Optimal. Leaf size=95 \[ -\frac {2 a (a+b x)^{3/2}}{3 b^2 (a-c)}+\frac {2 (a+b x)^{5/2}}{5 b^2 (a-c)}+\frac {2 c (c+b x)^{3/2}}{3 b^2 (a-c)}-\frac {2 (c+b x)^{5/2}}{5 b^2 (a-c)} \]
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Rubi [A]
time = 0.06, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2129, 45}
\begin {gather*} \frac {2 (a+b x)^{5/2}}{5 b^2 (a-c)}-\frac {2 a (a+b x)^{3/2}}{3 b^2 (a-c)}-\frac {2 (b x+c)^{5/2}}{5 b^2 (a-c)}+\frac {2 c (b x+c)^{3/2}}{3 b^2 (a-c)} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 2129
Rubi steps
\begin {align*} \int \frac {x}{\sqrt {a+b x}+\sqrt {c+b x}} \, dx &=-\frac {b \int x \sqrt {a+b x} \, dx}{-a b+b c}+\frac {b \int x \sqrt {c+b x} \, dx}{-a b+b c}\\ &=-\frac {b \int \left (-\frac {a \sqrt {a+b x}}{b}+\frac {(a+b x)^{3/2}}{b}\right ) \, dx}{-a b+b c}+\frac {b \int \left (-\frac {c \sqrt {c+b x}}{b}+\frac {(c+b x)^{3/2}}{b}\right ) \, dx}{-a b+b c}\\ &=-\frac {2 a (a+b x)^{3/2}}{3 b^2 (a-c)}+\frac {2 (a+b x)^{5/2}}{5 b^2 (a-c)}+\frac {2 c (c+b x)^{3/2}}{3 b^2 (a-c)}-\frac {2 (c+b x)^{5/2}}{5 b^2 (a-c)}\\ \end {align*}
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Mathematica [A]
time = 0.36, size = 101, normalized size = 1.06 \begin {gather*} -\frac {2 \sqrt {a+b x} \left (2 a^2+a c-3 c^2-a (c+b x)+6 c (c+b x)-3 (c+b x)^2\right )}{15 b^2 (a-c)}+\frac {2 \left (5 c (c+b x)^{3/2}-3 (c+b x)^{5/2}\right )}{15 b^2 (a-c)} \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 | \(\frac {\frac {2 \left (b x +a \right )^{\frac {5}{2}}}{5}-\frac {2 \left (b x +a \right )^{\frac {3}{2}} a}{3}}{\left (a -c \right ) b^{2}}-\frac {2 \left (\frac {\left (b x +c \right )^{\frac {5}{2}}}{5}-\frac {\left (b x +c \right )^{\frac {3}{2}} c}{3}\right )}{\left (a -c \right ) b^{2}}\) | \(66\) |
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 = 0.32, size = 70, normalized size = 0.74 \begin {gather*} \frac {2 \, {\left ({\left (3 \, b^{2} x^{2} + a b x - 2 \, a^{2}\right )} \sqrt {b x + a} - {\left (3 \, b^{2} x^{2} + b c x - 2 \, c^{2}\right )} \sqrt {b x + c}\right )}}{15 \, {\left (a b^{2} - b^{2} 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}{\sqrt {a + b x} + \sqrt {b x + c}}\, 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. 206 vs.
\(2 (79) = 158\).
time = 4.34, size = 206, normalized size = 2.17 \begin {gather*} -\frac {2 \, {\left ({\left ({\left (b x + a\right )} {\left (\frac {3 \, {\left (a b^{2} - b^{2} c\right )} {\left (b x + a\right )}}{a^{2} b^{3} - 2 \, a b^{3} c + b^{3} c^{2}} - \frac {6 \, a^{2} b^{2} - 7 \, a b^{2} c + b^{2} c^{2}}{a^{2} b^{3} - 2 \, a b^{3} c + b^{3} c^{2}}\right )} + \frac {3 \, a^{3} b^{2} - 4 \, a^{2} b^{2} c - a b^{2} c^{2} + 2 \, b^{2} c^{3}}{a^{2} b^{3} - 2 \, a b^{3} c + b^{3} c^{2}}\right )} \sqrt {b x + c} - \frac {3 \, {\left (b x + a\right )}^{\frac {5}{2}} - 5 \, {\left (b x + a\right )}^{\frac {3}{2}} a}{a b - b c}\right )}}{15 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.70, size = 129, normalized size = 1.36 \begin {gather*} \frac {2\,x^2\,\sqrt {a+b\,x}}{5\,\left (a-c\right )}-\frac {2\,x^2\,\sqrt {c+b\,x}}{5\,\left (a-c\right )}-\frac {4\,a^2\,\sqrt {a+b\,x}}{15\,b^2\,\left (a-c\right )}+\frac {4\,c^2\,\sqrt {c+b\,x}}{15\,b^2\,\left (a-c\right )}+\frac {2\,a\,x\,\sqrt {a+b\,x}}{15\,b\,\left (a-c\right )}-\frac {2\,c\,x\,\sqrt {c+b\,x}}{15\,b\,\left (a-c\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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