Optimal. Leaf size=147 \[ \frac {2 a^2 (a+b x)^{3/2}}{3 b^3 (a-c)}-\frac {4 a (a+b x)^{5/2}}{5 b^3 (a-c)}+\frac {2 (a+b x)^{7/2}}{7 b^3 (a-c)}-\frac {2 c^2 (c+b x)^{3/2}}{3 b^3 (a-c)}+\frac {4 c (c+b x)^{5/2}}{5 b^3 (a-c)}-\frac {2 (c+b x)^{7/2}}{7 b^3 (a-c)} \]
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Rubi [A]
time = 0.10, 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 = {2129, 45}
\begin {gather*} \frac {2 a^2 (a+b x)^{3/2}}{3 b^3 (a-c)}-\frac {2 c^2 (b x+c)^{3/2}}{3 b^3 (a-c)}+\frac {2 (a+b x)^{7/2}}{7 b^3 (a-c)}-\frac {4 a (a+b x)^{5/2}}{5 b^3 (a-c)}-\frac {2 (b x+c)^{7/2}}{7 b^3 (a-c)}+\frac {4 c (b x+c)^{5/2}}{5 b^3 (a-c)} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 2129
Rubi steps
\begin {align*} \int \frac {x^2}{\sqrt {a+b x}+\sqrt {c+b x}} \, dx &=-\frac {b \int x^2 \sqrt {a+b x} \, dx}{-a b+b c}+\frac {b \int x^2 \sqrt {c+b x} \, dx}{-a b+b c}\\ &=-\frac {b \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}{-a b+b c}+\frac {b \int \left (\frac {c^2 \sqrt {c+b x}}{b^2}-\frac {2 c (c+b x)^{3/2}}{b^2}+\frac {(c+b x)^{5/2}}{b^2}\right ) \, dx}{-a b+b c}\\ &=\frac {2 a^2 (a+b x)^{3/2}}{3 b^3 (a-c)}-\frac {4 a (a+b x)^{5/2}}{5 b^3 (a-c)}+\frac {2 (a+b x)^{7/2}}{7 b^3 (a-c)}-\frac {2 c^2 (c+b x)^{3/2}}{3 b^3 (a-c)}+\frac {4 c (c+b x)^{5/2}}{5 b^3 (a-c)}-\frac {2 (c+b x)^{7/2}}{7 b^3 (a-c)}\\ \end {align*}
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Mathematica [A]
time = 0.48, size = 140, normalized size = 0.95 \begin {gather*} \frac {2 \left (8 a^3 \sqrt {a+b x}-4 a^2 b x \sqrt {a+b x}+3 a b^2 x^2 \sqrt {a+b x}-8 c^3 \sqrt {c+b x}+4 b c^2 x \sqrt {c+b x}-3 b^2 c x^2 \sqrt {c+b x}+15 b^3 x^3 \left (\sqrt {a+b x}-\sqrt {c+b x}\right )\right )}{105 b^3 (a-c)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 90, normalized size = 0.61
method | result | size |
default | \(\frac {\frac {2 \left (b x +a \right )^{\frac {7}{2}}}{7}-\frac {4 a \left (b x +a \right )^{\frac {5}{2}}}{5}+\frac {2 a^{2} \left (b x +a \right )^{\frac {3}{2}}}{3}}{\left (a -c \right ) b^{3}}-\frac {2 \left (\frac {\left (b x +c \right )^{\frac {7}{2}}}{7}-\frac {2 \left (b x +c \right )^{\frac {5}{2}} c}{5}+\frac {c^{2} \left (b x +c \right )^{\frac {3}{2}}}{3}\right )}{\left (a -c \right ) b^{3}}\) | \(90\) |
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 = 0.36, size = 94, normalized size = 0.64 \begin {gather*} \frac {2 \, {\left ({\left (15 \, b^{3} x^{3} + 3 \, a b^{2} x^{2} - 4 \, a^{2} b x + 8 \, a^{3}\right )} \sqrt {b x + a} - {\left (15 \, b^{3} x^{3} + 3 \, b^{2} c x^{2} - 4 \, b c^{2} x + 8 \, c^{3}\right )} \sqrt {b x + c}\right )}}{105 \, {\left (a b^{3} - b^{3} 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^{2}}{\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. 390 vs.
\(2 (123) = 246\).
time = 3.67, size = 390, normalized size = 2.65 \begin {gather*} -\frac {2}{105} \, {\left ({\left (3 \, {\left (b x + a\right )} {\left (\frac {5 \, {\left (a^{2} b^{9} - 2 \, a b^{9} c + b^{9} c^{2}\right )} {\left (b x + a\right )}}{a^{3} b^{12} - 3 \, a^{2} b^{12} c + 3 \, a b^{12} c^{2} - b^{12} c^{3}} - \frac {15 \, a^{3} b^{9} - 31 \, a^{2} b^{9} c + 17 \, a b^{9} c^{2} - b^{9} c^{3}}{a^{3} b^{12} - 3 \, a^{2} b^{12} c + 3 \, a b^{12} c^{2} - b^{12} c^{3}}\right )} + \frac {45 \, a^{4} b^{9} - 96 \, a^{3} b^{9} c + 53 \, a^{2} b^{9} c^{2} + 2 \, a b^{9} c^{3} - 4 \, b^{9} c^{4}}{a^{3} b^{12} - 3 \, a^{2} b^{12} c + 3 \, a b^{12} c^{2} - b^{12} c^{3}}\right )} {\left (b x + a\right )} - \frac {15 \, a^{5} b^{9} - 33 \, a^{4} b^{9} c + 17 \, a^{3} b^{9} c^{2} - 3 \, a^{2} b^{9} c^{3} + 12 \, a b^{9} c^{4} - 8 \, b^{9} c^{5}}{a^{3} b^{12} - 3 \, a^{2} b^{12} c + 3 \, a b^{12} c^{2} - b^{12} c^{3}}\right )} \sqrt {b x + c} + \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 (a b^{3} - b^{3} c\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.95, size = 179, normalized size = 1.22 \begin {gather*} \frac {2\,x^3\,\sqrt {a+b\,x}}{7\,\left (a-c\right )}-\frac {2\,x^3\,\sqrt {c+b\,x}}{7\,\left (a-c\right )}+\frac {16\,a^3\,\sqrt {a+b\,x}}{105\,b^3\,\left (a-c\right )}-\frac {16\,c^3\,\sqrt {c+b\,x}}{105\,b^3\,\left (a-c\right )}+\frac {2\,a\,x^2\,\sqrt {a+b\,x}}{35\,b\,\left (a-c\right )}-\frac {8\,a^2\,x\,\sqrt {a+b\,x}}{105\,b^2\,\left (a-c\right )}-\frac {2\,c\,x^2\,\sqrt {c+b\,x}}{35\,b\,\left (a-c\right )}+\frac {8\,c^2\,x\,\sqrt {c+b\,x}}{105\,b^2\,\left (a-c\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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