Optimal. Leaf size=147 \[ \frac {2 a^2 (a+b x)^{3/2}}{3 b^3 (b-c)}-\frac {2 a^2 (a+c x)^{3/2}}{3 c^3 (b-c)}+\frac {2 (a+b x)^{7/2}}{7 b^3 (b-c)}-\frac {4 a (a+b x)^{5/2}}{5 b^3 (b-c)}-\frac {2 (a+c x)^{7/2}}{7 c^3 (b-c)}+\frac {4 a (a+c x)^{5/2}}{5 c^3 (b-c)} \]
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Rubi [A] time = 0.12, antiderivative size = 147, 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} \begin {gather*} \frac {2 a^2 (a+b x)^{3/2}}{3 b^3 (b-c)}-\frac {2 a^2 (a+c x)^{3/2}}{3 c^3 (b-c)}+\frac {2 (a+b x)^{7/2}}{7 b^3 (b-c)}-\frac {4 a (a+b x)^{5/2}}{5 b^3 (b-c)}-\frac {2 (a+c x)^{7/2}}{7 c^3 (b-c)}+\frac {4 a (a+c x)^{5/2}}{5 c^3 (b-c)} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 2103
Rubi steps
\begin {align*} \int \frac {x^3}{\sqrt {a+b x}+\sqrt {a+c x}} \, dx &=\frac {\int x^2 \sqrt {a+b x} \, dx}{b-c}-\frac {\int x^2 \sqrt {a+c x} \, dx}{b-c}\\ &=\frac {\int \left (\frac {a^2 \sqrt {a+b x}}{b^2}-\frac {2 a (a+b x)^{3/2}}{b^2}+\frac {(a+b x)^{5/2}}{b^2}\right ) \, dx}{b-c}-\frac {\int \left (\frac {a^2 \sqrt {a+c x}}{c^2}-\frac {2 a (a+c x)^{3/2}}{c^2}+\frac {(a+c x)^{5/2}}{c^2}\right ) \, dx}{b-c}\\ &=\frac {2 a^2 (a+b x)^{3/2}}{3 b^3 (b-c)}-\frac {4 a (a+b x)^{5/2}}{5 b^3 (b-c)}+\frac {2 (a+b x)^{7/2}}{7 b^3 (b-c)}-\frac {2 a^2 (a+c x)^{3/2}}{3 (b-c) c^3}+\frac {4 a (a+c x)^{5/2}}{5 (b-c) c^3}-\frac {2 (a+c x)^{7/2}}{7 (b-c) c^3}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 147, normalized size = 1.00 \begin {gather*} \frac {2 a^2 (a+b x)^{3/2}}{3 b^3 (b-c)}-\frac {2 a^2 (a+c x)^{3/2}}{3 c^3 (b-c)}+\frac {2 (a+b x)^{7/2}}{7 b^3 (b-c)}-\frac {4 a (a+b x)^{5/2}}{5 b^3 (b-c)}-\frac {2 (a+c x)^{7/2}}{7 c^3 (b-c)}+\frac {4 a (a+c x)^{5/2}}{5 c^3 (b-c)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.57, size = 208, normalized size = 1.41 \begin {gather*} -\frac {2 \left (35 a^2 (a+c x)^{3/2}+15 (a+c x)^{7/2}-42 a (a+c x)^{5/2}\right )}{105 c^3 (b-c)}-\frac {2 \sqrt {\frac {b (a+c x)}{c}-\frac {a b}{c}+a} \left (15 a^3 b^3-3 a^3 b^2 c-4 a^3 b c^2-8 a^3 c^3-45 a^2 b^3 (a+c x)+6 a^2 b^2 c (a+c x)+4 a^2 b c^2 (a+c x)+45 a b^3 (a+c x)^2-15 b^3 (a+c x)^3-3 a b^2 c (a+c x)^2\right )}{105 b^3 c^3 (b-c)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 122, normalized size = 0.83 \begin {gather*} \frac {2 \, {\left ({\left (15 \, b^{3} c^{3} x^{3} + 3 \, a b^{2} c^{3} x^{2} - 4 \, a^{2} b c^{3} x + 8 \, a^{3} c^{3}\right )} \sqrt {b x + a} - {\left (15 \, b^{3} c^{3} x^{3} + 3 \, a b^{3} c^{2} x^{2} - 4 \, a^{2} b^{3} c x + 8 \, a^{3} b^{3}\right )} \sqrt {c x + a}\right )}}{105 \, {\left (b^{4} c^{3} - b^{3} c^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.43, size = 451, normalized size = 3.07 \begin {gather*} -\frac {2}{105} \, \sqrt {a b^{2} + {\left (b x + a\right )} b c - a b c} {\left ({\left (3 \, {\left (b x + a\right )} {\left (\frac {5 \, {\left (b^{17} c^{5} {\left | b \right |} - 2 \, b^{16} c^{6} {\left | b \right |} + b^{15} c^{7} {\left | b \right |}\right )} {\left (b x + a\right )}}{b^{23} c^{5} - 3 \, b^{22} c^{6} + 3 \, b^{21} c^{7} - b^{20} c^{8}} + \frac {a b^{18} c^{4} {\left | b \right |} - 17 \, a b^{17} c^{5} {\left | b \right |} + 31 \, a b^{16} c^{6} {\left | b \right |} - 15 \, a b^{15} c^{7} {\left | b \right |}}{b^{23} c^{5} - 3 \, b^{22} c^{6} + 3 \, b^{21} c^{7} - b^{20} c^{8}}\right )} - \frac {4 \, a^{2} b^{19} c^{3} {\left | b \right |} - 2 \, a^{2} b^{18} c^{4} {\left | b \right |} - 53 \, a^{2} b^{17} c^{5} {\left | b \right |} + 96 \, a^{2} b^{16} c^{6} {\left | b \right |} - 45 \, a^{2} b^{15} c^{7} {\left | b \right |}}{b^{23} c^{5} - 3 \, b^{22} c^{6} + 3 \, b^{21} c^{7} - b^{20} c^{8}}\right )} {\left (b x + a\right )} + \frac {8 \, a^{3} b^{20} c^{2} {\left | b \right |} - 12 \, a^{3} b^{19} c^{3} {\left | b \right |} + 3 \, a^{3} b^{18} c^{4} {\left | b \right |} - 17 \, a^{3} b^{17} c^{5} {\left | b \right |} + 33 \, a^{3} b^{16} c^{6} {\left | b \right |} - 15 \, a^{3} b^{15} c^{7} {\left | b \right |}}{b^{23} c^{5} - 3 \, b^{22} c^{6} + 3 \, b^{21} c^{7} - b^{20} c^{8}}\right )} + \frac {2 \, {\left (15 \, {\left (b x + a\right )}^{\frac {7}{2}} - 42 \, {\left (b x + a\right )}^{\frac {5}{2}} a + 35 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2}\right )}}{105 \, {\left (b^{4} - b^{3} c\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 90, normalized size = 0.61 \begin {gather*} \frac {\frac {2 \left (b x +a \right )^{\frac {3}{2}} a^{2}}{3}-\frac {4 \left (b x +a \right )^{\frac {5}{2}} a}{5}+\frac {2 \left (b x +a \right )^{\frac {7}{2}}}{7}}{\left (b -c \right ) b^{3}}-\frac {2 \left (\frac {\left (c x +a \right )^{\frac {3}{2}} a^{2}}{3}-\frac {2 \left (c x +a \right )^{\frac {5}{2}} a}{5}+\frac {\left (c x +a \right )^{\frac {7}{2}}}{7}\right )}{\left (b -c \right ) c^{3}} \end {gather*}
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 \begin {gather*} \int \frac {x^{3}}{\sqrt {b x + a} + \sqrt {c x + a}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.92, size = 179, normalized size = 1.22 \begin {gather*} \frac {2\,x^3\,\sqrt {a+b\,x}}{7\,\left (b-c\right )}-\frac {2\,x^3\,\sqrt {a+c\,x}}{7\,\left (b-c\right )}+\frac {16\,a^3\,\sqrt {a+b\,x}}{105\,b^3\,\left (b-c\right )}-\frac {16\,a^3\,\sqrt {a+c\,x}}{105\,c^3\,\left (b-c\right )}+\frac {2\,a\,x^2\,\sqrt {a+b\,x}}{35\,b\,\left (b-c\right )}-\frac {8\,a^2\,x\,\sqrt {a+b\,x}}{105\,b^2\,\left (b-c\right )}-\frac {2\,a\,x^2\,\sqrt {a+c\,x}}{35\,c\,\left (b-c\right )}+\frac {8\,a^2\,x\,\sqrt {a+c\,x}}{105\,c^2\,\left (b-c\right )} \end {gather*}
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 \begin {gather*} \int \frac {x^{3}}{\sqrt {a + b x} + \sqrt {a + c x}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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