqrt {a+b x}+\sqrt {a+c x}} \, dx &=\frac {\int x \sqrt {a+b x} \, dx}{b-c}-\frac {\int x \sqrt {a+c x} \, dx}{b-c}\\ &=\frac {\int \left (-\frac {a \sqrt {a+b x}}{b}+\frac {(a+b x)^{3/2}}{b}\right ) \, dx}{b-c}-\frac {\int \left (-\frac {a \sqrt {a+c x}}{c}+\frac {(a+c x)^{3/2}}{c}\right ) \, dx}{b-c}\\ &=-\frac {2 a (a+b x)^{3/2}}{3 b^2 (b-c)}+\frac {2 (a+b x)^{5/2}}{5 b^2 (b-c)}+\frac {2 a (a+c x)^{3/2}}{3 (b-c) c^2}-\frac {2 (a+c x)^{5/2}}{5 (b-c) c^2}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 70, normalized size = 0.74 \[ \frac {2 \left (\frac {3 (a+b x)^{5/2}}{b^2}-\frac {5 a (a+b x)^{3/2}}{b^2}-\frac {3 (a+c x)^{5/2}}{c^2}+\frac {5 a (a+c x)^{3/2}}{c^2}\right )}{15 (b-c)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 92, normalized size = 0.97 \[ \frac {2 \, {\left ({\left (3 \, b^{2} c^{2} x^{2} + a b c^{2} x - 2 \, a^{2} c^{2}\right )} \sqrt {b x + a} - {\left (3 \, b^{2} c^{2} x^{2} + a b^{2} c x - 2 \, a^{2} b^{2}\right )} \sqrt {c x + a}\right )}}{15 \, {\left (b^{3} c^{2} - b^{2} c^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.39, size = 255, normalized size = 2.68 \[ -\frac {2}{15} \, \sqrt {a b^{2} + {\left (b x + a\right )} b c - a b c} {\left ({\left (b x + a\right )} {\left (\frac {3 \, {\left (b^{9} c^{3} {\left | b \right |} - b^{8} c^{4} {\left | b \right |}\right )} {\left (b x + a\right )}}{b^{14} c^{3} - 2 \, b^{13} c^{4} + b^{12} c^{5}} + \frac {a b^{10} c^{2} {\left | b \right |} - 7 \, a b^{9} c^{3} {\left | b \right |} + 6 \, a b^{8} c^{4} {\left | b \right |}}{b^{14} c^{3} - 2 \, b^{13} c^{4} + b^{12} c^{5}}\right )} - \frac {2 \, a^{2} b^{11} c {\left | b \right |} - a^{2} b^{10} c^{2} {\left | b \right |} - 4 \, a^{2} b^{9} c^{3} {\left | b \right |} + 3 \, a^{2} b^{8} c^{4} {\left | b \right |}}{b^{14} c^{3} - 2 \, b^{13} c^{4} + b^{12} c^{5}}\right )} + \frac {2 \, {\left (3 \, {\left (b x + a\right )}^{\frac {5}{2}} - 5 \, {\left (b x + a\right )}^{\frac {3}{2}} a\right )}}{15 \, {\left (b^{3} - b^{2} c\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 66, normalized size = 0.69 \[ \frac {-\frac {2 \left (b x +a \right )^{\frac {3}{2}} a}{3}+\frac {2 \left (b x +a \right )^{\frac {5}{2}}}{5}}{\left (b -c \right ) b^{2}}-\frac {2 \left (-\frac {\left (c x +a \right )^{\frac {3}{2}} a}{3}+\frac {\left (c x +a \right )^{\frac {5}{2}}}{5}\right )}{\left (b -c \right ) c^{2}} \]
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 \[ \int \frac {x^{2}}{\sqrt {b x + a} + \sqrt {c x + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.86, size = 129, normalized size = 1.36 \[ \frac {2\,x^2\,\sqrt {a+b\,x}}{5\,\left (b-c\right )}-\frac {2\,x^2\,\sqrt {a+c\,x}}{5\,\left (b-c\right )}-\frac {4\,a^2\,\sqrt {a+b\,x}}{15\,b^2\,\left (b-c\right )}+\frac {4\,a^2\,\sqrt {a+c\,x}}{15\,c^2\,\left (b-c\right )}+\frac {2\,a\,x\,\sqrt {a+b\,x}}{15\,b\,\left (b-c\right )}-\frac {2\,a\,x\,\sqrt {a+c\,x}}{15\,c\,\left (b-c\right )} \]
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 \[ \int \frac {x^{2}}{\sqrt {a + b x} + \sqrt {a + c x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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