Optimal. Leaf size=95 \[ -\frac {2 a (a+b x)^{3/2}}{3 b^2 (b-c)}+\frac {2 (a+b x)^{5/2}}{5 b^2 (b-c)}+\frac {2 a (a+c x)^{3/2}}{3 (b-c) c^2}-\frac {2 (a+c x)^{5/2}}{5 (b-c) c^2} \]
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Rubi [A]
time = 0.07, 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 = {2128, 45}
\begin {gather*} \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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 2128
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 = 0.74, size = 113, normalized size = 1.19 \begin {gather*} \frac {6 b^2 c^2 x^2 \left (\sqrt {a+b x}-\sqrt {a+c x}\right )+2 a b c x \left (c \sqrt {a+b x}-b \sqrt {a+c x}\right )+a^2 \left (-4 c^2 \sqrt {a+b x}+4 b^2 \sqrt {a+c x}\right )}{15 b^2 (b-c) c^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 66, normalized size = 0.69
method | result | size |
default | \(\frac {\frac {2 \left (b x +a \right )^{\frac {5}{2}}}{5}-\frac {2 \left (b x +a \right )^{\frac {3}{2}} a}{3}}{\left (b -c \right ) b^{2}}-\frac {2 \left (\frac {\left (c x +a \right )^{\frac {5}{2}}}{5}-\frac {\left (c x +a \right )^{\frac {3}{2}} a}{3}\right )}{\left (b -c \right ) c^{2}}\) | \(66\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 92, normalized size = 0.97 \begin {gather*} \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 )}} \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^{2}}{\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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 255 vs.
\(2 (79) = 158\).
time = 4.90, size = 255, normalized size = 2.68 \begin {gather*} -\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 )}} \end {gather*}
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 \begin {gather*} \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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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